+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basics/lists/list.ma".
-include "basics/deqsets.ma".
-include "binding/names.ma".
-include "binding/fp.ma".
-
-axiom alpha : Nset.
-notation "πΈ" non associative with precedence 90 for @{'alphabet}.
-interpretation "set of names" 'alphabet = alpha.
-
-inductive tp : Type[0] β
-| top : tp
-| arr : tp β tp β tp.
-inductive pretm : Type[0] β
-| var : nat β pretm
-| par : πΈ β pretm
-| abs : tp β pretm β pretm
-| app : pretm β pretm β pretm.
-
-let rec Nth T n (l:list T) on n β
- match l with
- [ nil β None ?
- | cons hd tl β match n with
- [ O β Some ? hd
- | S n0 β Nth T n0 tl ] ].
-
-let rec vclose_tm_aux u x k β match u with
- [ var n β if (leb k n) then var (S n) else u
- | par x0 β if (x0 == x) then (var k) else u
- | app v1 v2 β app (vclose_tm_aux v1 x k) (vclose_tm_aux v2 x k)
- | abs s v β abs s (vclose_tm_aux v x (S k)) ].
-definition vclose_tm β Ξ»u,x.vclose_tm_aux u x O.
-
-definition vopen_var β Ξ»n,x,k.match eqb n k with
- [ true β par x
- | false β match leb n k with
- [ true β var n
- | false β var (pred n) ] ].
-
-let rec vopen_tm_aux u x k β match u with
- [ var n β vopen_var n x k
- | par x0 β u
- | app v1 v2 β app (vopen_tm_aux v1 x k) (vopen_tm_aux v2 x k)
- | abs s v β abs s (vopen_tm_aux v x (S k)) ].
-definition vopen_tm β Ξ»u,x.vopen_tm_aux u x O.
-
-let rec FV u β match u with
- [ par x β [x]
- | app v1 v2 β FV v1@FV v2
- | abs s v β FV v
- | _ β [ ] ].
-
-definition lam β Ξ»x,s,u.abs s (vclose_tm u x).
-
-let rec Pi_map_tm p u on u β match u with
-[ par x β par (p x)
-| var _ β u
-| app v1 v2 β app (Pi_map_tm p v1) (Pi_map_tm p v2)
-| abs s v β abs s (Pi_map_tm p v) ].
-
-interpretation "permutation of tm" 'middot p x = (Pi_map_tm p x).
-
-notation "hvbox(uβxβ)"
- with precedence 45
- for @{ 'open $u $x }.
-
-(*
-notation "hvbox(uβxβ)"
- with precedence 45
- for @{ 'open $u $x }.
-notation "β΄ u β΅ x" non associative with precedence 90 for @{ 'open $u $x }.
-*)
-interpretation "ln term variable open" 'open u x = (vopen_tm u x).
-notation < "hvbox(Ξ½ x break . u)"
- with precedence 20
-for @{'nu $x $u }.
-notation > "Ξ½ list1 x sep , . term 19 u" with precedence 20
- for ${ fold right @{$u} rec acc @{'nu $x $acc)} }.
-interpretation "ln term variable close" 'nu x u = (vclose_tm u x).
-
-let rec tm_height u β match u with
-[ app v1 v2 β S (max (tm_height v1) (tm_height v2))
-| abs s v β S (tm_height v)
-| _ β O ].
-
-theorem le_n_O_rect_Type0 : βn:nat. n β€ O β βP: nat βType[0]. P O β P n.
-#n (cases n) // #a #abs cases (?:False) /2/ qed.
-
-theorem nat_rect_Type0_1 : βn:nat.βP:nat β Type[0].
-(βm.(βp. p < m β P p) β P m) β P n.
-#n #P #H
-cut (βq:nat. q β€ n β P q) /2/
-(elim n)
- [#q #HleO (* applica male *)
- @(le_n_O_rect_Type0 ? HleO)
- @H #p #ltpO cases (?:False) /2/ (* 3 *)
- |#p #Hind #q #HleS
- @H #a #lta @Hind @le_S_S_to_le /2/
- ]
-qed.
-
-lemma leb_false_to_lt : βn,m. leb n m = false β m < n.
-#n elim n
-[ #m normalize #H destruct(H)
-| #n0 #IH * // #m normalize #H @le_S_S @IH // ]
-qed.
-
-lemma nominal_eta_aux : βx,u.x β FV u β βk.vclose_tm_aux (vopen_tm_aux u x k) x k = u.
-#x #u elim u
-[ #n #_ #k normalize cases (decidable_eq_nat n k) #Hnk
- [ >Hnk >eqb_n_n normalize >(\b ?) //
- | >(not_eq_to_eqb_false β¦ Hnk) normalize cases (true_or_false (leb n k)) #Hleb
- [ >Hleb normalize >(?:leb k n = false) //
- @lt_to_leb_false @not_eq_to_le_to_lt /2/
- | >Hleb normalize >(?:leb k (pred n) = true) normalize
- [ cases (leb_false_to_lt β¦ Hleb) //
- | @le_to_leb_true cases (leb_false_to_lt β¦ Hleb) normalize /2/ ] ] ]
-| #y normalize #Hy >(\bf ?) // @(not_to_not β¦ Hy) //
-| #s #v #IH normalize #Hv #k >IH // @Hv
-| #v1 #v2 #IH1 #IH2 normalize #Hv1v2 #k
- >IH1 [ >IH2 // | @(not_to_not β¦ Hv1v2) @in_list_to_in_list_append_l ]
- @(not_to_not β¦ Hv1v2) @in_list_to_in_list_append_r ]
-qed.
-
-corollary nominal_eta : βx,u.x β FV u β (Ξ½x.uβxβ) = u.
-#x #u #Hu @nominal_eta_aux //
-qed.
-
-lemma eq_height_vopen_aux : βv,x,k.tm_height (vopen_tm_aux v x k) = tm_height v.
-#v #x elim v
-[ #n #k normalize cases (eqb n k) // cases (leb n k) //
-| #u #k %
-| #s #u #IH #k normalize >IH %
-| #u1 #u2 #IH1 #IH2 #k normalize >IH1 >IH2 % ]
-qed.
-
-corollary eq_height_vopen : βv,x.tm_height (vβxβ) = tm_height v.
-#v #x @eq_height_vopen_aux
-qed.
-
-theorem pretm_ind_plus_aux :
- βP:pretm β Type[0].
- (βx:πΈ.P (par x)) β
- (βn:β.P (var n)) β
- (βv1,v2. P v1 β P v2 β P (app v1 v2)) β
- βC:list πΈ.
- (βx,s,v.x β FV v β x β C β P (vβxβ) β P (lam x s (vβxβ))) β
- βn,u.tm_height u β€ n β P u.
-#P #Hpar #Hvar #Happ #C #Hlam #n change with ((Ξ»n.?) n); @(nat_rect_Type0_1 n ??)
-#m cases m
-[ #_ * /2/
- [ normalize #s #v #Hfalse cases (?:False) cases (not_le_Sn_O (tm_height v)) /2/
- | #v1 #v2 whd in β’ (?%?β?); #Hfalse cases (?:False) cases (not_le_Sn_O (max ??))
- [ #H @H @Hfalse|*:skip] ] ]
--m #m #IH * /2/
-[ #s #v whd in β’ (?%?β?); #Hv
- lapply (p_fresh β¦ (C@FV v)) letin y β (N_fresh β¦ (C@FV v)) #Hy
- >(?:abs s v = lam y s (vβyβ))
- [| whd in β’ (???%); >nominal_eta // @(not_to_not β¦ Hy) @in_list_to_in_list_append_r ]
- @Hlam
- [ @(not_to_not β¦ Hy) @in_list_to_in_list_append_r
- | @(not_to_not β¦ Hy) @in_list_to_in_list_append_l ]
- @IH [| @Hv | >eq_height_vopen % ]
-| #v1 #v2 whd in β’ (?%?β?); #Hv @Happ
- [ @IH [| @Hv | @le_max_1 ] | @IH [| @Hv | @le_max_2 ] ] ]
-qed.
-
-corollary pretm_ind_plus :
- βP:pretm β Type[0].
- (βx:πΈ.P (par x)) β
- (βn:β.P (var n)) β
- (βv1,v2. P v1 β P v2 β P (app v1 v2)) β
- βC:list πΈ.
- (βx,s,v.x β FV v β x β C β P (vβxβ) β P (lam x s (vβxβ))) β
- βu.P u.
-#P #Hpar #Hvar #Happ #C #Hlam #u @pretm_ind_plus_aux /2/
-qed.
-
-(* maps a permutation to a list of terms *)
-definition Pi_map_list : (πΈ β πΈ) β list πΈ β list πΈ β map πΈ πΈ .
-
-(* interpretation "permutation of name list" 'middot p x = (Pi_map_list p x).*)
-
-(*
-inductive tm : pretm β Prop β
-| tm_par : βx:πΈ.tm (par x)
-| tm_app : βu,v.tm u β tm v β tm (app u v)
-| tm_lam : βx,s,u.tm u β tm (lam x s u).
-
-inductive ctx_aux : nat β pretm β Prop β
-| ctx_var : βn,k.n < k β ctx_aux k (var n)
-| ctx_par : βx,k.ctx_aux k (par x)
-| ctx_app : βu,v,k.ctx_aux k u β ctx_aux k v β ctx_aux k (app u v)
-(* Γ¨ sostituibile da ctx_lam ? *)
-| ctx_abs : βs,u.ctx_aux (S k) u β ctx_aux k (abs s u).
-*)
-
-inductive tm_or_ctx (k:nat) : pretm β Type[0] β
-| toc_var : βn.n < k β tm_or_ctx k (var n)
-| toc_par : βx.tm_or_ctx k (par x)
-| toc_app : βu,v.tm_or_ctx k u β tm_or_ctx k v β tm_or_ctx k (app u v)
-| toc_lam : βx,s,u.tm_or_ctx k u β tm_or_ctx k (lam x s u).
-
-definition tm β Ξ»t.tm_or_ctx O t.
-definition ctx β Ξ»t.tm_or_ctx 1 t.
-
-definition check_tm β Ξ»u,k.
- pretm_ind_plus ?
- (Ξ»_.true)
- (Ξ»n.leb (S n) k)
- (Ξ»v1,v2,rv1,rv2.rv1 β§ rv2)
- [] (Ξ»x,s,v,px,pC,rv.rv)
- u.
-
-axiom pretm_ind_plus_app : βP,u,v,C,H1,H2,H3,H4.
- pretm_ind_plus P H1 H2 H3 C H4 (app u v) =
- H3 u v (pretm_ind_plus P H1 H2 H3 C H4 u) (pretm_ind_plus P H1 H2 H3 C H4 v).
-
-axiom pretm_ind_plus_lam : βP,x,s,u,C,px,pC,H1,H2,H3,H4.
- pretm_ind_plus P H1 H2 H3 C H4 (lam x s (uβxβ)) =
- H4 x s u px pC (pretm_ind_plus P H1 H2 H3 C H4 (uβxβ)).
-
-record TM : Type[0] β {
- pretm_of_TM :> pretm;
- tm_of_TM : check_tm pretm_of_TM O = true
-}.
-
-record CTX : Type[0] β {
- pretm_of_CTX :> pretm;
- ctx_of_CTX : check_tm pretm_of_CTX 1 = true
-}.
-
-inductive tm2 : pretm β Type[0] β
-| tm_par : βx.tm2 (par x)
-| tm_app : βu,v.tm2 u β tm2 v β tm2 (app u v)
-| tm_lam : βx,s,u.x β FV u β (βy.y β FV u β tm2 (uβyβ)) β tm2 (lam x s (uβxβ)).
-
-(*
-inductive tm' : pretm β Prop β
-| tm_par : βx.tm' (par x)
-| tm_app : βu,v.tm' u β tm' v β tm' (app u v)
-| tm_lam : βx,s,u,C.x β FV u β x β C β (βy.y β FV u β tm' (β΄uβ΅y)) β tm' (lam x s (β΄uβ΅x)).
-*)
-
-lemma pi_vclose_tm :
- βz1,z2,x,u.swap πΈ z1 z2Β·(Ξ½x.u) = (Ξ½ swap ? z1 z2 x.swap πΈ z1 z2 Β· u).
-#z1 #z2 #x #u
-change with (vclose_tm_aux ???) in match (vclose_tm ??);
-change with (vclose_tm_aux ???) in β’ (???%); lapply O elim u normalize //
-[ #n #k cases (leb k n) normalize %
-| #x0 #k cases (true_or_false (x0==z1)) #H1 >H1 normalize
- [ cases (true_or_false (x0==x)) #H2 >H2 normalize
- [ <(\P H2) >H1 normalize >(\b (refl ? z2)) %
- | >H1 normalize cases (true_or_false (x==z1)) #H3 >H3 normalize
- [ >(\P H3) in H2; >H1 #Hfalse destruct (Hfalse)
- | cases (true_or_false (x==z2)) #H4 >H4 normalize
- [ cases (true_or_false (z2==z1)) #H5 >H5 normalize //
- >(\P H5) in H4; >H3 #Hfalse destruct (Hfalse)
- | >(\bf ?) // @sym_not_eq @(\Pf H4) ]
- ]
- ]
- | cases (true_or_false (x0==x)) #H2 >H2 normalize
- [ <(\P H2) >H1 normalize >(\b (refl ??)) %
- | >H1 normalize cases (true_or_false (x==z1)) #H3 >H3 normalize
- [ cases (true_or_false (x0==z2)) #H4 >H4 normalize
- [ cases (true_or_false (z1==z2)) #H5 >H5 normalize //
- <(\P H5) in H4; <(\P H3) >H2 #Hfalse destruct (Hfalse)
- | >H4 % ]
- | cases (true_or_false (x0==z2)) #H4 >H4 normalize
- [ cases (true_or_false (x==z2)) #H5 >H5 normalize
- [ <(\P H5) in H4; >H2 #Hfalse destruct (Hfalse)
- | >(\bf ?) // @sym_not_eq @(\Pf H3) ]
- | cases (true_or_false (x==z2)) #H5 >H5 normalize
- [ >H1 %
- | >H2 % ]
- ]
- ]
- ]
- ]
-]
-qed.
-
-lemma pi_vopen_tm :
- βz1,z2,x,u.swap πΈ z1 z2Β·(uβxβ) = (swap πΈ z1 z2 Β· uβswap πΈ z1 z2 xβ).
-#z1 #z2 #x #u
-change with (vopen_tm_aux ???) in match (vopen_tm ??);
-change with (vopen_tm_aux ???) in β’ (???%); lapply O elim u normalize //
-#n #k cases (true_or_false (eqb n k)) #H1 >H1 normalize //
-cases (true_or_false (leb n k)) #H2 >H2 normalize //
-qed.
-
-lemma pi_lam :
- βz1,z2,x,s,u.swap πΈ z1 z2 Β· lam x s u = lam (swap πΈ z1 z2 x) s (swap πΈ z1 z2 Β· u).
-#z1 #z2 #x #s #u whd in β’ (???%); <(pi_vclose_tm β¦) %
-qed.
-
-lemma eqv_FV : βz1,z2,u.FV (swap πΈ z1 z2 Β· u) = Pi_map_list (swap πΈ z1 z2) (FV u).
-#z1 #z2 #u elim u //
-[ #s #v normalize //
-| #v1 #v2 normalize /2/ ]
-qed.
-
-lemma swap_inv_tm : βz1,z2,u.swap πΈ z1 z2 Β· (swap πΈ z1 z2 Β· u) = u.
-#z1 #z2 #u elim u [1,3,4:normalize //]
-#x whd in β’ (??%?); >swap_inv %
-qed.
-
-lemma eqv_in_list : βx,l,z1,z2.x β l β swap πΈ z1 z2 x β Pi_map_list (swap πΈ z1 z2) l.
-#x #l #z1 #z2 #Hin elim Hin
-[ #x0 #l0 %
-| #x1 #x2 #l0 #Hin #IH %2 @IH ]
-qed.
-
-lemma eqv_tm2 : βu.tm2 u β βz1,z2.tm2 ((swap ? z1 z2)Β·u).
-#u #Hu #z1 #z2 letin p β (swap ? z1 z2) elim Hu /2/
-#x #s #v #Hx #Hv #IH >pi_lam >pi_vopen_tm %3
-[ @(not_to_not β¦ Hx) -Hx #Hx
- <(swap_inv ? z1 z2 x) <(swap_inv_tm z1 z2 v) >eqv_FV @eqv_in_list //
-| #y #Hy <(swap_inv ? z1 z2 y)
- <pi_vopen_tm @IH @(not_to_not β¦ Hy) -Hy #Hy <(swap_inv ? z1 z2 y)
- >eqv_FV @eqv_in_list //
-]
-qed.
-
-lemma vclose_vopen_aux : βx,u,k.vopen_tm_aux (vclose_tm_aux u x k) x k = u.
-#x #u elim u normalize //
-[ #n #k cases (true_or_false (leb k n)) #H >H whd in β’ (??%?);
- [ cases (true_or_false (eqb (S n) k)) #H1 >H1
- [ <(eqb_true_to_eq β¦ H1) in H; #H lapply (leb_true_to_le β¦ H) -H #H
- cases (le_to_not_lt β¦ H) -H #H cases (H ?) %
- | whd in β’ (??%?); >lt_to_leb_false // @le_S_S /2/ ]
- | cases (true_or_false (eqb n k)) #H1 >H1 normalize
- [ >(eqb_true_to_eq β¦ H1) in H; #H lapply (leb_false_to_not_le β¦ H) -H
- * #H cases (H ?) %
- | >le_to_leb_true // @not_lt_to_le % #H2 >le_to_leb_true in H;
- [ #H destruct (H) | /2/ ]
- ]
- ]
-| #x0 #k cases (true_or_false (x0==x)) #H1 >H1 normalize // >(\P H1) >eqb_n_n % ]
-qed.
-
-lemma vclose_vopen : βx,u.((Ξ½x.u)βxβ) = u. #x #u @vclose_vopen_aux
-qed.
-
-(*
-theorem tm_to_tm : βt.tm' t β tm t.
-#t #H elim H
-*)
-
-lemma in_list_singleton : βT.βt1,t2:T.t1 β [t2] β t1 = t2.
-#T #t1 #t2 #H @(in_list_inv_ind ??? H) /2/
-qed.
-
-lemma fresh_vclose_tm_aux : βu,x,k.x β FV (vclose_tm_aux u x k).
-#u #x elim u //
-[ #n #k normalize cases (leb k n) normalize //
-| #x0 #k normalize cases (true_or_false (x0==x)) #H >H normalize //
- lapply (\Pf H) @not_to_not #Hin >(in_list_singleton ??? Hin) %
-| #v1 #v2 #IH1 #IH2 #k normalize % #Hin cases (in_list_append_to_or_in_list ???? Hin) /2/ ]
-qed.
-
-lemma fresh_vclose_tm : βu,x.x β FV (Ξ½x.u). //
-qed.
-
-lemma check_tm_true_to_toc : βu,k.check_tm u k = true β tm_or_ctx k u.
-#u @(pretm_ind_plus ???? [ ] ? u)
-[ #x #k #_ %2
-| #n #k change with (leb (S n) k) in β’ (??%?β?); #H % @leb_true_to_le //
-| #v1 #v2 #rv1 #rv2 #k change with (pretm_ind_plus ???????) in β’ (??%?β?);
- >pretm_ind_plus_app #H cases (andb_true ?? H) -H #Hv1 #Hv2 %3
- [ @rv1 @Hv1 | @rv2 @Hv2 ]
-| #x #s #v #Hx #_ #rv #k change with (pretm_ind_plus ???????) in β’ (??%?β?);
- >pretm_ind_plus_lam // #Hv %4 @rv @Hv ]
-qed.
-
-lemma toc_to_check_tm_true : βu,k.tm_or_ctx k u β check_tm u k = true.
-#u #k #Hu elim Hu //
-[ #n #Hn change with (leb (S n) k) in β’ (??%?); @le_to_leb_true @Hn
-| #v1 #v2 #Hv1 #Hv2 #IH1 #IH2 change with (pretm_ind_plus ???????) in β’ (??%?);
- >pretm_ind_plus_app change with (check_tm v1 k β§ check_tm v2 k) in β’ (??%?); /2/
-| #x #s #v #Hv #IH <(vclose_vopen x v) change with (pretm_ind_plus ???????) in β’ (??%?);
- >pretm_ind_plus_lam [| // | @fresh_vclose_tm ] >(vclose_vopen x v) @IH ]
-qed.
-
-lemma fresh_swap_tm : βz1,z2,u.z1 β FV u β z2 β FV u β swap πΈ z1 z2 Β· u = u.
-#z1 #z2 #u elim u
-[2: normalize in β’ (?β%β%β?); #x #Hz1 #Hz2 whd in β’ (??%?); >swap_other //
- [ @(not_to_not β¦ Hz2) | @(not_to_not β¦ Hz1) ] //
-|1: //
-| #s #v #IH normalize #Hz1 #Hz2 >IH // [@Hz2|@Hz1]
-| #v1 #v2 #IH1 #IH2 normalize #Hz1 #Hz2
- >IH1 [| @(not_to_not β¦ Hz2) @in_list_to_in_list_append_l | @(not_to_not β¦ Hz1) @in_list_to_in_list_append_l ]
- >IH2 // [@(not_to_not β¦ Hz2) @in_list_to_in_list_append_r | @(not_to_not β¦ Hz1) @in_list_to_in_list_append_r ]
-]
-qed.
-
-theorem tm_to_tm2 : βu.tm u β tm2 u.
-#t #Ht elim Ht
-[ #n #Hn cases (not_le_Sn_O n) #Hfalse cases (Hfalse Hn)
-| @tm_par
-| #u #v #Hu #Hv @tm_app
-| #x #s #u #Hu #IHu <(vclose_vopen x u) @tm_lam
- [ @fresh_vclose_tm
- | #y #Hy <(fresh_swap_tm x y (Ξ½x.u)) /2/ @fresh_vclose_tm ]
-]
-qed.
-
-theorem tm2_to_tm : βu.tm2 u β tm u.
-#u #pu elim pu /2/ #x #s #v #Hx #Hv #IH %4 @IH //
-qed.
-
-(* define PAR APP LAM *)
-definition PAR β Ξ»x.mk_TM (par x) ?. // qed.
-definition APP β Ξ»u,v:TM.mk_TM (app u v) ?.
-change with (pretm_ind_plus ???????) in match (check_tm ??); >pretm_ind_plus_app
-change with (check_tm ??) in match (pretm_ind_plus ???????); change with (check_tm ??) in match (pretm_ind_plus ???????) in β’ (??(??%)?);
-@andb_elim >(tm_of_TM u) >(tm_of_TM v) % qed.
-definition LAM β Ξ»x,s.Ξ»u:TM.mk_TM (lam x s u) ?.
-change with (pretm_ind_plus ???????) in match (check_tm ??); <(vclose_vopen x u)
->pretm_ind_plus_lam [| // | @fresh_vclose_tm ]
-change with (check_tm ??) in match (pretm_ind_plus ???????); >vclose_vopen
-@(tm_of_TM u) qed.
-
-axiom vopen_tm_down : βu,x,k.tm_or_ctx (S k) u β tm_or_ctx k (uβxβ).
-(* needs true_plus_false
-
-#u #x #k #Hu elim Hu
-[ #n #Hn normalize cases (true_or_false (eqb n O)) #H >H [%2]
- normalize >(?: leb n O = false) [|cases n in H; // >eqb_n_n #H destruct (H) ]
- normalize lapply Hn cases n in H; normalize [ #Hfalse destruct (Hfalse) ]
- #n0 #_ #Hn0 % @le_S_S_to_le //
-| #x0 %2
-| #v1 #v2 #Hv1 #Hv2 #IH1 #IH2 %3 //
-| #x0 #s #v #Hv #IH normalize @daemon
-]
-qed.
-*)
-
-definition vopen_TM β Ξ»u:CTX.Ξ»x.mk_TM (uβxβ) ?.
-@toc_to_check_tm_true @vopen_tm_down @check_tm_true_to_toc @ctx_of_CTX qed.
-
-axiom vclose_tm_up : βu,x,k.tm_or_ctx k u β tm_or_ctx (S k) (Ξ½x.u).
-
-definition vclose_TM β Ξ»u:TM.Ξ»x.mk_CTX (Ξ½x.u) ?.
-@toc_to_check_tm_true @vclose_tm_up @check_tm_true_to_toc @tm_of_TM qed.
-
-interpretation "ln wf term variable open" 'open u x = (vopen_TM u x).
-interpretation "ln wf term variable close" 'nu x u = (vclose_TM u x).
-
-theorem tm_alpha : βx,y,s,u.x β FV u β y β FV u β lam x s (uβxβ) = lam y s (uβyβ).
-#x #y #s #u #Hx #Hy whd in β’ (??%%); @eq_f >nominal_eta // >nominal_eta //
-qed.
-
-lemma TM_to_tm2 : βu:TM.tm2 u.
-#u @tm_to_tm2 @check_tm_true_to_toc @tm_of_TM qed.
-
-theorem TM_ind_plus_weak :
- βP:pretm β Type[0].
- (βx:πΈ.P (PAR x)) β
- (βv1,v2:TM.P v1 β P v2 β P (APP v1 v2)) β
- βC:list πΈ.
- (βx,s.βv:CTX.x β FV v β x β C β
- (βy.y β FV v β P (vβyβ)) β P (LAM x s (vβxβ))) β
- βu:TM.P u.
-#P #Hpar #Happ #C #Hlam #u elim (TM_to_tm2 u) //
-[ #v1 #v2 #pv1 #pv2 #IH1 #IH2 @(Happ (mk_TM β¦) (mk_TM β¦) IH1 IH2)
- @toc_to_check_tm_true @tm2_to_tm //
-| #x #s #v #Hx #pv #IH
- lapply (p_fresh β¦ (C@FV v)) letin x0 β (N_fresh β¦ (C@FV v)) #Hx0
- >(?:lam x s (vβxβ) = lam x0 s (vβx0β))
- [|@tm_alpha // @(not_to_not β¦ Hx0) @in_list_to_in_list_append_r ]
- @(Hlam x0 s (mk_CTX v ?) ??)
- [ <(nominal_eta β¦ Hx) @toc_to_check_tm_true @vclose_tm_up @tm2_to_tm @pv //
- | @(not_to_not β¦ Hx0) @in_list_to_in_list_append_r
- | @(not_to_not β¦ Hx0) @in_list_to_in_list_append_l
- | @IH ]
-]
-qed.
-
-lemma eq_mk_TM : βu,v.u = v β βpu,pv.mk_TM u pu = mk_TM v pv.
-#u #v #Heq >Heq #pu #pv %
-qed.
-
-lemma eq_P : βT:Type[0].βt1,t2:T.t1 = t2 β βP:T β Type[0].P t1 β P t2. // qed.
-
-theorem TM_ind_plus :
- βP:TM β Type[0].
- (βx:πΈ.P (PAR x)) β
- (βv1,v2:TM.P v1 β P v2 β P (APP v1 v2)) β
- βC:list πΈ.
- (βx,s.βv:CTX.x β FV v β x β C β
- (βy.y β FV v β P (vβyβ)) β P (LAM x s (vβxβ))) β
- βu:TM.P u.
-#P #Hpar #Happ #C #Hlam * #u #pu
->(?:mk_TM u pu =
- mk_TM u (toc_to_check_tm_true β¦ (tm2_to_tm β¦ (tm_to_tm2 β¦ (check_tm_true_to_toc β¦ pu))))) [|%]
-elim (tm_to_tm2 u ?) //
-[ #v1 #v2 #pv1 #pv2 #IH1 #IH2 @(Happ (mk_TM β¦) (mk_TM β¦) IH1 IH2)
-| #x #s #v #Hx #pv #IH
- lapply (p_fresh β¦ (C@FV v)) letin x0 β (N_fresh β¦ (C@FV v)) #Hx0
- lapply (Hlam x0 s (mk_CTX v ?) ???)
- [2: @(not_to_not β¦ Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_l @Hx0
- |4: @toc_to_check_tm_true <(nominal_eta x v) // @vclose_tm_up @tm2_to_tm @pv //
- | #y #Hy whd in match (vopen_TM ??);
- >(?:mk_TM (vβyβ) ? = mk_TM (vβyβ) (toc_to_check_tm_true (vβyβ) O (tm2_to_tm (vβyβ) (pv y Hy))))
- [@IH|%]
- | @(not_to_not β¦ Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_r @Hx0
- | @eq_P @eq_mk_TM whd in match (vopen_TM ??); @tm_alpha // @(not_to_not β¦ Hx0) @in_list_to_in_list_append_r ]
-]
-qed.
-
-notation
-"hvbox('nominal' u 'return' out 'with'
- [ 'xpar' ident x β f1
- | 'xapp' ident v1 ident v2 ident recv1 ident recv2 β f2
- | 'xlam' β¨ident y # Cβ© ident s ident w ident py1 ident py2 ident recw β f3 ])"
-with precedence 48
-for @{ TM_ind_plus $out (Ξ»${ident x}:?.$f1)
- (Ξ»${ident v1}:?.Ξ»${ident v2}:?.Ξ»${ident recv1}:?.Ξ»${ident recv2}:?.$f2)
- $C (Ξ»${ident y}:?.Ξ»${ident s}:?.Ξ»${ident w}:?.Ξ»${ident py1}:?.Ξ»${ident py2}:?.Ξ»${ident recw}:?.$f3)
- $u }.
-
-(* include "basics/jmeq.ma".*)
-
-definition subst β (Ξ»u:TM.Ξ»x,v.
- nominal u return (Ξ»_.TM) with
- [ xpar x0 β match x == x0 with [ true β v | false β PAR x0 ] (* u instead of PAR x0 does not work, u stays the same at every rec call! *)
- | xapp v1 v2 recv1 recv2 β APP recv1 recv2
- | xlam β¨y # x::FV vβ© s w py1 py2 recw β LAM y s (recw y py1) ]).
-
-lemma subst_def : βu,x,v.subst u x v =
- nominal u return (Ξ»_.TM) with
- [ xpar x0 β match x == x0 with [ true β v | false β PAR x0 ]
- | xapp v1 v2 recv1 recv2 β APP recv1 recv2
- | xlam β¨y # x::FV vβ© s w py1 py2 recw β LAM y s (recw y py1) ]. //
-qed.
-
-axiom TM_ind_plus_LAM :
- βx,s,u,out,f1,f2,C,f3,Hx1,Hx2.
- TM_ind_plus out f1 f2 C f3 (LAM x s (uβxβ)) =
- f3 x s u Hx1 Hx2 (Ξ»y,Hy.TM_ind_plus ? f1 f2 C f3 ?).
-
-axiom TM_ind_plus_APP :
- βu1,u2,out,f1,f2,C,f3.
- TM_ind_plus out f1 f2 C f3 (APP u1 u2) =
- f2 u1 u2 (TM_ind_plus out f1 f2 C f3 ?) (TM_ind_plus out f1 f2 C f3 ?).
-
-lemma eq_mk_CTX : βu,v.u = v β βpu,pv.mk_CTX u pu = mk_CTX v pv.
-#u #v #Heq >Heq #pu #pv %
-qed.
-
-lemma vclose_vopen_TM : βx.βu:TM.((Ξ½x.u)βxβ) = u.
-#x * #u #pu @eq_mk_TM @vclose_vopen qed.
-
-lemma nominal_eta_CTX : βx.βu:CTX.x β FV u β (Ξ½x.(uβxβ)) = u.
-#x * #u #pu #Hx @eq_mk_CTX @nominal_eta // qed.
-
-theorem TM_alpha : βx,y,s.βu:CTX.x β FV u β y β FV u β LAM x s (uβxβ) = LAM y s (uβyβ).
-#x #y #s #u #Hx #Hy @eq_mk_TM @tm_alpha // qed.
-
-axiom in_vopen_CTX : βx,y.βv:CTX.x β FV (vβyβ) β x = y β¨ x β FV v.
-
-theorem subst_fresh : βu,v:TM.βx.x β FV u β subst u x v = u.
-#u #v #x @(TM_ind_plus β¦ (x::FV v) β¦ u)
-[ #x0 normalize in β’ (%β?); #Hx normalize in β’ (??%?);
- >(\bf ?) [| @(not_to_not β¦ Hx) #Heq >Heq % ] %
-| #u1 #u2 #IH1 #IH2 normalize in β’ (%β?); #Hx
- >subst_def >TM_ind_plus_APP @eq_mk_TM @eq_f2 @eq_f
- [ <subst_def @IH1 @(not_to_not β¦ Hx) @in_list_to_in_list_append_l
- | <subst_def @IH2 @(not_to_not β¦ Hx) @in_list_to_in_list_append_r ]
-| #x0 #s #v0 #Hx0 #HC #IH #Hx >subst_def >TM_ind_plus_LAM [|@HC|@Hx0]
- @eq_f <subst_def @IH // @(not_to_not β¦ Hx) -Hx #Hx
- change with (FV (Ξ½x0.(v0βx0β))) in β’ (???%); >nominal_eta_CTX //
- cases (in_vopen_CTX β¦ Hx) // #Heq >Heq in HC; * #HC @False_ind @HC %
-]
-qed.
-
-example subst_LAM_same : βx,s,u,v. subst (LAM x s u) x v = LAM x s u.
-#x #s #u #v >subst_def <(vclose_vopen_TM x u)
-lapply (p_fresh β¦ (FV (Ξ½x.u)@x::FV v)) letin x0 β (N_fresh β¦ (FV (Ξ½x.u)@x::FV v)) #Hx0
->(TM_alpha x x0)
-[| @(not_to_not β¦ Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_l @Hx0 | @fresh_vclose_tm ]
->TM_ind_plus_LAM [| @(not_to_not β¦ Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_r @Hx0 | @(not_to_not β¦ Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_l @Hx0 ]
-@eq_f change with (subst ((Ξ½x.u)βx0β) x v) in β’ (??%?); @subst_fresh
-@(not_to_not β¦ Hx0) #Hx0' cases (in_vopen_CTX β¦ Hx0')
-[ #Heq >Heq @in_list_to_in_list_append_r %
-| #Hfalse @False_ind cases (fresh_vclose_tm u x) #H @H @Hfalse ]
-qed.
-
-(*
-notation > "Ξ ident x. ident T [ident x] β¦ P"
- with precedence 48 for @{'foo (Ξ»${ident x}.Ξ»${ident T}.$P)}.
-
-notation < "Ξ ident x. ident T [ident x] β¦ P"
- with precedence 48 for @{'foo (Ξ»${ident x}:$Q.Ξ»${ident T}:$R.$P)}.
-*)
-
-(*
-notation
-"hvbox('nominal' u 'with'
- [ 'xpar' ident x β f1
- | 'xapp' ident v1 ident v2 β f2
- | 'xlam' ident x # C s w β f3 ])"
-with precedence 48
-for @{ tm2_ind_plus ? (Ξ»${ident x}:$Tx.$f1)
- (Ξ»${ident v1}:$Tv1.Ξ»${ident v2}:$Tv2.Ξ»${ident pv1}:$Tpv1.Ξ»${ident pv2}:$Tpv2.Ξ»${ident recv1}:$Trv1.Ξ»${ident recv2}:$Trv2.$f2)
- $C (Ξ»${ident x}:$Tx.Ξ»${ident s}:$Ts.Ξ»${ident w}:$Tw.Ξ»${ident py1}:$Tpy1.Ξ»${ident py2}:$Tpy2.Ξ»${ident pw}:$Tpw.Ξ»${ident recw}:$Trw.$f3) $u (tm_to_tm2 ??) }.
-*)
-
-(*
-notation
-"hvbox('nominal' u 'with'
- [ 'xpar' ident x ^ f1
- | 'xapp' ident v1 ident v2 ^ f2 ])"
-(* | 'xlam' ident x # C s w ^ f3 ]) *)
-with precedence 48
-for @{ tm2_ind_plus ? (Ξ»${ident x}:$Tx.$f1)
- (Ξ»${ident v1}:$Tv1.Ξ»${ident v2}:$Tv2.Ξ»${ident pv1}:$Tpv1.Ξ»${ident pv2}:$Tpv2.Ξ»${ident recv1}:$Trv1.Ξ»${ident recv2}:$Trv2.$f2)
- $C (Ξ»${ident x}:$Tx.Ξ»${ident s}:$Ts.Ξ»${ident w}:$Tw.Ξ»${ident py1}:$Tpy1.Ξ»${ident py2}:$Tpy2.Ξ»${ident pw}:$Tpw.Ξ»${ident recw}:$Trw.$f3) $u (tm_to_tm2 ??) }.
-*)
-notation
-"hvbox('nominal' u 'with'
- [ 'xpar' ident x ^ f1
- | 'xapp' ident v1 ident v2 ^ f2 ])"
-with precedence 48
-for @{ tm2_ind_plus ? (Ξ»${ident x}:?.$f1)
- (Ξ»${ident v1}:$Tv1.Ξ»${ident v2}:$Tv2.Ξ»${ident pv1}:$Tpv1.Ξ»${ident pv2}:$Tpv2.Ξ»${ident recv1}:$Trv1.Ξ»${ident recv2}:$Trv2.$f2)
- $C (Ξ»${ident x}:?.Ξ»${ident s}:$Ts.Ξ»${ident w}:$Tw.Ξ»${ident py1}:$Tpy1.Ξ»${ident py2}:$Tpy2.Ξ»${ident pw}:$Tpw.Ξ»${ident recw}:$Trw.$f3) $u (tm_to_tm2 ??) }.
-
-axiom in_Env : πΈ Γ tp β Env β Prop.
-notation "X β G" non associative with precedence 45 for @{'lefttriangle $X $G}.
-interpretation "Env membership" 'lefttriangle x l = (in_Env x l).
-
-
-
-inductive judg : list tp β tm β tp β Prop β
-| t_var : βg,n,t.Nth ? n g = Some ? t β judg g (var n) t
-| t_app : βg,m,n,t,u.judg g m (arr t u) β judg g n t β judg g (app m n) u
-| t_abs : βg,t,m,u.judg (t::g) m u β judg g (abs t m) (arr t u).
-
-definition Env := list (πΈ Γ tp).
-
-axiom vclose_env : Env β list tp.
-axiom vclose_tm : Env β tm β tm.
-axiom Lam : πΈ β tp β tm β tm.
-definition Judg β Ξ»G,M,T.judg (vclose_env G) (vclose_tm G M) T.
-definition dom β Ξ»G:Env.map ?? (fst ??) G.
-
-definition sctx β πΈ Γ tm.
-axiom swap_tm : πΈ β πΈ β tm β tm.
-definition sctx_app : sctx β πΈ β tm β Ξ»M0,Y.let β©X,Mβͺ β M0 in swap_tm X Y M.
-
-axiom in_list : βA:Type[0].A β list A β Prop.
-interpretation "list membership" 'mem x l = (in_list ? x l).
-interpretation "list non-membership" 'notmem x l = (Not (in_list ? x l)).
-
-axiom in_Env : πΈ Γ tp β Env β Prop.
-notation "X β G" non associative with precedence 45 for @{'lefttriangle $X $G}.
-interpretation "Env membership" 'lefttriangle x l = (in_Env x l).
-
-(* axiom Lookup : πΈ β Env β option tp. *)
-
-(* forma alto livello del judgment
- t_abs* : βG,T,X,M,U.
- (βY β supp(M).Judg (β©Y,Tβͺ::G) (M[Y]) U) β
- Judg G (Lam X T (M[X])) (arr T U) *)
-
-(* prima dimostrare, poi perfezionare gli assiomi, poi dimostrarli *)
-
-axiom Judg_ind : βP:Env β tm β tp β Prop.
- (βX,G,T.β©X,Tβͺ β G β P G (par X) T) β
- (βG,M,N,T,U.
- Judg G M (arr T U) β Judg G N T β
- P G M (arr T U) β P G N T β P G (app M N) U) β
- (βG,T1,T2,X,M1.
- (βY.Y β (FV (Lam X T1 (sctx_app M1 X))) β Judg (β©Y,T1βͺ::G) (sctx_app M1 Y) T2) β
- (βY.Y β (FV (Lam X T1 (sctx_app M1 X))) β P (β©Y,T1βͺ::G) (sctx_app M1 Y) T2) β
- P G (Lam X T1 (sctx_app M1 X)) (arr T1 T2)) β
- βG,M,T.Judg G M T β P G M T.
-
-axiom t_par : βX,G,T.β©X,Tβͺ β G β Judg G (par X) T.
-axiom t_app2 : βG,M,N,T,U.Judg G M (arr T U) β Judg G N T β Judg G (app M N) U.
-axiom t_Lam : βG,X,M,T,U.Judg (β©X,Tβͺ::G) M U β Judg G (Lam X T M) (arr T U).
-
-definition subenv β Ξ»G1,G2.βx.x β G1 β x β G2.
-interpretation "subenv" 'subseteq G1 G2 = (subenv G1 G2).
-
-axiom daemon : βP:Prop.P.
-
-theorem weakening : βG1,G2,M,T.G1 β G2 β Judg G1 M T β Judg G2 M T.
-#G1 #G2 #M #T #Hsub #HJ lapply Hsub lapply G2 -G2 change with (βG2.?)
-@(Judg_ind β¦ HJ)
-[ #X #G #T0 #Hin #G2 #Hsub @t_par @Hsub //
-| #G #M0 #N #T0 #U #HM0 #HN #IH1 #IH2 #G2 #Hsub @t_app2
- [| @IH1 // | @IH2 // ]
-| #G #T1 #T2 #X #M1 #HM1 #IH #G2 #Hsub @t_Lam @IH
- [ (* trivial property of Lam *) @daemon
- | (* trivial property of subenv *) @daemon ]
-]
-qed.
-
-(* Serve un tipo Tm per i termini localmente chiusi e i suoi principi di induzione e
- ricorsione *)
\ No newline at end of file
projects / helm.git / blobdiff