+(**************************************************************************)
+(* ___ *)
+(* ||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 "𝔸" non associative with precedence 90 for @{'alphabet}.
+interpretation "set of names" 'alphabet = alpha.
+
+inductive tp : Type[0] ≝
+| top : tp
+| arr : tp → tp → tp.
+inductive pretm : Type[0] ≝
+| var : nat → pretm
+| par : 𝔸 → pretm
+| abs : tp → pretm → pretm
+| app : pretm → pretm → pretm.
+
+let rec Nth T n (l:list T) on n ≝
+ match l with
+ [ nil ⇒ None ?
+ | cons hd tl ⇒ match n with
+ [ O ⇒ Some ? hd
+ | S n0 ⇒ Nth T n0 tl ] ].
+
+let rec vclose_tm_aux u x k ≝ match u with
+ [ var n ⇒ if (leb k n) then var (S n) else u
+ | par x0 ⇒ if (x0 == x) then (var k) else u
+ | app v1 v2 ⇒ app (vclose_tm_aux v1 x k) (vclose_tm_aux v2 x k)
+ | abs s v ⇒ abs s (vclose_tm_aux v x (S k)) ].
+definition vclose_tm ≝ λu,x.vclose_tm_aux u x O.
+
+definition vopen_var ≝ λn,x,k.match eqb n k with
+ [ true ⇒ par x
+ | false ⇒ match leb n k with
+ [ true ⇒ var n
+ | false ⇒ var (pred n) ] ].
+
+let rec vopen_tm_aux u x k ≝ match u with
+ [ var n ⇒ vopen_var n x k
+ | par x0 ⇒ u
+ | app v1 v2 ⇒ app (vopen_tm_aux v1 x k) (vopen_tm_aux v2 x k)
+ | abs s v ⇒ abs s (vopen_tm_aux v x (S k)) ].
+definition vopen_tm ≝ λu,x.vopen_tm_aux u x O.
+
+let rec FV u ≝ match u with
+ [ par x ⇒ [x]
+ | app v1 v2 ⇒ FV v1@FV v2
+ | abs s v ⇒ FV v
+ | _ ⇒ [ ] ].
+
+definition lam ≝ λx,s,u.abs s (vclose_tm u x).
+
+let rec Pi_map_tm p u on u ≝ match u with
+[ par x ⇒ par (p x)
+| var _ ⇒ u
+| app v1 v2 ⇒ app (Pi_map_tm p v1) (Pi_map_tm p v2)
+| abs s v ⇒ abs s (Pi_map_tm p v) ].
+
+interpretation "permutation of tm" 'middot p x = (Pi_map_tm p x).
+
+notation "hvbox(u⌈x⌉)"
+ with precedence 45
+ for @{ 'open $u $x }.
+
+(*
+notation "hvbox(u⌈x⌉)"
+ with precedence 45
+ for @{ 'open $u $x }.
+notation "❴ u ❵ x" non associative with precedence 90 for @{ 'open $u $x }.
+*)
+interpretation "ln term variable open" 'open u x = (vopen_tm u x).
+notation < "hvbox(ν x break . u)"
+ with precedence 20
+for @{'nu $x $u }.
+notation > "ν 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 ≝ match u with
+[ app v1 v2 ⇒ S (max (tm_height v1) (tm_height v2))
+| abs s v ⇒ S (tm_height v)
+| _ ⇒ O ].
+
+theorem le_n_O_rect_Type0 : ∀n:nat. n ≤ O → ∀P: nat →Type[0]. P O → P n.
+#n (cases n) // #a #abs cases (?:False) /2/ qed.
+
+theorem nat_rect_Type0_1 : ∀n:nat.∀P:nat → Type[0].
+(∀m.(∀p. p < m → P p) → P m) → P n.
+#n #P #H
+cut (∀q:nat. q ≤ n → 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 : ∀n,m. leb n m = false → m < n.
+#n elim n
+[ #m normalize #H destruct(H)
+| #n0 #IH * // #m normalize #H @le_S_S @IH // ]
+qed.
+
+lemma nominal_eta_aux : ∀x,u.x ∉ FV u → ∀k.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 … 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 … Hleb) //
+ | @le_to_leb_true cases (leb_false_to_lt … Hleb) normalize /2/ ] ] ]
+| #y normalize #Hy >(\bf ?) // @(not_to_not … Hy) //
+| #s #v #IH normalize #Hv #k >IH // @Hv
+| #v1 #v2 #IH1 #IH2 normalize #Hv1v2 #k
+ >IH1 [ >IH2 // | @(not_to_not … Hv1v2) @in_list_to_in_list_append_l ]
+ @(not_to_not … Hv1v2) @in_list_to_in_list_append_r ]
+qed.
+
+corollary nominal_eta : ∀x,u.x ∉ FV u → (νx.u⌈x⌉) = u.
+#x #u #Hu @nominal_eta_aux //
+qed.
+
+lemma eq_height_vopen_aux : ∀v,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 : ∀v,x.tm_height (v⌈x⌉) = tm_height v.
+#v #x @eq_height_vopen_aux
+qed.
+
+theorem pretm_ind_plus_aux :
+ ∀P:pretm → Type[0].
+ (∀x:𝔸.P (par x)) →
+ (∀n:ℕ.P (var n)) →
+ (∀v1,v2. P v1 → P v2 → P (app v1 v2)) →
+ ∀C:list 𝔸.
+ (∀x,s,v.x ∉ FV v → x ∉ C → P (v⌈x⌉) → P (lam x s (v⌈x⌉))) →
+ ∀n,u.tm_height u ≤ n → P u.
+#P #Hpar #Hvar #Happ #C #Hlam #n change with ((λn.?) 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 ⊢ (?%?→?); #Hfalse cases (?:False) cases (not_le_Sn_O (max ??))
+ [ #H @H @Hfalse|*:skip] ] ]
+-m #m #IH * /2/
+[ #s #v whd in ⊢ (?%?→?); #Hv
+ lapply (p_fresh … (C@FV v)) letin y ≝ (N_fresh … (C@FV v)) #Hy
+ >(?:abs s v = lam y s (v⌈y⌉))
+ [| whd in ⊢ (???%); >nominal_eta // @(not_to_not … Hy) @in_list_to_in_list_append_r ]
+ @Hlam
+ [ @(not_to_not … Hy) @in_list_to_in_list_append_r
+ | @(not_to_not … Hy) @in_list_to_in_list_append_l ]
+ @IH [| @Hv | >eq_height_vopen % ]
+| #v1 #v2 whd in ⊢ (?%?→?); #Hv @Happ
+ [ @IH [| @Hv | @le_max_1 ] | @IH [| @Hv | @le_max_2 ] ] ]
+qed.
+
+corollary pretm_ind_plus :
+ ∀P:pretm → Type[0].
+ (∀x:𝔸.P (par x)) →
+ (∀n:ℕ.P (var n)) →
+ (∀v1,v2. P v1 → P v2 → P (app v1 v2)) →
+ ∀C:list 𝔸.
+ (∀x,s,v.x ∉ FV v → x ∉ C → P (v⌈x⌉) → P (lam x s (v⌈x⌉))) →
+ ∀u.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 : (𝔸 → 𝔸) → list 𝔸 → list 𝔸 ≝ map 𝔸 𝔸 .
+
+(* interpretation "permutation of name list" 'middot p x = (Pi_map_list p x).*)
+
+(*
+inductive tm : pretm → Prop ≝
+| tm_par : ∀x:𝔸.tm (par x)
+| tm_app : ∀u,v.tm u → tm v → tm (app u v)
+| tm_lam : ∀x,s,u.tm u → tm (lam x s u).
+
+inductive ctx_aux : nat → pretm → Prop ≝
+| ctx_var : ∀n,k.n < k → ctx_aux k (var n)
+| ctx_par : ∀x,k.ctx_aux k (par x)
+| ctx_app : ∀u,v,k.ctx_aux k u → ctx_aux k v → ctx_aux k (app u v)
+(* è sostituibile da ctx_lam ? *)
+| ctx_abs : ∀s,u.ctx_aux (S k) u → ctx_aux k (abs s u).
+*)
+
+inductive tm_or_ctx (k:nat) : pretm → Type[0] ≝
+| toc_var : ∀n.n < k → tm_or_ctx k (var n)
+| toc_par : ∀x.tm_or_ctx k (par x)
+| toc_app : ∀u,v.tm_or_ctx k u → tm_or_ctx k v → tm_or_ctx k (app u v)
+| toc_lam : ∀x,s,u.tm_or_ctx k u → tm_or_ctx k (lam x s u).
+
+definition tm ≝ λt.tm_or_ctx O t.
+definition ctx ≝ λt.tm_or_ctx 1 t.
+
+definition check_tm ≝ λu,k.
+ pretm_ind_plus ?
+ (λ_.true)
+ (λn.leb (S n) k)
+ (λv1,v2,rv1,rv2.rv1 ∧ rv2)
+ [] (λx,s,v,px,pC,rv.rv)
+ u.
+
+axiom pretm_ind_plus_app : ∀P,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 : ∀P,x,s,u,C,px,pC,H1,H2,H3,H4.
+ pretm_ind_plus P H1 H2 H3 C H4 (lam x s (u⌈x⌉)) =
+ H4 x s u px pC (pretm_ind_plus P H1 H2 H3 C H4 (u⌈x⌉)).
+
+record TM : Type[0] ≝ {
+ pretm_of_TM :> pretm;
+ tm_of_TM : check_tm pretm_of_TM O = true
+}.
+
+record CTX : Type[0] ≝ {
+ pretm_of_CTX :> pretm;
+ ctx_of_CTX : check_tm pretm_of_CTX 1 = true
+}.
+
+inductive tm2 : pretm → Type[0] ≝
+| tm_par : ∀x.tm2 (par x)
+| tm_app : ∀u,v.tm2 u → tm2 v → tm2 (app u v)
+| tm_lam : ∀x,s,u.x ∉ FV u → (∀y.y ∉ FV u → tm2 (u⌈y⌉)) → tm2 (lam x s (u⌈x⌉)).
+
+(*
+inductive tm' : pretm → Prop ≝
+| tm_par : ∀x.tm' (par x)
+| tm_app : ∀u,v.tm' u → tm' v → tm' (app u v)
+| tm_lam : ∀x,s,u,C.x ∉ FV u → x ∉ C → (∀y.y ∉ FV u → tm' (❴u❵y)) → tm' (lam x s (❴u❵x)).
+*)
+
+lemma pi_vclose_tm :
+ ∀z1,z2,x,u.swap 𝔸 z1 z2·(νx.u) = (ν swap ? z1 z2 x.swap 𝔸 z1 z2 · u).
+#z1 #z2 #x #u
+change with (vclose_tm_aux ???) in match (vclose_tm ??);
+change with (vclose_tm_aux ???) in ⊢ (???%); 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 :
+ ∀z1,z2,x,u.swap 𝔸 z1 z2·(u⌈x⌉) = (swap 𝔸 z1 z2 · u⌈swap 𝔸 z1 z2 x⌉).
+#z1 #z2 #x #u
+change with (vopen_tm_aux ???) in match (vopen_tm ??);
+change with (vopen_tm_aux ???) in ⊢ (???%); 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 :
+ ∀z1,z2,x,s,u.swap 𝔸 z1 z2 · lam x s u = lam (swap 𝔸 z1 z2 x) s (swap 𝔸 z1 z2 · u).
+#z1 #z2 #x #s #u whd in ⊢ (???%); <(pi_vclose_tm …) %
+qed.
+
+lemma eqv_FV : ∀z1,z2,u.FV (swap 𝔸 z1 z2 · u) = Pi_map_list (swap 𝔸 z1 z2) (FV u).
+#z1 #z2 #u elim u //
+[ #s #v normalize //
+| #v1 #v2 normalize /2/ ]
+qed.
+
+lemma swap_inv_tm : ∀z1,z2,u.swap 𝔸 z1 z2 · (swap 𝔸 z1 z2 · u) = u.
+#z1 #z2 #u elim u [1,3,4:normalize //]
+#x whd in ⊢ (??%?); >swap_inv %
+qed.
+
+lemma eqv_in_list : ∀x,l,z1,z2.x ∈ l → swap 𝔸 z1 z2 x ∈ Pi_map_list (swap 𝔸 z1 z2) l.
+#x #l #z1 #z2 #Hin elim Hin
+[ #x0 #l0 %
+| #x1 #x2 #l0 #Hin #IH %2 @IH ]
+qed.
+
+lemma eqv_tm2 : ∀u.tm2 u → ∀z1,z2.tm2 ((swap ? z1 z2)·u).
+#u #Hu #z1 #z2 letin p ≝ (swap ? z1 z2) elim Hu /2/
+#x #s #v #Hx #Hv #IH >pi_lam >pi_vopen_tm %3
+[ @(not_to_not … 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 … Hy) -Hy #Hy <(swap_inv ? z1 z2 y)
+ >eqv_FV @eqv_in_list //
+]
+qed.
+
+lemma vclose_vopen_aux : ∀x,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 ⊢ (??%?);
+ [ cases (true_or_false (eqb (S n) k)) #H1 >H1
+ [ <(eqb_true_to_eq … H1) in H; #H lapply (leb_true_to_le … H) -H #H
+ cases (le_to_not_lt … H) -H #H cases (H ?) %
+ | whd in ⊢ (??%?); >lt_to_leb_false // @le_S_S /2/ ]
+ | cases (true_or_false (eqb n k)) #H1 >H1 normalize
+ [ >(eqb_true_to_eq … H1) in H; #H lapply (leb_false_to_not_le … 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 : ∀x,u.((νx.u)⌈x⌉) = u. #x #u @vclose_vopen_aux
+qed.
+
+(*
+theorem tm_to_tm : ∀t.tm' t → tm t.
+#t #H elim H
+*)
+
+lemma in_list_singleton : ∀T.∀t1,t2:T.t1 ∈ [t2] → t1 = t2.
+#T #t1 #t2 #H @(in_list_inv_ind ??? H) /2/
+qed.
+
+lemma fresh_vclose_tm_aux : ∀u,x,k.x ∉ 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 : ∀u,x.x ∉ FV (νx.u). //
+qed.
+
+lemma check_tm_true_to_toc : ∀u,k.check_tm u k = true → tm_or_ctx k u.
+#u @(pretm_ind_plus ???? [ ] ? u)
+[ #x #k #_ %2
+| #n #k change with (leb (S n) k) in ⊢ (??%?→?); #H % @leb_true_to_le //
+| #v1 #v2 #rv1 #rv2 #k change with (pretm_ind_plus ???????) in ⊢ (??%?→?);
+ >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 ⊢ (??%?→?);
+ >pretm_ind_plus_lam // #Hv %4 @rv @Hv ]
+qed.
+
+lemma toc_to_check_tm_true : ∀u,k.tm_or_ctx k u → check_tm u k = true.
+#u #k #Hu elim Hu //
+[ #n #Hn change with (leb (S n) k) in ⊢ (??%?); @le_to_leb_true @Hn
+| #v1 #v2 #Hv1 #Hv2 #IH1 #IH2 change with (pretm_ind_plus ???????) in ⊢ (??%?);
+ >pretm_ind_plus_app change with (check_tm v1 k ∧ check_tm v2 k) in ⊢ (??%?); /2/
+| #x #s #v #Hv #IH <(vclose_vopen x v) change with (pretm_ind_plus ???????) in ⊢ (??%?);
+ >pretm_ind_plus_lam [| // | @fresh_vclose_tm ] >(vclose_vopen x v) @IH ]
+qed.
+
+lemma fresh_swap_tm : ∀z1,z2,u.z1 ∉ FV u → z2 ∉ FV u → swap 𝔸 z1 z2 · u = u.
+#z1 #z2 #u elim u
+[2: normalize in ⊢ (?→%→%→?); #x #Hz1 #Hz2 whd in ⊢ (??%?); >swap_other //
+ [ @(not_to_not … Hz2) | @(not_to_not … Hz1) ] //
+|1: //
+| #s #v #IH normalize #Hz1 #Hz2 >IH // [@Hz2|@Hz1]
+| #v1 #v2 #IH1 #IH2 normalize #Hz1 #Hz2
+ >IH1 [| @(not_to_not … Hz2) @in_list_to_in_list_append_l | @(not_to_not … Hz1) @in_list_to_in_list_append_l ]
+ >IH2 // [@(not_to_not … Hz2) @in_list_to_in_list_append_r | @(not_to_not … Hz1) @in_list_to_in_list_append_r ]
+]
+qed.
+
+theorem tm_to_tm2 : ∀u.tm u → 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 (νx.u)) /2/ @fresh_vclose_tm ]
+]
+qed.
+
+theorem tm2_to_tm : ∀u.tm2 u → tm u.
+#u #pu elim pu /2/ #x #s #v #Hx #Hv #IH %4 @IH //
+qed.
+
+(* define PAR APP LAM *)
+definition PAR ≝ λx.mk_TM (par x) ?. // qed.
+definition APP ≝ λu,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 ⊢ (??(??%)?);
+@andb_elim >(tm_of_TM u) >(tm_of_TM v) % qed.
+definition LAM ≝ λx,s.λu: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 : ∀u,x,k.tm_or_ctx (S k) u → tm_or_ctx k (u⌈x⌉).
+(* 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 ≝ λu:CTX.λx.mk_TM (u⌈x⌉) ?.
+@toc_to_check_tm_true @vopen_tm_down @check_tm_true_to_toc @ctx_of_CTX qed.
+
+axiom vclose_tm_up : ∀u,x,k.tm_or_ctx k u → tm_or_ctx (S k) (νx.u).
+
+definition vclose_TM ≝ λu:TM.λx.mk_CTX (νx.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 : ∀x,y,s,u.x ∉ FV u → y ∉ FV u → lam x s (u⌈x⌉) = lam y s (u⌈y⌉).
+#x #y #s #u #Hx #Hy whd in ⊢ (??%%); @eq_f >nominal_eta // >nominal_eta //
+qed.
+
+lemma TM_to_tm2 : ∀u:TM.tm2 u.
+#u @tm_to_tm2 @check_tm_true_to_toc @tm_of_TM qed.
+
+theorem TM_ind_plus_weak :
+ ∀P:pretm → Type[0].
+ (∀x:𝔸.P (PAR x)) →
+ (∀v1,v2:TM.P v1 → P v2 → P (APP v1 v2)) →
+ ∀C:list 𝔸.
+ (∀x,s.∀v:CTX.x ∉ FV v → x ∉ C →
+ (∀y.y ∉ FV v → P (v⌈y⌉)) → P (LAM x s (v⌈x⌉))) →
+ ∀u:TM.P u.
+#P #Hpar #Happ #C #Hlam #u elim (TM_to_tm2 u) //
+[ #v1 #v2 #pv1 #pv2 #IH1 #IH2 @(Happ (mk_TM …) (mk_TM …) IH1 IH2)
+ @toc_to_check_tm_true @tm2_to_tm //
+| #x #s #v #Hx #pv #IH
+ lapply (p_fresh … (C@FV v)) letin x0 ≝ (N_fresh … (C@FV v)) #Hx0
+ >(?:lam x s (v⌈x⌉) = lam x0 s (v⌈x0⌉))
+ [|@tm_alpha // @(not_to_not … Hx0) @in_list_to_in_list_append_r ]
+ @(Hlam x0 s (mk_CTX v ?) ??)
+ [ <(nominal_eta … Hx) @toc_to_check_tm_true @vclose_tm_up @tm2_to_tm @pv //
+ | @(not_to_not … Hx0) @in_list_to_in_list_append_r
+ | @(not_to_not … Hx0) @in_list_to_in_list_append_l
+ | @IH ]
+]
+qed.
+
+lemma eq_mk_TM : ∀u,v.u = v → ∀pu,pv.mk_TM u pu = mk_TM v pv.
+#u #v #Heq >Heq #pu #pv %
+qed.
+
+lemma eq_P : ∀T:Type[0].∀t1,t2:T.t1 = t2 → ∀P:T → Type[0].P t1 → P t2. // qed.
+
+theorem TM_ind_plus :
+ ∀P:TM → Type[0].
+ (∀x:𝔸.P (PAR x)) →
+ (∀v1,v2:TM.P v1 → P v2 → P (APP v1 v2)) →
+ ∀C:list 𝔸.
+ (∀x,s.∀v:CTX.x ∉ FV v → x ∉ C →
+ (∀y.y ∉ FV v → P (v⌈y⌉)) → P (LAM x s (v⌈x⌉))) →
+ ∀u:TM.P u.
+#P #Hpar #Happ #C #Hlam * #u #pu
+>(?:mk_TM u pu =
+ mk_TM u (toc_to_check_tm_true … (tm2_to_tm … (tm_to_tm2 … (check_tm_true_to_toc … pu))))) [|%]
+elim (tm_to_tm2 u ?) //
+[ #v1 #v2 #pv1 #pv2 #IH1 #IH2 @(Happ (mk_TM …) (mk_TM …) IH1 IH2)
+| #x #s #v #Hx #pv #IH
+ lapply (p_fresh … (C@FV v)) letin x0 ≝ (N_fresh … (C@FV v)) #Hx0
+ lapply (Hlam x0 s (mk_CTX v ?) ???)
+ [2: @(not_to_not … 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 (v⌈y⌉) ? = mk_TM (v⌈y⌉) (toc_to_check_tm_true (v⌈y⌉) O (tm2_to_tm (v⌈y⌉) (pv y Hy))))
+ [@IH|%]
+ | @(not_to_not … 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 … Hx0) @in_list_to_in_list_append_r ]
+]
+qed.
+
+notation
+"hvbox('nominal' u 'return' out 'with'
+ [ 'xpar' ident x ⇒ f1
+ | 'xapp' ident v1 ident v2 ident recv1 ident recv2 ⇒ f2
+ | 'xlam' ❨ident y # C❩ ident s ident w ident py1 ident py2 ident recw ⇒ f3 ])"
+with precedence 48
+for @{ TM_ind_plus $out (λ${ident x}:?.$f1)
+ (λ${ident v1}:?.λ${ident v2}:?.λ${ident recv1}:?.λ${ident recv2}:?.$f2)
+ $C (λ${ident y}:?.λ${ident s}:?.λ${ident w}:?.λ${ident py1}:?.λ${ident py2}:?.λ${ident recw}:?.$f3)
+ $u }.
+
+(* include "basics/jmeq.ma".*)
+
+definition subst ≝ (λu:TM.λx,v.
+ nominal u return (λ_.TM) with
+ [ xpar x0 ⇒ match x == x0 with [ true ⇒ v | false ⇒ PAR x0 ] (* u instead of PAR x0 does not work, u stays the same at every rec call! *)
+ | xapp v1 v2 recv1 recv2 ⇒ APP recv1 recv2
+ | xlam ❨y # x::FV v❩ s w py1 py2 recw ⇒ LAM y s (recw y py1) ]).
+
+lemma subst_def : ∀u,x,v.subst u x v =
+ nominal u return (λ_.TM) with
+ [ xpar x0 ⇒ match x == x0 with [ true ⇒ v | false ⇒ PAR x0 ]
+ | xapp v1 v2 recv1 recv2 ⇒ APP recv1 recv2
+ | xlam ❨y # x::FV v❩ s w py1 py2 recw ⇒ LAM y s (recw y py1) ]. //
+qed.
+
+axiom TM_ind_plus_LAM :
+ ∀x,s,u,out,f1,f2,C,f3,Hx1,Hx2.
+ TM_ind_plus out f1 f2 C f3 (LAM x s (u⌈x⌉)) =
+ f3 x s u Hx1 Hx2 (λy,Hy.TM_ind_plus ? f1 f2 C f3 ?).
+
+axiom TM_ind_plus_APP :
+ ∀u1,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 : ∀u,v.u = v → ∀pu,pv.mk_CTX u pu = mk_CTX v pv.
+#u #v #Heq >Heq #pu #pv %
+qed.
+
+lemma vclose_vopen_TM : ∀x.∀u:TM.((νx.u)⌈x⌉) = u.
+#x * #u #pu @eq_mk_TM @vclose_vopen qed.
+
+lemma nominal_eta_CTX : ∀x.∀u:CTX.x ∉ FV u → (νx.(u⌈x⌉)) = u.
+#x * #u #pu #Hx @eq_mk_CTX @nominal_eta // qed.
+
+theorem TM_alpha : ∀x,y,s.∀u:CTX.x ∉ FV u → y ∉ FV u → LAM x s (u⌈x⌉) = LAM y s (u⌈y⌉).
+#x #y #s #u #Hx #Hy @eq_mk_TM @tm_alpha // qed.
+
+axiom in_vopen_CTX : ∀x,y.∀v:CTX.x ∈ FV (v⌈y⌉) → x = y ∨ x ∈ FV v.
+
+theorem subst_fresh : ∀u,v:TM.∀x.x ∉ FV u → subst u x v = u.
+#u #v #x @(TM_ind_plus … (x::FV v) … u)
+[ #x0 normalize in ⊢ (%→?); #Hx normalize in ⊢ (??%?);
+ >(\bf ?) [| @(not_to_not … Hx) #Heq >Heq % ] %
+| #u1 #u2 #IH1 #IH2 normalize in ⊢ (%→?); #Hx
+ >subst_def >TM_ind_plus_APP @eq_mk_TM @eq_f2 @eq_f
+ [ <subst_def @IH1 @(not_to_not … Hx) @in_list_to_in_list_append_l
+ | <subst_def @IH2 @(not_to_not … 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 … Hx) -Hx #Hx
+ change with (FV (νx0.(v0⌈x0⌉))) in ⊢ (???%); >nominal_eta_CTX //
+ cases (in_vopen_CTX … Hx) // #Heq >Heq in HC; * #HC @False_ind @HC %
+]
+qed.
+
+example subst_LAM_same : ∀x,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 … (FV (νx.u)@x::FV v)) letin x0 ≝ (N_fresh … (FV (νx.u)@x::FV v)) #Hx0
+>(TM_alpha x x0)
+[| @(not_to_not … Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_l @Hx0 | @fresh_vclose_tm ]
+>TM_ind_plus_LAM [| @(not_to_not … Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_r @Hx0 | @(not_to_not … Hx0) -Hx0 #Hx0 @in_list_to_in_list_append_l @Hx0 ]
+@eq_f change with (subst ((νx.u)⌈x0⌉) x v) in ⊢ (??%?); @subst_fresh
+@(not_to_not … Hx0) #Hx0' cases (in_vopen_CTX … Hx0')
+[ #Heq >Heq @in_list_to_in_list_append_r %
+| #Hfalse @False_ind cases (fresh_vclose_tm u x) #H @H @Hfalse ]
+qed.
+
+(*
+notation > "Λ ident x. ident T [ident x] ↦ P"
+ with precedence 48 for @{'foo (λ${ident x}.λ${ident T}.$P)}.
+
+notation < "Λ ident x. ident T [ident x] ↦ P"
+ with precedence 48 for @{'foo (λ${ident x}:$Q.λ${ident T}:$R.$P)}.
+*)
+
+(*
+notation
+"hvbox('nominal' u 'with'
+ [ 'xpar' ident x ⇒ f1
+ | 'xapp' ident v1 ident v2 ⇒ f2
+ | 'xlam' ident x # C s w ⇒ f3 ])"
+with precedence 48
+for @{ tm2_ind_plus ? (λ${ident x}:$Tx.$f1)
+ (λ${ident v1}:$Tv1.λ${ident v2}:$Tv2.λ${ident pv1}:$Tpv1.λ${ident pv2}:$Tpv2.λ${ident recv1}:$Trv1.λ${ident recv2}:$Trv2.$f2)
+ $C (λ${ident x}:$Tx.λ${ident s}:$Ts.λ${ident w}:$Tw.λ${ident py1}:$Tpy1.λ${ident py2}:$Tpy2.λ${ident pw}:$Tpw.λ${ident recw}:$Trw.$f3) $u (tm_to_tm2 ??) }.
+*)
+
+(*
+notation
+"hvbox('nominal' u 'with'
+ [ 'xpar' ident x ^ f1
+ | 'xapp' ident v1 ident v2 ^ f2 ])"
+(* | 'xlam' ident x # C s w ^ f3 ]) *)
+with precedence 48
+for @{ tm2_ind_plus ? (λ${ident x}:$Tx.$f1)
+ (λ${ident v1}:$Tv1.λ${ident v2}:$Tv2.λ${ident pv1}:$Tpv1.λ${ident pv2}:$Tpv2.λ${ident recv1}:$Trv1.λ${ident recv2}:$Trv2.$f2)
+ $C (λ${ident x}:$Tx.λ${ident s}:$Ts.λ${ident w}:$Tw.λ${ident py1}:$Tpy1.λ${ident py2}:$Tpy2.λ${ident pw}:$Tpw.λ${ident recw}:$Trw.$f3) $u (tm_to_tm2 ??) }.
+*)
+notation
+"hvbox('nominal' u 'with'
+ [ 'xpar' ident x ^ f1
+ | 'xapp' ident v1 ident v2 ^ f2 ])"
+with precedence 48
+for @{ tm2_ind_plus ? (λ${ident x}:?.$f1)
+ (λ${ident v1}:$Tv1.λ${ident v2}:$Tv2.λ${ident pv1}:$Tpv1.λ${ident pv2}:$Tpv2.λ${ident recv1}:$Trv1.λ${ident recv2}:$Trv2.$f2)
+ $C (λ${ident x}:?.λ${ident s}:$Ts.λ${ident w}:$Tw.λ${ident py1}:$Tpy1.λ${ident py2}:$Tpy2.λ${ident pw}:$Tpw.λ${ident recw}:$Trw.$f3) $u (tm_to_tm2 ??) }.
+
+axiom in_Env : 𝔸 × tp → Env → Prop.
+notation "X ◃ G" non associative with precedence 45 for @{'lefttriangle $X $G}.
+interpretation "Env membership" 'lefttriangle x l = (in_Env x l).
+
+
+
+inductive judg : list tp → tm → tp → Prop ≝
+| t_var : ∀g,n,t.Nth ? n g = Some ? t → judg g (var n) t
+| t_app : ∀g,m,n,t,u.judg g m (arr t u) → judg g n t → judg g (app m n) u
+| t_abs : ∀g,t,m,u.judg (t::g) m u → judg g (abs t m) (arr t u).
+
+definition Env := list (𝔸 × tp).
+
+axiom vclose_env : Env → list tp.
+axiom vclose_tm : Env → tm → tm.
+axiom Lam : 𝔸 → tp → tm → tm.
+definition Judg ≝ λG,M,T.judg (vclose_env G) (vclose_tm G M) T.
+definition dom ≝ λG:Env.map ?? (fst ??) G.
+
+definition sctx ≝ 𝔸 × tm.
+axiom swap_tm : 𝔸 → 𝔸 → tm → tm.
+definition sctx_app : sctx → 𝔸 → tm ≝ λM0,Y.let 〈X,M〉 ≝ M0 in swap_tm X Y M.
+
+axiom in_list : ∀A:Type[0].A → list A → 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 : 𝔸 × tp → Env → Prop.
+notation "X ◃ G" non associative with precedence 45 for @{'lefttriangle $X $G}.
+interpretation "Env membership" 'lefttriangle x l = (in_Env x l).
+
+(* axiom Lookup : 𝔸 → Env → option tp. *)
+
+(* forma alto livello del judgment
+ t_abs* : ∀G,T,X,M,U.
+ (∀Y ∉ supp(M).Judg (〈Y,T〉::G) (M[Y]) U) →
+ Judg G (Lam X T (M[X])) (arr T U) *)
+
+(* prima dimostrare, poi perfezionare gli assiomi, poi dimostrarli *)
+
+axiom Judg_ind : ∀P:Env → tm → tp → Prop.
+ (∀X,G,T.〈X,T〉 ◃ G → P G (par X) T) →
+ (∀G,M,N,T,U.
+ Judg G M (arr T U) → Judg G N T →
+ P G M (arr T U) → P G N T → P G (app M N) U) →
+ (∀G,T1,T2,X,M1.
+ (∀Y.Y ∉ (FV (Lam X T1 (sctx_app M1 X))) → Judg (〈Y,T1〉::G) (sctx_app M1 Y) T2) →
+ (∀Y.Y ∉ (FV (Lam X T1 (sctx_app M1 X))) → P (〈Y,T1〉::G) (sctx_app M1 Y) T2) →
+ P G (Lam X T1 (sctx_app M1 X)) (arr T1 T2)) →
+ ∀G,M,T.Judg G M T → P G M T.
+
+axiom t_par : ∀X,G,T.〈X,T〉 ◃ G → Judg G (par X) T.
+axiom t_app2 : ∀G,M,N,T,U.Judg G M (arr T U) → Judg G N T → Judg G (app M N) U.
+axiom t_Lam : ∀G,X,M,T,U.Judg (〈X,T〉::G) M U → Judg G (Lam X T M) (arr T U).
+
+definition subenv ≝ λG1,G2.∀x.x ◃ G1 → x ◃ G2.
+interpretation "subenv" 'subseteq G1 G2 = (subenv G1 G2).
+
+axiom daemon : ∀P:Prop.P.
+
+theorem weakening : ∀G1,G2,M,T.G1 ⊆ G2 → Judg G1 M T → Judg G2 M T.
+#G1 #G2 #M #T #Hsub #HJ lapply Hsub lapply G2 -G2 change with (∀G2.?)
+@(Judg_ind … 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
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