From: Wilmer Ricciotti <ricciott@cs.unibo.it>
Date: Thu, 25 Apr 2013 07:15:53 +0000 (+0000)
Subject: An attempt at ostensiby named syntax.
X-Git-Tag: make_still_working~1178
X-Git-Url: http://matita.cs.unibo.it/gitweb/?a=commitdiff_plain;h=4a41f041869956a6c98418e6d06b3a2521a1fb2a;p=helm.git

An attempt at ostensiby named syntax.
---

diff --git a/matita/matita/lib/basics/core_notation.ma b/matita/matita/lib/basics/core_notation.ma
index 3f6eda5a7..dd672ee84 100644
--- a/matita/matita/lib/basics/core_notation.ma
+++ b/matita/matita/lib/basics/core_notation.ma
@@ -266,6 +266,9 @@ for @{ 'comprehension $s (\lambda ${ident i}. $p)}.
 notation "hvbox(a break ∈ b)" non associative with precedence 45
 for @{ 'mem $a $b }.
 
+notation "hvbox(a break ∉ b)" non associative with precedence 45
+for @{ 'notmem $a $b }.
+
 notation "hvbox(a break ≬ b)" non associative with precedence 45
 for @{ 'overlaps $a $b }. (* \between *)
 
diff --git a/matita/matita/lib/binding/db.ma b/matita/matita/lib/binding/db.ma
new file mode 100644
index 000000000..534874f21
--- /dev/null
+++ b/matita/matita/lib/binding/db.ma
@@ -0,0 +1,110 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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".
+
+axiom alpha : Type[0].
+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 tm : Type[0] ≝ 
+| var : nat → tm
+| par :  𝔸 → tm
+| abs : tp → tm → tm
+| app : tm → tm → tm.
+
+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 ] ].
+
+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).
+
+let rec FV M ≝ match M with 
+  [ par X ⇒ [X]
+  | app M1 M2 ⇒ FV M1@FV M2
+  | abs T M0 ⇒ FV M0
+  | _ ⇒ [ ] ].
+
+(* 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
diff --git a/matita/matita/lib/binding/fp.ma b/matita/matita/lib/binding/fp.ma
new file mode 100644
index 000000000..14ba2f774
--- /dev/null
+++ b/matita/matita/lib/binding/fp.ma
@@ -0,0 +1,296 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "binding/names.ma".
+
+(* permutations *)
+definition finite_perm : ∀X:Nset.(X → X) → Prop ≝ 
+ λX,f.injective X X f ∧ surjective X X f ∧ ∃l.∀x.x ∉ l → f x = x.
+
+(* maps a permutation to a list of parameters *)
+definition Pi_list : ∀X:Nset.(X → X) → list X → list X ≝ 
+ λX,p,xl.map ?? (λx.p x) xl.
+
+interpretation "permutation of X list" 'middot p x = (Pi_list p x).
+
+definition swap : ∀N:Nset.N → N → N → N ≝
+  λN,u,v,x.match (x == u) with
+    [true ⇒ v
+    |false ⇒ match (x == v) with
+       [true ⇒ u
+       |false ⇒ x]].
+       
+axiom swap_right : ∀N,x,y.swap N x y y = x.
+(*
+#N x y;nnormalize;nrewrite > (p_eqb3 ? y y …);//;
+nlapply (refl ? (y ≟ x));ncases (y ≟ x) in ⊢ (???% → %);nnormalize;//;
+#H1;napply p_eqb1;//;
+nqed.
+*)
+
+axiom swap_left : ∀N,x,y.swap N x y x = y.
+(*
+#N x y;nnormalize;nrewrite > (p_eqb3 ? x x …);//;
+nqed.
+*)
+
+axiom swap_other : ∀N,x,y,z.x ≠ z → y ≠ z → swap N x y z = z.
+(*
+#N x y z H1 H2;nnormalize;nrewrite > (p_eqb4 …);
+##[nrewrite > (p_eqb4 …);//;@;ncases H2;/2/;
+##|@;ncases H1;/2/
+##]
+nqed.
+*)
+
+axiom swap_inv : ∀N,x,y,z.swap N x y (swap N x y z) = z.
+(*
+#N x y z;nlapply (refl ? (x ≟ z));ncases (x ≟ z) in ⊢ (???% → ?);#H1
+##[nrewrite > (p_eqb1 … H1);nrewrite > (swap_left …);//;
+##|nlapply (refl ? (y ≟ z));ncases (y ≟ z) in ⊢ (???% → ?);#H2
+   ##[nrewrite > (p_eqb1 … H2);nrewrite > (swap_right …);//;
+   ##|nrewrite > (swap_other …) in ⊢ (??(????%)?);/2/;
+      nrewrite > (swap_other …);/2/;
+   ##]
+##]
+nqed.
+*)
+
+axiom swap_fp : ∀N,x1,x2.finite_perm ? (swap N x1 x2).
+(*
+#N x1 x2;@
+##[@
+   ##[nwhd;#xa xb;nnormalize;nlapply (refl ? (xa ≟ x1));
+      ncases (xa ≟ x1) in ⊢ (???% → %);#H1
+      ##[nrewrite > (p_eqb1 … H1);nlapply (refl ? (xb ≟ x1));
+         ncases (xb ≟ x1) in ⊢ (???% → %);#H2
+         ##[nrewrite > (p_eqb1 … H2);//
+         ##|nlapply (refl ? (xb ≟ x2));
+            ncases (xb ≟ x2) in ⊢ (???% → %);#H3
+            ##[nnormalize;#H4;nrewrite > H4 in H3;
+               #H3;nrewrite > H3 in H2;#H2;ndestruct (H2)
+            ##|nnormalize;#H4;nrewrite > H4 in H3;
+               nrewrite > (p_eqb3 …);//;#H5;ndestruct (H5)
+            ##]
+         ##]
+      ##|nlapply (refl ? (xa ≟ x2));
+         ncases (xa ≟ x2) in ⊢ (???% → %);#H2
+         ##[nrewrite > (p_eqb1 … H2);nlapply (refl ? (xb ≟ x1));
+            ncases (xb ≟ x1) in ⊢ (???% → %);#H3
+            ##[nnormalize;#H4;nrewrite > H4 in H3;
+               #H3;nrewrite > (p_eqb1 … H3);@
+            ##|nnormalize;nlapply (refl ? (xb ≟ x2));
+               ncases (xb ≟ x2) in ⊢ (???% → %);#H4
+               ##[nrewrite > (p_eqb1 … H4);//
+               ##|nnormalize;#H5;nrewrite > H5 in H3;
+                  nrewrite > (p_eqb3 …);//;#H6;ndestruct (H6);
+               ##]
+            ##]
+         ##|nnormalize;nlapply (refl ? (xb ≟ x1));
+            ncases (xb ≟ x1) in ⊢ (???% → %);#H3
+            ##[nnormalize;#H4;nrewrite > H4 in H2;nrewrite > (p_eqb3 …);//;
+               #H5;ndestruct (H5)
+            ##|nlapply (refl ? (xb ≟ x2));
+               ncases (xb ≟ x2) in ⊢ (???% → %);#H4
+               ##[nnormalize;#H5;nrewrite > H5 in H1;nrewrite > (p_eqb3 …);//;
+                  #H6;ndestruct (H6)
+               ##|nnormalize;//
+               ##]
+            ##]
+         ##]
+      ##]
+   ##|nwhd;#z;nnormalize;nlapply (refl ? (z ≟ x1));
+      ncases (z ≟ x1) in ⊢ (???% → %);#H1
+      ##[nlapply (refl ? (z ≟ x2));
+         ncases (z ≟ x2) in ⊢ (???% → %);#H2
+         ##[@ z;nrewrite > H1;nrewrite > H2;napply p_eqb1;//
+         ##|@ x2;nrewrite > (p_eqb4 …);
+            ##[nrewrite > (p_eqb3 …);//;
+               nnormalize;napply p_eqb1;//
+            ##|nrewrite < (p_eqb1 … H1);@;#H3;nrewrite > H3 in H2;
+               nrewrite > (p_eqb3 …);//;#H2;ndestruct (H2)
+            ##]
+         ##]
+      ##|nlapply (refl ? (z ≟ x2));
+         ncases (z ≟ x2) in ⊢ (???% → %);#H2
+         ##[@ x1;nrewrite > (p_eqb3 …);//;
+            napply p_eqb1;nnormalize;//
+         ##|@ z;nrewrite > H1;nrewrite > H2;@;
+         ##]
+      ##]
+   ##]
+##|@ [x1;x2];#x0 H1;nrewrite > (swap_other …)
+   ##[@
+   ##|@;#H2;nrewrite > H2 in H1;*;#H3;napply H3;/2/;
+   ##|@;#H2;nrewrite > H2 in H1;*;#H3;napply H3;//;
+   ##]
+##]
+nqed.
+*)
+
+axiom inj_swap : ∀N,u,v.injective ?? (swap N u v).
+(*
+#N u v;ncases (swap_fp N u v);*;#H1 H2 H3;//;
+nqed.
+*)
+
+axiom surj_swap : ∀N,u,v.surjective ?? (swap N u v).
+(*
+#N u v;ncases (swap_fp N u v);*;#H1 H2 H3;//;
+nqed.
+*)
+
+axiom finite_swap : ∀N,u,v.∃l.∀x.x ∉ l → swap N u v x = x.
+(*
+#N u v;ncases (swap_fp N u v);*;#H1 H2 H3;//;
+nqed.
+*)
+
+(* swaps two lists of parameters 
+definition Pi_swap_list : ∀xl,xl':list X.X → X ≝ 
+ λxl,xl',x.foldr2 ??? (λu,v,r.swap ? u v r) x xl xl'.
+
+nlemma fp_swap_list : 
+  ∀xl,xl'.finite_perm ? (Pi_swap_list xl xl').
+#xl xl';@
+##[@;
+   ##[ngeneralize in match xl';nelim xl
+      ##[nnormalize;//;
+      ##|#x0 xl0;#IH xl'';nelim xl''
+         ##[nnormalize;//
+         ##|#x1 xl1 IH1 y0 y1;nchange in ⊢ (??%% → ?) with (swap ????);
+            #H1;nlapply (inj_swap … H1);#H2;
+            nlapply (IH … H2);//
+         ##]
+      ##]
+   ##|ngeneralize in match xl';nelim xl
+      ##[nnormalize;#_;#z;@z;@
+      ##|#x' xl0 IH xl'';nelim xl''
+         ##[nnormalize;#z;@z;@
+         ##|#x1 xl1 IH1 z;
+            nchange in ⊢ (??(λ_.???%)) with (swap ????);
+            ncases (surj_swap X x' x1 z);#x2 H1;
+            ncases (IH xl1 x2);#x3 H2;@ x3;
+            nrewrite < H2;napply H1
+         ##]
+      ##]
+   ##]
+##|ngeneralize in match xl';nelim xl
+   ##[#;@ [];#;@
+   ##|#x0 xl0 IH xl'';ncases xl''
+      ##[@ [];#;@
+      ##|#x1 xl1;ncases (IH xl1);#xl2 H1;
+         ncases (finite_swap X x0 x1);#xl3 H2;
+         @ (xl2@xl3);#x2 H3;
+         nchange in ⊢ (??%?) with (swap ????);
+         nrewrite > (H1 …);
+         ##[nrewrite > (H2 …);//;@;#H4;ncases H3;#H5;napply H5;
+            napply in_list_to_in_list_append_r;//
+         ##|@;#H4;ncases H3;#H5;napply H5;
+            napply in_list_to_in_list_append_l;//
+         ##]
+      ##]
+   ##]
+##]
+nqed.
+
+(* the 'reverse' swap of lists of parameters 
+   composing Pi_swap_list and Pi_swap_list_r yields the identity function
+   (see the Pi_swap_list_inv lemma) *)
+ndefinition Pi_swap_list_r : ∀xl,xl':list X. Pi ≝ 
+ λxl,xl',x.foldl2 ??? (λr,u,v.swap ? u v r ) x xl xl'.
+
+nlemma fp_swap_list_r : 
+  ∀xl,xl'.finite_perm ? (Pi_swap_list_r xl xl').
+#xl xl';@
+##[@;
+   ##[ngeneralize in match xl';nelim xl
+      ##[nnormalize;//;
+      ##|#x0 xl0;#IH xl'';nelim xl''
+         ##[nnormalize;//
+         ##|#x1 xl1 IH1 y0 y1;nwhd in ⊢ (??%% → ?);
+            #H1;nlapply (IH … H1);#H2;
+            napply (inj_swap … H2);
+         ##]
+      ##]
+   ##|ngeneralize in match xl';nelim xl
+      ##[nnormalize;#_;#z;@z;@
+      ##|#x' xl0 IH xl'';nelim xl''
+         ##[nnormalize;#z;@z;@
+         ##|#x1 xl1 IH1 z;nwhd in ⊢ (??(λ_.???%));
+            ncases (IH xl1 z);#x2 H1;
+            ncases (surj_swap X x' x1 x2);#x3 H2;
+            @ x3;nrewrite < H2;napply H1;
+         ##]
+      ##]
+   ##]
+##|ngeneralize in match xl';nelim xl
+   ##[#;@ [];#;@
+   ##|#x0 xl0 IH xl'';ncases xl''
+      ##[@ [];#;@
+      ##|#x1 xl1;
+         ncases (IH xl1);#xl2 H1;
+         ncases (finite_swap X x0 x1);#xl3 H2;
+         @ (xl2@xl3);#x2 H3;nwhd in ⊢ (??%?);
+         nrewrite > (H2 …);
+         ##[nrewrite > (H1 …);//;@;#H4;ncases H3;#H5;napply H5;
+            napply in_list_to_in_list_append_l;//
+         ##|@;#H4;ncases H3;#H5;napply H5;
+            napply in_list_to_in_list_append_r;//
+         ##]
+      ##]
+   ##]
+##]
+nqed.
+
+nlemma Pi_swap_list_inv :
+ ∀xl1,xl2,x.
+ Pi_swap_list xl1 xl2 (Pi_swap_list_r xl1 xl2 x) = x.
+#xl;nelim xl
+##[#;@
+##|#x1 xl1 IH xl';ncases xl'
+   ##[#;@
+   ##|#x2 xl2;#x;
+      nchange in ⊢ (??%?) with 
+        (swap ??? (Pi_swap_list ?? 
+                  (Pi_swap_list_r ?? (swap ????))));
+      nrewrite > (IH xl2 ?);napply swap_inv;
+   ##]
+##]
+nqed.
+
+nlemma Pi_swap_list_fresh :
+  ∀x,xl1,xl2.x ∉ xl1 → x ∉ xl2 → Pi_swap_list xl1 xl2 x = x.
+#x xl1;nelim xl1
+##[#;@
+##|#x3 xl3 IH xl2' H1;ncases xl2'
+   ##[#;@
+   ##|#x4 xl4 H2;ncut (x ∉ xl3 ∧ x ∉ xl4);
+      ##[@
+         ##[@;#H3;ncases H1;#H4;napply H4;/2/;
+         ##|@;#H3;ncases H2;#H4;napply H4;/2/
+         ##]
+      ##] *; #H1' H2';
+      nchange in ⊢ (??%?) with (swap ????);
+      nrewrite > (swap_other …)
+      ##[napply IH;//;
+      ##|nchange in ⊢ (?(???%)) with (Pi_swap_list ???);
+         nrewrite > (IH …);//;@;#H3;ncases H2;#H4;napply H4;//;
+      ##|nchange in ⊢ (?(???%)) with (Pi_swap_list ???);
+         nrewrite > (IH …);//;@;#H3;ncases H1;#H4;napply H4;//
+      ##]
+   ##]
+##]
+nqed.
+*)
\ No newline at end of file
diff --git a/matita/matita/lib/binding/ln.ma b/matita/matita/lib/binding/ln.ma
new file mode 100644
index 000000000..ca7f574ed
--- /dev/null
+++ b/matita/matita/lib/binding/ln.ma
@@ -0,0 +1,716 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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
diff --git a/matita/matita/lib/binding/ln_concrete.ma b/matita/matita/lib/binding/ln_concrete.ma
new file mode 100644
index 000000000..bdafdb1a0
--- /dev/null
+++ b/matita/matita/lib/binding/ln_concrete.ma
@@ -0,0 +1,683 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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".
+
+definition alpha : Nset ≝ X. check alpha
+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 whd in ⊢ (??%?); >(\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 in ⊢ (%→?→?); #Hy whd in ⊢ (?→??%?); >(\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 (S (max ??))) /2/ ] ]
+-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 | // ] | @IH [| @Hv | // ] ] ]
+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.
+
+record TM : Type[0] ≝ {
+  pretm_of_TM :> pretm;
+  tm_of_TM : tm pretm_of_TM
+}.
+
+record CTX : Type[0] ≝ {
+  pretm_of_CTX :> pretm;
+  ctx_of_CTX : ctx pretm_of_CTX
+}.
+
+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)).
+*)
+
+axiom swap_inj : ∀N.∀z1,z2,x,y.swap N z1 z2 x = swap N z1 z2 y → x = y.
+
+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
+[3:whd in ⊢ (?→?→(?→ ??%%)→?→??%%); //
+|4:whd in ⊢ (?→?→(?→??%%)→(?→??%%)→?→??%%); //
+| #n #k whd in ⊢ (??(??%)%); cases (leb k n) normalize %
+| #x0 #k cases (true_or_false (x0==z1)) #H1 >H1 whd in ⊢ (??%%);
+  [ cases (true_or_false (x0==x)) #H2 >H2 whd in ⊢ (??(??%)%);
+    [ <(\P H2) >H1 whd in ⊢ (??(??%)%); >(\b ?) // >(\b ?) //
+    | >H2 whd in match (swap ????); >H1
+      whd in match (if false then var k else ?);
+      whd in match (if true then z2 else ?); >(\bf ?)
+      [ >(\P H1) >swap_left %
+      | <(swap_inv ? z1 z2 z2) in ⊢ (?(??%?)); % #H3
+        lapply (swap_inj … H3) >swap_right #H4 <H4 in H2; >H1 #H destruct (H) ]
+    ]
+  | >(?:(swap ? z1 z2 x0 == swap ? z1 z2 x) = (x0 == x))
+    [| cases (true_or_false (x0==x)) #H2 >H2
+      [ >(\P H2) @(\b ?) %
+      | @(\bf ?) % #H >(swap_inj … H) in H2; >(\b ?) // #H0 destruct (H0) ] ]
+    cases (true_or_false (x0==x)) #H2 >H2 whd in ⊢ (??(??%)%); 
+    [ <(\P H2) >H1 >(\b (refl ??)) %
+    | >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 //
+[2: #s #v whd in ⊢ ((?→??%%)→?→??%%); //
+|3: #v1 #v2 whd in ⊢ ((?→??%%)→(?→??%%)→?→??%%); /2/ ]
+#n #k whd in ⊢ (??(??%)%); cases (true_or_false (eqb n k)) #H1 >H1 //
+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 #H @H
+| #v1 #v2 whd in ⊢ (??%%→??%%→??%%); #H1 #H2 >H1 >H2
+  whd in ⊢ (???(????%)); /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: #n %
+|3: #s #v whd in ⊢ (?→??%%); //
+|4: #v1 #v2 #Hv1 #Hv2 whd in ⊢ (??%%); // ]
+#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 [1,3,4: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 whd in ⊢ (??(?%??)?); 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 whd in ⊢ (?(???(?%))); 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) -Hin #Hin
+  [ cases (IH1 k) -IH1 #IH1 @IH1 @Hin | cases (IH2 k) -IH2 #IH2 @IH2 @Hin ]
+qed.
+
+lemma fresh_vclose_tm : ∀u,x.x ∉ FV (νx.u). //
+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.
+
+definition PAR ≝ λx.mk_TM (par x) ?. // qed.
+definition APP ≝ λu,v:TM.mk_TM (app u v) ?./2/ qed.
+definition LAM ≝ λx,s.λu:TM.mk_TM (lam x s u) ?./2/ 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⌉) (vopen_tm_down …). @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) (vclose_tm_up …). @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.
+
+theorem TM_ind_plus : 
+(* non si può dare il principio in modo dipendente (almeno utilizzando tm2)
+   la "prova" purtroppo è in Type e non si può garantire che sia esattamente
+   quella che ci aspetteremmo
+ *)
+ ∀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 #pu elim (tm_to_tm2 u pu) //
+[ #v1 #v2 #pv1 #pv2 #IH1 #IH2 @(Happ (mk_TM …) (mk_TM …)) /2/
+| #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) @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.
+
+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 ⇒ u ]
+  | 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 fasfd : ∀s,v. pretm_of_TM (subst (LAM O s (PAR 1)) O v) = pretm_of_TM (LAM O s (PAR 1)).
+#s #v normalize in ⊢ (??%?);
+
+
+theorem tm2_ind_plus : 
+(* non si può dare il principio in modo dipendente (almeno utilizzando tm2) *)
+ ∀P:pretm → Type[0].
+   (∀x:𝔸.P (par x)) → 
+   (∀v1,v2.tm2 v1 → tm2 v2 → P v1 → P v2 → P (app v1 v2)) → 
+   ∀C:list 𝔸.
+   (∀x,s,v.x ∉ FV v → x ∉ C → (∀y.y ∉ FV v → tm2 (v⌈y⌉)) → 
+     (∀y.y ∉ FV v → P (v⌈y⌉)) → P (lam x s (v⌈x⌉))) →
+ ∀u.tm2 u → P u.
+#P #Hpar #Happ #C #Hlam #u #pu elim pu /2/
+#x #s #v #px #pv #IH 
+lapply (p_fresh … (C@FV v)) letin y ≝ (N_fresh … (C@FV v)) #Hy
+>(?:lam x s (v⌈x⌉) = lam y s (v⌈y⌉)) [| @tm_alpha // @(not_to_not … Hy) @in_list_to_in_list_append_r ]
+@Hlam /2/ lapply Hy -Hy @not_to_not #Hy
+[ @in_list_to_in_list_append_r @Hy | @in_list_to_in_list_append_l @Hy ]
+qed.
+
+definition check_tm ≝ 
+  λu.pretm_ind_plus ? (λ_.true) (λ_.false) 
+    (λv1,v2,r1,r2.r1 ∧ r2) [ ] (λx,s,v,pv1,pv2,rv.rv) u.
+
+(*
+lemma check_tm_complete : ∀u.tm u → check_tm u = true.
+#u #pu @(tm2_ind_plus … [ ] … (tm_to_tm2 ? pu)) //
+[ #v1 #v2 #pv1 #pv2 #IH1 #IH2
+| #x #s #v #Hx1 #Hx2 #Hv #IH
+*)
+
+notation 
+"hvbox('nominal' u 'return' out 'with' 
+       [ 'xpar' ident x ⇒ f1 
+       | 'xapp' ident v1 ident v2 ident pv1 ident pv2 ident recv1 ident recv2 ⇒ f2 
+       | 'xlam' ❨ident y # C❩ ident s ident w ident py1 ident py2 ident pw ident recw ⇒ f3 ])"
+with precedence 48 
+for @{ tm2_ind_plus $out (λ${ident x}:?.$f1)
+       (λ${ident v1}:?.λ${ident v2}:?.λ${ident pv1}:?.λ${ident pv2}:?.λ${ident recv1}:?.λ${ident recv2}:?.$f2)
+       $C (λ${ident y}:?.λ${ident s}:?.λ${ident w}:?.λ${ident py1}:?.λ${ident py2}:?.λ${ident pw}:?.λ${ident recw}:?.$f3)
+       ? (tm_to_tm2 ? $u) }.
+(* notation 
+"hvbox('nominal' u 'with' 
+       [ 'xlam' ident x # C ident s ident w ⇒ f3 ])"
+with precedence 48 
+for @{ tm2_ind_plus ???
+ $C (λ${ident x}:?.λ${ident s}:?.λ${ident w}:?.λ${ident py1}:?.λ${ident py2}:?.
+     λ${ident pw}:?.λ${ident recw}:?.$f3) $u (tm_to_tm2 ??) }.
+*)
+
+
+definition subst ≝ (λu.λpu:tm u.λx,v.
+  nominal pu return (λ_.pretm) with 
+  [ xpar x0 ⇒ match x == x0 with [ true ⇒ v | false ⇒ u ]
+  | xapp v1 v2 pv1 pv2 recv1 recv2 ⇒ app recv1 recv2
+  | xlam ❨y # x::FV v❩ s w py1 py2 pw recw ⇒ lam y s (recw y py1) ]).
+  
+lemma fasfd : ∀x,s,u,p1,v. subst (lam x s u) p1 x v = lam x s u.
+#x #s #u #p1 #v
+
+
+definition subst ≝ λu.λpu:tm u.λx,y.
+  tm2_ind_plus ?
+  (* par x0 *)              (λx0.match x == x0 with [ true ⇒ v | false ⇒ u ])
+  (* app v1 v2 *)           (λv1,v2,pv1,pv2,recv1,recv2.app recv1 recv2)
+  (* lam y#(x::FV v) s w *) (x::FV v) (λy,s,w,py1,py2,pw,recw.lam y s (recw y py1)) 
+  u (tm_to_tm2 … pu).
+check subst
+definition subst ≝ λu.λpu:tm u.λx,v.
+  nominal u with 
+  [ xlam y # (x::FV v) s w ^ ? ].
+
+(*
+notation > "Λ ident x. ident T [ident x] ↦ P"
+  with precedence 48 for @{'foo (λ${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 ??) }.
+
+
+definition subst ≝ λu.λpu:tm u.λx,v.
+  nominal u with 
+  [ xpar x0 ^ match x == x0 with [ true ⇒ v | false ⇒ u ]
+  | xapp v1 v2 ^ ? ].
+  | xlam y # (x::FV v) s w ^ ? ].
+  
+  
+  (* par x0 *)              (λx0.match x == x0 with [ true ⇒ v | false ⇒ u ])
+  (* app v1 v2 *)           (λv1,v2,pv1,pv2,recv1,recv2.app recv1 recv2)
+  (* lam y#(x::FV v) s w *) (x::FV v) (λy,s,w,py1,py2,pw,recw.lam y s (recw y py1)) 
+  u (tm_to_tm2 … pu).
+ 
+ 
+*)
+definition subst ≝ λu.λpu:tm u.λx,v.
+  tm2_ind_plus ?
+  (* par x0 *)              (λx0.match x == x0 with [ true ⇒ v | false ⇒ u ])
+  (* app v1 v2 *)           (λv1,v2,pv1,pv2,recv1,recv2.app recv1 recv2)
+  (* lam y#(x::FV v) s w *) (x::FV v) (λy,s,w,py1,py2,pw,recw.lam y s (recw y py1)) 
+  u (tm_to_tm2 … pu).
+
+check subst
+ 
+
+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).
+
+let rec FV M ≝ match M with 
+  [ par X ⇒ [X]
+  | app M1 M2 ⇒ FV M1@FV M2
+  | abs T M0 ⇒ FV M0
+  | _ ⇒ [ ] ].
+
+(* 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
diff --git a/matita/matita/lib/binding/names.ma b/matita/matita/lib/binding/names.ma
new file mode 100644
index 000000000..bc5f07d43
--- /dev/null
+++ b/matita/matita/lib/binding/names.ma
@@ -0,0 +1,78 @@
+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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/logic.ma".
+include "basics/lists/in.ma".
+include "basics/types.ma".
+
+(*interpretation "list membership" 'mem x l = (in_list ? x l).*)
+
+record Nset : Type[1] ≝
+{
+  (* carrier is specified as a coercion: when an object X of type Nset is
+     given, but something of type Type is expected, Matita will insert a
+     hidden coercion: the user sees "X", but really means "carrier X"     *)
+  carrier :> DeqSet;
+  N_fresh : list carrier → carrier;
+  p_fresh : ∀l.N_fresh l ∉ l
+}.
+
+definition maxlist ≝
+ λl.foldr ?? (λx,acc.max x acc) 0 l.
+
+definition natfresh ≝ λl.S (maxlist l).
+
+lemma le_max_1 : ∀x,y.x ≤ max x y. /2/
+qed.
+
+lemma le_max_2 : ∀x,y.y ≤ max x y. /2/
+qed.
+
+lemma le_maxlist : ∀l,x.x ∈ l → x ≤ maxlist l.
+#l elim l
+[#x #Hx @False_ind cases (not_in_list_nil ? x) #H1 /2/
+|#y #tl #IH #x #H1 change with (max ??) in ⊢ (??%);
+ cases (in_list_cons_case ???? H1);#H2;
+ [ >H2 @le_max_1
+ | whd in ⊢ (??%); lapply (refl ? (leb y (maxlist tl)));
+   cases (leb y (maxlist tl)) in ⊢ (???% → %);#H3
+   [ @IH //
+   | lapply (IH ? H2) #H4
+     lapply (leb_false_to_not_le … H3) #H5
+     lapply (not_le_to_lt … H5) #H6
+     @(transitive_le … H4)
+     @(transitive_le … H6) %2 %
+   ]
+ ]
+]
+qed.
+
+(* prove freshness for nat *)
+lemma lt_l_natfresh_l : ∀l,x.x ∈ l → x < natfresh l.
+#l #x #H1 @le_S_S /2/
+qed.
+
+(*naxiom p_Xfresh : ∀l.∀x:Xcarr.x ∈ l → x ≠ ntm (Xfresh l) ∧ x ≠ ntp (Xfresh l).*)
+lemma p_natfresh : ∀l.natfresh l ∉ l.
+#l % #H1 lapply (lt_l_natfresh_l … H1) #H2
+cases (lt_to_not_eq … H2) #H3 @H3 %
+qed.
+
+include "basics/finset.ma".
+
+definition X : Nset ≝ mk_Nset DeqNat ….
+[ @natfresh
+| @p_natfresh
+]
+qed.
\ No newline at end of file