+++ /dev/null
-(**************************************************************************)
-(* ___ *)
-(* ||M|| *)
-(* ||A|| A project by Andrea Asperti *)
-(* ||T|| *)
-(* ||I|| Developers: *)
-(* ||T|| The HELM team. *)
-(* ||A|| http://helm.cs.unibo.it *)
-(* \ / *)
-(* \ / This file is distributed under the terms of the *)
-(* v GNU General Public License Version 2 *)
-(* *)
-(**************************************************************************)
-
-include "basics/lists/list.ma".
-
-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
projects / helm.git / blobdiff