some renaming to free the baseuri cic:/matita/lambda

[helm.git] / matita / matita / lib / pts_dummy_new / rc_eval.ma

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+(**************************************************************************)
+(*       ___                                                              *)
+(*      ||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 "lambda/rc_hsat.ma".
+
+(* THE EVALUATION *************************************************************)
+
+(* The arity of a term t in an environment E *)
+let rec aa E t on t ≝ match t with
+   [ Sort _     ⇒ SORT
+   | Rel i      ⇒ nth i … E SORT
+   | App M N    ⇒ pred (aa E M)
+   | Lambda N M ⇒ let Q ≝ aa E N in ABST Q (aa (Q::E) M)
+   | Prod N M   ⇒ aa ((aa E N)::E) M
+   | D M        ⇒ aa E M
+   ].
+
+interpretation "arity assignment (term)" 'Eval1 t E = (aa E t).
+
+(* The arity of a type context *)
+let rec caa E G on G ≝ match G with
+   [ nil      ⇒ E
+   | cons t F ⇒ let D ≝ caa E F in 〚t〛_[D] :: D
+   ].
+
+interpretation "arity assignment (type context)" 'Eval1 G E = (caa E G).
+
+lemma aa_app: ∀M,N,E. 〚App M N〛_[E] = pred (〚M〛_[E]).
+// qed.
+
+lemma aa_lambda: ∀M,N,E. 〚Lambda N M〛_[E] = ABST (〚N〛_[E]) (〚M〛_[〚N〛_[E]::E]).
+// qed.
+
+lemma aa_prod: ∀M,N,E. 〚Prod N M〛_[E] = 〚M〛_[〚N〛_[E]::E].
+// qed.
+
+lemma aa_rel_lt: ∀D,E,i. (S i) ≤ |E| → 〚Rel i〛_[E @ D] = 〚Rel i〛_[E].
+#D #E (elim E) -E [1: #i #Hie (elim (not_le_Sn_O i)) #Hi (elim (Hi Hie)) ]
+#C #F #IHE #i (elim i) -i // #i #_ #Hie @IHE @le_S_S_to_le @Hie
+qed.
+
+lemma aa_rel_ge: ∀D,E,i. (S i) ≰ |E| →
+                   〚Rel i〛_[E @ D] = 〚Rel (i-|E|)〛_[D].
+#D #E (elim E) -E [ normalize // ]
+#C #F #IHE #i (elim i) -i [1: -IHE #Hie (elim Hie) -Hie #Hie (elim (Hie ?)) /2/ ]
+normalize #i #_ #Hie @IHE /2/
+qed.
+
+(* weakeing and thinning lemma arity assignment *)
+(* NOTE: >commutative_plus comes from |a::b| ↦ S |b| rather than |b| + 1 *)
+lemma aa_lift: ∀E,Ep,t,Ek.
+                 〚lift t (|Ek|) (|Ep|)〛_[Ek @ Ep @ E] = 〚t〛_[Ek @ E].
+#E #Ep #t (elim t) -t
+   [ #n //
+   | #i #Ek @(leb_elim (S i) (|Ek|)) #Hik
+     [ >(lift_rel_lt … Hik) >(aa_rel_lt … Hik) >(aa_rel_lt … Hik) //
+     | >(lift_rel_ge … Hik) >(aa_rel_ge … Hik) <associative_append
+       >(aa_rel_ge …) (>length_append)
+       [ >arith2 // @not_lt_to_le /2/ | @(arith3 … Hik) ]
+     ]
+   | /4/
+   | #N #M #IHN #IHM #Ek >lift_lambda >aa_lambda
+     >commutative_plus >(IHM (? :: ?)) /3/
+   | #N #M #IHN #IHM #Ek >lift_prod >aa_prod
+     >commutative_plus >(IHM (? :: ?)) /3/
+   | #M #IHM #Ek @IHM
+   ]
+qed.
+
+(* substitution lemma arity assignment *)
+(* NOTE: >commutative_plus comes from |a::b| ↦ S |b| rather than |b| + 1 *)
+lemma aa_subst: ∀v,E,t,Ek. 〚t[|Ek|≝v]〛_[Ek @ E] = 〚t〛_[Ek @ 〚v〛_[E]::E].
+#v #E #t (elim t) -t
+   [ //
+   | #i #Ek @(leb_elim (S i) (|Ek|)) #H1ik
+     [ >(aa_rel_lt … H1ik) >(subst_rel1 … H1ik) >(aa_rel_lt … H1ik) //
+     | @(eqb_elim i (|Ek|)) #H2ik
+       [ >(aa_rel_ge … H1ik) >H2ik -H2ik H1ik >subst_rel2
+         >(aa_lift ? ? ? ([])) <minus_n_n /2/
+       | (lapply (arith4 … H1ik H2ik)) -H1ik H2ik #Hik
+         (>(subst_rel3 … Hik)) (>aa_rel_ge) [2: /2/ ] 
+          <(associative_append ? ? ([?]) ?) 
+           >aa_rel_ge >length_append (>commutative_plus)
+           [ <minus_plus // | @not_le_to_not_le_S_S /2/ ]
+       ]
+     ]
+   | //
+   | #N #M #IHN #IHM #Ek >subst_lambda > aa_lambda
+     >commutative_plus >(IHM (? :: ?)) /3/
+   | #N #M #IHN #IHM #Ek >subst_prod > aa_prod
+     >commutative_plus >(IHM (? :: ?)) /4/
+   | #M #IHM #Ek @IHM
+qed.
+
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+
+(*
+(* extensional equality of the interpretations *)
+definition eveq: T → T → Prop ≝ λt1,t2. ∀E,K. 〚t1〛_[E, K] ≅ 〚t2〛_[E, K].
+
+interpretation 
+   "extensional equality of the type interpretations"
+   'napart t1 t2 = (eveq t1 t2).
+*)
+
+axiom ev_lift_0_S: ∀t,p,C,E,K. 〚lift t 0 (S p)〛_[C::E, K] ≅ 〚lift t 0 p〛_[E, K].
+
+theorem tj_ev: ∀G,t,u. G ⊢t:u → ∀E,l. l ∈ 〚G〛_[E] → t[l] ∈ 〚u[l]〛_[[], []].
+#G #t #u #tjtu (elim tjtu) -G t u tjtu
+   [ #i #j #_ #E #l #_ >tsubst_sort >tsubst_sort /2 by SAT0_sort/
+   | #G #u #n #tju #IHu #E #l (elim l) -l (normalize)
+     [ #_ /2 by SAT1_rel/
+     | #hd #tl #_ #H (elim H) -H #Hhd #Htl
+       >lift_0 >delift // >lift_0
+       
+       
+       
+       (@mem_rceq_trans) [3: @symmetric_rceq @ev_lift_0_S | skip ] 
+
+*)
+(* 
+theorem tj_sn: ∀G,A,B. G ⊢A:B → SN A.
+#G #A #B #tjAB lapply (tj_trc … tjAB (nil ?) (nil ?)) -tjAB (normalize) /3/
+qed.
+*)
+
+(*
+theorem tev_rel_S: ∀i,R,H. 〚Rel (S i)〛_(R::H) = 〚Rel i〛_(H).
+// qed.
+*)
+(*
+theorem append_cons: ∀(A:Type[0]). ∀(l1,l2:list A). ∀a.
+                     (a :: l1) @ l2 = a :: (l1 @ l2).
+// qed.
+*)