+++ /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 "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.
-*)
projects / helm.git / blobdiff