| Prod: T → T → T (* type, body *)
.
-nlet rec lift_aux t k p ≝
+nlet rec lift t k p ≝
match t with
[ Sort n ⇒ Sort n
| Rel n ⇒ if_then_else T (leb (S n) k) (Rel n) (Rel (n+p))
- | App m n ⇒ App (lift_aux m k p) (lift_aux n k p)
- | Lambda m n ⇒ Lambda (lift_aux m k p) (lift_aux n (k+1) p)
- | Prod m n ⇒ Prod (lift_aux m k p) (lift_aux n (k+1) p)
+ | App m n ⇒ App (lift m k p) (lift n k p)
+ | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
+ | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
].
-ndefinition lift ≝ λt.λp.lift_aux t 0 p.
+(*
+ndefinition lift ≝ λt.λp.lift_aux t 0 p.*)
-notation "↑ \sup n ( M )" non associative with precedence 70 for @{'Lift $n $M}.
-notation "↑ \sub k \sup n ( M )" non associative with precedence 70 for @{'Lift_aux $n $k $M}.
+notation "↑ \sup n ( M )" non associative with precedence 70 for @{'Lift O $M}.
+notation "↑ \sub k \sup n ( M )" non associative with precedence 70 for @{'Lift $n $k $M}.
-interpretation "Lift" 'Lift n M = (lift M n).
-interpretation "Lift_aux" 'Lift_aux n k M = (lift_aux M k n).
+(* interpretation "Lift" 'Lift n M = (lift M n). *)
+interpretation "Lift" 'Lift n k M = (lift M k n).
-nlet rec subst_aux t k a ≝
+nlet rec subst t k a ≝
match t with
[ Sort n ⇒ Sort n
| Rel n ⇒ if_then_else T (leb (S n) k) (Rel n)
- (if_then_else T (eqb n k) (lift a n) (Rel (n-1)))
- | App m n ⇒ App (subst_aux m k a) (subst_aux n k a)
- | Lambda m n ⇒ Lambda (subst_aux m k a) (subst_aux n (k+1) a)
- | Prod m n ⇒ Prod (subst_aux m k a) (subst_aux n (k+1) a)
+ (if_then_else T (eqb n k) (lift a 0 n) (Rel (n-1)))
+ | App m n ⇒ App (subst m k a) (subst n k a)
+ | Lambda m n ⇒ Lambda (subst m k a) (subst n (k+1) a)
+ | Prod m n ⇒ Prod (subst m k a) (subst n (k+1) a)
].
+(* meglio non definire
ndefinition subst ≝ λa.λt.subst_aux t 0 a.
-
notation "M [ N ]" non associative with precedence 90 for @{'Subst $N $M}.
-notation "M [ k ← N]" non associative with precedence 90 for @{'Subst_aux $M $k $N}.
+*)
+
+notation "M [ k ← N]" non associative with precedence 90 for @{'Subst $M $k $N}.
-interpretation "Subst" 'Subst N M = (subst N M).
-interpretation "Subst_aux" 'Subst_aux M k N = (subst_aux M k N).
+(* interpretation "Subst" 'Subst N M = (subst N M). *)
+interpretation "Subst" 'Subst M k N = (subst M k N).
(*** properties of lift and subst ***)
-nlemma lift_aux_0: ∀t:T.∀k. lift_aux t k 0 = t.
+nlemma lift_0: ∀t:T.∀k. lift t k 0 = t.
#t; nelim t; nnormalize; //; #n; #k; ncases (leb (S n) k);
nnormalize;//;nqed.
-nlemma lift_0: ∀t:T. lift t 0 = t.
-#t; nelim t; nnormalize; //; nqed.
+(* nlemma lift_0: ∀t:T. lift t 0 = t.
+#t; nelim t; nnormalize; //; nqed. *)
-nlemma lift_sort: ∀i,k. lift (Sort i) k = Sort i.
+nlemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
//; nqed.
-nlemma lift_rel: ∀i,k. lift (Rel i) k = Rel (i+k).
+nlemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
//; nqed.
-nlemma lift_rel1: ∀i.lift (Rel i) 1 = Rel (S i).
-#i; nchange with (lift (Rel i) 1 = Rel (1 + i)); //; nqed.
+nlemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
+#i; nchange with (lift (Rel i) 0 1 = Rel (1 + i)); //; nqed.
-nlemma lift_lift_aux: ∀t.∀i,j.j ≤ i → ∀h,k.
-lift_aux (lift_aux t k i) (j+k) h = lift_aux t k (i+h).
+nlemma lift_lift: ∀t.∀i,j.j ≤ i → ∀h,k.
+ lift (lift t k i) (j+k) h = lift t k (i+h).
#t; #i; #j; #h; nelim t; nnormalize; //; #n; #h;#k;
napply (leb_elim (S n) k); #Hnk;nnormalize;
##[nrewrite > (le_to_leb_true (S n) (j+k) ?);nnormalize;/2/;
##]
nqed.
-nlemma lift_lift_aux1: ∀t.∀i,j,k. lift_aux (lift_aux t k j) k i = lift_aux t k (j+i).
+nlemma lift_lift1: ∀t.∀i,j,k.
+ lift(lift t k j) k i = lift t k (j+i).
#t;/3/; nqed.
-nlemma lift_lift_aux2: ∀t.∀i,j,k. lift_aux (lift_aux t k j) (j+k) i = lift_aux t k (j+i).
+nlemma lift_lift2: ∀t.∀i,j,k.
+ lift (lift t k j) (j+k) i = lift t k (j+i).
#t; /2/; nqed.
+(*
nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
-nnormalize; //; nqed.
+nnormalize; //; nqed. *)
-nlemma subst_lift_aux_k: ∀A,B.∀k.
- subst_aux (lift_aux B k 1) k A = B.
+nlemma subst_lift_k: ∀A,B.∀k. subst (lift B k 1) k A = B.
#A; #B; nelim B; nnormalize; /2/; #n; #k;
napply (leb_elim (S n) k); nnormalize; #Hnk;
##[nrewrite > (le_to_leb_true ?? Hnk);nnormalize;//;
##]
nqed.
+(*
nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
-nnormalize; //; nqed.
-
-nlemma subst_aux_sort: ∀A.∀n,k. subst_aux (Sort n) k A = Sort n.
-//; nqed.
+nnormalize; //; nqed. *)
-nlemma subst_sort: ∀A.∀n. subst A (Sort n) = Sort n.
+nlemma subst_sort: ∀A.∀n,k. subst (Sort n) k A = Sort n.
//; nqed.
-nlemma subst_rel: ∀A.subst A (Rel O) = A.
+nlemma subst_rel: ∀A.subst (Rel 0) 0 A = A.
nnormalize; //; nqed.
nlemma subst_rel1: ∀A.∀k,i. i < k →
- subst_aux (Rel i) k A = Rel i.
+ subst (Rel i) k A = Rel i.
#A; #k; #i; nnormalize; #ltik;
nrewrite > (le_to_leb_true (S i) k ?); //; nqed.
-nlemma subst_rel2: ∀A.∀k. subst_aux (Rel k) k A = lift A k.
+nlemma subst_rel2: ∀A.∀k.
+ subst (Rel k) k A = lift A 0 k.
#A; #k; nnormalize;
nrewrite > (lt_to_leb_false (S k) k ?); //;
nrewrite > (eq_to_eqb_true … (refl …)); //;
nqed.
nlemma subst_rel3: ∀A.∀k,i. k < i →
- subst_aux (Rel i) k A = Rel (i-1).
+ subst (Rel i) k A = Rel (i-1).
#A; #k; #i; nnormalize; #ltik;
nrewrite > (lt_to_leb_false (S i) k ?); /2/;
nrewrite > (not_eq_to_eqb_false i k ?); //;
napply nmk; #eqik; nelim (lt_to_not_eq … (ltik …)); /2/;
nqed.
-nlemma lift_subst_aux_ijk: ∀A,B.∀i,j,k.
- lift_aux (subst_aux B (j+k) A) k i = subst_aux (lift_aux B k i) (j+k+i) A.
+nlemma lift_subst_ijk: ∀A,B.∀i,j,k.
+ lift (subst B (j+k) A) k i = subst (lift B k i) (j+k+i) A.
#A; #B; #i; #j; nelim B; nnormalize; /2/; #n; #k;
napply (leb_elim (S n) (j + k)); nnormalize; #Hnjk;
##[nelim (leb (S n) k);
##[nrewrite > (lt_to_leb_false (S n) k ?);
##[ncut (j+k+i = n+i);##[//;##] #Heq;
nrewrite > Heq; nrewrite > (subst_rel2 A ?);
- nnormalize; napplyS lift_lift_aux;//;
+ nnormalize; napplyS lift_lift;//;
##|/2/;
##]
##|ncut (j + k < n);
nqed.
ntheorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
- subst_aux (lift_aux B i (S k)) j A = (lift_aux B i k).
+ subst (lift B i (S k)) j A = (lift B i k).
#A; #B; nelim B; nnormalize; /2/;
##[##2,3,4: #T; #T0; #Hind1; #Hind2; #i; #j; #k; #leij; #lejk;
napply eq_f2;/2/; napply Hind2;
(********************* substitution lemma ***********************)
nlemma subst_lemma: ∀A,B,C.∀k,i.
- subst_aux (subst_aux A i B) (k+i) C =
- subst_aux (subst_aux A (S (k+i)) C) i (subst_aux B k C).
+ subst (subst A i B) (k+i) C =
+ subst (subst A (S (k+i)) C) i (subst B k C).
#A; #B; #C; #k; nelim A; nnormalize;//; (* WOW *)
#n; #i; napply (leb_elim (S n) i); #Hle;
##[ncut (n < k+i); ##[/2/##] #ltn; (* lento *)
nrewrite > (le_to_leb_true i (k+i) ?); //;
nrewrite > (subst_rel2 …); nnormalize;
napply symmetric_eq;
- napplyS (lift_subst_aux_ijk C B i k O);
+ napplyS (lift_subst_ijk C B i k O);
##|napply (leb_elim (S (n-1)) (k+i)); #nk;
##[nrewrite > (subst_rel1 C (k+i) (n-1) nk);
nrewrite > (le_to_leb_true n (k+i) ?);