Keeping only lift_aux e subst_aux (renamed to lift and subst).

[helm.git] / helm / software / matita / nlibrary / PTS / subst.ma

diff --git a/helm/software/matita/nlibrary/PTS/subst.ma b/helm/software/matita/nlibrary/PTS/subst.ma
index 7ca2fdc03f059c0df5b292fdc3f204567f9e3143..a37074b4d306a60b85f7909bcecc2637b7849d82 100644 (file)
--- a/helm/software/matita/nlibrary/PTS/subst.ma
+++ b/helm/software/matita/nlibrary/PTS/subst.ma
@@ -22,61 +22,64 @@ ninductive T : Type ≝
   | 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/;
@@ -86,17 +89,19 @@ napply (leb_elim (S n) k); #Hnk;nnormalize;
   ##]
 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;//;
@@ -108,39 +113,38 @@ napply (leb_elim (S n) k); nnormalize; #Hnk;
   ##]
 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);
@@ -151,7 +155,7 @@ napply (leb_elim (S n) (j + k)); nnormalize; #Hnjk;
     ##[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);
@@ -171,7 +175,7 @@ napply (leb_elim (S n) (j + k)); nnormalize; #Hnjk;
 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;
@@ -194,8 +198,8 @@ nqed.
 (********************* 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 *)
@@ -208,7 +212,7 @@ nlemma subst_lemma: ∀A,B,C.∀k,i.
        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) ?);