- notation is now in a separate file

[helm.git] / matita / matita / lib / lambda / ext.ma

diff --git a/matita/matita/lib/lambda/ext.ma b/matita/matita/lib/lambda/ext.ma
index b933ed7b6df0d760a3dca8f45c83e64b92cd4612..ee674a9cd3c4776314739fc97e7cdfe241322f89 100644 (file)
--- a/matita/matita/lib/lambda/ext.ma
+++ b/matita/matita/lib/lambda/ext.ma
@@ -13,17 +13,37 @@
 (**************************************************************************)
 
 include "lambda/types.ma".
+include "lambda/lambda_notation.ma".
 
 (* MATTER CONCERNING STRONG NORMALIZATION TO BE PUT ELSEWHERE *****************)
 
-(* from sn.ma *****************************************************************)
+(* arithmetics ****************************************************************)
+
+theorem arith1: ∀x,y. (S x) ≰ (S y) → x ≰ y.
+#x #y #HS @nmk (elim HS) -HS /3/
+qed.
+
+theorem arith2: ∀i,p,k. k ≤ i → i + p - (k + p) = i - k.
+#i #p #k #H @plus_to_minus
+>commutative_plus >(commutative_plus k) >associative_plus @eq_f /2/
+qed.
+
+theorem arith3: ∀x,y,z. x ≰ y → x + z ≰ y + z.
+#x #y #z #H @nmk (elim H) -H /3/
+qed.
+
+(* lists **********************************************************************)
 
 (* all(P,l) holds when P holds for all members of l *)
-let rec all (P:T→Prop) l on l ≝ match l with 
-   [ nil ⇒ True
-   | cons A D ⇒ P A ∧ all P D
+let rec all (A:Type[0]) (P:A→Prop) l on l ≝ match l with 
+   [ nil        ⇒ True
+   | cons hd tl ⇒ P hd ∧ all A P tl
    ].
 
+theorem all_append: ∀A,P,l2,l1. all A P l1 → all A P l2 → all A P (l1 @ l2).
+#A #P #l2 #l1 (elim l1) -l1 (normalize) // #hd #tl #IH1 #H (elim H) /3/
+qed.
+
 (* all(?,P,l1,l2) holds when P holds for all paired members of l1 and l2 *)
 let rec all2 (A:Type[0]) (P:A→A→Prop) l1 l2 on l1 ≝ match l1 with
    [ nil          ⇒ l2 = nil ?
@@ -33,6 +53,12 @@ let rec all2 (A:Type[0]) (P:A→A→Prop) l1 l2 on l1 ≝ match l1 with
       ]
    ].
 
+theorem length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
+#A #l2 #l1 (elim l1) -l1 (normalize) //
+qed.
+
+(* terms **********************************************************************)
+
 (* Appl F l generalizes App applying F to a list of arguments
  * The head of l is applied first
  *)
@@ -41,6 +67,21 @@ let rec Appl F l on l ≝ match l with
    | cons A D ⇒ Appl (App F A) D  
    ].
 
+theorem appl_append: ∀N,l,M. Appl M (l @ [N]) = App (Appl M l) N.
+#N #l (elim l) -l // #hd #tl #IHl #M >IHl //
+qed.
+
+(* FG: not needed for now 
+(* nautral terms *)
+inductive neutral: T → Prop ≝
+   | neutral_sort: ∀n.neutral (Sort n)
+   | neutral_rel: ∀i.neutral (Rel i)
+   | neutral_app: ∀M,N.neutral (App M N)
+.
+*)
+
+(* substitution ***************************************************************)
+
 (* FG: do we need this? 
 definition lift0 ≝ λp,k,M . lift M p k. (**) (* remove definition *)
 
@@ -50,25 +91,6 @@ theorem lift_appl: ∀p,k,l,F. lift (Appl F l) p k =
 qed.
 *)
 
-(* from rc.ma *****************************************************************)
-
-theorem arith1: ∀x,y. (S x) ≰ (S y) → x ≰ y.
-#x #y #HS @nmk (elim HS) -HS /3/
-qed.
-
-theorem arith2: ∀i,p,k. k ≤ i → i + p - (k + p) = i - k.
-#i #p #k #H @plus_to_minus
->commutative_plus >(commutative_plus k) >associative_plus @eq_f /2/
-qed.
-
-theorem arith3: ∀x,y,z. x ≰ y → x + z ≰ y + z.
-#x #y #z #H @nmk (elim H) -H /3/
-qed.
-
-theorem length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
-#A #l2 #l1 (elim l1) -l1 (normalize) //
-qed.
-
 theorem lift_rel_lt: ∀i,p,k. (S i) ≤ k → lift (Rel i) k p = Rel i.
 #i #p #k #Hik normalize >(le_to_leb_true … Hik) //
 qed.
@@ -98,9 +120,7 @@ let rec substc M l on l ≝ match l with
    | cons A D ⇒ (lift (substc M[0≝A] D) 0 1)
    ]. 
 
-notation "M [ l ]" non associative with precedence 90 for @{'Substc $M $l}.
-
-interpretation "Substc" 'Substc M l = (substc M l).
+interpretation "Substc" 'Subst1 M l = (substc M l).
 
 (* this is just to test that substitution works as expected
 theorem test1: ∀A,B,C. (App (App (Rel 0) (Rel 1)) (Rel 2))[A::B::C::nil ?] = 
@@ -119,11 +139,3 @@ qed.
 theorem substc_sort: ∀n,l. (Sort n)[l] = Sort n.
 //
 qed.
-(* FG: not needed for now 
-(* nautral terms *)
-inductive neutral: T → Prop ≝
-   | neutral_sort: ∀n.neutral (Sort n)
-   | neutral_rel: ∀i.neutral (Rel i)
-   | neutral_app: ∀M,N.neutral (App M N)
-.
-*)