(* *)
(**************************************************************************)
-include "lambda/subst.ma".
include "basics/list.ma".
-include "lambda/lambda_notation.ma".
(* MATTER CONCERNING STRONG NORMALIZATION TO BE PUT ELSEWHERE *****************)
(* lists **********************************************************************)
+lemma length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
+#A #l2 #l1 elim l1 -l1; normalize //
+qed.
+
(* all(?,P,l) holds when P holds for all members of l *)
let rec all (A:Type[0]) (P:A→Prop) l on l ≝ match l with
[ nil ⇒ True
].
lemma all_hd: ∀A:Type[0]. ∀P:A→Prop. ∀a. P a → ∀l. all … P l → P (hd … l a).
-#A #P #a #Ha #l elim l -l [ #_ @Ha | #b #l #_ #Hl elim Hl // ]
+#A #P #a #Ha #l elim l -l [ #_ @Ha | #b #l #_ #Hl elim Hl -Hl // ]
qed.
lemma all_tl: ∀A:Type[0]. ∀P:A→Prop. ∀l. all … P l → all … P (tail … l).
-#A #P #l elim l -l // #b #l #IH #Hl elim Hl //
+#A #P #l elim l -l // #b #l #IH #Hl elim Hl -Hl //
qed.
lemma all_nth: ∀A:Type[0]. ∀P:A→Prop. ∀a. P a → ∀i,l. all … P l → P (nth i … l a).
qed.
lemma 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/
+#A #P #l2 #l1 elim l1 -l1; normalize // #hd #tl #IH1 #H elim H -H /3/
qed.
(* all2(?,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
+let rec all2 (A,B:Type[0]) (P:A→B→Prop) l1 l2 on l1 ≝ match l1 with
[ nil ⇒ l2 = nil ?
| cons hd1 tl1 ⇒ match l2 with
[ nil ⇒ False
- | cons hd2 tl2 ⇒ P hd1 hd2 ∧ all2 A P tl1 tl2
+ | cons hd2 tl2 ⇒ P hd1 hd2 ∧ all2 A B P tl1 tl2
]
].
-lemma length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
-#A #l2 #l1 (elim l1) -l1 (normalize) //
-qed.
-
-(* terms **********************************************************************)
+lemma all2_length: ∀A,B:Type[0]. ∀P:A→B→Prop.
+ ∀l1,l2. all2 … P l1 l2 → |l1|=|l2|.
+#A #B #P #l1 elim l1 -l1 [ #l2 #H >H // ]
+#x1 #l1 #IH1 #l2 elim l2 -l2 [ #false elim false ]
+#x2 #l2 #_ #H elim H -H; normalize /3/
+qed.
-(* Appl F l generalizes App applying F to a list of arguments
- * The head of l is applied first
- *)
-let rec Appl F l on l ≝ match l with
- [ nil ⇒ F
- | cons A D ⇒ Appl (App F A) D
- ].
-
-lemma appl_append: ∀N,l,M. Appl M (l @ [N]) = App (Appl M l) N.
-#N #l (elim l) -l // #hd #tl #IHl #M >IHl //
+lemma all2_hd: ∀A,B:Type[0]. ∀P:A→B→Prop. ∀a,b. P a b →
+ ∀l1,l2. all2 … P l1 l2 → P (hd … l1 a) (hd … l2 b).
+#A #B #P #a #b #Hab #l1 elim l1 -l1 [ #l2 #H2 >H2 @Hab ]
+#x1 #l1 #_ #l2 elim l2 -l2 [ #false elim false ]
+#x2 #l2 #_ #H elim H -H //
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 *)
-
-lemma lift_appl: ∀p,k,l,F. lift (Appl F l) p k =
- Appl (lift F p k) (map … (lift0 p k) l).
-#p #k #l (elim l) -l /2/ #A #D #IHl #F >IHl //
+lemma all2_tl: ∀A,B:Type[0]. ∀P:A→B→Prop.
+ ∀l1,l2. all2 … P l1 l2 → all2 … P (tail … l1) (tail … l2).
+#A #B #P #l1 elim l1 -l1 [ #l2 #H >H // ]
+#x1 #l1 #_ #l2 elim l2 -l2 [ #false elim false ]
+#x2 #l2 #_ #H elim H -H //
qed.
-*)
-lemma 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) //
+lemma all2_nth: ∀A,B:Type[0]. ∀P:A→B→Prop. ∀a,b. P a b →
+ ∀i,l1,l2. all2 … P l1 l2 → P (nth i … l1 a) (nth i … l2 b).
+#A #B #P #a #b #Hab #i elim i -i [ @all2_hd // | #i #IH #l1 #l2 #H @IH /2/ ]
qed.
-lemma lift_rel_ge: ∀i,p,k. (S i) ≰ k → lift (Rel i) k p = Rel (i+p).
-#i #p #k #Hik normalize >(lt_to_leb_false (S i) k) /2/
-qed.
-
-lemma lift_app: ∀M,N,k,p.
- lift (App M N) k p = App (lift M k p) (lift N k p).
-// qed.
-
-lemma lift_lambda: ∀N,M,k,p. lift (Lambda N M) k p =
- Lambda (lift N k p) (lift M (k + 1) p).
-// qed.
-
-lemma lift_prod: ∀N,M,k,p.
- lift (Prod N M) k p = Prod (lift N k p) (lift M (k + 1) p).
-// qed.
-
-lemma subst_app: ∀M,N,k,L. (App M N)[k≝L] = App M[k≝L] N[k≝L].
-// qed.
-
-lemma subst_lambda: ∀N,M,k,L. (Lambda N M)[k≝L] = Lambda N[k≝L] M[k+1≝L].
-// qed.
-
-lemma subst_prod: ∀N,M,k,L. (Prod N M)[k≝L] = Prod N[k≝L] M[k+1≝L].
-// qed.
-
-
-axiom lift_subst_lt: ∀A,B,i,j,k. lift (B[j≝A]) (j+k) i =
- (lift B (j+k+1) i)[j≝lift A k i].
-
-(* telescopic delifting substitution of l in M.
- * Rel 0 is replaced with the head of l
- *)
-let rec tsubst M l on l ≝ match l with
- [ nil ⇒ M
- | cons A D ⇒ (tsubst M[0≝A] D)
- ].
-
-interpretation "telescopic substitution" 'Subst1 M l = (tsubst M l).
-
-lemma tsubst_refl: ∀l,t. (lift t 0 (|l|))[l] = t.
-#l (elim l) -l (normalize) // #hd #tl #IHl #t cut (S (|tl|) = |tl| + 1) // (**) (* eliminate cut *)
+lemma all2_append: ∀A,B,P,l2,m2. all2 A B P l2 m2 →
+ ∀l1,m1. all2 A B P l1 m1 → all2 A B P (l1 @ l2) (m1 @ m2).
+#A #B #P #l2 #m2 #H2 #l1 (elim l1) -l1 [ #m1 #H >H @H2 ]
+#x1 #l1 #IH1 #m2 elim m2 -m2 [ #false elim false ]
+#x2 #m2 #_ #H elim H -H /3/
qed.
-lemma tsubst_sort: ∀n,l. (Sort n)[l] = Sort n.
-//
+lemma all2_symmetric: ∀A. ∀P:A→A→Prop. symmetric … P → symmetric … (all2 … P).
+#A #P #HP #l1 elim l1 -l1 [ #l2 #H >H // ]
+#x1 #l1 #IH1 #l2 elim l2 -l2 [ #false elim false ]
+#x2 #l2 #_ #H elim H -H /3/
qed.
projects / helm.git / blobdiff