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4 (* ||A|| A project by Andrea Asperti *)
6 (* ||I|| Developers: *)
7 (* ||T|| The HELM team. *)
8 (* ||A|| http://helm.cs.unibo.it *)
10 (* \ / This file is distributed under the terms of the *)
11 (* v GNU General Public License Version 2 *)
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15 include "lambda/subst.ma".
16 include "basics/list.ma".
17 include "lambda/lambda_notation.ma".
19 (* MATTER CONCERNING STRONG NORMALIZATION TO BE PUT ELSEWHERE *****************)
21 (* arithmetics ****************************************************************)
23 lemma arith1: ∀x,y. (S x) ≰ (S y) → x ≰ y.
24 #x #y #HS @nmk (elim HS) -HS /3/
27 lemma arith2: ∀i,p,k. k ≤ i → i + p - (k + p) = i - k.
28 #i #p #k #H @plus_to_minus
29 >commutative_plus >(commutative_plus k) >associative_plus @eq_f /2/
32 lemma arith3: ∀x,y,z. x ≰ y → x + z ≰ y + z.
33 #x #y #z #H @nmk (elim H) -H /3/
36 lemma arith4: ∀x,y. S x ≰ y → x≠y → y < x.
37 #x #y #H1 #H2 lapply (not_le_to_lt … H1) -H1 #H1 @not_eq_to_le_to_lt /2/
40 lemma arith5: ∀x,y. x < y → S (y - 1) ≰ x.
41 #x #y #H @lt_to_not_le <minus_Sn_m /2/
44 (* lists **********************************************************************)
46 lemma length_append: ∀A. ∀(l2,l1:list A). |l1@l2| = |l1| + |l2|.
47 #A #l2 #l1 elim l1 -l1; normalize //
50 (* all(?,P,l) holds when P holds for all members of l *)
51 let rec all (A:Type[0]) (P:A→Prop) l on l ≝ match l with
53 | cons hd tl ⇒ P hd ∧ all A P tl
56 lemma all_hd: ∀A:Type[0]. ∀P:A→Prop. ∀a. P a → ∀l. all … P l → P (hd … l a).
57 #A #P #a #Ha #l elim l -l [ #_ @Ha | #b #l #_ #Hl elim Hl -Hl // ]
60 lemma all_tl: ∀A:Type[0]. ∀P:A→Prop. ∀l. all … P l → all … P (tail … l).
61 #A #P #l elim l -l // #b #l #IH #Hl elim Hl -Hl //
64 lemma all_nth: ∀A:Type[0]. ∀P:A→Prop. ∀a. P a → ∀i,l. all … P l → P (nth i … l a).
65 #A #P #a #Ha #i elim i -i [ @all_hd // | #i #IH #l #Hl @IH /2/ ]
68 lemma all_append: ∀A,P,l2,l1. all A P l1 → all A P l2 → all A P (l1 @ l2).
69 #A #P #l2 #l1 elim l1 -l1; normalize // #hd #tl #IH1 #H elim H -H /3/
72 (* all2(?,P,l1,l2) holds when P holds for all paired members of l1 and l2 *)
73 let rec all2 (A,B:Type[0]) (P:A→B→Prop) l1 l2 on l1 ≝ match l1 with
75 | cons hd1 tl1 ⇒ match l2 with
77 | cons hd2 tl2 ⇒ P hd1 hd2 ∧ all2 A B P tl1 tl2
81 lemma all2_length: ∀A,B:Type[0]. ∀P:A→B→Prop.
82 ∀l1,l2. all2 … P l1 l2 → |l1|=|l2|.
83 #A #B #P #l1 elim l1 -l1 [ #l2 #H >H // ]
84 #x1 #l1 #IH1 #l2 elim l2 -l2 [ #false elim false ]
85 #x2 #l2 #_ #H elim H -H; normalize /3/
88 lemma all2_hd: ∀A,B:Type[0]. ∀P:A→B→Prop. ∀a,b. P a b →
89 ∀l1,l2. all2 … P l1 l2 → P (hd … l1 a) (hd … l2 b).
90 #A #B #P #a #b #Hab #l1 elim l1 -l1 [ #l2 #H2 >H2 @Hab ]
91 #x1 #l1 #_ #l2 elim l2 -l2 [ #false elim false ]
92 #x2 #l2 #_ #H elim H -H //
95 lemma all2_tl: ∀A,B:Type[0]. ∀P:A→B→Prop.
96 ∀l1,l2. all2 … P l1 l2 → all2 … P (tail … l1) (tail … l2).
97 #A #B #P #l1 elim l1 -l1 [ #l2 #H >H // ]
98 #x1 #l1 #_ #l2 elim l2 -l2 [ #false elim false ]
99 #x2 #l2 #_ #H elim H -H //
102 lemma all2_nth: ∀A,B:Type[0]. ∀P:A→B→Prop. ∀a,b. P a b →
103 ∀i,l1,l2. all2 … P l1 l2 → P (nth i … l1 a) (nth i … l2 b).
104 #A #B #P #a #b #Hab #i elim i -i [ @all2_hd // | #i #IH #l1 #l2 #H @IH /2/ ]
107 lemma all2_append: ∀A,B,P,l2,m2. all2 A B P l2 m2 →
108 ∀l1,m1. all2 A B P l1 m1 → all2 A B P (l1 @ l2) (m1 @ m2).
109 #A #B #P #l2 #m2 #H2 #l1 (elim l1) -l1 [ #m1 #H >H @H2 ]
110 #x1 #l1 #IH1 #m2 elim m2 -m2 [ #false elim false ]
111 #x2 #m2 #_ #H elim H -H /3/
114 lemma all2_symmetric: ∀A. ∀P:A→A→Prop. symmetric … P → symmetric … (all2 … P).
115 #A #P #HP #l1 elim l1 -l1 [ #l2 #H >H // ]
116 #x1 #l1 #IH1 #l2 elim l2 -l2 [ #false elim false ]
117 #x2 #l2 #_ #H elim H -H /3/
120 (* terms **********************************************************************)
122 (* Appl F l generalizes App applying F to a list of arguments
123 * The head of l is applied first
125 let rec Appl F l on l ≝ match l with
127 | cons A D ⇒ Appl (App F A) D
130 lemma appl_append: ∀N,l,M. Appl M (l @ [N]) = App (Appl M l) N.
131 #N #l elim l -l // #hd #tl #IHl #M >IHl //
134 (* FG: not needed for now
136 inductive neutral: T → Prop ≝
137 | neutral_sort: ∀n.neutral (Sort n)
138 | neutral_rel: ∀i.neutral (Rel i)
139 | neutral_app: ∀M,N.neutral (App M N)
143 (* substitution ***************************************************************)
145 (* FG: do we need this?
146 definition lift0 ≝ λp,k,M . lift M p k. (**) (* remove definition *)
148 lemma lift_appl: ∀p,k,l,F. lift (Appl F l) p k =
149 Appl (lift F p k) (map … (lift0 p k) l).
150 #p #k #l (elim l) -l /2/ #A #D #IHl #F >IHl //
154 lemma lift_rel_lt: ∀i,p,k. (S i) ≤ k → lift (Rel i) k p = Rel i.
155 #i #p #k #Hik normalize >(le_to_leb_true … Hik) //
158 lemma lift_rel_ge: ∀i,p,k. (S i) ≰ k → lift (Rel i) k p = Rel (i+p).
159 #i #p #k #Hik normalize >(lt_to_leb_false (S i) k) /2/
162 lemma lift_app: ∀M,N,k,p.
163 lift (App M N) k p = App (lift M k p) (lift N k p).
166 lemma lift_lambda: ∀N,M,k,p. lift (Lambda N M) k p =
167 Lambda (lift N k p) (lift M (k + 1) p).
170 lemma lift_prod: ∀N,M,k,p.
171 lift (Prod N M) k p = Prod (lift N k p) (lift M (k + 1) p).
174 lemma subst_app: ∀M,N,k,L. (App M N)[k≝L] = App M[k≝L] N[k≝L].
177 lemma subst_lambda: ∀N,M,k,L. (Lambda N M)[k≝L] = Lambda N[k≝L] M[k+1≝L].
180 lemma subst_prod: ∀N,M,k,L. (Prod N M)[k≝L] = Prod N[k≝L] M[k+1≝L].
184 axiom lift_subst_lt: ∀A,B,i,j,k. lift (B[j≝A]) (j+k) i =
185 (lift B (j+k+1) i)[j≝lift A k i].
187 (* telescopic delifting substitution of l in M.
188 * Rel 0 is replaced with the head of l
190 let rec tsubst M l on l ≝ match l with
192 | cons A D ⇒ (tsubst M[0≝A] D)
195 interpretation "telescopic substitution" 'Subst1 M l = (tsubst M l).
197 lemma tsubst_refl: ∀l,t. (lift t 0 (|l|))[l] = t.
198 #l elim l -l; normalize // #hd #tl #IHl #t cut (S (|tl|) = |tl| + 1) // (**) (* eliminate cut *)
201 lemma tsubst_sort: ∀n,l. (Sort n)[l] = Sort n.