2 ||M|| This file is part of HELM, an Hypertextual, Electronic
3 ||A|| Library of Mathematics, developed at the Computer Science
4 ||T|| Department of the University of Bologna, Italy.
7 ||A|| This file is distributed under the terms of the
8 \ / GNU General Public License Version 2
10 V_______________________________________________________________ *)
12 include "lambda/terms.ma".
14 (* arguments: k is the depth (starts from 0), p is the height (starts from 0) *)
18 | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n) (Rel (n+p))
19 | App m n ⇒ App (lift m k p) (lift n k p)
20 | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
21 | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
22 | D n ⇒ D (lift n k p)
26 ndefinition lift ≝ λt.λp.lift_aux t 0 p.
28 notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift O $M}.
29 notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}.
31 (* interpretation "Lift" 'Lift n M = (lift M n). *)
32 interpretation "Lift" 'Lift n k M = (lift M k n).
37 | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n)
38 (if_then_else T (eqb n k) (lift a 0 n) (Rel (n-1)))
39 | App m n ⇒ App (subst m k a) (subst n k a)
40 | Lambda m n ⇒ Lambda (subst m k a) (subst n (k+1) a)
41 | Prod m n ⇒ Prod (subst m k a) (subst n (k+1) a)
42 | D n ⇒ D (subst n k a)
45 (* meglio non definire
46 ndefinition subst ≝ λa.λt.subst_aux t 0 a.
47 notation "M [ N ]" non associative with precedence 90 for @{'Subst $N $M}.
50 (* interpretation "Subst" 'Subst N M = (subst N M). *)
51 interpretation "Subst" 'Subst1 M k N = (subst M k N).
53 (*** properties of lift and subst ***)
55 lemma lift_0: ∀t:T.∀k. lift t k 0 = t.
56 #t (elim t) normalize // #n #k cases (leb (S n) k) normalize //
59 (* nlemma lift_0: ∀t:T. lift t 0 = t.
60 #t; nelim t; nnormalize; //; nqed. *)
62 lemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
65 lemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
68 lemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
69 #i (change with (lift (Rel i) 0 1 = Rel (1 + i))) //
72 lemma lift_rel_lt : ∀n,k,i. i < k → lift (Rel i) k n = Rel i.
73 #n #k #i #ltik change with
74 (if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel i)
75 >(le_to_leb_true … ltik) //
78 lemma lift_rel_ge : ∀n,k,i. k ≤ i → lift (Rel i) k n = Rel (i+n).
79 #n #k #i #leki change with
80 (if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel (i+n))
81 >lt_to_leb_false // @le_S_S //
84 lemma lift_lift: ∀t.∀m,j.j ≤ m → ∀n,k.
85 lift (lift t k m) (j+k) n = lift t k (m+n).
86 #t #i #j #h (elim t) normalize // #n #h #k
87 @(leb_elim (S n) k) #Hnk normalize
88 [>(le_to_leb_true (S n) (j+k) ?) normalize /2/
89 |>(lt_to_leb_false (S n+i) (j+k) ?)
90 normalize // @le_S_S >(commutative_plus j k)
91 @le_plus // @not_lt_to_le /2/
95 lemma lift_lift_up: ∀n,m,t,k,i.
96 lift (lift t i m) (m+k+i) n = lift (lift t (k+i) n) i m.
98 [1,3,4,5,6: normalize //
99 |#p #k #i @(leb_elim i p);
100 [#leip >lift_rel_ge // @(leb_elim (k+i) p);
101 [#lekip >lift_rel_ge;
102 [>lift_rel_ge // >lift_rel_ge // @(transitive_le … leip) //
103 |>associative_plus >commutative_plus @monotonic_le_plus_l //
105 |#lefalse (cut (p < k+i)) [@not_le_to_lt //] #ltpki
106 >lift_rel_lt; [|>associative_plus >commutative_plus @monotonic_lt_plus_r //]
107 >lift_rel_lt // >lift_rel_ge //
109 |#lefalse (cut (p < i)) [@not_le_to_lt //] #ltpi
110 >lift_rel_lt // >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
111 >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
117 lemma lift_lift1: ∀t.∀i,j,k.
118 lift(lift t k j) k i = lift t k (j+i).
121 lemma lift_lift2: ∀t.∀i,j,k.
122 lift (lift t k j) (j+k) i = lift t k (j+i).
126 nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
127 nnormalize; //; nqed. *)
129 lemma subst_lift_k: ∀A,B.∀k. (lift B k 1)[k ≝ A] = B.
130 #A #B (elim B) normalize /2/ #n #k
131 @(leb_elim (S n) k) normalize #Hnk
132 [>(le_to_leb_true ?? Hnk) normalize //
133 |>(lt_to_leb_false (S (n + 1)) k ?) normalize
134 [>(not_eq_to_eqb_false (n+1) k ?) normalize /2/
135 |@le_S (applyS (not_le_to_lt (S n) k Hnk))
141 nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
142 nnormalize; //; nqed. *)
144 lemma subst_sort: ∀A.∀n,k.(Sort n) [k ≝ A] = Sort n.
147 lemma subst_rel: ∀A.(Rel 0) [0 ≝ A] = A.
150 lemma subst_rel1: ∀A.∀k,i. i < k →
151 (Rel i) [k ≝ A] = Rel i.
152 #A #k #i normalize #ltik >(le_to_leb_true (S i) k) //
155 lemma subst_rel2: ∀A.∀k.
156 (Rel k) [k ≝ A] = lift A 0 k.
157 #A #k normalize >(lt_to_leb_false (S k) k) // >(eq_to_eqb_true … (refl …)) //
160 lemma subst_rel3: ∀A.∀k,i. k < i →
161 (Rel i) [k ≝ A] = Rel (i-1).
162 #A #k #i normalize #ltik >(lt_to_leb_false (S i) k) /2/
163 >(not_eq_to_eqb_false i k) // @sym_not_eq @lt_to_not_eq //
166 lemma lift_subst_ijk: ∀A,B.∀i,j,k.
167 lift (B [j+k := A]) k i = (lift B k i) [j+k+i ≝ A].
168 #A #B #i #j (elim B) normalize /2/ #n #k
169 @(leb_elim (S n) (j + k)) normalize #Hnjk
170 [(elim (leb (S n) k))
171 [>(subst_rel1 A (j+k+i) n) /2/
172 |>(subst_rel1 A (j+k+i) (n+i)) /2/
174 |@(eqb_elim n (j+k)) normalize #Heqnjk
175 [>(lt_to_leb_false (S n) k);
176 [(cut (j+k+i = n+i)) [//] #Heq
177 >Heq >(subst_rel2 A ?) normalize (applyS lift_lift) //
181 [@not_eq_to_le_to_lt;
182 [/2/ |@le_S_S_to_le @not_le_to_lt /2/ ]
184 (cut (O < n)) [/2/] #posn (cut (k ≤ n)) [/2/] #lekn
185 >(lt_to_leb_false (S (n-1)) k) normalize
186 [>(lt_to_leb_false … (le_S_S … lekn))
187 >(subst_rel3 A (j+k+i) (n+i)); [/3/ |/2/]
188 |@le_S_S; (* /3/; 65 *) (applyS monotonic_pred) @le_plus_b //
194 lemma lift_subst_up: ∀M,N,n,i,j.
195 lift M[i≝N] (i+j) n = (lift M (i+j+1) n)[i≝ (lift N j n)].
198 |#p #N #n #i #j (cases (true_or_false (leb p i)))
199 [#lepi (cases (le_to_or_lt_eq … (leb_true_to_le … lepi)))
200 [#ltpi >(subst_rel1 … ltpi)
201 (cut (p < i+j)) [@(lt_to_le_to_lt … ltpi) //] #ltpij
202 >(lift_rel_lt … ltpij); >(lift_rel_lt ?? p ?);
203 [>subst_rel1 // | @(lt_to_le_to_lt … ltpij) //]
204 |#eqpi >eqpi >subst_rel2 >lift_rel_lt;
205 [>subst_rel2 >(plus_n_O (i+j))
207 |@(le_to_lt_to_lt ? (i+j)) //
210 |#lefalse (cut (i < p)) [@not_le_to_lt /2/] #ltip
211 (cut (0 < p)) [@(le_to_lt_to_lt … ltip) //] #posp
212 >(subst_rel3 … ltip) (cases (true_or_false (leb (S p) (i+j+1))))
213 [#Htrue (cut (p < i+j+1)) [@(leb_true_to_le … Htrue)] #Hlt
215 [>lift_rel_lt // >(subst_rel3 … ltip) // | @lt_plus_to_minus //]
216 |#Hfalse >lift_rel_ge;
218 [>subst_rel3; [@eq_f /2/ | @(lt_to_le_to_lt … ltip) //]
219 |@not_lt_to_le @(leb_false_to_not_le … Hfalse)
221 |@le_plus_to_minus_r @not_lt_to_le
222 @(leb_false_to_not_le … Hfalse)
226 |#P #Q #HindP #HindQ #N #n #i #j normalize
227 @eq_f2; [@HindP |@HindQ ]
228 |#P #Q #HindP #HindQ #N #n #i #j normalize
229 @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
230 <associative_plus @HindQ]
231 |#P #Q #HindP #HindQ #N #n #i #j normalize
232 @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
233 <associative_plus @HindQ]
234 |#P #HindP #N #n #i #j normalize
239 theorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
240 (lift B i (S k)) [j ≝ A] = lift B i k.
241 #A #B (elim B) normalize /2/
242 [2,3,4: #T #T0 #Hind1 #Hind2 #i #j #k #leij #lejk
243 @eq_f2 /2/ @Hind2 (applyS (monotonic_le_plus_l 1)) //
244 |5:#T #Hind #i #j #k #leij #lejk @eq_f @Hind //
245 |#n #i #j #k #leij #ltjk @(leb_elim (S n) i) normalize #len
246 [>(le_to_leb_true (S n) j) /2/
247 |>(lt_to_leb_false (S (n+S k)) j);
248 [normalize >(not_eq_to_eqb_false (n+S k) j)normalize
249 /2/ @(not_to_not …len) #H @(le_plus_to_le_r k) normalize //
250 |@le_S_S @(transitive_le … ltjk) @le_plus // @not_lt_to_le /2/
256 (********************* substitution lemma ***********************)
258 lemma subst_lemma: ∀A,B,C.∀k,i.
259 (A [i ≝ B]) [k+i ≝ C] =
260 (A [S (k+i) := C]) [i ≝ B [k ≝ C]].
261 #A #B #C #k (elim A) normalize // (* WOW *)
262 #n #i @(leb_elim (S n) i) #Hle
263 [(cut (n < k+i)) [/2/] #ltn (* lento *) (cut (n ≤ k+i)) [/2/] #len
264 >(subst_rel1 C (k+i) n ltn) >(le_to_leb_true n (k+i) len) >(subst_rel1 … Hle) //
265 |@(eqb_elim n i) #eqni
266 [>eqni >(le_to_leb_true i (k+i)) // >(subst_rel2 …);
267 normalize @sym_eq (applyS (lift_subst_ijk C B i k O))
268 |@(leb_elim (S (n-1)) (k+i)) #nk
269 [>(subst_rel1 C (k+i) (n-1) nk) >(le_to_leb_true n (k+i));
270 [>(subst_rel3 ? i n) // @not_eq_to_le_to_lt;
271 [/2/ |@not_lt_to_le /2/]
272 |@(transitive_le … nk) //
274 |(cut (i < n)) [@not_eq_to_le_to_lt; [/2/] @(not_lt_to_le … Hle)]
275 #ltin (cut (O < n)) [/2/] #posn
276 @(eqb_elim (n-1) (k+i)) #H
277 [>H >(subst_rel2 C (k+i)) >(lt_to_leb_false n (k+i));
278 [>(eq_to_eqb_true n (S(k+i)));
279 [normalize |<H (applyS plus_minus_m_m) // ]
280 (generalize in match ltin)
281 <H @(lt_O_n_elim … posn) #m #leim >delift normalize /2/
282 |<H @(lt_O_n_elim … posn) #m normalize //
285 [@not_eq_to_le_to_lt; [@sym_not_eq @H |@(not_lt_to_le … nk)]]
286 #Hlt >(lt_to_leb_false n (k+i));
287 [>(not_eq_to_eqb_false n (S(k+i)));
288 [>(subst_rel3 C (k+i) (n-1) Hlt);
289 >(subst_rel3 ? i (n-1)) // @(le_to_lt_to_lt … Hlt) //
290 |@(not_to_not … H) #Hn >Hn normalize //
292 |@(transitive_lt … Hlt) @(lt_O_n_elim … posn) normalize //