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 "lambdaN/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 k n) (Rel (n+p)) (Rel n)
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 m ⇒ D (lift n k p) (lift m 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 k n)
38 (if_then_else T (eqb k n) (lift a 0 n) (Rel (n-1))) (Rel n)
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 m ⇒ D (subst n k a) (subst m 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 k n) 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 k i) (Rel (i+n)) (Rel i) = Rel i)
75 >(lt_to_leb_false … 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 k i) (Rel (i+n)) (Rel i) = Rel (i+n))
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 k n) #Hnk normalize
88 [>(le_to_leb_true (j+k) (n+i) ?)
89 normalize // >(commutative_plus j k) @le_plus //
90 |>(lt_to_leb_false (j+k) n ?) normalize //
91 @(transitive_le ? k) // @not_le_to_lt //
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 k n) normalize #Hnk
132 [cut (k ≤ n+1) [@transitive_le //] #H
133 >(le_to_leb_true … H) normalize
134 >(not_eq_to_eqb_false k (n+1)) normalize /2/
135 |>(lt_to_leb_false … (not_le_to_lt … Hnk)) normalize //
140 nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
141 nnormalize; //; nqed. *)
143 lemma subst_sort: ∀A.∀n,k.(Sort n) [k ≝ A] = Sort n.
146 lemma subst_rel: ∀A.(Rel 0) [0 ≝ A] = A.
149 lemma subst_rel1: ∀A.∀k,i. i < k →
150 (Rel i) [k ≝ A] = Rel i.
151 #A #k #i normalize #ltik >(lt_to_leb_false … ltik) //
154 lemma subst_rel2: ∀A.∀k.
155 (Rel k) [k ≝ A] = lift A 0 k.
156 #A #k normalize >(le_to_leb_true k k) // >(eq_to_eqb_true … (refl …)) //
159 lemma subst_rel3: ∀A.∀k,i. k < i →
160 (Rel i) [k ≝ A] = Rel (i-1).
161 #A #k #i normalize #ltik >(le_to_leb_true k i) /2/
162 >(not_eq_to_eqb_false k i) // @lt_to_not_eq //
165 lemma lift_subst_ijk: ∀A,B.∀i,j,k.
166 lift (B [j+k := A]) k i = (lift B k i) [j+k+i ≝ A].
167 #A #B #i #j (elim B) normalize /2/ #n #k
168 @(leb_elim (j+k) n) normalize #Hnjk
169 [@(eqb_elim (j+k) n) normalize #Heqnjk
170 [>(le_to_leb_true k n) //
171 (cut (j+k+i = n+i)) [//] #Heq
172 >Heq >(subst_rel2 A ?) (applyS lift_lift) //
174 [@not_eq_to_le_to_lt; /2/] #ltjkn
175 (cut (O < n)) [/2/] #posn (cut (k ≤ n)) [/2/] #lekn
176 >(le_to_leb_true k (n-1)) normalize
177 [>(le_to_leb_true … lekn)
178 >(subst_rel3 A (j+k+i) (n+i)); [/3/ |/2/]
179 |(applyS monotonic_pred) @le_plus_b //
183 [>(subst_rel1 A (j+k+i) (n+i)) // @monotonic_lt_plus_l /2/
184 |>(subst_rel1 A (j+k+i) n) // @(lt_to_le_to_lt ? (j+k)) /2/
189 lemma lift_subst_up: ∀M,N,n,i,j.
190 lift M[i≝N] (i+j) n = (lift M (i+j+1) n)[i≝ (lift N j n)].
193 |#p #N #n #i #j (cases (true_or_false (leb p i)))
194 [#lepi (cases (le_to_or_lt_eq … (leb_true_to_le … lepi)))
195 [#ltpi >(subst_rel1 … ltpi)
196 (cut (p < i+j)) [@(lt_to_le_to_lt … ltpi) //] #ltpij
197 >(lift_rel_lt … ltpij); >(lift_rel_lt ?? p ?);
198 [>subst_rel1 // | @(lt_to_le_to_lt … ltpij) //]
199 |#eqpi >eqpi >subst_rel2 >lift_rel_lt;
200 [>subst_rel2 >(plus_n_O (i+j))
202 |@(le_to_lt_to_lt ? (i+j)) //
205 |#lefalse (cut (i < p)) [@not_le_to_lt /2/] #ltip
206 (cut (0 < p)) [@(le_to_lt_to_lt … ltip) //] #posp
207 >(subst_rel3 … ltip) (cases (true_or_false (leb (S p) (i+j+1))))
208 [#Htrue (cut (p < i+j+1)) [@(leb_true_to_le … Htrue)] #Hlt
210 [>lift_rel_lt // >(subst_rel3 … ltip) // | @lt_plus_to_minus //]
211 |#Hfalse >lift_rel_ge;
213 [>subst_rel3; [@eq_f /2/ | @(lt_to_le_to_lt … ltip) //]
214 |@not_lt_to_le @(leb_false_to_not_le … Hfalse)
216 |@le_plus_to_minus_r @not_lt_to_le
217 @(leb_false_to_not_le … Hfalse)
221 |#P #Q #HindP #HindQ #N #n #i #j normalize
222 @eq_f2; [@HindP |@HindQ ]
223 |#P #Q #HindP #HindQ #N #n #i #j normalize
224 @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
225 <associative_plus @HindQ]
226 |#P #Q #HindP #HindQ #N #n #i #j normalize
227 @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
228 <associative_plus @HindQ]
229 |#P #Q #HindP #HindQ #N #n #i #j normalize
230 @eq_f2; [@HindP |@HindQ ]
234 theorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
235 (lift B i (S k)) [j ≝ A] = lift B i k.
236 #A #B (elim B) normalize /2/
237 [2,3,4,5: #T #T0 #Hind1 #Hind2 #i #j #k #leij #lejk
238 @eq_f2 /2/ @Hind2 (applyS (monotonic_le_plus_l 1)) //
239 |#n #i #j #k #leij #ltjk @(leb_elim i n) normalize #len
241 [<plus_n_Sm @le_S_S @(transitive_le … ltjk) /2/] #H
242 >(le_to_leb_true j (n+S k));
243 [normalize >(not_eq_to_eqb_false j (n+S k)) normalize /2/
246 |>(lt_to_leb_false j n) // @(lt_to_le_to_lt … leij)
252 (********************* substitution lemma ***********************)
254 lemma subst_lemma: ∀A,B,C.∀k,i.
255 (A [i ≝ B]) [k+i ≝ C] =
256 (A [(k+i)+1:= C]) [i ≝ B [k ≝ C]].
257 #A #B #C #k (elim A) normalize // (* WOW *)
258 #n #i @(leb_elim i n) #Hle
259 [@(eqb_elim i n) #eqni
260 [<eqni >(lt_to_leb_false (k+i+1) i) // >(subst_rel2 …);
261 normalize @sym_eq (applyS (lift_subst_ijk C B i k O))
263 [cases (le_to_or_lt_eq … Hle) // #eqin @False_ind /2/] #ltin
264 (cut (O < n)) [@(le_to_lt_to_lt … ltin) //] #posn
265 normalize @(leb_elim (k+i) (n-1)) #nk
266 [@(eqb_elim (k+i) (n-1)) #H normalize
267 [cut (k+i+1 = n); [/2/] #H1
268 >(le_to_leb_true (k+i+1) n) /2/
269 >(eq_to_eqb_true … H1) normalize
270 (generalize in match ltin)
271 @(lt_O_n_elim … posn) #m #leim >delift // /2/
272 |(cut (k+i < n-1)) [@not_eq_to_le_to_lt; //] #Hlt
273 >(le_to_leb_true (k+i+1) n);
274 [>(not_eq_to_eqb_false (k+i+1) n);
275 [>(subst_rel3 ? i (n-1));
276 // @(le_to_lt_to_lt … Hlt) //
277 |@(not_to_not … H) #Hn /2/
279 |@le_minus_to_plus_r //
282 |>(not_le_to_leb_false (k+i+1) n);
283 [>(subst_rel3 ? i n) normalize //
284 |@(not_to_not … nk) #H @le_plus_to_minus_r //
288 |(cut (n < k+i)) [@(lt_to_le_to_lt ? i) /2/] #ltn (* lento *)
289 (* (cut (n ≤ k+i)) [/2/] #len *)
290 >(subst_rel1 C (k+i) n ltn) >(lt_to_leb_false (k+i+1) n);
291 [>subst_rel1 /2/ | @(transitive_lt …ltn) // ]