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 "arithmetics/nat.ma".
14 inductive T : Type[0] ≝
15 | Sort: nat → T (* starts from 0 *)
16 | Rel: nat → T (* starts from ... ? *)
17 | App: T → T → T (* function, argument *)
18 | Lambda: T → T → T (* type, body *)
19 | Prod: T → T → T (* type, body *)
20 | D: T → T (* dummifier *)
23 (* arguments: k is the depth (starts from 0), p is the height (starts from 0) *)
27 | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n) (Rel (n+p))
28 | App m n ⇒ App (lift m k p) (lift n k p)
29 | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
30 | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
31 | D n ⇒ D (lift n k p)
35 ndefinition lift ≝ λt.λp.lift_aux t 0 p.
37 notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift O $M}.
38 notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}.
40 (* interpretation "Lift" 'Lift n M = (lift M n). *)
41 interpretation "Lift" 'Lift n k M = (lift M k n).
46 | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n)
47 (if_then_else T (eqb n k) (lift a 0 n) (Rel (n-1)))
48 | App m n ⇒ App (subst m k a) (subst n k a)
49 | Lambda m n ⇒ Lambda (subst m k a) (subst n (k+1) a)
50 | Prod m n ⇒ Prod (subst m k a) (subst n (k+1) a)
51 | D n ⇒ D (subst n k a)
54 (* meglio non definire
55 ndefinition subst ≝ λa.λt.subst_aux t 0 a.
56 notation "M [ N ]" non associative with precedence 90 for @{'Subst $N $M}.
59 notation "M [ k := N ]" non associative with precedence 90 for @{'Subst $M $k $N}.
61 (* interpretation "Subst" 'Subst N M = (subst N M). *)
62 interpretation "Subst" 'Subst M k N = (subst M k N).
64 (*** properties of lift and subst ***)
66 lemma lift_0: ∀t:T.∀k. lift t k 0 = t.
67 #t (elim t) normalize // #n #k cases (leb (S n) k) normalize //
70 (* nlemma lift_0: ∀t:T. lift t 0 = t.
71 #t; nelim t; nnormalize; //; nqed. *)
73 lemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
76 lemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
79 lemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
80 #i (change with (lift (Rel i) 0 1 = Rel (1 + i))) //
83 lemma lift_rel_lt : ∀n,k,i. i < k → lift (Rel i) k n = Rel i.
84 #n #k #i #ltik change with
85 (if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel i)
86 >(le_to_leb_true … ltik) //
89 lemma lift_rel_ge : ∀n,k,i. k ≤ i → lift (Rel i) k n = Rel (i+n).
90 #n #k #i #leki change with
91 (if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel (i+n))
92 >lt_to_leb_false // @le_S_S //
95 lemma lift_lift: ∀t.∀m,j.j ≤ m → ∀n,k.
96 lift (lift t k m) (j+k) n = lift t k (m+n).
97 #t #i #j #h (elim t) normalize // #n #h #k
98 @(leb_elim (S n) k) #Hnk normalize
99 [>(le_to_leb_true (S n) (j+k) ?) normalize /2/
100 |>(lt_to_leb_false (S n+i) (j+k) ?)
101 normalize // @le_S_S >(commutative_plus j k)
102 @le_plus // @not_lt_to_le /2/
106 lemma lift_lift_up: ∀n,m,t,k,i.
107 lift (lift t i m) (m+k+i) n = lift (lift t (k+i) n) i m.
109 [1,3,4,5,6: normalize //
110 |#p #k #i @(leb_elim i p);
111 [#leip >lift_rel_ge // @(leb_elim (k+i) p);
112 [#lekip >lift_rel_ge;
113 [>lift_rel_ge // >lift_rel_ge // @(transitive_le … leip) //
114 |>associative_plus >commutative_plus @monotonic_le_plus_l //
116 |#lefalse (cut (p < k+i)) [@not_le_to_lt //] #ltpki
117 >lift_rel_lt; [|>associative_plus >commutative_plus @monotonic_lt_plus_r //]
118 >lift_rel_lt // >lift_rel_ge //
120 |#lefalse (cut (p < i)) [@not_le_to_lt //] #ltpi
121 >lift_rel_lt // >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
122 >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
128 lemma lift_lift1: ∀t.∀i,j,k.
129 lift(lift t k j) k i = lift t k (j+i).
132 lemma lift_lift2: ∀t.∀i,j,k.
133 lift (lift t k j) (j+k) i = lift t k (j+i).
137 nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
138 nnormalize; //; nqed. *)
140 lemma subst_lift_k: ∀A,B.∀k. (lift B k 1)[k ≝ A] = B.
141 #A #B (elim B) normalize /2/ #n #k
142 @(leb_elim (S n) k) normalize #Hnk
143 [>(le_to_leb_true ?? Hnk) normalize //
144 |>(lt_to_leb_false (S (n + 1)) k ?) normalize
145 [>(not_eq_to_eqb_false (n+1) k ?) normalize /2/
146 |@le_S (applyS (not_le_to_lt (S n) k Hnk))
152 nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
153 nnormalize; //; nqed. *)
155 lemma subst_sort: ∀A.∀n,k.(Sort n) [k ≝ A] = Sort n.
158 lemma subst_rel: ∀A.(Rel 0) [0 ≝ A] = A.
161 lemma subst_rel1: ∀A.∀k,i. i < k →
162 (Rel i) [k ≝ A] = Rel i.
163 #A #k #i normalize #ltik >(le_to_leb_true (S i) k) //
166 lemma subst_rel2: ∀A.∀k.
167 (Rel k) [k ≝ A] = lift A 0 k.
168 #A #k normalize >(lt_to_leb_false (S k) k) // >(eq_to_eqb_true … (refl …)) //
171 lemma subst_rel3: ∀A.∀k,i. k < i →
172 (Rel i) [k ≝ A] = Rel (i-1).
173 #A #k #i normalize #ltik >(lt_to_leb_false (S i) k) /2/
174 >(not_eq_to_eqb_false i k) // @sym_not_eq @lt_to_not_eq //
177 lemma lift_subst_ijk: ∀A,B.∀i,j,k.
178 lift (B [j+k := A]) k i = (lift B k i) [j+k+i ≝ A].
179 #A #B #i #j (elim B) normalize /2/ #n #k
180 @(leb_elim (S n) (j + k)) normalize #Hnjk
181 [(elim (leb (S n) k))
182 [>(subst_rel1 A (j+k+i) n) /2/
183 |>(subst_rel1 A (j+k+i) (n+i)) /2/
185 |@(eqb_elim n (j+k)) normalize #Heqnjk
186 [>(lt_to_leb_false (S n) k);
187 [(cut (j+k+i = n+i)) [//] #Heq
188 >Heq >(subst_rel2 A ?) normalize (applyS lift_lift) //
192 [@not_eq_to_le_to_lt;
193 [/2/ |@le_S_S_to_le @not_le_to_lt /2/ ]
195 (cut (O < n)) [/2/] #posn (cut (k ≤ n)) [/2/] #lekn
196 >(lt_to_leb_false (S (n-1)) k) normalize
197 [>(lt_to_leb_false … (le_S_S … lekn))
198 >(subst_rel3 A (j+k+i) (n+i)); [/3/ |/2/]
199 |@le_S_S; (* /3/; 65 *) (applyS monotonic_pred) @le_plus_b //
205 lemma lift_subst_up: ∀M,N,n,i,j.
206 lift M[i≝N] (i+j) n = (lift M (i+j+1) n)[i≝ (lift N j n)].
209 |#p #N #n #i #j (cases (true_or_false (leb p i)))
210 [#lepi (cases (le_to_or_lt_eq … (leb_true_to_le … lepi)))
211 [#ltpi >(subst_rel1 … ltpi)
212 (cut (p < i+j)) [@(lt_to_le_to_lt … ltpi) //] #ltpij
213 >(lift_rel_lt … ltpij); >(lift_rel_lt ?? p ?);
214 [>subst_rel1 // | @(lt_to_le_to_lt … ltpij) //]
215 |#eqpi >eqpi >subst_rel2 >lift_rel_lt;
216 [>subst_rel2 >(plus_n_O (i+j))
218 |@(le_to_lt_to_lt ? (i+j)) //
221 |#lefalse (cut (i < p)) [@not_le_to_lt /2/] #ltip
222 (cut (0 < p)) [@(le_to_lt_to_lt … ltip) //] #posp
223 >(subst_rel3 … ltip) (cases (true_or_false (leb (S p) (i+j+1))))
224 [#Htrue (cut (p < i+j+1)) [@(leb_true_to_le … Htrue)] #Hlt
226 [>lift_rel_lt // >(subst_rel3 … ltip) // | @lt_plus_to_minus //]
227 |#Hfalse >lift_rel_ge;
229 [>subst_rel3; [@eq_f /2/ | @(lt_to_le_to_lt … ltip) //]
230 |@not_lt_to_le @(leb_false_to_not_le … Hfalse)
232 |@le_plus_to_minus_r @not_lt_to_le
233 @(leb_false_to_not_le … Hfalse)
237 |#P #Q #HindP #HindQ #N #n #i #j normalize
238 @eq_f2; [@HindP |@HindQ ]
239 |#P #Q #HindP #HindQ #N #n #i #j normalize
240 @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
241 <associative_plus @HindQ]
242 |#P #Q #HindP #HindQ #N #n #i #j normalize
243 @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
244 <associative_plus @HindQ]
245 |#P #HindP #N #n #i #j normalize
250 theorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
251 (lift B i (S k)) [j ≝ A] = lift B i k.
252 #A #B (elim B) normalize /2/
253 [2,3,4: #T #T0 #Hind1 #Hind2 #i #j #k #leij #lejk
254 @eq_f2 /2/ @Hind2 (applyS (monotonic_le_plus_l 1)) //
255 |5:#T #Hind #i #j #k #leij #lejk @eq_f @Hind //
256 |#n #i #j #k #leij #ltjk @(leb_elim (S n) i) normalize #len
257 [>(le_to_leb_true (S n) j) /2/
258 |>(lt_to_leb_false (S (n+S k)) j);
259 [normalize >(not_eq_to_eqb_false (n+S k) j)normalize
260 /2/ @(not_to_not …len) #H @(le_plus_to_le_r k) normalize //
261 |@le_S_S @(transitive_le … ltjk) @le_plus // @not_lt_to_le /2/
267 (********************* substitution lemma ***********************)
269 lemma subst_lemma: ∀A,B,C.∀k,i.
270 (A [i ≝ B]) [k+i ≝ C] =
271 (A [S (k+i) := C]) [i ≝ B [k ≝ C]].
272 #A #B #C #k (elim A) normalize // (* WOW *)
273 #n #i @(leb_elim (S n) i) #Hle
274 [(cut (n < k+i)) [/2/] #ltn (* lento *) (cut (n ≤ k+i)) [/2/] #len
275 >(subst_rel1 C (k+i) n ltn) >(le_to_leb_true n (k+i) len) >(subst_rel1 … Hle) //
276 |@(eqb_elim n i) #eqni
277 [>eqni >(le_to_leb_true i (k+i)) // >(subst_rel2 …);
278 normalize @sym_eq (applyS (lift_subst_ijk C B i k O))
279 |@(leb_elim (S (n-1)) (k+i)) #nk
280 [>(subst_rel1 C (k+i) (n-1) nk) >(le_to_leb_true n (k+i));
281 [>(subst_rel3 ? i n) // @not_eq_to_le_to_lt;
282 [/2/ |@not_lt_to_le /2/]
283 |@(transitive_le … nk) //
285 |(cut (i < n)) [@not_eq_to_le_to_lt; [/2/] @(not_lt_to_le … Hle)]
286 #ltin (cut (O < n)) [/2/] #posn
287 @(eqb_elim (n-1) (k+i)) #H
288 [>H >(subst_rel2 C (k+i)) >(lt_to_leb_false n (k+i));
289 [>(eq_to_eqb_true n (S(k+i)));
290 [normalize |<H (applyS plus_minus_m_m) // ]
291 (generalize in match ltin)
292 <H @(lt_O_n_elim … posn) #m #leim >delift normalize /2/
293 |<H @(lt_O_n_elim … posn) #m normalize //
296 [@not_eq_to_le_to_lt; [@sym_not_eq @H |@(not_lt_to_le … nk)]]
297 #Hlt >(lt_to_leb_false n (k+i));
298 [>(not_eq_to_eqb_false n (S(k+i)));
299 [>(subst_rel3 C (k+i) (n-1) Hlt);
300 >(subst_rel3 ? i (n-1)) // @(le_to_lt_to_lt … Hlt) //
301 |@(not_to_not … H) #Hn >Hn normalize //
303 |@(transitive_lt … Hlt) @(lt_O_n_elim … posn) normalize //