--- /dev/null
+(*
+ ||M|| This file is part of HELM, an Hypertextual, Electronic
+ ||A|| Library of Mathematics, developed at the Computer Science
+ ||T|| Department of the University of Bologna, Italy.
+ ||I||
+ ||T||
+ ||A|| This file is distributed under the terms of the
+ \ / GNU General Public License Version 2
+ \ /
+ V_______________________________________________________________ *)
+
+include "arithmetics/nat.ma".
+
+inductive T : Type[0] ≝
+ | Sort: nat → T (* starts from 0 *)
+ | Rel: nat → T (* starts from ... ? *)
+ | App: T → T → T (* function, argument *)
+ | Lambda: T → T → T (* type, body *)
+ | Prod: T → T → T (* type, body *)
+ | D: T → T (* dummifier *)
+.
+
+(* arguments: k is the depth (starts from 0), p is the height (starts from 0) *)
+let rec lift t k p ≝
+ match t with
+ [ Sort n ⇒ Sort n
+ | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n) (Rel (n+p))
+ | App m n ⇒ App (lift m k p) (lift n k p)
+ | Lambda m n ⇒ Lambda (lift m k p) (lift n (k+1) p)
+ | Prod m n ⇒ Prod (lift m k p) (lift n (k+1) p)
+ | D n ⇒ D (lift n k p)
+ ].
+
+(*
+ndefinition lift ≝ λt.λp.lift_aux t 0 p.
+
+notation "↑ ^ n ( M )" non associative with precedence 40 for @{'Lift O $M}.
+notation "↑ _ k ^ n ( M )" non associative with precedence 40 for @{'Lift $n $k $M}.
+*)
+(* interpretation "Lift" 'Lift n M = (lift M n). *)
+interpretation "Lift" 'Lift n k M = (lift M k n).
+
+let rec subst t k a ≝
+ match t with
+ [ Sort n ⇒ Sort n
+ | Rel n ⇒ if_then_else T (leb (S n) k) (Rel n)
+ (if_then_else T (eqb n k) (lift a 0 n) (Rel (n-1)))
+ | App m n ⇒ App (subst m k a) (subst n k a)
+ | Lambda m n ⇒ Lambda (subst m k a) (subst n (k+1) a)
+ | Prod m n ⇒ Prod (subst m k a) (subst n (k+1) a)
+ | D n ⇒ D (subst n k a)
+ ].
+
+(* meglio non definire
+ndefinition subst ≝ λa.λt.subst_aux t 0 a.
+notation "M [ N ]" non associative with precedence 90 for @{'Subst $N $M}.
+*)
+
+notation "M [ k := N ]" non associative with precedence 90 for @{'Subst $M $k $N}.
+
+(* interpretation "Subst" 'Subst N M = (subst N M). *)
+interpretation "Subst" 'Subst M k N = (subst M k N).
+
+(*** properties of lift and subst ***)
+
+lemma lift_0: ∀t:T.∀k. lift t k 0 = t.
+#t (elim t) normalize // #n #k cases (leb (S n) k) normalize //
+qed.
+
+(* nlemma lift_0: ∀t:T. lift t 0 = t.
+#t; nelim t; nnormalize; //; nqed. *)
+
+lemma lift_sort: ∀i,k,n. lift (Sort i) k n = Sort i.
+// qed.
+
+lemma lift_rel: ∀i,n. lift (Rel i) 0 n = Rel (i+n).
+// qed.
+
+lemma lift_rel1: ∀i.lift (Rel i) 0 1 = Rel (S i).
+#i (change with (lift (Rel i) 0 1 = Rel (1 + i))) //
+qed.
+
+lemma lift_lift: ∀t.∀i,j.j ≤ i → ∀h,k.
+ lift (lift t k i) (j+k) h = lift t k (i+h).
+#t #i #j #h (elim t) normalize // #n #h #k
+@(leb_elim (S n) k) #Hnk normalize
+ [>(le_to_leb_true (S n) (j+k) ?) normalize /2/
+ |>(lt_to_leb_false (S n+i) (j+k) ?)
+ normalize // @le_S_S >(commutative_plus j k)
+ @le_plus // @not_lt_to_le /2/
+ ]
+qed.
+
+lemma lift_lift1: ∀t.∀i,j,k.
+ lift(lift t k j) k i = lift t k (j+i).
+/2/ qed.
+
+lemma lift_lift2: ∀t.∀i,j,k.
+ lift (lift t k j) (j+k) i = lift t k (j+i).
+/2/ qed.
+
+(*
+nlemma lift_lift: ∀t.∀i,j. lift (lift t j) i = lift t (j+i).
+nnormalize; //; nqed. *)
+
+lemma subst_lift_k: ∀A,B.∀k. (lift B k 1)[k ≝ A] = B.
+#A #B (elim B) normalize /2/ #n #k
+@(leb_elim (S n) k) normalize #Hnk
+ [>(le_to_leb_true ?? Hnk) normalize //
+ |>(lt_to_leb_false (S (n + 1)) k ?) normalize
+ [>(not_eq_to_eqb_false (n+1) k ?) normalize /2/
+ |@le_S (applyS (not_le_to_lt (S n) k Hnk))
+ ]
+ ]
+qed.
+
+(*
+nlemma subst_lift: ∀A,B. subst A (lift B 1) = B.
+nnormalize; //; nqed. *)
+
+lemma subst_sort: ∀A.∀n,k.(Sort n) [k ≝ A] = Sort n.
+// qed.
+
+lemma subst_rel: ∀A.(Rel 0) [0 ≝ A] = A.
+normalize // qed.
+
+lemma subst_rel1: ∀A.∀k,i. i < k →
+ (Rel i) [k ≝ A] = Rel i.
+#A #k #i normalize #ltik >(le_to_leb_true (S i) k) //
+qed.
+
+lemma subst_rel2: ∀A.∀k.
+ (Rel k) [k ≝ A] = lift A 0 k.
+#A #k normalize >(lt_to_leb_false (S k) k) // >(eq_to_eqb_true … (refl …)) //
+qed.
+
+lemma subst_rel3: ∀A.∀k,i. k < i →
+ (Rel i) [k ≝ A] = Rel (i-1).
+#A #k #i normalize #ltik >(lt_to_leb_false (S i) k) /2/
+>(not_eq_to_eqb_false i k) // @sym_not_eq @lt_to_not_eq //
+qed.
+
+lemma lift_subst_ijk: ∀A,B.∀i,j,k.
+ lift (B [j+k := A]) k i = (lift B k i) [j+k+i ≝ A].
+#A #B #i #j (elim B) normalize /2/ #n #k
+@(leb_elim (S n) (j + k)) normalize #Hnjk
+ [(elim (leb (S n) k))
+ [>(subst_rel1 A (j+k+i) n) /2/
+ |>(subst_rel1 A (j+k+i) (n+i)) /2/
+ ]
+ |@(eqb_elim n (j+k)) normalize #Heqnjk
+ [>(lt_to_leb_false (S n) k);
+ [(cut (j+k+i = n+i)) [//] #Heq
+ >Heq >(subst_rel2 A ?) normalize (applyS lift_lift) //
+ |/2/
+ ]
+ |(cut (j + k < n))
+ [@not_eq_to_le_to_lt;
+ [/2/ |@le_S_S_to_le @not_le_to_lt /2/ ]
+ |#ltjkn
+ (cut (O < n)) [/2/] #posn (cut (k ≤ n)) [/2/] #lekn
+ >(lt_to_leb_false (S (n-1)) k) normalize
+ [>(lt_to_leb_false … (le_S_S … lekn))
+ >(subst_rel3 A (j+k+i) (n+i)); [/3/ |/2/]
+ |@le_S_S; (* /3/; 65 *) (applyS monotonic_pred) @le_plus_b //
+ ]
+ ]
+ ]
+qed.
+
+theorem delift : ∀A,B.∀i,j,k. i ≤ j → j ≤ i + k →
+ (lift B i (S k)) [j ≝ A] = lift B i k.
+#A #B (elim B) normalize /2/
+ [2,3,4: #T #T0 #Hind1 #Hind2 #i #j #k #leij #lejk
+ @eq_f2 /2/ @Hind2 (applyS (monotonic_le_plus_l 1)) //
+ |5:#T #Hind #i #j #k #leij #lejk @eq_f @Hind //
+ |#n #i #j #k #leij #ltjk @(leb_elim (S n) i) normalize #len
+ [>(le_to_leb_true (S n) j) /2/
+ |>(lt_to_leb_false (S (n+S k)) j);
+ [normalize >(not_eq_to_eqb_false (n+S k) j)normalize
+ /2/ @(not_to_not …len) #H @(le_plus_to_le_r k) normalize //
+ |@le_S_S @(transitive_le … ltjk) @le_plus // @not_lt_to_le /2/
+ ]
+ ]
+ ]
+qed.
+
+(********************* substitution lemma ***********************)
+
+lemma subst_lemma: ∀A,B,C.∀k,i.
+ (A [i ≝ B]) [k+i ≝ C] =
+ (A [S (k+i) := C]) [i ≝ B [k ≝ C]].
+#A #B #C #k (elim A) normalize // (* WOW *)
+#n #i @(leb_elim (S n) i) #Hle
+ [(cut (n < k+i)) [/2/] #ltn (* lento *) (cut (n ≤ k+i)) [/2/] #len
+ >(subst_rel1 C (k+i) n ltn) >(le_to_leb_true n (k+i) len) >(subst_rel1 … Hle) //
+ |@(eqb_elim n i) #eqni
+ [>eqni >(le_to_leb_true i (k+i)) // >(subst_rel2 …);
+ normalize @sym_eq (applyS (lift_subst_ijk C B i k O))
+ |@(leb_elim (S (n-1)) (k+i)) #nk
+ [>(subst_rel1 C (k+i) (n-1) nk) >(le_to_leb_true n (k+i));
+ [>(subst_rel3 ? i n) // @not_eq_to_le_to_lt;
+ [/2/ |@not_lt_to_le /2/]
+ |@(transitive_le … nk) //
+ ]
+ |(cut (i < n)) [@not_eq_to_le_to_lt; [/2/] @(not_lt_to_le … Hle)]
+ #ltin (cut (O < n)) [/2/] #posn
+ @(eqb_elim (n-1) (k+i)) #H
+ [>H >(subst_rel2 C (k+i)) >(lt_to_leb_false n (k+i));
+ [>(eq_to_eqb_true n (S(k+i)));
+ [normalize |<H (applyS plus_minus_m_m) // ]
+ (generalize in match ltin)
+ <H @(lt_O_n_elim … posn) #m #leim >delift normalize /2/
+ |<H @(lt_O_n_elim … posn) #m normalize //
+ ]
+ |(cut (k+i < n-1))
+ [@not_eq_to_le_to_lt; [@sym_not_eq @H |@(not_lt_to_le … nk)]]
+ #Hlt >(lt_to_leb_false n (k+i));
+ [>(not_eq_to_eqb_false n (S(k+i)));
+ [>(subst_rel3 C (k+i) (n-1) Hlt);
+ >(subst_rel3 ? i (n-1)) // @(le_to_lt_to_lt … Hlt) //
+ |@(not_to_not … H) #Hn >Hn normalize //
+ ]
+ |@(transitive_lt … Hlt) @(lt_O_n_elim … posn) normalize //
+ ]
+ ]
+ ]
+ ]
+ ]
+qed.
projects / helm.git / blobdiff