\ /
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).
+include "lambda/lift.ma".
let rec subst t k a ≝
match t with
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_rel_lt : ∀n,k,i. i < k → lift (Rel i) k n = Rel i.
-#n #k #i #ltik change with
-(if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel i)
->(le_to_leb_true … ltik) //
-qed.
-
-lemma lift_rel_ge : ∀n,k,i. k ≤ i → lift (Rel i) k n = Rel (i+n).
-#n #k #i #leki change with
-(if_then_else ? (leb (S i) k) (Rel i) (Rel (i+n)) = Rel (i+n))
->lt_to_leb_false // @le_S_S //
-qed.
-
-lemma lift_lift: ∀t.∀m,j.j ≤ m → ∀n,k.
- lift (lift t k m) (j+k) n = lift t k (m+n).
-#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_lift_up: ∀n,m,t,k,i.
- lift (lift t i m) (m+k+i) n = lift (lift t (k+i) n) i m.
-#n #m #N (elim N)
- [1,3,4,5,6: normalize //
- |#p #k #i @(leb_elim i p);
- [#leip >lift_rel_ge // @(leb_elim (k+i) p);
- [#lekip >lift_rel_ge;
- [>lift_rel_ge // >lift_rel_ge // @(transitive_le … leip) //
- |>associative_plus >commutative_plus @monotonic_le_plus_l //
- ]
- |#lefalse (cut (p < k+i)) [@not_le_to_lt //] #ltpki
- >lift_rel_lt; [|>associative_plus >commutative_plus @monotonic_lt_plus_r //]
- >lift_rel_lt // >lift_rel_ge //
- ]
- |#lefalse (cut (p < i)) [@not_le_to_lt //] #ltpi
- >lift_rel_lt // >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
- >lift_rel_lt; [|@(lt_to_le_to_lt … ltpi) //]
- >lift_rel_lt //
- ]
- ]
-qed.
-
-lemma lift_lift1: ∀t.∀i,j,k.
- lift(lift t k j) k i = lift t k (j+i).
-/2/ qed.
+interpretation "Subst" 'Subst1 M k N = (subst M k N).
-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. *)
+(*** properties of subst ***)
lemma subst_lift_k: ∀A,B.∀k. (lift B k 1)[k ≝ A] = B.
#A #B (elim B) normalize /2/ #n #k
]
qed.
+lemma lift_subst_up_O: ∀v,t,k,p. (lift t (k+1) p)[O≝lift v k p] = lift t[O≝v] k p.
+// 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/
]
]
qed.
+
+lemma subst_lemma_comm: ∀A,B,C.∀k,i.
+ (A [i ≝ B]) [i+k ≝ C] = (A [i+k+1 := C]) [i ≝ B [k ≝ C]].
+#A #B #C #k #i >commutative_plus >subst_lemma //
+qed.
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