\ /
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.
+interpretation "Subst" 'Subst1 M k N = (subst M k N).
-(* 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. *)
+(*** 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: ∀M,N,n,i,j.
+ lift M[i≝N] (i+j) n = (lift M (i+j+1) n)[i≝ (lift N j n)].
+#M (elim M)
+ [//
+ |#p #N #n #i #j (cases (true_or_false (leb p i)))
+ [#lepi (cases (le_to_or_lt_eq … (leb_true_to_le … lepi)))
+ [#ltpi >(subst_rel1 … ltpi)
+ (cut (p < i+j)) [@(lt_to_le_to_lt … ltpi) //] #ltpij
+ >(lift_rel_lt … ltpij); >(lift_rel_lt ?? p ?);
+ [>subst_rel1 // | @(lt_to_le_to_lt … ltpij) //]
+ |#eqpi >eqpi >subst_rel2 >lift_rel_lt;
+ [>subst_rel2 >(plus_n_O (i+j))
+ applyS lift_lift_up
+ |@(le_to_lt_to_lt ? (i+j)) //
+ ]
+ ]
+ |#lefalse (cut (i < p)) [@not_le_to_lt /2/] #ltip
+ (cut (0 < p)) [@(le_to_lt_to_lt … ltip) //] #posp
+ >(subst_rel3 … ltip) (cases (true_or_false (leb (S p) (i+j+1))))
+ [#Htrue (cut (p < i+j+1)) [@(leb_true_to_le … Htrue)] #Hlt
+ >lift_rel_lt;
+ [>lift_rel_lt // >(subst_rel3 … ltip) // | @lt_plus_to_minus //]
+ |#Hfalse >lift_rel_ge;
+ [>lift_rel_ge;
+ [>subst_rel3; [@eq_f /2/ | @(lt_to_le_to_lt … ltip) //]
+ |@not_lt_to_le @(leb_false_to_not_le … Hfalse)
+ ]
+ |@le_plus_to_minus_r @not_lt_to_le
+ @(leb_false_to_not_le … Hfalse)
+ ]
+ ]
+ ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |@HindQ ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
+ <associative_plus @HindQ]
+ |#P #Q #HindP #HindQ #N #n #i #j normalize
+ @eq_f2; [@HindP |>associative_plus >(commutative_plus j 1)
+ <associative_plus @HindQ]
+ |#P #HindP #N #n #i #j normalize
+ @eq_f @HindP
+ ]
+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.
projects / helm.git / blobdiff