-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.