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).
+interpretation "Lift" 'Lift n k M = (lift M k n).
let rec subst t k a ≝
match t with
#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).
+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/
]
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.
]
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.
+
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/