1.
(APP1) append(Empty, l) = l (APP2) append(Cons(x, xs), l) = Cons(x, append(xs, l))
(REV1) reverse(Empty) = Empty (REV2) reverse(Cons(x, xs)) = append(reverse(xs), Cons(x, Empty))
2.
(M1) mirror(Nil) = Nil (M2) mirror(Node(e, l, r)) = Node(e, mirror(r), mirror(l))
(F1) flatten(Nil) = Empty (F2) flatten(Node(e, l, r)) = Cons(e, append(flatten(l), flatten(r)))
3.
(UPD1) update(MEmpty, k, v) = Insert((k, v), MEmpty)
(UPD2) update(Insert((k, v), m), k', v') = if k = k' then Insert((k, v'), m)
else Insert((k, v), update(m, k', v'))
(DEL1) delete(MEmpty, k) = MEmpty
(DEL2) delete(Insert((k, v), m), k') = if k = k' then m
else Insert((k, v), delete(m, k'))
4.
cazul de baza:
mirror(Nil) = Nil (M1) => size(Nil) = size(mirror(Nil))
ipoteza inductiei: size(l) = size(mirror(l)), size(r) = size(mirror(r))
size(Node(e, l, r)) = 1 + size(l) + size(r) (S2)
= 1 + size(r) + size(l) (comutativitatea adunării)
= 1 + size(mirror(r)) + size(mirror(l)) (ipoteza inductiei)
= size(Node(e, mirror(r), mirror(l))) (S2)
= size(mirror(Node(e, l, r)) (M2)
cazul de baza:
size(Nil) = (S1) = 0 = (L1) = length(Empty) = (F1) = length(flatten(Nil))
ipoteza inductiei: size(l) = length(flatten(l)), size(r) = length(flatten(r))
size(Node(e, l, r)) = 1 + size(l) + size(r) (S2)
= 1 + length(flatten(l)) + length(flatten(r)) (ipoteza inductiei)
= 1 + length(append(flatten(l), flatten(r)) (vom demonstra)
= length(Cons(e, append(flatten(l), flatten(r))) (L2)
= length(flatten(Node(e, l, r))) (F2)
mai trebuie sa demonstram pasul intermediar: demonstram ca
length(append(a, b)) = length(a) + length(b) (LAPP)
prin inductie dupa a
cazul de baza:
length(append(Empty, b)) = length(b) (APP1)
= 0 + length(b)
= length(Empty) + length(b)
ipoteza inductiei: length(append(xs, b)) = length(xs) + length(b)
pasul inductiei:
length(append(Cons(x, xs), b)) = length(Cons(x, append(xs, b))) (APP2)
= 1 + length(append(xs, b)) (L2)
= 1 + length(xs) + length(b) (ip. inductiei)
= length(xs) + 1 + length(b) (comutativitatea adunării)
= length(Cons(x, xs)) + length(b) (L2)
cazul de baza:
append(Empty, Empty) = Empty (APP1)
ipoteza inductiei: append(xs, Empty) = s
append(Cons(x, xs), Empty) = Cons(x, append(xs, Empty)) (APP2)
= Cons(x, s) (ip. inductiei)
facem inductie structurala dupa l_1. Vom nota append(a, b) cu a ++ b ca sa ne fie mai usor
cazul de baza:
Empty ++ (l_2 ++ l_3)) = l_2 ++ l_3 (APP1)
= (Empty ++ l_2) ++ l_3 (APP1)
ipoteza inductiei: l_1 ++ (l_2 ++ l_3) = (l_1 ++ l_2) ++ l_3
pasul inductiei:
Cons(x, l_1) ++ (l_2 ++ l_3) = Cons(x, l_1 ++ (l_2 ++ l_3)) (APP2)
= Cons(x, (l_1 ++ l_2) ++ l_3) (ip. inductiei)
= Cons(x, l_1 ++ l_2) ++ l_3 (APP2)
= (Cons(x, l_1) ++ l_2) ++ l_3 (APP2)
facem inductie structurala dupa l_1
cazul de baza:
length(append(Empty, l_2)) = length(l_2) (APP1)
= length(append(l_2, Empty)) (4.c)
ipoteza inductiei: length(append(l_1, l_2)) = length(append(l_2, l_1))
pasul inductiei:
length(append(Cons(x, l_1), l_2)) = length(Cons(x, append(l_1, l_2))) (APP2)
= 1 + length(append(l_1, l_2)) (L2)
= 1 + length(l_1) + length(l_2) (LAPP)
= length(l_2) + 1 + length(l_1) (comutativitatea adunării)
= length(l_2) + length(Cons(x, l_1)) (L2)
= length(append(l_2, Cons(x, l_1))) (LAPP)
facem inductie structurala dupa l_1
cazul de baza:
reverse(append(Empty, l_2)) = reverse(l_2) (APP1)
= append(reverse(l_2), Empty) (4.c)
= append(reverse(l_2), reverse(Empty)) (REV1)
ipoteza inductiei: reverse(append(l_1, l_2)) = append(reverse(l_2), reverse(l_1))
pasul inductiei:
reverse(append(Cons(x, l_1), l_2)) = reverse(Cons(x, append(l_1, l_2))) (APP2)
= append(reverse(append(l_1, l_2)), Cons(x, Empty)) (REV2)
= append(append(reverse(l_2), reverse(l_1)), Cons(x, Empty)) (ip. inductivă)
= append(reverse(l_2), append(reverse(l_1), Cons(x, Empty))) (4.d)
= append(reverse(l_2), reverse(Cons(x, l_1))) (REV2)