lean2/library/data/fin.lean

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2014-11-18 07:44:57 +00:00
import data.nat logic.cast
open nat
inductive fin : nat → Type :=
fz : Π n, fin (succ n),
fs : Π {n}, fin n → fin (succ n)
namespace fin
definition z_cases_on (C : fin zero → Type) (p : fin zero) : C p :=
have aux : Π (C : Type) (n : nat) (p : fin n), n = zero → C, from
λ C n p, fin.rec_on p
(λ n h, nat.no_confusion h)
(λ n f ih h, nat.no_confusion h),
aux (C p) zero p rfl
definition nz_cases_on {C : Π n, fin (succ n) → Type}
(H₁ : Π n, C n (fz n))
(H₂ : Π n (f : fin n), C n (fs f))
{n : nat}
(f : fin (succ n)) : C n f :=
have aux : Π (n₁ : nat) (f₁ : fin n₁) (heq₁ : n₁ = succ n) (f : fin (succ n)) (heq₂ : f₁ == f), C n f, from
λ n₁ f₁, fin.rec_on f₁
(λ (n₁ : nat) (heq₁ : succ n₁ = succ n),
have heq₁' : n₁ = n, from nat.no_confusion heq₁ (λ e, e),
eq.rec_on heq₁' (λ (f : fin (succ n₁)) (heq₂ : fz n₁ == f),
have heq₂' : fz n₁ = f, from heq.to_eq heq₂,
have Cfz : C n₁ (fz n₁), from H₁ n₁,
eq.rec_on heq₂' Cfz))
(λ (n₁ : nat) (f₁ : fin n₁) (ih : _) (heq₁ : succ n₁ = succ n),
have heq₁' : n₁ = n, from nat.no_confusion heq₁ (λ e, e),
eq.rec_on heq₁' (λ (f : fin (succ n₁)) (heq₂ : @fs n₁ f₁ == f),
have heq₂' : @fs n₁ f₁ = f, from heq.to_eq heq₂,
have Cfs : C n₁ (@fs n₁ f₁), from H₂ n₁ f₁,
eq.rec_on heq₂' Cfs)),
aux (succ n) f rfl f !heq.refl
definition to_nat {n : nat} (f : fin n) : nat :=
fin.rec_on f
(λ n, zero)
(λ n f r, succ r)
theorem to_nat.lt {n : nat} (f : fin n) : to_nat f < n :=
fin.rec_on f
(λ n, calc
to_nat (fz n) = 0 : rfl
... < succ n : succ_pos n)
(λ n f ih, calc
to_nat (fs f) = succ (to_nat f) : rfl
... < succ n : succ_lt ih)
definition lift {n : nat} (f : fin n) : Π m, fin (m + n) :=
fin.rec_on f
(λ n m, fz (m + n))
(λ n f ih m, fs (ih m))
theorem to_nat.lift {n : nat} (f : fin n) : ∀m, to_nat f = to_nat (lift f m) :=
fin.rec_on f
(λ n m, rfl)
(λ n f ih m, calc
to_nat (fs f) = succ (to_nat f) : rfl
... = succ (to_nat (lift f m)) : ih
... = to_nat (lift (fs f) m) : rfl)
private definition of_nat_core (p : nat) : fin (succ p) :=
nat.rec_on p
(fz zero)
(λ a r, fs r)
private theorem to_nat.of_nat_core (p : nat) : to_nat (of_nat_core p) = p :=
nat.induction_on p
rfl
(λ p₁ ih, calc
to_nat (of_nat_core (succ p₁)) = succ (to_nat (of_nat_core p₁)) : rfl
... = succ p₁ : ih)
private lemma of_nat_eq {p : nat} {n : nat} (H : p < n) : n - succ p + succ p = n :=
add_sub_ge_left (eq.subst (lt_def p n) H)
definition of_nat (p : nat) (n : nat) : p < n → fin n :=
λ H : p < n,
eq.rec_on (of_nat_eq H) (lift (of_nat_core p) (n - succ p))
theorem of_nat_def (p : nat) (n : nat) (H : p < n) : of_nat p n H = eq.rec_on (of_nat_eq H) (lift (of_nat_core p) (n - succ p)) :=
rfl
theorem of_nat_heq (p : nat) (n : nat) (H : p < n) : of_nat p n H == lift (of_nat_core p) (n - succ p) :=
heq.symm (eq_rec_to_heq (eq.symm (of_nat_def p n H)))
theorem to_nat.of_nat (p : nat) (n : nat) (H : p < n) : to_nat (of_nat p n H) = p :=
have aux₁ : to_nat (of_nat p n H) == to_nat (lift (of_nat_core p) (n - succ p)), from
hcongr_arg2 @to_nat (eq.symm (of_nat_eq H)) (of_nat_heq p n H),
have aux₂ : to_nat (lift (of_nat_core p) (n - succ p)) = p, from calc
to_nat (lift (of_nat_core p) (n - succ p)) = to_nat (of_nat_core p) : to_nat.lift
... = p : to_nat.of_nat_core,
heq.to_eq (heq.trans aux₁ (heq.from_eq aux₂))
end fin