michael/lean2
1
0
Fork 0
Code Issues Pull requests Projects Releases Packages Wiki Activity Actions

refactor(library/data/fin): redo fin using recursive equations

Browse source
This commit is contained in:
Leonardo de Moura 2015-01-07 17:37:02 -08:00
parent 5c118127ae
commit 98412ee942
1 changed files with 52 additions and 71 deletions
Show stats Download patch file Download diff file Expand all files Collapse all files

123
library/data/fin.lean
View file

@ -7,7 +7,6 @@ Author: Leonardo de Moura
Finite ordinals.
-/
import data.nat logic.cast
open nat
@ -16,79 +15,61 @@ fz : Π n, fin (succ n),
fs : Π {n}, fin n → fin (succ n)
namespace fin
definition to_nat : Π {n}, fin n → nat,
@to_nat ⌞n+1⌟ (fz n) := zero,
@to_nat ⌞n+1⌟ (fs f) := succ (to_nat f)
definition z_cases_on (C : fin zero → Type) (p : fin zero) : C p :=
by cases p
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 :=
begin
cases f with (n', n', f'),
apply (H₁ n'),
apply (H₂ n' f')
end
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 n : nat} (H : p < n) : n - succ p + succ p = n :=
add_sub_ge_left (succ_le_of_lt 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)) :=
theorem to_nat_fz (n : nat) : to_nat (fz n) = zero :=
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_fs {n : nat} (f : fin n) : to_nat (fs f) = succ (to_nat f) :=
rfl
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.of_eq aux₂))
theorem to_nat_lt : Π {n} (f : fin n), to_nat f < n,
to_nat_lt (fz n) := calc
to_nat (fz n) = 0 : rfl
... < n+1 : succ_pos n,
to_nat_lt (@fs n f) := calc
to_nat (fs f) = (to_nat f)+1 : rfl
... < n+1 : succ_lt (to_nat_lt f)
definition lift : Π {n : nat}, fin n → Π (m : nat), fin (m + n),
@lift ⌞n+1⌟ (fz n) m := fz (m + n),
@lift ⌞n+1⌟ (@fs n f) m := fs (lift f m)
theorem lift_fz (n m : nat) : lift (fz n) m = fz (m + n) :=
rfl
theorem lift_fs {n : nat} (f : fin n) (m : nat) : lift (fs f) m = fs (lift f m) :=
rfl
theorem to_nat_lift : ∀ {n : nat} (f : fin n) (m : nat), to_nat f = to_nat (lift f m),
to_nat_lift (fz n) m := rfl,
to_nat_lift (@fs n f) m := calc
to_nat (fs f) = (to_nat f) + 1 : rfl
... = (to_nat (lift f m)) + 1 : to_nat_lift f
... = to_nat (lift (fs f) m) : rfl
definition of_nat : Π (p : nat) (n : nat), p < n → fin n,
of_nat 0 0 h := absurd h (not_lt_zero zero),
of_nat 0 (n+1) h := fz n,
of_nat (p+1) 0 h := absurd h (not_lt_zero (succ p)),
of_nat (p+1) (n+1) h := fs (of_nat p n (lt.of_succ_lt_succ h))
theorem of_nat_zero_succ (n : nat) (h : 0 < n+1) : of_nat 0 (n+1) h = fz n :=
rfl
theorem of_nat_succ_succ (p n : nat) (h : p+1 < n+1) :
of_nat (p+1) (n+1) h = fs (of_nat p n (lt.of_succ_lt_succ h)) :=
rfl
theorem to_nat_of_nat : ∀ (p : nat) (n : nat) (h : p < n), to_nat (of_nat p n h) = p,
to_nat_of_nat 0 0 h := absurd h (not_lt_zero 0),
to_nat_of_nat 0 (n+1) h := rfl,
to_nat_of_nat (p+1) 0 h := absurd h (not_lt_zero (p+1)),
to_nat_of_nat (p+1) (n+1) h := calc
to_nat (of_nat (p+1) (n+1) h)
= succ (to_nat (of_nat p n _)) : rfl
... = succ p : to_nat_of_nat p n _
end fin