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refactor(library/data/fin): cleanup pattern matching equations

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This commit is contained in:
Leonardo de Moura 2015-03-05 14:42:42 -08:00
parent 28487ede3b
commit 57ec52b6f1
1 changed files with 18 additions and 18 deletions
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36
library/data/fin.lean
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@ -16,8 +16,8 @@ inductive fin : nat → Type :=
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)
| ⌞n+1⌟ (fz n) := zero
| ⌞n+1⌟ (fs f) := succ (to_nat f)
theorem to_nat_fz (n : nat) : to_nat (fz n) = zero :=
rfl
@ -26,16 +26,16 @@ namespace fin
rfl
theorem to_nat_lt : Π {n} (f : fin n), to_nat f < n
| to_nat_lt (fz n) := calc
| (n+1) (fz n) := calc
to_nat (fz n) = 0 : rfl
... < n+1 : succ_pos n
| to_nat_lt (@fs n f) := calc
| (n+1) (fs f) := calc
to_nat (fs f) = (to_nat f)+1 : rfl
... < n+1 : succ_lt_succ (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)
| ⌞n+1⌟ (fz n) m := fz (m + n)
| ⌞n+1⌟ (fs f) m := fs (lift f m)
theorem lift_fz (n m : nat) : lift (fz n) m = fz (m + n) :=
rfl
@ -44,17 +44,17 @@ namespace fin
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
| (n+1) (fz n) m := rfl
| (n+1) (fs 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))
| 0 0 h := absurd h (not_lt_zero zero)
| 0 (n+1) h := fz n
| (p+1) 0 h := absurd h (not_lt_zero (succ p))
| (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
@ -64,17 +64,17 @@ namespace fin
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
| 0 0 h := absurd h (not_lt_zero 0)
| 0 (n+1) h := rfl
| (p+1) 0 h := absurd h (not_lt_zero (p+1))
| (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 _}
theorem of_nat_to_nat : ∀ {n : nat} (f : fin n) (h : to_nat f < n), of_nat (to_nat f) n h = f
| of_nat_to_nat (fz n) h := rfl
| of_nat_to_nat (@fs n f) h := calc
| (n+1) (fz n) h := rfl
| (n+1) (fs f) h := calc
of_nat (to_nat (fs f)) (succ n) h = fs (of_nat (to_nat f) n _) : rfl
... = fs f : {of_nat_to_nat f _}