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chore(library/data/{nat,real}): rename and move iterate function

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Rob Lewis 2015-09-15 20:29:54 -04:00 • committed by Leonardo de Moura
parent ea3915f279
commit ee257a7c6c
2 changed files with 16 additions and 13 deletions
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9
library/data/nat/basic.lean
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@ -319,3 +319,12 @@ section migrate_algebra
end migrate_algebra
end nat
section
open nat
definition iterate {A : Type} (op : A → A) : ℕ → A → A
| 0 := λ a, a
| (succ k) := λ a, op (iterate k a)
notation f`^[`n`]` := iterate f n
end

20
library/data/real/complete.lean
View file

@ -532,12 +532,6 @@ local postfix `~` := nat_of_pnat
local notation 2 := (1 : ℚ) + 1
parameter X : ℝ → Prop
-- this definition belongs somewhere else. Where?
private definition rpt {A : Type} (op : A → A) : ℕ → A → A
| rpt 0 := λ a, a
| rpt (succ k) := λ a, op (rpt k a)
definition ub (x : ℝ) := ∀ y : ℝ, X y → y ≤ x
definition is_sup (x : ℝ) := ub x ∧ ∀ y : ℝ, ub y → x ≤ y
@ -599,9 +593,9 @@ private theorem over_spec : ub over :=
apply over_spec1
end
private noncomputable definition under_seq := λ n : ℕ, pr1 (rpt bisect n (under, over)) -- A
private noncomputable definition under_seq := λ n : ℕ, pr1 (iterate bisect n (under, over)) -- A
private noncomputable definition over_seq := λ n : ℕ, pr2 (rpt bisect n (under, over)) -- B
private noncomputable definition over_seq := λ n : ℕ, pr2 (iterate bisect n (under, over)) -- B
private noncomputable definition avg_seq := λ n : ℕ, avg (over_seq n) (under_seq n) -- C
@ -613,7 +607,7 @@ private theorem over_0 : over_seq 0 = over := rfl
private theorem under_0 : under_seq 0 = under := rfl
private theorem succ_helper (n : ℕ) :
avg (pr1 (rpt bisect n (under, over))) (pr2 (rpt bisect n (under, over))) = avg_seq n :=
avg (pr1 (iterate bisect n (under, over))) (pr2 (iterate bisect n (under, over))) = avg_seq n :=
by rewrite avg_symm
private theorem under_succ (n : ℕ) : under_seq (succ n) =
@ -621,11 +615,11 @@ private theorem under_succ (n : ℕ) : under_seq (succ n) =
begin
cases em (ub (avg_seq n)) with [Hub, Hub],
rewrite [if_pos Hub],
have H : pr1 (bisect (rpt bisect n (under, over))) = under_seq n, by
have H : pr1 (bisect (iterate bisect n (under, over))) = under_seq n, by
rewrite [↑under_seq, ↑bisect at {2}, -succ_helper at Hub, if_pos Hub],
apply H,
rewrite [if_neg Hub],
have H : pr1 (bisect (rpt bisect n (under, over))) = avg_seq n, by
have H : pr1 (bisect (iterate bisect n (under, over))) = avg_seq n, by
rewrite [↑bisect at {2}, -succ_helper at Hub, if_neg Hub, avg_symm],
apply H
end
@ -635,11 +629,11 @@ private theorem over_succ (n : ℕ) : over_seq (succ n) =
begin
cases em (ub (avg_seq n)) with [Hub, Hub],
rewrite [if_pos Hub],
have H : pr2 (bisect (rpt bisect n (under, over))) = avg_seq n, by
have H : pr2 (bisect (iterate bisect n (under, over))) = avg_seq n, by
rewrite [↑bisect at {2}, -succ_helper at Hub, if_pos Hub, avg_symm],
apply H,
rewrite [if_neg Hub],
have H : pr2 (bisect (rpt bisect n (under, over))) = over_seq n, by
have H : pr2 (bisect (iterate bisect n (under, over))) = over_seq n, by
rewrite [↑over_seq, ↑bisect at {2}, -succ_helper at Hub, if_neg Hub],
apply H
end