3ab0e07ba9
This commit also removes the fake_simplifier. It doesn't work anymore because simp is now a reserved word.
121 lines
4.8 KiB
Text
121 lines
4.8 KiB
Text
import logic data.nat.sub algebra.relation data.prod
|
||
|
||
open nat relation relation.iff_ops prod
|
||
open decidable
|
||
open eq.ops
|
||
|
||
namespace nat
|
||
|
||
-- A general recursion principle
|
||
-- -----------------------------
|
||
--
|
||
-- Data:
|
||
--
|
||
-- dom, codom : Type
|
||
-- default : codom
|
||
-- measure : dom → ℕ
|
||
-- rec_val : dom → (dom → codom) → codom
|
||
--
|
||
-- and a proof
|
||
--
|
||
-- rec_decreasing : ∀m, m ≥ measure x, rec_val x f = rec_val x (restrict f m)
|
||
--
|
||
-- ... which says that the recursive call only depends on f at values with measure less than x,
|
||
-- in the sense that changing other values to the default value doesn't change the result.
|
||
--
|
||
-- The result is a function f = rec_measure, satisfying
|
||
--
|
||
-- f x = rec_val x f
|
||
|
||
definition restrict {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom)
|
||
(m : ℕ) (x : dom) :=
|
||
if measure x < m then f x else default
|
||
|
||
theorem restrict_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ) (f : dom → codom)
|
||
(m : ℕ) (x : dom) (H : measure x < m) :
|
||
restrict default measure f m x = f x :=
|
||
if_pos H
|
||
|
||
theorem restrict_not_lt_eq {dom codom : Type} (default : codom) (measure : dom → ℕ)
|
||
(f : dom → codom) (m : ℕ) (x : dom) (H : ¬ measure x < m) :
|
||
restrict default measure f m x = default :=
|
||
if_neg H
|
||
|
||
definition rec_measure_aux {dom codom : Type} (default : codom) (measure : dom → ℕ)
|
||
(rec_val : dom → (dom → codom) → codom) : ℕ → dom → codom :=
|
||
nat.rec (λx, default) (λm g x, if measure x < succ m then rec_val x g else default)
|
||
|
||
definition rec_measure {dom codom : Type} (default : codom) (measure : dom → ℕ)
|
||
(rec_val : dom → (dom → codom) → codom) (x : dom) : codom :=
|
||
rec_measure_aux default measure rec_val (succ (measure x)) x
|
||
|
||
attribute decidable [multiple-instances]
|
||
|
||
theorem rec_measure_aux_spec {dom codom : Type} (default : codom) (measure : dom → ℕ)
|
||
(rec_val : dom → (dom → codom) → codom)
|
||
(rec_decreasing : ∀g1 g2 x, (∀z, measure z < measure x → g1 z = g2 z) →
|
||
rec_val x g1 = rec_val x g2)
|
||
(m : ℕ) :
|
||
let f' := rec_measure_aux default measure rec_val in
|
||
let f := rec_measure default measure rec_val in
|
||
∀x, f' m x = restrict default measure f m x :=
|
||
let f' := rec_measure_aux default measure rec_val in
|
||
let f := rec_measure default measure rec_val in
|
||
nat.case_strong_induction_on m
|
||
(take x,
|
||
have H1 : f' 0 x = default, from rfl,
|
||
have H2 : ¬ measure x < 0, from !not_lt_zero,
|
||
have H3 : restrict default measure f 0 x = default, from if_neg H2,
|
||
show f' 0 x = restrict default measure f 0 x, from H1 ⬝ H3⁻¹)
|
||
(take m,
|
||
assume IH: ∀n, n ≤ m → ∀x, f' n x = restrict default measure f n x,
|
||
take x : dom,
|
||
show f' (succ m) x = restrict default measure f (succ m) x, from
|
||
by_cases -- (measure x < succ m)
|
||
(assume H1 : measure x < succ m,
|
||
assert H2a : ∀z, measure z < measure x → f' m z = f z,
|
||
proof
|
||
take z,
|
||
assume Hzx : measure z < measure x,
|
||
calc
|
||
f' m z = restrict default measure f m z : IH m !le.refl z
|
||
... = f z : !restrict_lt_eq (lt_of_lt_of_le Hzx (le_of_lt_succ H1))
|
||
∎,
|
||
have H2 : f' (succ m) x = rec_val x f,
|
||
proof
|
||
calc
|
||
f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
|
||
... = rec_val x (f' m) : if_pos H1
|
||
... = rec_val x f : rec_decreasing (f' m) f x H2a
|
||
∎,
|
||
let m' := measure x in
|
||
assert H3a : ∀z, measure z < m' → f' m' z = f z,
|
||
proof
|
||
take z,
|
||
assume Hzx : measure z < measure x,
|
||
calc
|
||
f' m' z = restrict default measure f m' z : IH _ (le_of_lt_succ H1) _
|
||
... = f z : !restrict_lt_eq Hzx
|
||
qed,
|
||
have H3 : restrict default measure f (succ m) x = rec_val x f,
|
||
proof
|
||
calc
|
||
restrict default measure f (succ m) x = f x : if_pos H1
|
||
... = f' (succ m') x : !eq.refl
|
||
... = if measure x < succ m' then rec_val x (f' m') else default : rfl
|
||
... = rec_val x (f' m') : if_pos !lt_succ_self
|
||
... = rec_val x f : rec_decreasing _ _ _ H3a
|
||
qed,
|
||
show f' (succ m) x = restrict default measure f (succ m) x,
|
||
from H2 ⬝ H3⁻¹)
|
||
(assume H1 : ¬ measure x < succ m,
|
||
have H2 : f' (succ m) x = default, from
|
||
calc
|
||
f' (succ m) x = if measure x < succ m then rec_val x (f' m) else default : rfl
|
||
... = default : if_neg H1,
|
||
have H3 : restrict default measure f (succ m) x = default,
|
||
from if_neg H1,
|
||
show f' (succ m) x = restrict default measure f (succ m) x,
|
||
from H2 ⬝ H3⁻¹))
|
||
|
||
end nat
|