lean2/library/init/wf.lean
Leonardo de Moura 17f2c240e1 refactor(library/init/wf): cleanup wf proofs using tactics
add dependent elimination for acc.
2015-06-06 16:58:24 -07:00

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/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import init.relation init.tactic
inductive acc {A : Type} (R : A → A → Prop) : A → Prop :=
intro : ∀x, (∀ y, R y x → acc R y) → acc R x
namespace acc
variables {A : Type} {R : A → A → Prop}
definition inv {x y : A} (H₁ : acc R x) (H₂ : R y x) : acc R y :=
acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
-- dependent elimination for acc
protected definition drec [recursor]
{C : Π (a : A), acc R a → Type}
(h₁ : Π (x : A) (acx : Π (y : A), R y x → acc R y),
(Π (y : A) (ryx : R y x), C y (acx y ryx)) → C x (acc.intro x acx))
{a : A} (h₂ : acc R a) : C a h₂ :=
begin
refine acc.rec _ h₂ h₂,
intro x acx ih h₂,
exact h₁ x acx (λ y ryx, ih y ryx (acx y ryx))
end
end acc
inductive well_founded [class] {A : Type} (R : A → A → Prop) : Prop :=
intro : (∀ a, acc R a) → well_founded R
namespace well_founded
definition apply [coercion] {A : Type} {R : A → A → Prop} (wf : well_founded R) : ∀a, acc R a :=
take a, well_founded.rec_on wf (λp, p) a
section
parameters {A : Type} {R : A → A → Prop}
local infix `≺`:50 := R
hypothesis [Hwf : well_founded R]
theorem recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
acc.rec_on (Hwf a) (λ x₁ ac₁ iH, H x₁ iH)
theorem induction {C : A → Prop} (a : A) (H : ∀x, (∀y, y ≺ x → C y) → C x) : C a :=
recursion a H
variable {C : A → Type}
variable F : Πx, (Πy, y ≺ x → C y) → C x
definition fix_F (x : A) (a : acc R x) : C x :=
acc.rec_on a (λ x₁ ac₁ iH, F x₁ iH)
theorem fix_F_eq (x : A) (r : acc R x) :
fix_F F x r = F x (λ (y : A) (p : y ≺ x), fix_F F y (acc.inv r p)) :=
begin
induction r using acc.drec,
reflexivity -- proof is trivial due to proof irrelevance
end
end
variables {A : Type} {C : A → Type} {R : A → A → Prop}
-- Well-founded fixpoint
definition fix [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) : C x :=
fix_F F x (Hwf x)
-- Well-founded fixpoint satisfies fixpoint equation
theorem fix_eq [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) :
fix F x = F x (λy h, fix F y) :=
fix_F_eq F x (Hwf x)
end well_founded
open well_founded
-- Empty relation is well-founded
definition empty.wf {A : Type} : well_founded empty_relation :=
well_founded.intro (λ (a : A),
acc.intro a (λ (b : A) (lt : false), false.rec _ lt))
-- Subrelation of a well-founded relation is well-founded
namespace subrelation
section
parameters {A : Type} {R Q : A → A → Prop}
parameters (H₁ : subrelation Q R)
parameters (H₂ : well_founded R)
definition accessible {a : A} (ac : acc R a) : acc Q a :=
using H₁,
begin
induction ac with x ax ih, constructor,
exact λ (y : A) (lt : Q y x), ih y (H₁ lt)
end
definition wf : well_founded Q :=
well_founded.intro (λ a, accessible (H₂ a))
end
end subrelation
-- The inverse image of a well-founded relation is well-founded
namespace inv_image
section
parameters {A B : Type} {R : B → B → Prop}
parameters (f : A → B)
parameters (H : well_founded R)
private definition acc_aux {b : B} (ac : acc R b) : ∀ x, f x = b → acc (inv_image R f) x :=
begin
induction ac with x acx ih,
intro z e, constructor,
intro y lt, subst x,
exact ih (f y) lt y rfl
end
definition accessible {a : A} (ac : acc R (f a)) : acc (inv_image R f) a :=
acc_aux ac a rfl
definition wf : well_founded (inv_image R f) :=
well_founded.intro (λ a, accessible (H (f a)))
end
end inv_image
-- The transitive closure of a well-founded relation is well-founded
namespace tc
section
parameters {A : Type} {R : A → A → Prop}
local notation `R⁺` := tc R
definition accessible {z} (ac: acc R z) : acc R⁺ z :=
begin
induction ac with x acx ih,
constructor, intro y lt,
induction lt with a b rab a b c rab rbc ih₁ ih₂,
{exact ih a rab},
{exact acc.inv (ih₂ acx ih) rab}
end
definition wf (H : well_founded R) : well_founded R⁺ :=
well_founded.intro (λ a, accessible (H a))
end
end tc