68 lines
2.5 KiB
Text
68 lines
2.5 KiB
Text
-- Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
|
-- Released under Apache 2.0 license as described in the file LICENSE.
|
|
-- Author: Leonardo de Moura
|
|
import logic.eq
|
|
|
|
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
|
|
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
|
|
|
|
context
|
|
parameters {A : Type} {R : A → A → Prop}
|
|
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 indunction {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)) :=
|
|
have gen : Π 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)), from
|
|
acc.rec_on r
|
|
(λ x₁ ac iH (r₁ : acc R x₁),
|
|
-- The proof is straightforward after we replace r₁ with acc.intro (to "unblock" evaluation).
|
|
calc fix_F F x₁ r₁
|
|
= fix_F F x₁ (acc.intro x₁ ac) : proof_irrel r₁
|
|
... = F x₁ (λ y ay, fix_F F y (acc.inv r₁ ay)) : rfl),
|
|
gen r
|
|
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) :=
|
|
calc
|
|
-- The proof is straightforward, it just uses fix_F_eq and proof irrelevance
|
|
fix F x
|
|
= F x (λy h, fix_F F y (acc.inv (Hwf x) h)) : fix_F_eq F x (Hwf x)
|
|
... = F x (λy h, fix F y) : rfl -- proof irrelevance is used here
|
|
|
|
end well_founded
|