143 lines
4.8 KiB
Text
143 lines
4.8 KiB
Text
/-
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Copyright (c) 2014 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Author: Leonardo de Moura
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-/
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prelude
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import init.relation
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inductive acc {A : Type} (R : A → A → Prop) : A → Prop :=
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intro : ∀x, (∀ y, R y x → acc R y) → acc R x
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namespace acc
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variables {A : Type} {R : A → A → Prop}
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definition inv {x y : A} (H₁ : acc R x) (H₂ : R y x) : acc R y :=
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acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂) H₂
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end acc
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inductive well_founded [class] {A : Type} (R : A → A → Prop) : Prop :=
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intro : (∀ a, acc R a) → well_founded R
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namespace well_founded
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definition apply [coercion] {A : Type} {R : A → A → Prop} (wf : well_founded R) : ∀a, acc R a :=
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take a, well_founded.rec_on wf (λp, p) a
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section
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parameters {A : Type} {R : A → A → Prop}
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local infix `≺`:50 := R
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hypothesis [Hwf : well_founded R]
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theorem recursion {C : A → Type} (a : A) (H : Πx, (Πy, y ≺ x → C y) → C x) : C a :=
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acc.rec_on (Hwf a) (λ x₁ ac₁ iH, H x₁ iH)
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theorem induction {C : A → Prop} (a : A) (H : ∀x, (∀y, y ≺ x → C y) → C x) : C a :=
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recursion a H
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variable {C : A → Type}
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variable F : Πx, (Πy, y ≺ x → C y) → C x
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definition fix_F (x : A) (a : acc R x) : C x :=
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acc.rec_on a (λ x₁ ac₁ iH, F x₁ iH)
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theorem fix_F_eq (x : A) (r : acc R x) :
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fix_F F x r = F x (λ (y : A) (p : y ≺ x), fix_F F y (acc.inv r p)) :=
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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
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acc.rec_on r
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(λ x₁ ac iH (r₁ : acc R x₁),
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-- The proof is straightforward after we replace r₁ with acc.intro (to "unblock" evaluation).
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calc fix_F F x₁ r₁
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= fix_F F x₁ (acc.intro x₁ ac) : proof_irrel r₁
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... = F x₁ (λ y ay, fix_F F y (acc.inv r₁ ay)) : rfl),
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gen r
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end
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variables {A : Type} {C : A → Type} {R : A → A → Prop}
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-- Well-founded fixpoint
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definition fix [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) : C x :=
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fix_F F x (Hwf x)
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-- Well-founded fixpoint satisfies fixpoint equation
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theorem fix_eq [Hwf : well_founded R] (F : Πx, (Πy, R y x → C y) → C x) (x : A) :
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fix F x = F x (λy h, fix F y) :=
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calc
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-- The proof is straightforward, it just uses fix_F_eq and proof irrelevance
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fix F x
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= F x (λy h, fix_F F y (acc.inv (Hwf x) h)) : fix_F_eq F x (Hwf x)
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... = F x (λy h, fix F y) : rfl -- proof irrelevance is used here
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end well_founded
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open well_founded
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-- Empty relation is well-founded
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definition empty.wf {A : Type} : well_founded empty_relation :=
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well_founded.intro (λ (a : A),
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acc.intro a (λ (b : A) (lt : false), false.rec _ lt))
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-- Subrelation of a well-founded relation is well-founded
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namespace subrelation
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section
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parameters {A : Type} {R Q : A → A → Prop}
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parameters (H₁ : subrelation Q R)
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parameters (H₂ : well_founded R)
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definition accessible {a : A} (ac : acc R a) : acc Q a :=
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acc.rec_on ac
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(λ (x : A) (ax : _) (iH : ∀ (y : A), R y x → acc Q y),
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acc.intro x (λ (y : A) (lt : Q y x), iH y (H₁ lt)))
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definition wf : well_founded Q :=
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well_founded.intro (λ a, accessible (H₂ a))
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end
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end subrelation
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-- The inverse image of a well-founded relation is well-founded
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namespace inv_image
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section
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parameters {A B : Type} {R : B → B → Prop}
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parameters (f : A → B)
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parameters (H : well_founded R)
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definition accessible {a : A} (ac : acc R (f a)) : acc (inv_image R f) a :=
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have gen : ∀x, f x = f a → acc (inv_image R f) x, from
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acc.rec_on ac
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(λx acx (iH : ∀y, R y x → (∀z, f z = y → acc (inv_image R f) z))
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(z : A) (eq₁ : f z = x),
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acc.intro z (λ (y : A) (lt : R (f y) (f z)),
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iH (f y) (eq.rec_on eq₁ lt) y rfl)),
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gen a rfl
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definition wf : well_founded (inv_image R f) :=
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well_founded.intro (λ a, accessible (H (f a)))
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end
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end inv_image
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-- The transitive closure of a well-founded relation is well-founded
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namespace tc
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section
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parameters {A : Type} {R : A → A → Prop}
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local notation `R⁺` := tc R
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definition accessible {z} (ac: acc R z) : acc R⁺ z :=
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acc.rec_on ac
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(λ x acx (iH : ∀y, R y x → acc R⁺ y),
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acc.intro x (λ (y : A) (lt : R⁺ y x),
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have gen : x = x → acc R⁺ y, from
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tc.rec_on lt
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(λa b (H : R a b) (Heq : b = x),
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iH a (eq.rec_on Heq H))
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(λa b c (H₁ : R⁺ a b) (H₂ : R⁺ b c)
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(iH₁ : b = x → acc R⁺ a)
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(iH₂ : c = x → acc R⁺ b)
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(Heq : c = x),
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acc.inv (iH₂ Heq) H₁),
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gen rfl))
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definition wf (H : well_founded R) : well_founded R⁺ :=
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well_founded.intro (λ a, accessible (H a))
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end
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end tc
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