63 lines
2.3 KiB
Text
63 lines
2.3 KiB
Text
-- 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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import logic.eq
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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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definition well_founded {A : Type} (R : A → A → Prop) :=
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∀a, acc R a
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namespace well_founded
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context
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parameters {A : Type} {R : A → A → Prop}
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infix `≺`:50 := R
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definition acc_inv {x y : A} (H₁ : acc R x) (H₂ : y ≺ x) : acc R y :=
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have gen : y ≺ x → acc R y, from
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acc.rec_on H₁ (λ x₁ ac₁ iH H₂, ac₁ y H₂),
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gen H₂
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hypothesis Hwf : well_founded R
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theorem well_founded_rec {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 well_founded_ind {C : A → Prop} (a : A) (H : ∀x, (∀y, y ≺ x → C y) → C x) : C a :=
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well_founded_rec 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 Hwf F x = F x (λy h, fix Hwf 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 Hwf 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 Hwf F y) : rfl -- proof irrelevance is used here
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end well_founded
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