------------------------------------------------------------------------ -- The Agda standard library -- -- Well-founded induction ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} module Induction.WellFounded where open import Data.Product.Base using (Σ; _,_; proj₁; proj₂) open import Function.Base using (_∘_; flip; _on_) open import Induction open import Level using (Level; _⊔_) open import Relation.Binary.Core using (Rel) open import Relation.Binary.Definitions using (Symmetric; Asymmetric; Irreflexive; _Respects₂_; _Respectsʳ_; _Respects_) open import Relation.Binary.PropositionalEquality.Core using (_≡_; refl) open import Relation.Binary.Consequences using (asym⇒irr) open import Relation.Unary open import Relation.Nullary.Negation.Core using (¬_) private variable a b ℓ ℓ₁ ℓ₂ r : Level A : Set a B : Set b ------------------------------------------------------------------------ -- Definitions -- When using well-founded recursion you can recurse arbitrarily, as -- long as the arguments become smaller, and "smaller" is -- well-founded. WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _ WfRec _<_ P x = ∀ {y} → y < x → P y -- The accessibility predicate: x is accessible if everything which is -- smaller than x is also accessible (inductively). data Acc {A : Set a} (_<_ : Rel A ℓ) (x : A) : Set (a ⊔ ℓ) where acc : (rs : WfRec _<_ (Acc _<_) x) → Acc _<_ x -- The accessibility predicate encodes what it means to be -- well-founded; if all elements are accessible, then _<_ is -- well-founded. WellFounded : Rel A ℓ → Set _ WellFounded _<_ = ∀ x → Acc _<_ x ------------------------------------------------------------------------ -- Basic properties acc-inverse : ∀ {_<_ : Rel A ℓ} {x : A} (q : Acc _<_ x) → WfRec _<_ (Acc _<_) x acc-inverse (acc rs) y<x = rs y<x module _ {_≈_ : Rel A ℓ₁} {_<_ : Rel A ℓ₂} where Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_ Acc-resp-flip-≈ respʳ x≈y (acc rec) = acc λ z<y → rec (respʳ x≈y z<y) Acc-resp-≈ : Symmetric _≈_ → _<_ Respectsʳ _≈_ → (Acc _<_) Respects _≈_ Acc-resp-≈ sym respʳ x≈y wf = Acc-resp-flip-≈ (respʳ ∘ sym) x≈y wf ------------------------------------------------------------------------ -- Well-founded induction for the subset of accessible elements: module Some {_<_ : Rel A r} {ℓ} where wfRecBuilder : SubsetRecursorBuilder (Acc _<_) (WfRec _<_ {ℓ = ℓ}) wfRecBuilder P f x (acc rs) = λ y<x → f _ (wfRecBuilder P f _ (rs y<x)) wfRec : SubsetRecursor (Acc _<_) (WfRec _<_) wfRec = subsetBuild wfRecBuilder unfold-wfRec : (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P) {x : A} (q : Acc _<_ x) → wfRec P f x q ≡ f x λ y<x → wfRec P f _ (acc-inverse q y<x) unfold-wfRec P f (acc rs) = refl ------------------------------------------------------------------------ -- Well-founded induction for all elements, assuming they are all -- accessible: module All {_<_ : Rel A r} (wf : WellFounded _<_) ℓ where wfRecBuilder : RecursorBuilder (WfRec _<_ {ℓ = ℓ}) wfRecBuilder P f x = Some.wfRecBuilder P f x (wf x) wfRec : Recursor (WfRec _<_) wfRec = build wfRecBuilder wfRec-builder = wfRecBuilder module FixPoint {_<_ : Rel A r} (wf : WellFounded _<_) (P : Pred A ℓ) (f : WfRec _<_ P ⊆′ P) (f-ext : (x : A) {IH IH′ : WfRec _<_ P x} → (∀ {y} y<x → IH {y} y<x ≡ IH′ y<x) → f x IH ≡ f x IH′) where some-wfrec-Irrelevant : Pred A _ some-wfrec-Irrelevant x = ∀ q q′ → Some.wfRec P f x q ≡ Some.wfRec P f x q′ some-wfRec-irrelevant : ∀ x → some-wfrec-Irrelevant x some-wfRec-irrelevant = All.wfRec wf _ some-wfrec-Irrelevant λ { x IH (acc rs) (acc rs′) → f-ext x λ y<x → IH y<x (rs y<x) (rs′ y<x) } open All wf ℓ wfRecBuilder-wfRec : ∀ {x y} y<x → wfRecBuilder P f x y<x ≡ wfRec P f y wfRecBuilder-wfRec {x} {y} y<x with acc rs ← wf x = some-wfRec-irrelevant y (rs y<x) (wf y) unfold-wfRec : ∀ {x} → wfRec P f x ≡ f x λ _ → wfRec P f _ unfold-wfRec {x} = f-ext x wfRecBuilder-wfRec ------------------------------------------------------------------------ -- Well-founded relations are asymmetric and irreflexive. module _ {_<_ : Rel A r} where acc⇒asym : ∀ {x y} → Acc _<_ x → x < y → ¬ (y < x) acc⇒asym {x} hx = Some.wfRec (λ x → ∀ {y} → x < y → ¬ (y < x)) (λ _ hx x<y y<x → hx y<x y<x x<y) _ hx wf⇒asym : WellFounded _<_ → Asymmetric _<_ wf⇒asym wf = acc⇒asym (wf _) wf⇒irrefl : {_≈_ : Rel A ℓ} → _<_ Respects₂ _≈_ → Symmetric _≈_ → WellFounded _<_ → Irreflexive _≈_ _<_ wf⇒irrefl r s wf = asym⇒irr r s (wf⇒asym wf) ------------------------------------------------------------------------ -- It might be useful to establish proofs of Acc or Well-founded using -- combinators such as the ones below (see, for instance, -- "Constructing Recursion Operators in Intuitionistic Type Theory" by -- Lawrence C Paulson). module Subrelation {_<₁_ : Rel A ℓ₁} {_<₂_ : Rel A ℓ₂} (<₁⇒<₂ : ∀ {x y} → x <₁ y → x <₂ y) where accessible : Acc _<₂_ ⊆ Acc _<₁_ accessible (acc rs) = acc λ y<x → accessible (rs (<₁⇒<₂ y<x)) wellFounded : WellFounded _<₂_ → WellFounded _<₁_ wellFounded wf = λ x → accessible (wf x) -- DEPRECATED in v1.4. -- Please use proofs in `Relation.Binary.Construct.On` instead. module InverseImage {_<_ : Rel B ℓ} (f : A → B) where accessible : ∀ {x} → Acc _<_ (f x) → Acc (_<_ on f) x accessible (acc rs) = acc λ fy<fx → accessible (rs fy<fx) wellFounded : WellFounded _<_ → WellFounded (_<_ on f) wellFounded wf = λ x → accessible (wf (f x)) well-founded = wellFounded {-# WARNING_ON_USAGE accessible "Warning: accessible was deprecated in v1.4. \ \Please use accessible from `Relation.Binary.Construct.On` instead." #-} {-# WARNING_ON_USAGE wellFounded "Warning: wellFounded was deprecated in v1.4. \ \Please use wellFounded from `Relation.Binary.Construct.On` instead." #-} -- DEPRECATED in v1.5. -- Please use `TransClosure` from `Relation.Binary.Construct.Closure.Transitive` instead. module TransitiveClosure {A : Set a} (_<_ : Rel A ℓ) where infix 4 _<⁺_ data _<⁺_ : Rel A (a ⊔ ℓ) where [_] : ∀ {x y} (x<y : x < y) → x <⁺ y trans : ∀ {x y z} (x<y : x <⁺ y) (y<z : y <⁺ z) → x <⁺ z downwardsClosed : ∀ {x y} → Acc _<⁺_ y → x <⁺ y → Acc _<⁺_ x downwardsClosed (acc rs) x<y = acc λ z<x → rs (trans z<x x<y) mutual accessible : ∀ {x} → Acc _<_ x → Acc _<⁺_ x accessible acc-x = acc (accessible′ acc-x) accessible′ : ∀ {x} → Acc _<_ x → WfRec _<⁺_ (Acc _<⁺_) x accessible′ (acc rs) [ y<x ] = accessible (rs y<x) accessible′ acc-x (trans y<z z<x) = downwardsClosed (accessible′ acc-x z<x) y<z wellFounded : WellFounded _<_ → WellFounded _<⁺_ wellFounded wf = λ x → accessible (wf x) {-# WARNING_ON_USAGE _<⁺_ "Warning: _<⁺_ was deprecated in v1.5. \ \Please use TransClosure from Relation.Binary.Construct.Closure.Transitive instead." #-} -- DEPRECATED in v1.3. -- Please use `×-Lex` from `Data.Product.Relation.Binary.Lex.Strict` instead. module Lexicographic {A : Set a} {B : A → Set b} (RelA : Rel A ℓ₁) (RelB : ∀ x → Rel (B x) ℓ₂) where infix 4 _<_ data _<_ : Rel (Σ A B) (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂) where left : ∀ {x₁ y₁ x₂ y₂} (x₁<x₂ : RelA x₁ x₂) → (x₁ , y₁) < (x₂ , y₂) right : ∀ {x y₁ y₂} (y₁<y₂ : RelB x y₁ y₂) → (x , y₁) < (x , y₂) mutual accessible : ∀ {x y} → Acc RelA x → (∀ {x} → WellFounded (RelB x)) → Acc _<_ (x , y) accessible accA wfB = acc (accessible′ accA (wfB _) wfB) accessible′ : ∀ {x y} → Acc RelA x → Acc (RelB x) y → (∀ {x} → WellFounded (RelB x)) → WfRec _<_ (Acc _<_) (x , y) accessible′ (acc rsA) _ wfB (left x′<x) = accessible (rsA x′<x) wfB accessible′ accA (acc rsB) wfB (right y′<y) = acc (accessible′ accA (rsB y′<y) wfB) wellFounded : WellFounded RelA → (∀ {x} → WellFounded (RelB x)) → WellFounded _<_ wellFounded wfA wfB p = accessible (wfA (proj₁ p)) wfB well-founded = wellFounded {-# WARNING_ON_USAGE _<_ "Warning: _<_ was deprecated in v1.3. \ \Please use `×-Lex` from `Data.Product.Relation.Binary.Lex.Strict` instead." #-} {-# WARNING_ON_USAGE accessible "Warning: accessible was deprecated in v1.3." #-} {-# WARNING_ON_USAGE accessible′ "Warning: accessible′ was deprecated in v1.3." #-} {-# WARNING_ON_USAGE wellFounded "Warning: wellFounded was deprecated in v1.3. \ \Please use `×-wellFounded` from `Data.Product.Relation.Binary.Lex.Strict` instead." #-} ------------------------------------------------------------------------ -- DEPRECATED NAMES ------------------------------------------------------------------------ -- Please use the new names as continuing support for the old names is -- not guaranteed. -- Version 1.0 module Inverse-image = InverseImage module Transitive-closure = TransitiveClosure