------------------------------------------------------------------------ -- The Agda standard library -- -- Syntax for the building blocks of equational reasoning modules ------------------------------------------------------------------------ {-# OPTIONS --cubical-compatible --safe #-} -- List of `Reasoning` modules that do not use this framework and so -- need to be updated manually if the syntax changes. -- -- Data/Vec/Relation/Binary/Equality/Cast -- Relation/Binary/HeterogeneousEquality -- Effect/Monad/Partiality -- Effect/Monad/Partiality/All -- Codata/Guarded/Stream/Relation/Binary/Pointwise -- Function/Reasoning module Relation.Binary.Reasoning.Syntax where open import Level using (Level; _⊔_; suc) open import Relation.Nullary.Decidable.Core using (Dec; True; toWitness) open import Relation.Nullary.Negation.Core using (contradiction) open import Relation.Binary.Core using (Rel; REL; _⇒_) open import Relation.Binary.Definitions using (_Respectsʳ_; Asymmetric; Trans; Sym; Reflexive) open import Relation.Binary.PropositionalEquality.Core as ≡ using (_≡_) private variable a ℓ₁ ℓ₂ ℓ₃ ℓ₄ : Level A B C : Set a x y z : A ------------------------------------------------------------------------ -- Syntax for beginning a reasoning chain ------------------------------------------------------------------------ ------------------------------------------------------------------------ -- Basic begin syntax module begin-syntax (R : REL A B ℓ₁) {S : REL A B ℓ₂} (reflexive : R ⇒ S) where infix 1 begin_ begin_ : R x y → S x y begin_ = reflexive ------------------------------------------------------------------------ -- Begin subrelation syntax -- Sometimes we want to support sub-relations with the -- same reasoning operators as the main relations (e.g. perform equality -- proofs with non-strict reasoning operators). This record bundles all -- the parts needed to extract the sub-relation proofs. record SubRelation {A : Set a} (R : Rel A ℓ₁) ℓ₂ ℓ₃ : Set (a ⊔ ℓ₁ ⊔ suc ℓ₂ ⊔ suc ℓ₃) where field S : Rel A ℓ₂ IsS : R x y → Set ℓ₃ IsS? : ∀ (xRy : R x y) → Dec (IsS xRy) extract : ∀ {xRy : R x y} → IsS xRy → S x y module begin-subrelation-syntax (R : Rel A ℓ₁) (sub : SubRelation R ℓ₂ ℓ₃) where open SubRelation sub infix 1 begin_ begin_ : ∀ {x y} (xRy : R x y) → {s : True (IsS? xRy)} → S x y begin_ r {s} = extract (toWitness s) -- Begin equality syntax module begin-equality-syntax (R : Rel A ℓ₁) (sub : SubRelation R ℓ₂ ℓ₃) where open begin-subrelation-syntax R sub public renaming (begin_ to begin-equality_) -- Begin apartness syntax module begin-apartness-syntax (R : Rel A ℓ₁) (sub : SubRelation R ℓ₂ ℓ₃) where open begin-subrelation-syntax R sub public renaming (begin_ to begin-apartness_) -- Begin strict syntax module begin-strict-syntax (R : Rel A ℓ₁) (sub : SubRelation R ℓ₂ ℓ₃) where open begin-subrelation-syntax R sub public renaming (begin_ to begin-strict_) ------------------------------------------------------------------------ -- Begin membership syntax module begin-membership-syntax (R : Rel A ℓ₁) (_∈_ : REL B A ℓ₂) (resp : _∈_ Respectsʳ R) where infix 1 step-∈ step-∈ : ∀ (x : B) {xs ys} → R xs ys → x ∈ xs → x ∈ ys step-∈ x = resp syntax step-∈ x xs⊆ys x∈xs = x ∈⟨ x∈xs ⟩ xs⊆ys ------------------------------------------------------------------------ -- Begin contradiction syntax -- Used with asymmetric subrelations to derive a contradiction from a -- proof that an element is related to itself. module begin-contradiction-syntax (R : Rel A ℓ₁) (sub : SubRelation R ℓ₂ ℓ₃) (asym : Asymmetric (SubRelation.S sub)) where open SubRelation sub infix 1 begin-contradiction_ begin-contradiction_ : ∀ (xRx : R x x) {s : True (IsS? xRx)} → ∀ {b} {B : Set b} → B begin-contradiction_ {x} r {s} = contradiction x<x (asym x<x) where x<x : S x x x<x = extract (toWitness s) ------------------------------------------------------------------------ -- Syntax for continuing a chain of reasoning steps ------------------------------------------------------------------------ -- Note that the arguments to the `step`s are not provided in their -- "natural" order and syntax declarations are later used to re-order -- them. This is because the `step` ordering allows the type-checker to -- better infer the middle argument `y` from the `_IsRelatedTo_` -- argument (see issue 622). -- -- This has two practical benefits. First it speeds up type-checking by -- approximately a factor of 5. Secondly it allows the combinators to be -- used with macros that use reflection, e.g. `Tactic.RingSolver`, where -- they need to be able to extract `y` using reflection. ------------------------------------------------------------------------ -- Syntax for unidirectional relations -- See https://github.com/agda/agda-stdlib/issues/2150 for a possible -- simplification. module _ {R : REL A B ℓ₂} (S : REL B C ℓ₁) (T : REL A C ℓ₃) (step : Trans R S T) where forward : ∀ (x : A) {y z} → S y z → R x y → T x z forward x yRz x∼y = step {x} x∼y yRz -- Arbitrary relation syntax module ∼-syntax where infixr 2 step-∼ step-∼ = forward syntax step-∼ x yRz x∼y = x ∼⟨ x∼y ⟩ yRz -- Preorder syntax module ≲-syntax where infixr 2 step-≲ step-≲ = forward syntax step-≲ x yRz x≲y = x ≲⟨ x≲y ⟩ yRz -- Partial order syntax module ≤-syntax where infixr 2 step-≤ step-≤ = forward syntax step-≤ x yRz x≤y = x ≤⟨ x≤y ⟩ yRz -- Strict partial order syntax module <-syntax where infixr 2 step-< step-< = forward syntax step-< x yRz x<y = x <⟨ x<y ⟩ yRz -- Subset order syntax module ⊆-syntax where infixr 2 step-⊆ step-⊆ = forward syntax step-⊆ x yRz x⊆y = x ⊆⟨ x⊆y ⟩ yRz -- Strict subset order syntax module ⊂-syntax where infixr 2 step-⊂ step-⊂ = forward syntax step-⊂ x yRz x⊂y = x ⊂⟨ x⊂y ⟩ yRz -- Square subset order syntax module ⊑-syntax where infixr 2 step-⊑ step-⊑ = forward syntax step-⊑ x yRz x⊑y = x ⊑⟨ x⊑y ⟩ yRz -- Strict square subset order syntax module ⊏-syntax where infixr 2 step-⊏ step-⊏ = forward syntax step-⊏ x yRz x⊏y = x ⊏⟨ x⊏y ⟩ yRz -- Divisibility syntax module ∣-syntax where infixr 2 step-∣ step-∣ = forward syntax step-∣ x yRz x∣y = x ∣⟨ x∣y ⟩ yRz -- Single-step syntax module ⟶-syntax where infixr 2 step-⟶ step-⟶ = forward syntax step-⟶ x yRz x∣y = x ⟶⟨ x∣y ⟩ yRz -- Multi-step syntax module ⟶*-syntax where infixr 2 step-⟶* step-⟶* = forward syntax step-⟶* x yRz x∣y = x ⟶*⟨ x∣y ⟩ yRz ------------------------------------------------------------------------ -- Syntax for bidirectional relations module _ {U : REL B A ℓ₄} (sym : Sym U R) where backward : ∀ x {y z} → S y z → U y x → T x z backward x yRz x≈y = forward x yRz (sym x≈y) -- Setoid equality syntax module ≈-syntax where infixr 2 step-≈-⟩ step-≈-⟨ step-≈-⟩ = forward step-≈-⟨ = backward syntax step-≈-⟩ x yRz x≈y = x ≈⟨ x≈y ⟩ yRz syntax step-≈-⟨ x yRz y≈x = x ≈⟨ y≈x ⟨ yRz -- Deprecated infixr 2 step-≈ step-≈˘ step-≈ = step-≈-⟩ {-# WARNING_ON_USAGE step-≈ "Warning: step-≈ was deprecated in v2.0. Please use step-≈-⟩ instead." #-} step-≈˘ = step-≈-⟨ {-# WARNING_ON_USAGE step-≈˘ "Warning: step-≈˘ and _≈˘⟨_⟩_ was deprecated in v2.0. Please use step-≈-⟨ and _≈⟨_⟨_ instead." #-} syntax step-≈˘ x yRz y≈x = x ≈˘⟨ y≈x ⟩ yRz -- Container equality syntax module ≋-syntax where infixr 2 step-≋-⟩ step-≋-⟨ step-≋-⟩ = forward step-≋-⟨ = backward syntax step-≋-⟩ x yRz x≋y = x ≋⟨ x≋y ⟩ yRz syntax step-≋-⟨ x yRz y≋x = x ≋⟨ y≋x ⟨ yRz -- Don't remove until https://github.com/agda/agda/issues/5617 fixed. infixr 2 step-≋ step-≋˘ step-≋ = step-≋-⟩ {-# WARNING_ON_USAGE step-≋ "Warning: step-≋ was deprecated in v2.0. Please use step-≋-⟩ instead." #-} step-≋˘ = step-≋-⟨ {-# WARNING_ON_USAGE step-≋˘ "Warning: step-≋˘ and _≋˘⟨_⟩_ was deprecated in v2.0. Please use step-≋-⟨ and _≋⟨_⟨_ instead." #-} syntax step-≋˘ x yRz y≋x = x ≋˘⟨ y≋x ⟩ yRz -- Other equality syntax module ≃-syntax where infixr 2 step-≃-⟩ step-≃-⟨ step-≃-⟩ = forward step-≃-⟨ = backward syntax step-≃-⟩ x yRz x≃y = x ≃⟨ x≃y ⟩ yRz syntax step-≃-⟨ x yRz y≃x = x ≃⟨ y≃x ⟨ yRz -- Apartness relation syntax module #-syntax where infixr 2 step-#-⟩ step-#-⟨ step-#-⟩ = forward step-#-⟨ = backward syntax step-#-⟩ x yRz x#y = x #⟨ x#y ⟩ yRz syntax step-#-⟨ x yRz y#x = x #⟨ y#x ⟨ yRz -- Don't remove until https://github.com/agda/agda/issues/5617 fixed. infixr 2 step-# step-#˘ step-# = step-#-⟩ {-# WARNING_ON_USAGE step-# "Warning: step-# was deprecated in v2.0. Please use step-#-⟩ instead." #-} step-#˘ = step-#-⟨ {-# WARNING_ON_USAGE step-#˘ "Warning: step-#˘ and _#˘⟨_⟩_ was deprecated in v2.0. Please use step-#-⟨ and _#⟨_⟨_ instead." #-} syntax step-#˘ x yRz y#x = x #˘⟨ y#x ⟩ yRz -- Bijection syntax module ⤖-syntax where infixr 2 step-⤖ step-⬻ step-⤖ = forward step-⬻ = backward syntax step-⤖ x yRz x⤖y = x ⤖⟨ x⤖y ⟩ yRz syntax step-⬻ x yRz y⤖x = x ⬻⟨ y⤖x ⟩ yRz -- Inverse syntax module ↔-syntax where infixr 2 step-↔-⟩ step-↔-⟨ step-↔-⟩ = forward step-↔-⟨ = backward syntax step-↔-⟩ x yRz x↔y = x ↔⟨ x↔y ⟩ yRz syntax step-↔-⟨ x yRz y↔x = x ↔⟨ y↔x ⟨ yRz -- Inverse syntax module ↭-syntax where infixr 2 step-↭-⟩ step-↭-⟨ step-↭-⟩ = forward step-↭-⟨ = backward syntax step-↭-⟩ x yRz x↭y = x ↭⟨ x↭y ⟩ yRz syntax step-↭-⟨ x yRz y↭x = x ↭⟨ y↭x ⟨ yRz -- Don't remove until https://github.com/agda/agda/issues/5617 fixed. infixr 2 step-↭ step-↭˘ step-↭ = forward {-# WARNING_ON_USAGE step-↭ "Warning: step-↭ was deprecated in v2.0. Please use step-↭-⟩ instead." #-} step-↭˘ = backward {-# WARNING_ON_USAGE step-↭˘ "Warning: step-↭˘ and _↭˘⟨_⟩_ was deprecated in v2.0. Please use step-↭-⟨ and _↭⟨_⟨_ instead." #-} syntax step-↭˘ x yRz y↭x = x ↭˘⟨ y↭x ⟩ yRz ------------------------------------------------------------------------ -- Propositional equality -- Crucially often the step function cannot just be `subst` or pattern -- match on `refl` as we often want to compute which constructor the -- relation begins with, in order for the implicit subrelation -- arguments to resolve. See `≡-noncomputable-syntax` below if this -- is not required. module ≡-syntax (R : REL A B ℓ₁) (step : Trans _≡_ R R) where infixr 2 step-≡-⟩ step-≡-∣ step-≡-⟨ step-≡-⟩ = forward R R step step-≡-∣ : ∀ x {y} → R x y → R x y step-≡-∣ x xRy = xRy step-≡-⟨ = backward R R step ≡.sym syntax step-≡-⟩ x yRz x≡y = x ≡⟨ x≡y ⟩ yRz syntax step-≡-∣ x xRy = x ≡⟨⟩ xRy syntax step-≡-⟨ x yRz y≡x = x ≡⟨ y≡x ⟨ yRz -- Don't remove until https://github.com/agda/agda/issues/5617 fixed. infixr 2 step-≡ step-≡˘ step-≡ = step-≡-⟩ {-# WARNING_ON_USAGE step-≡ "Warning: step-≡ was deprecated in v2.0. Please use step-≡-⟩ instead." #-} step-≡˘ = step-≡-⟨ {-# WARNING_ON_USAGE step-≡˘ "Warning: step-≡˘ and _≡˘⟨_⟩_ was deprecated in v2.0. Please use step-≡-⟨ and _≡⟨_⟨_ instead." #-} syntax step-≡˘ x yRz y≡x = x ≡˘⟨ y≡x ⟩ yRz -- Unlike ≡-syntax above, chains of reasoning using this syntax will not -- reduce when proofs of propositional equality which are not definitionally -- equal to `refl` are passed. module ≡-noncomputing-syntax (R : REL A B ℓ₁) where private step : Trans _≡_ R R step ≡.refl xRy = xRy open ≡-syntax R step public ------------------------------------------------------------------------ -- Syntax for ending a chain of reasoning ------------------------------------------------------------------------ module end-syntax (R : Rel A ℓ₁) (reflexive : Reflexive R) where infix 3 _∎ _∎ : ∀ x → R x x x ∎ = reflexive