------------------------------------------------------------------------ -- The Agda standard library -- -- The basic code for equational reasoning with a single relation ------------------------------------------------------------------------ {-# OPTIONS --without-K --safe #-} open import Relation.Binary module Relation.Binary.Reasoning.Base.Single {a ℓ} {A : Set a} (_∼_ : Rel A ℓ) (refl : Reflexive _∼_) (trans : Transitive _∼_) where -- TODO: the following part is copied from Relation.Binary.Reasoning.Base.Partial -- in order to avoid larger refactors. We will refactor this part later -- so taht we use the same framework as Relation.Binary.Reasoning.Base.Partial. open import Level using (_⊔_) open import Relation.Binary.PropositionalEquality.Core as P using (_≡_) infix 4 _IsRelatedTo_ ------------------------------------------------------------------------ -- Definition of "related to" -- This seemingly unnecessary type is used to make it possible to -- infer arguments even if the underlying equality evaluates. data _IsRelatedTo_ (x y : A) : Set ℓ where relTo : (x∼y : x ∼ y) → x IsRelatedTo y ------------------------------------------------------------------------ -- Reasoning combinators -- 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. infix 1 begin_ infixr 2 step-∼ step-≡ step-≡˘ infixr 2 _≡⟨⟩_ infix 3 _∎ -- Beginning of a proof begin_ : ∀ {x y} → x IsRelatedTo y → x ∼ y begin relTo x∼y = x∼y -- Standard step with the relation step-∼ : ∀ x {y z} → y IsRelatedTo z → x ∼ y → x IsRelatedTo z step-∼ _ (relTo y∼z) x∼y = relTo (trans x∼y y∼z) -- Step with a non-trivial propositional equality step-≡ : ∀ x {y z} → y IsRelatedTo z → x ≡ y → x IsRelatedTo z step-≡ _ x∼z P.refl = x∼z -- Step with a flipped non-trivial propositional equality step-≡˘ : ∀ x {y z} → y IsRelatedTo z → y ≡ x → x IsRelatedTo z step-≡˘ _ x∼z P.refl = x∼z -- Step with a trivial propositional equality _≡⟨⟩_ : ∀ x {y} → x IsRelatedTo y → x IsRelatedTo y _ ≡⟨⟩ x∼y = x∼y -- Termination _∎ : ∀ x → x IsRelatedTo x x ∎ = relTo refl -- Syntax declarations syntax step-∼ x y∼z x∼y = x ∼⟨ x∼y ⟩ y∼z syntax step-≡ x y≡z x≡y = x ≡⟨ x≡y ⟩ y≡z syntax step-≡˘ x y≡z y≡x = x ≡˘⟨ y≡x ⟩ y≡z