125 lines
3.7 KiB
Agda
125 lines
3.7 KiB
Agda
{-# OPTIONS --prop --rewriting #-}
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module PropUtil where
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-- ⊥ is a data with no constructor
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-- ⊤ is a record with one always-available constructor
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data ⊥ : Prop where
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record ⊤ : Prop where
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constructor tt
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data _∨_ : Prop → Prop → Prop where
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inj₁ : {P Q : Prop} → P → P ∨ Q
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inj₂ : {P Q : Prop} → Q → P ∨ Q
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record _∧_ (P Q : Prop) : Prop where
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constructor ⟨_,_⟩
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field
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p : P
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q : Q
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infixr 10 _∧_
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infixr 11 _∨_
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-- ∧ elimination
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proj₁ : {P Q : Prop} → P ∧ Q → P
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proj₁ pq = _∧_.p pq
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proj₂ : {P Q : Prop} → P ∧ Q → Q
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proj₂ pq = _∧_.q pq
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-- ∨ elimination
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dis : {P Q S : Prop} → (P ∨ Q) → (P → S) → (Q → S) → S
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dis (inj₁ p) ps qs = ps p
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dis (inj₂ q) ps qs = qs q
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-- ¬ is a shorthand for « → ⊥ »
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¬ : Prop → Prop
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¬ P = P → ⊥
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-- ⊥ elimination
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case⊥ : {P : Prop} → ⊥ → P
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case⊥ ()
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-- ⇔ shorthand
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_⇔_ : Prop → Prop → Prop
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P ⇔ Q = (P → Q) ∧ (Q → P)
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-- Syntactic sugar for writing applications
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infixr 200 _$_
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_$_ : {T U : Prop} → (T → U) → T → U
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h $ t = h t
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open import Agda.Primitive
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postulate _≈_ : ∀{ℓ}{A : Set ℓ}(a : A) → A → Set ℓ
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{-# BUILTIN REWRITE _≈_ #-}
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infix 3 _≡_
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data _≡_ {ℓ}{A : Set ℓ}(a : A) : A → Prop ℓ where
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refl : a ≡ a
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≡sym : {ℓ : Level} → {A : Set ℓ}→ {a a' : A} → a ≡ a' → a' ≡ a
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≡sym refl = refl
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≡tran : {ℓ : Level} {A : Set ℓ} → {a a' a'' : A} → a ≡ a' → a' ≡ a'' → a ≡ a''
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≡tran refl refl = refl
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postulate ≡fun : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : Set ℓ'} → {f g : A → B} → ((x : A) → (f x ≡ g x)) → f ≡ g
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postulate ≡fun' : {ℓ ℓ' : Level} → {A : Set ℓ} → {B : A → Set ℓ'} → {f g : (a : A) → B a} → ((x : A) → (f x ≡ g x)) → f ≡ g
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postulate subst : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Set ℓ'){a a' : A} → a ≡ a' → P a → P a'
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postulate substP : ∀{ℓ}{A : Set ℓ}{ℓ'}(P : A → Prop ℓ'){a a' : A} → a ≡ a' → P a → P a'
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postulate substrefl : ∀{ℓ}{A : Set ℓ}{ℓ'}{P : A → Set ℓ'}{a : A}{e : a ≡ a}{p : P a} → subst P e p ≈ p
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{-# REWRITE substrefl #-}
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cong : {ℓ ℓ' : Level}{A : Set ℓ}{B : Set ℓ'} → (f : A → B) → {a a' : A} → a ≡ a' → f a ≡ f a'
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cong f refl = refl
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cong₂ : {ℓ ℓ' ℓ'' : Level}{A : Set ℓ}{B : Set ℓ'}{C : Set ℓ''} → (f : A → B → C) → {a a' : A} {b b' : B} → a ≡ a' → b ≡ b' → f a b ≡ f a' b'
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cong₂ f refl refl = refl
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{-# BUILTIN EQUALITY _≡_ #-}
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data Nat : Set where
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zero : Nat
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succ : Nat → Nat
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{-# BUILTIN NATURAL Nat #-}
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variable
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ℓ ℓ' : Level
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record _×_ (A : Set ℓ) (B : Set ℓ) : Set ℓ where
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constructor _,×_
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field
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a : A
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b : B
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record _×'_ (A : Set ℓ) (B : Prop ℓ) : Set ℓ where
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constructor _,×'_
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field
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a : A
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b : B
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record _×''_ (A : Set ℓ) (B : A → Prop ℓ') : Set (ℓ ⊔ ℓ') where
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constructor _,×''_
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field
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a : A
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b : B a
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proj×₁ : {A B : Set} → (A × B) → A
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proj×₁ p = _×_.a p
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proj×₂ : {A B : Set} → (A × B) → B
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proj×₂ p = _×_.b p
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proj×'₁ : {A : Set} → {B : Prop} → (A ×' B) → A
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proj×'₁ p = _×'_.a p
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proj×'₂ : {A : Set} → {B : Prop} → (A ×' B) → B
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proj×'₂ p = _×'_.b p
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proj×''₁ : {A : Set} → {B : A → Prop} → (A ×'' B) → A
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proj×''₁ p = _×''_.a p
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proj×''₂ : {A : Set} → {B : A → Prop} → (p : A ×'' B) → B (proj×''₁ p)
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proj×''₂ p = _×''_.b p
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