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data Σ(A : Set) (B : A → Set) : Set where | |
ι : (a : A) → B a → Σ A B |
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data _⊕_ (A B : Set) : Set where | |
i₁ : A → A ⊕ B | |
i₂ : B → A ⊕ B |
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data _×_ (A B : Set) : Set where | |
[_,_] : A → B → A × B |
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open import Nat | |
-- If f(m+1) = f(m) for all m, then f(n) = f(0) for all n | |
constthm : (f : ℕ → ℕ) → ((m : ℕ) → (f (succ m)) == (f m)) → (n : ℕ) → (f n) == (f 0) | |
constthm f _ 0 = refl (f 0) | |
constthm f adjEq (succ n) = (adjEq n) transEq (constthm f adjEq n) |
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transport : {A : Set} → {B : Set} → {x y : A} → (f : A → B) → x == y → f(x) == f(y) | |
transport f (refl a) = refl (f a) | |
symm : {A : Set} → {x y : A} → x == y → y == x | |
symm (refl a) = refl a | |
_transEq_ : {A : Set} → {x y z : A} → x == y → y == z → x == z | |
(refl a) transEq (refl .a) = refl a |
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data _==_ {A : Set} : A → A → Set where | |
refl : (a : A) → a == a |
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1odd : Even 1 → False | |
1odd () |
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2even : Even 2 | |
2even = +2even zeroeven |
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data Even : ℕ → Set where | |
zeroeven : Even zero | |
+2even : {n : ℕ} → Even n → Even (succ (succ n)) |
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vacuous : {A : Set} → False → A | |
vacuous () |