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|
------------------------------------------------------------------------
-- The Agda standard library
--
-- Some examples of how to use non-trivial induction over the natural
-- numbers.
------------------------------------------------------------------------
module README.Data.Nat.Induction where
open import Data.Nat
open import Data.Nat.Induction
open import Data.Product.Base using (_,_)
open import Function.Base using (_∘_)
open import Induction.WellFounded
open import Relation.Binary.PropositionalEquality
private
n<′1+n : ∀ {n} → n <′ suc n
n<′1+n = ≤′-refl
n<′2+n : ∀ {n} → n <′ suc (suc n)
n<′2+n = ≤′-step ≤′-refl
-- Doubles its input.
twice : ℕ → ℕ
twice = rec _ λ
{ zero _ → zero
; (suc n) twice-n → suc (suc twice-n)
}
-- Halves its input (rounding downwards).
--
-- The step function is mentioned in a proof below, so it has been
-- given a name. (The mutual keyword is used to avoid having to give
-- a type signature for the step function.)
mutual
half₁-step = λ
{ zero _ → zero
; (suc zero) _ → zero
; (suc (suc n)) (_ , half₁n , _) → suc half₁n
}
half₁ : ℕ → ℕ
half₁ = cRec _ half₁-step
-- An alternative implementation of half₁.
mutual
half₂-step = λ
{ zero _ → zero
; (suc zero) _ → zero
; (suc (suc n)) rec → suc (rec n<′2+n)
}
half₂ : ℕ → ℕ
half₂ = <′-rec _ half₂-step
-- The application half₁ (2 + n) is definitionally equal to
-- 1 + half₁ n. Perhaps it is instructive to see why.
half₁-2+ : ∀ n → half₁ (2 + n) ≡ 1 + half₁ n
half₁-2+ n = begin
half₁ (2 + n) ≡⟨⟩
cRec _ half₁-step (2 + n) ≡⟨⟩
half₁-step (2 + n) (cRecBuilder _ half₁-step (2 + n)) ≡⟨⟩
half₁-step (2 + n)
(let ih = cRecBuilder _ half₁-step (1 + n) in
half₁-step (1 + n) ih , ih) ≡⟨⟩
half₁-step (2 + n)
(let ih = cRecBuilder _ half₁-step n in
half₁-step (1 + n) (half₁-step n ih , ih) , half₁-step n ih , ih) ≡⟨⟩
1 + half₁-step n (cRecBuilder _ half₁-step n) ≡⟨⟩
1 + cRec _ half₁-step n ≡⟨⟩
1 + half₁ n ∎
where open ≡-Reasoning
-- The application half₂ (2 + n) is definitionally equal to
-- 1 + half₂ n. Perhaps it is instructive to see why.
half₂-2+ : ∀ n → half₂ (2 + n) ≡ 1 + half₂ n
half₂-2+ n = begin
half₂ (2 + n) ≡⟨⟩
<′-rec _ half₂-step (2 + n) ≡⟨⟩
half₂-step (2 + n) (<′-recBuilder _ half₂-step (2 + n)) ≡⟨⟩
1 + <′-recBuilder _ half₂-step (2 + n) n<′2+n ≡⟨⟩
1 + Some.wfRecBuilder _ half₂-step (2 + n)
(<′-wellFounded (2 + n)) n<′2+n ≡⟨⟩
1 + Some.wfRecBuilder _ half₂-step (2 + n)
(acc (<′-wellFounded′ (2 + n))) n<′2+n ≡⟨⟩
1 + half₂-step n
(Some.wfRecBuilder _ half₂-step n
(<′-wellFounded′ (2 + n) n<′2+n)) ≡⟨⟩
1 + half₂-step n
(Some.wfRecBuilder _ half₂-step n
(<′-wellFounded′ (1 + n) n<′1+n)) ≡⟨⟩
1 + half₂-step n
(Some.wfRecBuilder _ half₂-step n (<′-wellFounded n)) ≡⟨⟩
1 + half₂-step n (<′-recBuilder _ half₂-step n) ≡⟨⟩
1 + <′-rec _ half₂-step n ≡⟨⟩
1 + half₂ n ∎
where open ≡-Reasoning
-- Some properties that the functions above satisfy, proved using
-- cRec.
half₁-+₁ : ∀ n → half₁ (twice n) ≡ n
half₁-+₁ = cRec _ λ
{ zero _ → refl
; (suc zero) _ → refl
; (suc (suc n)) (_ , half₁twice-n≡n , _) →
cong (suc ∘ suc) half₁twice-n≡n
}
half₂-+₁ : ∀ n → half₂ (twice n) ≡ n
half₂-+₁ = cRec _ λ
{ zero _ → refl
; (suc zero) _ → refl
; (suc (suc n)) (_ , half₁twice-n≡n , _) →
cong (suc ∘ suc) half₁twice-n≡n
}
-- Some properties that the functions above satisfy, proved using
-- <′-rec.
half₁-+₂ : ∀ n → half₁ (twice n) ≡ n
half₁-+₂ = <′-rec _ λ
{ zero _ → refl
; (suc zero) _ → refl
; (suc (suc n)) rec →
cong (suc ∘ suc) (rec n<′2+n)
}
half₂-+₂ : ∀ n → half₂ (twice n) ≡ n
half₂-+₂ = <′-rec _ λ
{ zero _ → refl
; (suc zero) _ → refl
; (suc (suc n)) rec →
cong (suc ∘ suc) (rec n<′2+n)
}
|
