Mercurial > hg > Members > atton > delta_monad
diff agda/delta.agda @ 131:d205ff1e406f InfiniteDeltaWithMonad
Cleanup proofs
author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
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date | Tue, 03 Feb 2015 12:57:13 +0900 |
parents | e6499a50ccbd |
children | 2bf1fa6d2006 |
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--- a/agda/delta.agda Tue Feb 03 12:46:20 2015 +0900 +++ b/agda/delta.agda Tue Feb 03 12:57:13 2015 +0900 @@ -1,12 +1,12 @@ +open import Level +open import Relation.Binary.PropositionalEquality +open ≡-Reasoning + open import list open import basic open import nat open import laws -open import Level -open import Relation.Binary.PropositionalEquality -open ≡-Reasoning - module delta where data Delta {l : Level} (A : Set l) : (Nat -> (Set l)) where @@ -38,9 +38,6 @@ delta-eta {n = O} x = mono x delta-eta {n = (S n)} x = delta x (delta-eta {n = n} x) - - - delta-mu : {l : Level} {A : Set l} {n : Nat} -> (Delta (Delta A (S n)) (S n)) -> Delta A (S n) delta-mu (mono x) = x delta-mu (delta x d) = delta (headDelta x) (delta-mu (delta-fmap tailDelta d)) @@ -48,10 +45,6 @@ delta-bind : {l : Level} {A B : Set l} {n : Nat} -> (Delta A (S n)) -> (A -> Delta B (S n)) -> Delta B (S n) delta-bind d f = delta-mu (delta-fmap f d) ---delta-bind (mono x) f = f x ---delta-bind (delta x d) f = delta (headDelta (f x)) (tailDelta (f x)) - - {- -- Monad (Haskell) delta-return : {l : Level} {A : Set l} -> A -> Delta A (S O) @@ -61,67 +54,4 @@ (x : Delta A n) -> (f : A -> (Delta B n)) -> (Delta B n) d >>= f = delta-bind d f --} - -{- --- proofs - --- sub-proofs - -n-tail-plus : {l : Level} {A : Set l} -> (n : Nat) -> ((n-tail {l} {A} n) ∙ tailDelta) ≡ n-tail (S n) -n-tail-plus O = refl -n-tail-plus (S n) = begin - n-tail (S n) ∙ tailDelta ≡⟨ refl ⟩ - (tailDelta ∙ (n-tail n)) ∙ tailDelta ≡⟨ refl ⟩ - tailDelta ∙ ((n-tail n) ∙ tailDelta) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-plus n) ⟩ - n-tail (S (S n)) - ∎ - -n-tail-add : {l : Level} {A : Set l} {d : Delta A} -> (n m : Nat) -> (n-tail {l} {A} n) ∙ (n-tail m) ≡ n-tail (n + m) -n-tail-add O m = refl -n-tail-add (S n) O = begin - n-tail (S n) ∙ n-tail O ≡⟨ refl ⟩ - n-tail (S n) ≡⟨ cong (\n -> n-tail n) (nat-add-right-zero (S n))⟩ - n-tail (S n + O) - ∎ -n-tail-add {l} {A} {d} (S n) (S m) = begin - n-tail (S n) ∙ n-tail (S m) ≡⟨ refl ⟩ - (tailDelta ∙ (n-tail n)) ∙ n-tail (S m) ≡⟨ refl ⟩ - tailDelta ∙ ((n-tail n) ∙ n-tail (S m)) ≡⟨ cong (\t -> tailDelta ∙ t) (n-tail-add {l} {A} {d} n (S m)) ⟩ - tailDelta ∙ (n-tail (n + (S m))) ≡⟨ refl ⟩ - n-tail (S (n + S m)) ≡⟨ refl ⟩ - n-tail (S n + S m) ∎ - -tail-delta-to-mono : {l : Level} {A : Set l} -> (n : Nat) -> (x : A) -> - (n-tail n) (mono x) ≡ (mono x) -tail-delta-to-mono O x = refl -tail-delta-to-mono (S n) x = begin - n-tail (S n) (mono x) ≡⟨ refl ⟩ - tailDelta (n-tail n (mono x)) ≡⟨ refl ⟩ - tailDelta (n-tail n (mono x)) ≡⟨ cong (\t -> tailDelta t) (tail-delta-to-mono n x) ⟩ - tailDelta (mono x) ≡⟨ refl ⟩ - mono x ∎ - -head-delta-natural-transformation : {l : Level} {A B : Set l} - -> (f : A -> B) -> (d : Delta A) -> headDelta (delta-fmap f d) ≡ f (headDelta d) -head-delta-natural-transformation f (mono x) = refl -head-delta-natural-transformation f (delta x d) = refl - -n-tail-natural-transformation : {l : Level} {A B : Set l} - -> (n : Nat) -> (f : A -> B) -> (d : Delta A) -> n-tail n (delta-fmap f d) ≡ delta-fmap f (n-tail n d) -n-tail-natural-transformation O f d = refl -n-tail-natural-transformation (S n) f (mono x) = begin - n-tail (S n) (delta-fmap f (mono x)) ≡⟨ refl ⟩ - n-tail (S n) (mono (f x)) ≡⟨ tail-delta-to-mono (S n) (f x) ⟩ - (mono (f x)) ≡⟨ refl ⟩ - delta-fmap f (mono x) ≡⟨ cong (\d -> delta-fmap f d) (sym (tail-delta-to-mono (S n) x)) ⟩ - delta-fmap f (n-tail (S n) (mono x)) ∎ -n-tail-natural-transformation (S n) f (delta x d) = begin - n-tail (S n) (delta-fmap f (delta x d)) ≡⟨ refl ⟩ - n-tail (S n) (delta (f x) (delta-fmap f d)) ≡⟨ cong (\t -> t (delta (f x) (delta-fmap f d))) (sym (n-tail-plus n)) ⟩ - ((n-tail n) ∙ tailDelta) (delta (f x) (delta-fmap f d)) ≡⟨ refl ⟩ - n-tail n (delta-fmap f d) ≡⟨ n-tail-natural-transformation n f d ⟩ - delta-fmap f (n-tail n d) ≡⟨ refl ⟩ - delta-fmap f (((n-tail n) ∙ tailDelta) (delta x d)) ≡⟨ cong (\t -> delta-fmap f (t (delta x d))) (n-tail-plus n) ⟩ - delta-fmap f (n-tail (S n) (delta x d)) ∎ --} +-} \ No newline at end of file