Mercurial > hg > Members > atton > delta_monad
changeset 77:4b16b485a4b2
Split nat definition
| author | Yasutaka Higa <e115763@ie.u-ryukyu.ac.jp> |
|---|---|
| date | Mon, 01 Dec 2014 11:58:35 +0900 |
| parents | c7076f9bbaed |
| children | f02391a7402f |
| files | agda/delta.agda agda/nat.agda |
| diffstat | 2 files changed, 41 insertions(+), 7 deletions(-) [+] |
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--- a/agda/delta.agda Mon Dec 01 11:47:52 2014 +0900 +++ b/agda/delta.agda Mon Dec 01 11:58:35 2014 +0900 @@ -83,7 +83,7 @@ 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) (int-add-right-zero (S n))⟩ + 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 @@ -156,7 +156,7 @@ n-tail n (n-tail (S m) (delta x ds)) ≡⟨ cong (\t -> n-tail n (t (delta x ds))) (sym (n-tail-plus m)) ⟩ n-tail n (((n-tail m) ∙ tailDelta) (delta x ds)) ≡⟨ refl ⟩ n-tail n ((n-tail m) ds) ≡⟨ cong (\t -> t ds) (n-tail-add {d = ds} n m) ⟩ - n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (int-add-assoc n m) ⟩ + n-tail (n + m) ds ≡⟨ cong (\n -> n-tail n ds) (nat-add-sym n m) ⟩ n-tail (m + n) ds ≡⟨ refl ⟩ ((n-tail (m + n)) ∙ tailDelta) (delta x ds) ≡⟨ cong (\t -> t (delta x ds)) (n-tail-plus (m + n))⟩ n-tail (S (m + n)) (delta x ds) ≡⟨ refl ⟩ @@ -169,7 +169,7 @@ (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ bind (delta d ds) (n-tail (S m)) ≡⟨ refl ⟩ - bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (int-add-right-zero (S m)) ⟩ + bind (n-tail O (delta d ds)) (n-tail (S m)) ≡⟨ cong (\n -> bind (n-tail O (delta d ds)) (n-tail n)) (nat-add-right-zero (S m)) ⟩ bind (n-tail O (delta d ds)) (n-tail (S m + O)) ∎ monad-law-1-5 (S m) (S n) (delta d ds) = begin @@ -178,7 +178,7 @@ ((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ (n-tail n) (bind ds (tailDelta ∙ (n-tail (S m)))) ≡⟨ refl ⟩ (n-tail n) (bind ds (n-tail (S (S m)))) ≡⟨ monad-law-1-5 (S (S m)) n ds ⟩ - bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (int-add-right m n)) ⟩ + bind ((n-tail n) ds) (n-tail (S (S (m + n)))) ≡⟨ cong (\nm -> bind ((n-tail n) ds) (n-tail nm)) (sym (nat-right-increment (S m) n)) ⟩ bind ((n-tail n) ds) (n-tail (S m + S n)) ≡⟨ refl ⟩ bind (((n-tail n) ∙ tailDelta) (delta d ds)) (n-tail (S m + S n)) ≡⟨ cong (\t -> bind (t (delta d ds)) (n-tail (S m + S n))) (n-tail-plus n) ⟩ bind (n-tail (S n) (delta d ds)) (n-tail (S m + S n)) @@ -209,7 +209,7 @@ monad-law-1-4 (S m) n (mono dd) = begin headDelta (n-tail n (bind (mono dd) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (n-tail n ((n-tail (S m)) dd))≡⟨ cong (\t -> headDelta (t dd)) (n-tail-add {d = dd} n (S m)) ⟩ - headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (int-add-assoc n (S m)) ⟩ + headDelta (n-tail (n + S m) dd) ≡⟨ cong (\n -> headDelta ((n-tail n) dd)) (nat-add-sym n (S m)) ⟩ headDelta (n-tail (S m + n) dd) ≡⟨ refl ⟩ headDelta (n-tail (S m + n) (headDelta (mono dd))) ≡⟨ cong (\de -> headDelta (n-tail (S m + n) (headDelta de))) (sym (tail-delta-to-mono n dd)) ⟩ headDelta (n-tail (S m + n) (headDelta (n-tail n (mono dd)))) @@ -218,7 +218,7 @@ headDelta (n-tail O (bind (delta d ds) (n-tail (S m)))) ≡⟨ refl ⟩ headDelta (bind (delta d ds) (n-tail (S m))) ≡⟨ refl ⟩ headDelta (delta (headDelta ((n-tail (S m) d))) (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ - headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (int-add-right-zero (S m)) ⟩ + headDelta (n-tail (S m) d) ≡⟨ cong (\n -> headDelta ((n-tail n) d)) (nat-add-right-zero (S m)) ⟩ headDelta (n-tail (S m + O) d) ≡⟨ refl ⟩ headDelta (n-tail (S m + O) (headDelta (delta d ds))) ≡⟨ refl ⟩ headDelta (n-tail (S m + O) (headDelta (n-tail O (delta d ds)))) @@ -229,7 +229,7 @@ headDelta ((((n-tail n) ∙ tailDelta) (delta (headDelta ((n-tail (S m)) d)) (bind ds (tailDelta ∙ (n-tail (S m))))))) ≡⟨ refl ⟩ headDelta (n-tail n (bind ds (tailDelta ∙ (n-tail (S m))))) ≡⟨ refl ⟩ headDelta (n-tail n (bind ds (n-tail (S (S m))))) ≡⟨ monad-law-1-4 (S (S m)) n ds ⟩ - headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (int-add-right m n)) ⟩ + headDelta (n-tail ((S (S m) + n)) (headDelta (n-tail n ds))) ≡⟨ cong (\nm -> headDelta ((n-tail nm) (headDelta (n-tail n ds)))) (sym (nat-right-increment (S m) n)) ⟩ headDelta (n-tail (S m + S n) (headDelta (n-tail n ds))) ≡⟨ refl ⟩ headDelta (n-tail (S m + S n) (headDelta (((n-tail n) ∙ tailDelta) (delta d ds)))) ≡⟨ cong (\t -> headDelta (n-tail (S m + S n) (headDelta (t (delta d ds))))) (n-tail-plus n) ⟩ headDelta (n-tail (S m + S n) (headDelta (n-tail (S n) (delta d ds))))
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/agda/nat.agda Mon Dec 01 11:58:35 2014 +0900 @@ -0,0 +1,34 @@ +open import Relation.Binary.PropositionalEquality +open ≡-Reasoning + +module nat where + +data Nat : Set where + O : Nat + S : Nat -> Nat + +_+_ : Nat -> Nat -> Nat +O + n = n +(S m) + n = S (m + n) + +nat-add-right-zero : (n : Nat) -> n ≡ n + O +nat-add-right-zero O = refl +nat-add-right-zero (S n) = begin + S n ≡⟨ cong (\n -> S n) (nat-add-right-zero n) ⟩ + S (n + O) ≡⟨ refl ⟩ + S n + O + ∎ + +nat-right-increment : (n m : Nat) -> n + S m ≡ S (n + m) +nat-right-increment O m = refl +nat-right-increment (S n) m = cong S (nat-right-increment n m) + +nat-add-sym : (n m : Nat) -> n + m ≡ m + n +nat-add-sym O O = refl +nat-add-sym O (S m) = cong S (nat-add-sym O m) +nat-add-sym (S n) O = cong S (nat-add-sym n O) +nat-add-sym (S n) (S m) = begin + S n + S m ≡⟨ refl ⟩ + S (n + S m) ≡⟨ cong S (nat-add-sym n (S m)) ⟩ + S ((S m) + n) ≡⟨ sym (nat-right-increment (S m) n) ⟩ + S m + S n ∎