Binary file Paper/soto.pdf has changed

--- a/Paper/soto.tex	Fri Nov 05 15:19:08 2021 +0900
+++ b/Paper/soto.tex	Sat Nov 06 19:45:49 2021 +0900
@@ -86,6 +86,7 @@
 \input{tex/cbc_agda.tex}% cbc と agda(gearsagda)
 \input{tex/hoare.tex} % hoare logic の説明
 \input{tex/while_loop.tex} % while loop の実装と検証(簡単に)
+\input{tex/tree_desc.tex}% Gears Agda における木構造の設計
 % red black tree の実装(こいつも簡単に)
 % red black tree の検証
 % まとめ

--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/bt/bt.agda	Sat Nov 06 19:45:49 2021 +0900
@@ -0,0 +1,171 @@
+module bt where
+
+open import Data.Nat
+open import Level renaming (zero to Z ; suc to succ)
+open import Data.List
+
+
+open import Data.Nat.Properties as NatProp -- <-cmp
+open import Relation.Binary
+
+open import Function as F hiding (const)
+
+data bt {n : Level} (A : Set n) : Set n where
+  leaf : bt A
+  node : (key-t : ℕ) → (value : A) →
+    (ltree : bt A ) → (rtree : bt A ) → bt A
+
+record Env {n : Level} (A : Set n) : Set n where
+  field
+    varn : ℕ
+    varv : A
+    vart : bt A
+    varl : List (bt A)
+open Env
+{-
+bt-depth : {n : Level} {A : Set n} → (env : Env A ) → ℕ
+bt-depth env  with vart env
+... | leaf = zero
+... | node key value ltree rtree = Data.Nat._⊔_ (bt-depth-c ltree ) (bt-depth-c rtree ) where
+    bt-depth-c : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
+    bt-depth-c leaf = suc zero
+    bt-depth-c (node key value lt rt) = suc (Data.Nat._⊔_ (bt-depth-c lt ) ( bt-depth-c rt))
+-}
+
+bt-depth : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
+bt-depth leaf = 0
+bt-depth (node key value t t₁) = Data.Nat._⊔_ (bt-depth t ) (bt-depth t₁ )
+
+bt-depth1 : {n : Level} {A : Set n} → (env : Env A ) → ℕ
+bt-depth1 env  with vart env
+... | tree = bt-depth-c tree where
+    bt-depth-c : {n : Level} {A : Set n} → (tree : bt A ) → ℕ
+    bt-depth-c leaf = zero
+    bt-depth-c (node key value lt rt) = suc (Data.Nat._⊔_ (bt-depth-c lt ) ( bt-depth-c rt))
+
+test-depth :  ℕ
+test-depth = bt-depth (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf)))
+
+test-env : {n : Level} {A : Set n} →  Env ℕ
+test-env = record {varn = 0; varv = 4 ; vart = (node 3 2 leaf (node 1 1 leaf (node 3 5 leaf leaf))); varl = []}
+
+test-depth1 :  ℕ
+test-depth1 = bt-depth1 record {varn = 0; varv = 4;  vart = (node 3 0 leaf (node 1 1 leaf (node 3 5 leaf leaf))); varl = []}
+
+find : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A ) → List (bt A)
+           → (next : bt A → List (bt A) → t )
+           → (exit : bt A → List (bt A) → t ) → t
+find key leaf st _ exit = exit leaf st
+find key (node key₁ v tree tree₁) st next exit with <-cmp key key₁
+find key n st _ exit | tri≈ ¬a b ¬c = exit n st
+find key n@(node key₁ v tree tree₁) st next _ | tri< a ¬b ¬c = next tree (n ∷ st)
+find key n@(node key₁ v tree tree₁) st next _ | tri> ¬a ¬b c = next tree₁ (n ∷ st)
+
+find1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
+           → (next : (env : Env A ) → t )
+           → (exit : (env : Env A ) → t ) → t
+find1 env next exit = find1-c (varn env) (vart env) env next exit where
+    find1-c : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (tree : bt A) → (env : Env A )
+           → (next : (env : Env A ) → t )
+           → (exit : (env : Env A ) → t ) → t
+    find1-c key leaf env next exit = exit env
+    find1-c key n@(node key-t value ltree rtree) env next exit with <-cmp key key-t
+    ... | tri< a ¬b ¬c = find1-c key ltree record env {vart = ltree ; varl = (n ∷ (varl env))} next exit
+    ... | tri≈ ¬a b ¬c = exit record env {vart = n}
+    ... | tri> ¬a ¬b c = find1-c key rtree record env {vart = rtree ; varl = (n ∷ (varl env))} next exit
+
+{-
+... | leaf = exit env
+... | n@(node key-t value ltree rtree) with <-cmp (varn env) key-t
+... | tri≈ ¬a b ¬c = exit record env {vart = n}
+... | tri< a ¬b ¬c = find1 record env {vart = ltree ; varl = (n ∷ (varl env))} next exit
+... | tri> ¬a ¬b c = find1 record env {vart = rtree ; varl = (n ∷ (varl env))} next exit
+-}
+
+{-# TERMINATING #-}
+find-loop : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → bt A → List (bt A)  → (exit : bt A → List (bt A) → t) → t
+find-loop {n} {m} {A} {t} key tree st exit = find-loop1 tree st where
+    find-loop1 : bt A → List (bt A) → t
+    find-loop1 tree st = find key tree st find-loop1 exit
+
+find-loop-1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
+    → (exit : (env : Env A ) → t) → t
+find-loop-1 env exit = find1 env exit exit
+
+fin-cg : {n m : Level} {A : Set n} {t : Set m} → t → (env : Env A ) → Env A
+fin-cg t = (λ env → env )
+
+test-cg = find-loop-1 record {varn = 5; varv = 4; vart = (node 2 2 (node 1 1 leaf leaf) (node 4 4 leaf (node 5 5 leaf leaf))); varl = []} (λ env → env )
+
+replaceNode : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → (bt A → t) → t
+replaceNode k v leaf next = next (node k v leaf leaf)
+replaceNode k v (node key value t t₁) next = next (node k v t t₁)
+
+replaceNode1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
+    → (next : Env A → t) → t
+replaceNode1 env next with vart env
+... | leaf = next record env {vart = (node (varn env) (varv env) leaf leaf) }
+... | node key-t value ltree rtree = next record env {vart = (node (varn env) (varv env) ltree rtree) }
+
+replace : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → List (bt A)
+    → (next : ℕ → A → bt A → List (bt A) → t )
+    → (exit : bt A → t) → t
+replace key value tree [] next exit = exit tree
+replace key value tree (leaf ∷ st) next exit = next key value tree st
+replace key value tree (node key₁ value₁ left right ∷ st) next exit with <-cmp key key₁
+... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) st
+... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st
+... | tri> ¬a ¬b c = next key value (node key₁ value₁ left tree ) st
+
+replace1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
+    → (next : Env A → t )
+    → (exit : Env A → t) → t
+replace1 env next exit = replace1-c (varl env) env next exit where
+    replace1-c : {n m : Level} {A : Set n} {t : Set m} → List (bt A) → (env : Env A )
+        → (next : Env A → t )
+        → (exit : Env A → t) → t
+    replace1-c st env next exit with st
+    ... | [] = exit env
+    ... | leaf ∷ st1 = replace1-c st1 env next exit
+    ... | n@(node key-t value ltree rtree) ∷ st1 with <-cmp (varn env) (key-t)
+    ... | tri< a ¬b ¬c = replace1-c st1 record env{vart = (node key-t value (vart env) rtree); varl = st1} next exit
+    ... | tri≈ ¬a b ¬c = replace1-c st1 record env{vart = (node key-t (varv env) ltree rtree); varl = st1} next exit
+    ... | tri> ¬a ¬b c = replace1-c st1 record env{vart = (node key-t value ltree (vart env)); varl = st1} next exit
+
+{-
+replace1 env next exit = next env
+replace1 env tree (node key₁ value₁ left right ∷ st) next exit with <-cmp (varn env) key₁
+... | tri< a ¬b ¬c = next key value (node key₁ value₁ tree right ) st
+... | tri≈ ¬a b ¬c = next key value (node key₁ value  left right ) st
+... | tri> ¬a ¬b c = next key value (node key₁ value₁ left tree ) st
+-}
+
+{-# TERMINATING #-}
+replace-loop : {n m : Level} {A : Set n} {t : Set m} → (key : ℕ) → (value : A) → bt A → List (bt A)  → (exit : bt A → t) → t
+replace-loop {_} {_} {A} {t} key value tree st exit = replace-loop1 key value tree st where
+    replace-loop1 : (key : ℕ) → (value : A) → bt A → List (bt A) → t
+    replace-loop1 key value tree st = replace key value tree st replace-loop1  exit
+
+insertTree : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : bt A → t ) → t
+insertTree tree key value exit = find-loop key tree []
+  $ λ t st → replaceNode key value t -- find-loop next
+  $ λ t1 → replace-loop key value t1 st exit -- replaceNode exit
+
+replace-loop1 : {n m : Level} {A : Set n} {t : Set m} → (env : Env A )
+    → (exit : (env : Env A ) → t) → t
+replace-loop1 env exit = replace1 env exit exit
+
+insertTree1 : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : Env A  → t ) → t
+insertTree1 tree key value exit = find-loop-1 record{varn = key; varv = value; vart = tree ; varl = []}
+  $ λ env → replaceNode1 env
+    $ λ env → replace-loop1 env exit
+
+insertTree11 : {n m : Level} {A : Set n} {t : Set m} → (tree : bt A) → (key : ℕ) → (value : A) → (next : Env A  → t ) → t
+insertTree11 tree key value exit = find-loop-1 record{varn = key; varv = value; vart = tree ; varl = []}
+  $ λ env → replaceNode1 env exit
+
+test-insert = insertTree1 (node 2 2 (node 1 1 leaf leaf) (node 4 4 leaf (node 5 5 leaf leaf))) 0 5 (λ env → env )
+
+test-insert1 = insertTree11 (node 2 2 (node 1 1 leaf leaf) (node 4 4 leaf (node 5 5 leaf leaf))) 0 5 (λ env → env )
+
+