# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1544965300 -32400
# Node ID b95a3cf9727cc161c0ca21493ab240a9c2bdb907
# Parent  23cce74379180aea764be54d87da81844c68941d
add Gears1

diff -r 23cce7437918 -r b95a3cf9727c whileTestGears1.agda
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/whileTestGears1.agda	Sun Dec 16 22:01:40 2018 +0900
@@ -0,0 +1,111 @@
+module whileTestGears1 where
+
+open import Function
+open import Data.Nat
+open import Data.Bool hiding ( _≟_ )
+open import Level renaming ( suc to succ ; zero to Zero )
+open import Relation.Nullary using (¬_; Dec; yes; no)
+open import Relation.Binary.PropositionalEquality
+
+open import utilities
+open  _/\_
+
+record Env  : Set where
+  field
+    varn : ℕ
+    vari : ℕ
+open Env
+
+whileTest : {l : Level} {t : Set l} -> (c10 : ℕ) → (Code : Env -> t) -> t
+whileTest c10 next = next (record {varn = c10 ; vari = 0} )
+
+{-# TERMINATING #-}
+whileLoop : {l : Level} {t : Set l} -> Env -> (Code : Env -> t) -> t
+whileLoop env next with lt 0 (varn env)
+whileLoop env next | false = next env
+whileLoop env next | true =
+    whileLoop (record {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
+
+test1 : Env
+test1 = whileTest 10 (λ env → whileLoop env (λ env1 → env1 ))
+
+
+proof1 : whileTest 10 (λ env → whileLoop env (λ e → (vari e) ≡ 10 ))
+proof1 = refl
+
+--                                                                              ↓PostCondition
+whileTest' : {l : Level} {t : Set l}  -> {c10 :  ℕ } → (Code : (env : Env)  -> ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -> t) -> t
+whileTest' {_} {_} {c10} next = next env proof2
+  where
+    env : Env
+    env = record {vari = 0 ; varn = c10}
+    proof2 : ((vari env) ≡ 0) /\ ((varn env) ≡ c10) -- PostCondition
+    proof2 = record {pi1 = refl ; pi2 = refl}
+
+open import Data.Empty
+open import Data.Nat.Properties
+
+
+{-# TERMINATING #-} --                                                  ↓PreCondition(Invaliant)
+whileLoop' : {l : Level} {t : Set l} -> (env : Env) -> {c10 :  ℕ } → ((varn env) + (vari env) ≡ c10) -> (Code : Env -> t) -> t
+whileLoop' env proof next with  ( suc zero  ≤? (varn  env) )
+whileLoop' env proof next | no p = next env 
+whileLoop' env {c10} proof next | yes p = whileLoop' env1 (proof3 p ) next
+    where
+      env1 = record {varn = (varn  env) - 1 ; vari = (vari env) + 1}
+      1<0 : 1 ≤ zero → ⊥
+      1<0 ()
+      proof3 : (suc zero  ≤ (varn  env))  → varn env1 + vari env1 ≡ c10
+      proof3 (s≤s lt) with varn  env
+      proof3 (s≤s z≤n) | zero = ⊥-elim (1<0 p)
+      proof3 (s≤s (z≤n {n'}) ) | suc n =  let open ≡-Reasoning  in
+          begin
+             n' + (vari env + 1) 
+          ≡⟨ cong ( λ z → n' + z ) ( +-sym  {vari env} {1} )  ⟩
+             n' + (1 + vari env ) 
+          ≡⟨ sym ( +-assoc (n')  1 (vari env) ) ⟩
+             (n' + 1) + vari env 
+          ≡⟨ cong ( λ z → z + vari env )  +1≡suc  ⟩
+             (suc n' ) + vari env 
+          ≡⟨⟩
+             varn env + vari env
+          ≡⟨ proof  ⟩
+             c10
+          ∎
+
+-- Condition to Invaliant
+conversion1 : {l : Level} {t : Set l } → (env : Env) -> {c10 :  ℕ } → ((vari env) ≡ 0) /\ ((varn env) ≡ c10)
+               -> (Code : (env1 : Env) -> (varn env1 + vari env1 ≡ c10) -> t) -> t
+conversion1 env {c10} p1 next = next env proof4
+   where
+      proof4 : varn env + vari env ≡ c10
+      proof4 = let open ≡-Reasoning  in
+          begin
+            varn env + vari env
+          ≡⟨ cong ( λ n → n + vari env ) (pi2 p1 ) ⟩
+            c10 + vari env
+          ≡⟨ cong ( λ n → c10 + n ) (pi1 p1 ) ⟩
+            c10 + 0
+          ≡⟨ +-sym {c10} {0} ⟩
+            c10
+          ∎
+
+
+proofGears : {c10 :  ℕ } → Set
+proofGears {c10} = whileTest' {_} {_} {c10} (λ n p1 →  conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 →  ( vari n2 ≡ c10 )))) 
+
+proofGearsMeta : {c10 :  ℕ } → whileTest' {_} {_} {c10} (λ n p1 →  conversion1 n p1 (λ n1 p2 → whileLoop' n1 p2 (λ n2 →  ( vari n2 ≡ c10 )))) 
+proofGearsMeta {c10} = {!!}
+
+
+
+whileTest0 : {l : Level} {t m : Set l} -> (c10 : ℕ) → (Code : m -> Env -> t) -> t
+whileTest0 c10 next = next {!!} (record {varn = c10 ; vari = 0} )
+
+{-# TERMINATING #-}
+whileLoop0 : {l : Level} {t m : Set l} -> m -> Env -> (Code : m -> Env -> t) -> t
+whileLoop0 m env next with lt 0 (varn env)
+whileLoop0 m env next | false = next m env
+whileLoop0 m env next | true =
+    whileLoop0 m (record {varn = (varn env) - 1 ; vari = (vari env) + 1}) next
+
diff -r 23cce7437918 -r b95a3cf9727c whileTestPrim.agda
--- a/whileTestPrim.agda	Sun Dec 16 19:31:36 2018 +0900
+++ b/whileTestPrim.agda	Sun Dec 16 22:01:40 2018 +0900
@@ -251,12 +251,9 @@
           ∎
         )
      lemma51 : (z : Env ) → neg (λ z → lt zero (varn z)) z ≡ true → varn z ≡ zero
-     lemma51 z cond with lt zero (varn z) | (suc zero) ≤? (varn z)
-     lemma51 z () | false | yes p
-     lemma51 z () | true | yes p
-     lemma51 z refl | _ | no ¬p with varn z
-     lemma51 z refl | _ | no ¬p | zero = refl
-     lemma51 z refl | _ | no ¬p | suc x = ⊥-elim ( ¬p (s≤s z≤n ) )
+     lemma51 z cond with varn z
+     lemma51 z refl | zero = refl
+     lemma51 z () | suc x
      lemma5 : {c10 : ℕ} →  Tautology ((λ e → Equal (varn e + vari e) c10) and (neg (λ z → lt zero (varn z)))) termCond
      lemma5 {c10} env = impl⇒ ( λ cond → let open ≡-Reasoning  in
          begin