--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/gears-while.agda	Tue Sep 08 18:38:08 2020 +0900
@@ -0,0 +1,51 @@
+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)
+    proof2 = record {pi1 = refl ; pi2 = refl}    
+
+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
+          ∎
+
+{-# TERMINATING #-}
+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
+          ∎