--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/whileTestGears1.agda	Sat Apr 08 17:00:53 2023 +0900
@@ -0,0 +1,106 @@
+module whileTestGears1 where
+
+open import Function
+open import Data.Nat renaming ( _∸_ to _-_)
+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 import Data.Empty
+open import Data.Nat.Properties
+open import Data.Unit hiding ( _≟_ ;  _≤?_ ; _≤_)
+
+lemma1 : {i : ℕ} →  ¬ 1 ≤ i → i ≡ 0
+lemma1 {zero} not = refl
+lemma1 {suc i} not = ⊥-elim ( not (s≤s z≤n) )
+
+open import Relation.Binary.Definitions
+
+nat-≤> : { x y : ℕ } → x ≤ y → y < x → ⊥
+nat-≤>  (s≤s x<y) (s≤s y<x) = nat-≤> x<y y<x
+lemma3 : {i j : ℕ} → 0 ≡ i → j < i → ⊥
+lemma3 refl ()
+lemma5 : {i j : ℕ} → i < 1 → j < i → ⊥
+lemma5 (s≤s z≤n) ()
+
+open  _/\_
+
+record Env ( c : ℕ ) : Set where
+  field
+    varn : ℕ
+    vari : ℕ
+    n+i=c : varn + vari ≡ c
+open Env
+
+TerminatingLoopS : {l : Level} {t : Set l} (Index : Set ) → ( reduce : Index → ℕ)
+   → (loop : (r : Index)  → (next : (r1 : Index)  → reduce r1 < reduce r  → t ) → t)
+   → (r : Index) → t 
+TerminatingLoopS {_} {t} Index reduce loop  r with <-cmp 0 (reduce r)
+... | tri≈ ¬a b ¬c = loop r (λ r1 lt → ⊥-elim (lemma3 b lt) ) 
+... | tri< a ¬b ¬c = loop r (λ r1 lt1 → TerminatingLoop1 (reduce r) r r1 (≤-step lt1) lt1 ) where 
+    TerminatingLoop1 : (j : ℕ) → (r r1 : Index) → reduce r1 < suc j  → reduce r1 < reduce r → t
+    TerminatingLoop1 zero r r1 n≤j lt = loop r1 (λ r2 lt1 → ⊥-elim (lemma5 n≤j lt1)) 
+    TerminatingLoop1 (suc j) r r1  n≤j lt with <-cmp (reduce r1) (suc j)
+    ... | tri< a ¬b ¬c = TerminatingLoop1 j r r1 a lt 
+    ... | tri≈ ¬a b ¬c = loop r1 (λ r2 lt1 → TerminatingLoop1 j r1 r2 (subst (λ k → reduce r2 < k ) b lt1 ) lt1 )
+    ... | tri> ¬a ¬b c =  ⊥-elim ( nat-≤> c n≤j )   
+
+whileTestSpec1 : (c10 : ℕ) →  (e1 : Env c10 ) → vari e1 ≡ c10 → ⊤
+whileTestSpec1 _ _ x = tt
+
+whileLoopSeg : {l : Level} {t : Set l} → (c10 :  ℕ ) → (env : Env c10 ) 
+   → (next : (e1 : Env c10 ) → varn e1 < varn env → t)  
+   → (exit : (e1 : Env c10 ) → vari e1 ≡ c10 → t) → t
+whileLoopSeg c10 env next exit with  ( suc zero  ≤? (varn  env) )
+whileLoopSeg c10 env next exit | no p = exit env ( begin
+       vari env            ≡⟨ refl ⟩
+       0 + vari env        ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩
+       varn env + vari env ≡⟨ n+i=c env ⟩
+       c10 ∎ ) where open ≡-Reasoning  
+whileLoopSeg c10 env next exit | yes p = next env1  (proof4 (varn env) p) where
+      env1 = record {varn = (varn  env) - 1 ; vari = (vari env) + 1 ; n+i=c = proof3 p } where
+          1<0 : 1 ≤ zero → ⊥
+          1<0 ()
+          proof3 : (suc zero  ≤ (varn  env))  → ((varn env) - 1) + (vari env + 1)  ≡ 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  ≡⟨  n+i=c env ⟩
+                 c10
+              ∎
+      proof4 : (i : ℕ) → 1 ≤ i  → i - 1 < i
+      proof4 zero ()
+      proof4 (suc i) lt = begin
+          suc (suc i - 1 ) ≤⟨ ≤-refl ⟩
+          suc i ∎ where open ≤-Reasoning 
+
+proofGearsTermS : (c10 :  ℕ ) → ⊤
+proofGearsTermS c10 = 
+    TerminatingLoopS (Env c10) (λ env → varn env) (λ n2 loop → whileLoopSeg c10 n2 loop (λ ne pe → whileTestSpec1 c10 ne pe ) )
+        record { varn = 0 ; vari = c10 ; n+i=c = refl }  
+
+proofGearsExec : (c10 :  ℕ ) → ℕ
+proofGearsExec c10 = 
+    TerminatingLoopS (Env c10) (λ env → varn env) (λ n2 loop → whileLoopSeg c10 n2 loop (λ ne pe → vari ne ) )
+        record { varn = 0 ; vari = c10 ; n+i=c = refl }  
+
+test = proofGearsExec 20
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