# HG changeset patch
# User Shinji KONO <kono@ie.u-ryukyu.ac.jp>
# Date 1635686112 -32400
# Node ID c2bc4ee841af844b8b92d2488554b47cadc4083b
# Parent  accd3d99cc8662c03dc4fac1524303ceb2f32b5d
...

diff -r accd3d99cc86 -r c2bc4ee841af utilities.agda
--- a/utilities.agda	Sun Oct 31 16:25:46 2021 +0900
+++ b/utilities.agda	Sun Oct 31 22:15:12 2021 +0900
@@ -3,6 +3,7 @@
 
 open import Function
 open import Data.Nat
+open import Data.Nat.Properties
 open import Data.Product
 open import Data.Bool  hiding ( _≟_ ; _≤?_)
 open import Level renaming ( suc to succ ; zero to Zero )
@@ -25,10 +26,10 @@
 
 open  _/\_
 
-_-_ : ℕ → ℕ → ℕ 
-x - zero  = x
-zero - _  = zero
-(suc x) - (suc y)  = x - y
+-- _-_ : ℕ → ℕ → ℕ 
+-- x - zero  = x
+-- zero - _  = zero
+-- (suc x) - (suc y)  = x - y
 
 +zero : { y : ℕ } → y + zero  ≡ y
 +zero {zero} = refl
@@ -65,7 +66,7 @@
           ∎ )
 
 
-minus-plus : { x y : ℕ } → (suc x - 1) + (y + 1) ≡ suc x + y
+minus-plus : { x y : ℕ } → (suc x ∸ 1) + (y + 1) ≡ suc x + y
 minus-plus {zero} {y} = +-sym {y} {1}
 minus-plus {suc x} {y} =  cong ( λ z → suc z ) (minus-plus {x} {y})
 
@@ -158,10 +159,14 @@
 Equal→≡ {x} {y} refl | yes refl = refl
 Equal→≡ {x} {y} () | no ¬p 
 
+open import Data.Empty
+
 Equal+1 : { x y : ℕ } →  Equal x y ≡ Equal (suc x) (suc y)
-Equal+1 {x} {y} with  x ≟ y
-Equal+1 {x} {.x} | yes refl = {!!}
-Equal+1 {x} {y} | no ¬p = {!!}
+Equal+1 {x} {y} with  x ≟ y | suc x ≟ suc y
+Equal+1 {x} {.x} | yes refl | yes refl = refl
+Equal+1 {x} {.x} | yes refl | no ¬p = ⊥-elim (¬p refl)
+Equal+1 {x} {y} | no ¬p | no _  = refl
+Equal+1 {x} {y} | no ¬p | yes refl  = ⊥-elim (¬p refl)
 
 open import Data.Empty 
 
diff -r accd3d99cc86 -r c2bc4ee841af whileTestGears1.agda
--- a/whileTestGears1.agda	Sun Oct 31 16:25:46 2021 +0900
+++ b/whileTestGears1.agda	Sun Oct 31 22:15:12 2021 +0900
@@ -1,7 +1,7 @@
 module whileTestGears1 where
 
 open import Function
-open import Data.Nat
+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)
@@ -111,12 +111,15 @@
        0 + vari env        ≡⟨ cong (λ k → k + vari env) (sym (lemma1 p )) ⟩
        varn env + vari env ≡⟨ proof ⟩
        c10 ∎ ) where open ≡-Reasoning  
-whileLoopSeg {_} {_} {c10} env proof next exit | yes p = next env1 (proof3 p ) proof4 where
+whileLoopSeg {_} {_} {c10} env proof next exit | yes p = next env1 (proof3 p ) (proof4 (varn env) p) where
       env1 = record {varn = (varn  env) - 1 ; vari = (vari env) + 1}
       1<0 : 1 ≤ zero → ⊥
       1<0 ()
-      proof4 : varn env1 < varn env
-      proof4 = {!!}
+      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 
       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)
@@ -133,22 +136,24 @@
 
 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 ≤ zero → j < i → ⊥
+lemma5 z≤n ()
 
 TerminatingLoop : {l : Level} {t : Set l} {c10 :  ℕ } → (i : ℕ) → (env : Env) → i ≡ varn env
    →  varn env + vari env ≡ c10 
    →  (exit : (e1 : Env )→ vari e1 ≡ c10 → t) → t
 TerminatingLoop {_} {t} {c10} i env refl p exit with <-cmp 0 i
-... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} env p (λ e1 eq lt → ⊥-elim (lemma3 e1 b lt) ) exit where
-    lemma3 : (e1 : Env) → 0 ≡ varn env → varn e1 < varn env → ⊥
-    lemma3 e refl ()
-... | tri< a ¬b ¬c = whileLoopSeg {_} {t} {c10} env p (TerminatingLoop1 i) exit where
-    TerminatingLoop1 : (j : ℕ) → (e1 : Env) → varn e1 + vari e1 ≡ c10 → varn e1 < varn env → t
-    TerminatingLoop1 zero e1 eq lt = whileLoopSeg {_} {t} {c10} env p {!!} exit 
-    TerminatingLoop1 (suc j) e1 eq lt with <-cmp j (varn e1)
-    ... | tri< (s≤s a) ¬b ¬c = TerminatingLoop1 j e1 {!!} {!!}
-    ... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} e1 {!!} lemma4 exit where
+... | tri≈ ¬a b ¬c = whileLoopSeg {_} {t} {c10} env p (λ e1 eq lt → ⊥-elim (lemma3 b lt) ) exit 
+... | tri< a ¬b ¬c = whileLoopSeg {_} {t} {c10} env p (λ e1 p1 lt1 → TerminatingLoop1 i env e1 (<⇒≤ lt1) p1 lt1 ) exit where --  varn e1 ≤ varn env
+    TerminatingLoop1 : (j : ℕ) → (env e1 : Env) → varn e1 ≤ j  → varn e1 + vari e1 ≡ c10 → varn e1 < varn env → t
+    TerminatingLoop1 zero env e1 n≤j eq lt = whileLoopSeg {_} {t} {c10} e1 eq (λ e2 eq lt1 → ⊥-elim (lemma5 n≤j lt1)) exit  
+    TerminatingLoop1 (suc j) env e1 n≤j eq lt with <-cmp (varn e1) (suc j)
+    ... | tri< (s≤s a) ¬b ¬c = TerminatingLoop1 j env e1 a eq lt 
+    ... | tri≈ ¬a refl ¬c = whileLoopSeg {_} {t} {c10} e1 eq lemma4 exit where
        lemma4 : (e2 : Env) → varn e2 + vari e2 ≡ c10 → varn e2 < varn e1 → t
-       lemma4 e2 eq lt = TerminatingLoop1 j {!!} {!!} {!!}
-    ... | tri> ¬a ¬b c =  ⊥-elim ( nat-≤> lt {!!} )
+       lemma4 e2 eq lt1 = TerminatingLoop1 j e1 e2 {!!} eq lt1 -- varn e2 ≤ j
+    ... | tri> ¬a ¬b c =  ⊥-elim ( nat-≤> n≤j c )