--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/whileLoopPSemSound.agda.replaced	Tue Sep 08 18:38:08 2020 +0900
@@ -0,0 +1,20 @@
+loopPPSem : (input output : Envc ) @$\rightarrow$@  output @$\equiv$@ loopPP (varn input)  input refl
+  @$\rightarrow$@ (whileTestStateP s2 input ) @$\rightarrow$@ (whileTestStateP s2 input ) implies (whileTestStateP sf output)
+loopPPSem input output refl s2p = loopPPSemInduct (varn input) input  refl refl s2p
+  where
+    lem : (n : @$\mathbb{N}$@) @$\rightarrow$@ (env : Envc) @$\rightarrow$@ n + suc (vari env) @$\equiv$@ suc (n + vari env)
+    lem n env = +-suc (n) (vari env)
+    loopPPSemInduct : (n : @$\mathbb{N}$@) @$\rightarrow$@ (current : Envc) @$\rightarrow$@ (eq : n @$\equiv$@ varn current) @$\rightarrow$@  (loopeq : output @$\equiv$@ loopPP n current eq)
+      @$\rightarrow$@ (whileTestStateP s2 current ) @$\rightarrow$@ (whileTestStateP s2 current ) implies (whileTestStateP sf output)
+    loopPPSemInduct zero current refl loopeq refl rewrite loopeq = proof (@$\lambda$@ x @$\rightarrow$@ refl)
+    loopPPSemInduct (suc n) current refl loopeq refl rewrite (sym (lem n current)) =
+        whileLoopPSem current refl
+            (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
+            (@$\lambda$@ output x @$\rightarrow$@ loopPPSemInduct n (record { c10 = n + suc (vari current) ; varn = n ; vari = suc (vari current) }) refl loopeq refl)
+
+
+whileLoopPSemSound : {l : Level} @$\rightarrow$@ (input output : Envc )
+  @$\rightarrow$@ (varn input + vari input @$\equiv$@ c10 input)
+  @$\rightarrow$@  output @$\equiv$@ loopPP (varn input) input refl
+  @$\rightarrow$@ (varn input + vari input @$\equiv$@ c10 input) implies (vari output @$\equiv$@ c10 output)
+whileLoopPSemSound {l} input output pre eq = loopPPSem input output eq pre