--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/whileLoopPSem.agda.replaced	Tue Sep 08 18:38:08 2020 +0900
@@ -0,0 +1,21 @@
+whileLoopPSem : {l : Level} {t : Set l}   @$\rightarrow$@ (input : Envc ) @$\rightarrow$@ (vari input) + (varn input) @$\equiv$@ (c10 input)
+@$\rightarrow$@ (next : (output : Envc ) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output) + (varn output) @$\equiv$@ (c10 output))  @$\rightarrow$@ t)
+@$\rightarrow$@ (exit : (output : Envc ) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output @$\equiv$@ c10 output))  @$\rightarrow$@ t) @$\rightarrow$@ t
+whileLoopPSem env s next exit with varn env | s
+... | zero | _ = exit env (proof (@$\lambda$@ z @$\rightarrow$@ z))
+... | (suc varn ) | refl = next ( record env { varn = varn ; vari = suc (vari env) } ) (proof @$\lambda$@ x @$\rightarrow$@ +-suc varn (vari env) )
+
+
+loopPPSem : (input output : Envc ) @$\rightarrow$@  output @$\equiv$@ loopPP (varn input)  input refl
+  @$\rightarrow$@ (vari input) + (varn input) @$\equiv$@ (c10 input) @$\rightarrow$@ ((vari input) + (varn input) @$\equiv$@ (c10 input) ) implies ((vari output @$\equiv$@ c10 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)
+      @$\rightarrow$@ ((vari current) + (varn current) @$\equiv$@ (c10 current) ) @$\rightarrow$@ ((vari current) + (varn current) @$\equiv$@ (c10 current) ) implies ((vari output @$\equiv$@ c10 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)