patch 8.0.0416: setting v:progpath is not quite right

Problem:    Setting v:progpath is not quite right.
Solution:   On MS-Windows add the extension. On Unix use the full path for a
            relative directory. (partly by James McCoy, closes #1531)
diff --git a/src/main.c b/src/main.c
index 2575e82..8669e33 100644
--- a/src/main.c
+++ b/src/main.c
@@ -3533,21 +3533,31 @@
 set_progpath(char_u *argv0)
 {
     char_u *val = argv0;
+#ifdef WIN32
+    char_u *path = NULL;
+#else
     char_u buf[MAXPATHL];
+#endif
 
     /* A relative path containing a "/" will become invalid when using ":cd",
      * turn it into a full path.
      * On MS-Windows "vim.exe" is found in the current directory, thus also do
      * it when there is no path and the file exists. */
-    if ( !mch_isFullName(argv0)
+    if (!mch_isFullName(argv0))
+    {
 # ifdef WIN32
-	    && mch_can_exe(argv0, NULL, TRUE)
+	if (mch_can_exe(argv0, &path, FALSE) && path != NULL)
+	    val = path;
 # else
-	    && gettail(argv0) != argv0
+	if (gettail(argv0) != argv0
+			   && vim_FullName(argv0, buf, MAXPATHL, TRUE) != FAIL)
+	    val = buf;
 # endif
-	    && vim_FullName(argv0, buf, MAXPATHL, TRUE) != FAIL)
-	val = buf;
+    }
     set_vim_var_string(VV_PROGPATH, val, -1);
+#ifdef WIN32
+    vim_free(path);
+#endif
 }
 
 #endif /* NO_VIM_MAIN */
diff --git a/src/os_unix.c b/src/os_unix.c
index 6220044..6c5955b 100644
--- a/src/os_unix.c
+++ b/src/os_unix.c
@@ -3103,7 +3103,7 @@
 	{
 	    if (path != NULL)
 	    {
-		if (name[0] == '.')
+		if (name[0] != '/')
 		    *path = FullName_save(name, TRUE);
 		else
 		    *path = vim_strsave(name);
@@ -3142,7 +3142,7 @@
 	{
 	    if (path != NULL)
 	    {
-		if (buf[0] == '.')
+		if (buf[0] != '/')
 		    *path = FullName_save(buf, TRUE);
 		else
 		    *path = vim_strsave(buf);
diff --git a/src/os_win32.c b/src/os_win32.c
index a304842..dd3e6de 100644
--- a/src/os_win32.c
+++ b/src/os_win32.c
@@ -1902,17 +1902,31 @@
 #endif
 
 /*
- * Return TRUE if "name" is in $PATH.
+ * If "use_path" is TRUE: Return TRUE if "name" is in $PATH.
+ * If "use_path" is FALSE: Return TRUE if "name" exists.
+ * When returning TRUE and "path" is not NULL save the path and set "*path" to
+ * the allocated memory.
  * TODO: Should somehow check if it's really executable.
  */
     static int
-executable_exists(char *name, char_u **path)
+executable_exists(char *name, char_u **path, int use_path)
 {
     char	*dum;
     char	fname[_MAX_PATH];
     char	*curpath, *newpath;
     long	n;
 
+    if (!use_path)
+    {
+	if (mch_getperm(name) != -1 && !mch_isdir(name))
+	{
+	    if (path != NULL)
+		*path = vim_strsave((char_u *)name);
+	    return TRUE;
+	}
+	return FALSE;
+    }
+
 #ifdef FEAT_MBYTE
     if (enc_codepage >= 0 && (int)GetACP() != enc_codepage)
     {
@@ -2038,7 +2052,7 @@
 	    vimrun_path = (char *)vim_strsave(vimrun_location);
 	    s_dont_use_vimrun = FALSE;
 	}
-	else if (executable_exists("vimrun.exe", NULL))
+	else if (executable_exists("vimrun.exe", NULL, TRUE))
 	    s_dont_use_vimrun = FALSE;
 
 	/* Don't give the warning for a missing vimrun.exe right now, but only
@@ -2052,7 +2066,7 @@
      * If "finstr.exe" doesn't exist, use "grep -n" for 'grepprg'.
      * Otherwise the default "findstr /n" is used.
      */
-    if (!executable_exists("findstr.exe", NULL))
+    if (!executable_exists("findstr.exe", NULL, TRUE))
 	set_option_value((char_u *)"grepprg", 0, (char_u *)"grep -n", 0);
 
 #ifdef FEAT_CLIPBOARD
@@ -3358,9 +3372,10 @@
 }
 
 /*
- * Return 1 if "name" can be executed, 0 if not.
+ * Return TRUE if "name" can be executed, FALSE if not.
  * If "use_path" is FALSE only check if "name" is executable.
- * Return -1 if unknown.
+ * When returning TRUE and "path" is not NULL save the path and set "*path" to
+ * the allocated memory.
  */
     int
 mch_can_exe(char_u *name, char_u **path, int use_path)
@@ -3371,17 +3386,12 @@
 
     if (len >= _MAX_PATH)	/* safety check */
 	return FALSE;
-    if (!use_path)
-    {
-	/* TODO: check if file is really executable. */
-	return mch_getperm(name) != -1 && !mch_isdir(name);
-    }
 
     /* If there already is an extension try using the name directly.  Also do
      * this with a Unix-shell like 'shell'. */
     if (vim_strchr(gettail(name), '.') != NULL
 			       || strstr((char *)gettail(p_sh), "sh") != NULL)
-	if (executable_exists((char *)name, path))
+	if (executable_exists((char *)name, path, use_path))
 	    return TRUE;
 
     /*
@@ -3403,7 +3413,7 @@
 	}
 	else
 	    copy_option_part(&p, buf + len, _MAX_PATH - len, ";");
-	if (executable_exists((char *)buf, path))
+	if (executable_exists((char *)buf, path, use_path))
 	    return TRUE;
     }
     return FALSE;
diff --git a/src/version.c b/src/version.c
index dbb7df2..98fa4a8 100644
--- a/src/version.c
+++ b/src/version.c
@@ -765,6 +765,8 @@
 static int included_patches[] =
 {   /* Add new patch number below this line */
 /**/
+    416,
+/**/
     415,
 /**/
     414,