The sensible walker has 3 uphill gears: slowly, very slowly and admiring the view!

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Yes.  A dog is regarded in law as a "normal accompaniment" (like a camera).  The law does not actually specify whether it should be on a lead, or not, but merely that it should be under control.  If you are near livestock, you should always keep it on the lead.  If a landowner sees your dog worrying livestock, or if he thinks it may be about to, he is legally entitled to shoot it.