Re: [isabelle] Proving nth on a sorted list monotone



HOL is a logic of total functions. nth xs i is defined, but unless i <
length xs, you have no idea what the value is. Hence you can only prove
monotonicity with that precondition.

Best
Tobias

Sigurd Torkel Meldgaard schrieb:
> Hi Isabelle-users
> 
> I am stumblingly trying to prove that nth is monotone on a sorted
> list, I got this far:
> 
> lemma sorted_mono[simp]:
>   assumes
>   "sorted (xs::int list)"
>   shows "xs ≠ [] ⟶ mono (λ a . xs ! a)"
> proof (induct xs rule: sorted.induct)
>   case 1 show ?case by simp
>   next
>     case (2 element) show ?case
>     proof (simp add: mono_def)
> 
> And here I am stuck, do I have to explicitly assume that the indices
> are within the length of xs, or how does a proof with a partial
> function work?
> 
> All the best
> 
> /Sigurd






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