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Triple Rev

Is there a function rev of type "nat list => nat list" that applied thrice on a list of nats returns the original list again.

i.e. find a function rev and prove the following lemma:

rev (rev (rev xs)) = xs

Making sure you do not use the identity function, if your input is longer than 1.

length xs >= 2 ==> rev xs ~= xs

Resources

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Definitions File

theory Defs
  imports Main
begin  
  
end

Template File

theory Submission
  imports Defs
begin

lemma showmewhatyougot: "\<exists>f::(nat list \<Rightarrow> nat list). (\<forall> xs. (length xs \<ge> 2 \<longrightarrow> f xs \<noteq> xs))
           \<and> (\<forall> xs. (f (f (f xs))) = xs)"
sorry


end

Check File

theory Check
  imports Submission
begin

lemma  "\<exists>f::(nat list \<Rightarrow> nat list). (\<forall> xs. (length xs \<ge> 2 \<longrightarrow> f xs \<noteq> xs))
           \<and> (\<forall> xs. (f (f (f xs))) = xs)"
by (rule Submission.showmewhatyougot)


end

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