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Continuous Confusion

Carl is constantly confused about the continuity of some of his functions. He builds a large theory upon some tiny assumptions. But in the end he is not really sure whether they are actually true.

After continuously working on his theories for some hours he finally is at peace when he can reduce his unknowns to only one pending assumption, and has a beautiful and peaceful sleep

Can you help him to show that his assumptions are correct, or doom him by showing that they are all wrong?

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Definitions File

theory Defs
  imports "HOL-Library.Extended_Nat"
begin 

end

Template File

theory Submission
imports Defs
begin


lemma enat_add_cont1: 
   "(SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<ge> (SUP b:B. f b + g b)"  
  apply (rule Sup_least)
  sorry

lemma enat_add_cont1_not: 
  shows "~(\<forall>f g B. (SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<ge> (SUP b:B. f b + g b))" 
  sorry

lemma enat_add_cont2: 
    "(SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<le> (SUP b:B. f b + g b)"  
  sorry


lemma enat_add_cont2_not: 
  shows "~(\<forall>f g B. (SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<le> (SUP b:B. f b + g b))" 
  sorry

lemma enat_add_cont:
    "(SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) = (SUP b:B. f b + g b)"  
  sorry


lemma enat_add_cont_not: 
  shows "~(\<forall>f g B. (SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) = (SUP b:B. f b + g b))" 
  sorry 

end

Check File

theory Check
imports  Submission
begin

lemma A:
  shows "(SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<ge> (SUP b:B. f b + g b)"  
  apply(fact enat_add_cont1) done

(* or *)

lemma A:
  shows "~(\<forall>f g B. (SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<ge> (SUP b:B. f b + g b))" 
  apply(fact enat_add_cont1_not) done



 
lemma B:
    "(SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<le> (SUP b:B. f b + g b)"  
  apply(fact enat_add_cont2) done

(* or *)

lemma B: "~(\<forall>f g B. (SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) \<le> (SUP b:B. f b + g b))" 
  apply(fact enat_add_cont2_not) done


lemma C:
    "(SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) = (SUP b:B. f b + g b)"  
  apply(fact enat_add_cont) done 

(* or *)

lemma C: "~(\<forall>f g B. (SUP b:B. (f::(enat\<Rightarrow>enat)) b) + (SUP b:B. g b) = (SUP b:B. f b + g b))" 
  apply(fact enat_add_cont_not) done 

end

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