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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?

Resources

Definitions File

theory Defs
imports "HOL-Library.Extended_Nat"
begin

end

Template File

theory Submission
imports Defs
begin

"(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

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

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

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

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

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)"

(* 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))"

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

(* 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))"

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