I agree Our site saves small pieces of text information (cookies) on your device in order to deliver better content and for statistical purposes. You can disable the usage of cookies by changing the settings of your browser. By browsing our website without changing the browser settings you grant us permission to store that information on your device.
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?
theory Defs imports "HOL-Library.Extended_Nat" begin end
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
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
theory Defs imports "HOL-Library.Extended_Nat" begin end
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
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
theory Defs imports "HOL-Library.Extended_Nat" begin end
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
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