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# Complete Lattices

This is the task corresponding to exercise 5. Complete Lattices.

## Resources

### Definitions File

```theory Defs
imports Main
begin

definition lub :: "'a set set \<Rightarrow> 'a set set \<Rightarrow> 'a set \<Rightarrow> bool" where
"lub L M X = ((\<forall>Y \<in> M. Y \<subseteq> X) \<and> (\<forall>X' \<in> L. (\<forall>Y \<in> M. Y \<subseteq> X') \<longrightarrow> X \<subseteq> X'))"

definition glb :: "'a set set \<Rightarrow> 'a set set \<Rightarrow> 'a set \<Rightarrow> bool" where
"glb L M X = ((\<forall>Y \<in> M. X \<subseteq> Y) \<and> (\<forall>X' \<in> L. (\<forall>Y \<in> M. X' \<subseteq> Y) \<longrightarrow> X' \<subseteq> X))"

definition cl :: "'a set set \<Rightarrow> bool" where
"cl L = (\<forall>M \<subseteq> L. \<exists>X \<in> L. glb L M X)"

end
```

### Template File

```theory Submission
imports Defs
begin

theorem cl_lub
assumes "cl L"
and "M \<subseteq> L"
shows "\<exists>X. lub L M X"
sorry

end
```

### Check File

```theory Check
imports Submission
begin

lemma "\<lbrakk>cl L; M \<subseteq> L\<rbrakk> \<Longrightarrow>\<exists>X. lub L M X"
by (rule Submission.cl_lub)

end
```

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