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-chapter \<open>Maybe type\<close>
-
-theory Maybe
-imports Prelude
-
-begin
-
-text \<open>Defined as a sum.\<close>
-
-definition "Maybe A \<equiv> A \<or> \<top>"
-definition "none A \<equiv> inr A \<top> tt"
-definition "some A a \<equiv> inl A \<top> a"
-
-lemma
- MaybeF: "A: U i \<Longrightarrow> Maybe A: U i" and
- Maybe_none: "A: U i \<Longrightarrow> none A: Maybe A" and
- Maybe_some: "a: A \<Longrightarrow> some A a: Maybe A"
- unfolding Maybe_def none_def some_def by typechk+
-
-Definition MaybeInd:
- assumes
- "A: U i"
- "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
- "c\<^sub>0: C (none A)"
- "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
- "m: Maybe A"
- shows "C m"
- using assms[unfolded Maybe_def none_def some_def, type]
- apply (elim m)
- apply fact
- apply (elim, fact)
- done
-
-Lemma Maybe_comp_none:
- assumes
- "A: U i"
- "c\<^sub>0: C (none A)"
- "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
- "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
- shows "MaybeInd A C c\<^sub>0 f (none A) \<equiv> c\<^sub>0"
- using assms
- unfolding Maybe_def MaybeInd_def none_def some_def
- by compute
-
-Lemma Maybe_comp_some:
- assumes
- "A: U i"
- "a: A"
- "c\<^sub>0: C (none A)"
- "\<And>a. a: A \<Longrightarrow> f a: C (some A a)"
- "\<And>m. m: Maybe A \<Longrightarrow> C m: U i"
- shows "MaybeInd A C c\<^sub>0 f (some A a) \<equiv> f a"
- using assms
- unfolding Maybe_def MaybeInd_def none_def some_def
- by compute
-
-lemmas
- [form] = MaybeF and
- [intr, intro] = Maybe_none Maybe_some and
- [comp] = Maybe_comp_none Maybe_comp_some and
- MaybeE [elim "?m"] = MaybeInd[rotated 4]
-lemmas
- Maybe_cases [cases] = MaybeE
-
-abbreviation "MaybeRec A C \<equiv> MaybeInd A (K C)"
-
-definition none_i ("none") where [implicit]: "none \<equiv> Maybe.none {}"
-definition some_i ("some") where [implicit]: "some a \<equiv> Maybe.some {} a"
-
-translations
- "none" \<leftharpoondown> "CONST Maybe.none A"
- "some a" \<leftharpoondown> "CONST Maybe.some A a"
-
-
-end