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diff --git a/spartan/lib/Maybe.thy b/spartan/lib/Maybe.thy
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+chapter \<open>Maybe type\<close>
+
+theory Maybe
+imports More_Types
+
+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+
+
+Lemma (derive) 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 "?MaybeInd A (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) m: C m"
+  supply assms[unfolded Maybe_def none_def some_def]
+  apply (elim m)
+  \<guillemotright> unfolding Maybe_def .
+  \<guillemotright> by (rule \<open>_ \<Longrightarrow> _: C (inl _ _ _)\<close>)
+  \<guillemotright> by elim (rule \<open>_: C (inr _ _ _)\<close>)
+  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 (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (none A) \<equiv> c\<^sub>0"
+  supply assms[unfolded Maybe_def some_def none_def]
+  unfolding MaybeInd_def none_def by reduce
+
+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 (\<lambda>m. C m) c\<^sub>0 (\<lambda>a. f a) (some A a) \<equiv> f a"
+  supply assms[unfolded Maybe_def some_def none_def]
+  unfolding MaybeInd_def some_def by (reduce add: Maybe_def)
+
+lemmas
+  [intros] = MaybeF Maybe_none Maybe_some and
+  [comps] = Maybe_comp_none Maybe_comp_some and
+  MaybeE [elims "?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