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diff --git a/lib/isabelle/State_monad_extras.thy b/lib/isabelle/State_monad_extras.thy
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+theory State_monad_extras
+  imports State_monad
+begin
+
+lemma bind_ext_cong[fundef_cong]:
+  assumes m: "m1 s = m2 s"
+    and f: "\<And>a s'. (Value a, s') \<in> set (m2 s) \<Longrightarrow> f1 a s' = f2 a s'"
+  shows "(bind m1 f1) s = (bind m2 f2) s"
+proof -
+  have "List.concat (map (\<lambda>x. case x of (Value a, s') \<Rightarrow> f1 a s' | (Exception0 e, s') \<Rightarrow> [(Exception0 e, s')]) (m2 s)) =
+        List.concat (map (\<lambda>x. case x of (Value a, s') \<Rightarrow> f2 a s' | (Exception0 e, s') \<Rightarrow> [(Exception0 e, s')]) (m2 s))"
+    using f by (intro arg_cong[where f = List.concat]) (auto intro: map_ext split: result.splits)
+  then show ?thesis using m by (auto simp: bind_def)
+qed
+
+lemma bind_cong[fundef_cong]:
+  assumes m: "m1 = m2"
+    and f: "\<And>s a s'. (Value a, s') \<in> set (m2 s) \<Longrightarrow> f1 a s' = f2 a s'"
+  shows "bind m1 f1 = bind m2 f2"
+  using assms by (blast intro: bind_ext_cong)
+
+lemma bind_return[simp]: "bind (return x) m = m x"
+  by (auto simp add: bind_def return_def)
+
+end