src/HOL/Codatatype/BNF_FP.thy

--- a/src/HOL/Codatatype/BNF_FP.thy	Wed Sep 12 06:27:36 2012 +0200
+++ b/src/HOL/Codatatype/BNF_FP.thy	Wed Sep 12 06:27:48 2012 +0200
@@ -14,6 +14,87 @@
   "defaults"
 begin
 
+lemma case_unit: "(case u of () => f) = f"
+by (cases u) (hypsubst, rule unit.cases)
+
+(* FIXME: needed? *)
+lemma all_unit_eq: "(\<And>x. PROP P x) \<equiv> PROP P ()" by simp
+
+(* FIXME: needed? *)
+lemma all_prod_eq: "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))" by clarsimp
+
+(* FIXME: needed? *)
+lemma False_imp_eq: "(False \<Longrightarrow> P) \<equiv> Trueprop True"
+by presburger
+
+(* FIXME: needed? *)
+lemma all_point_1: "(\<And>z. z = b \<Longrightarrow> phi z) \<equiv> Trueprop (phi b)"
+by presburger
+
+lemma rev_bspec: "a \<in> A \<Longrightarrow> \<forall>z \<in> A. P z \<Longrightarrow> P a"
+by simp
+
+lemma Un_cong: "\<lbrakk>A = B; C = D\<rbrakk> \<Longrightarrow> A \<union> C = B \<union> D"
+by simp
+
+definition convol ("<_ , _>") where
+"<f , g> \<equiv> %a. (f a, g a)"
+
+lemma fst_convol:
+"fst o <f , g> = f"
+apply(rule ext)
+unfolding convol_def by simp
+
+lemma snd_convol:
+"snd o <f , g> = g"
+apply(rule ext)
+unfolding convol_def by simp
+
+lemma pointfree_idE: "f o g = id \<Longrightarrow> f (g x) = x"
+unfolding o_def fun_eq_iff by simp
+
+lemma o_bij:
+  assumes gf: "g o f = id" and fg: "f o g = id"
+  shows "bij f"
+unfolding bij_def inj_on_def surj_def proof safe
+  fix a1 a2 assume "f a1 = f a2"
+  hence "g ( f a1) = g (f a2)" by simp
+  thus "a1 = a2" using gf unfolding fun_eq_iff by simp
+next
+  fix b
+  have "b = f (g b)"
+  using fg unfolding fun_eq_iff by simp
+  thus "EX a. b = f a" by blast
+qed
+
+lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
+
+lemma sum_case_step:
+  "sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
+  "sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
+by auto
+
+lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+by simp
+
+lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
+by blast
+
+lemma obj_sumE_f:
+"\<lbrakk>\<forall>x. s = f (Inl x) \<longrightarrow> P; \<forall>x. s = f (Inr x) \<longrightarrow> P\<rbrakk> \<Longrightarrow> \<forall>x. s = f x \<longrightarrow> P"
+by (metis sum.exhaust)
+
+lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
+by (cases s) auto
+
+lemma obj_sum_step:
+  "\<lbrakk>\<forall>x. s = f (Inr (Inl x)) \<longrightarrow> P; \<forall>x. s = f (Inr (Inr x)) \<longrightarrow> P\<rbrakk> \<Longrightarrow> \<forall>x. s = f (Inr x) \<longrightarrow> P"
+by (metis obj_sumE)
+
+lemma sum_case_if:
+"sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"
+by simp
+
 ML_file "Tools/bnf_fp_util.ML"
 ML_file "Tools/bnf_fp_sugar_tactics.ML"
 ML_file "Tools/bnf_fp_sugar.ML"