src/HOL/BNF/BNF_FP_Basic.thy
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(*  Title:      HOL/BNF/BNF_FP_Basic.thy
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    Author:     Dmitriy Traytel, TU Muenchen
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    Author:     Jasmin Blanchette, TU Muenchen
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    Copyright   2012
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Basic fixed point operations on bounded natural functors.
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*)
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header {* Basic Fixed Point Operations on Bounded Natural Functors *}
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theory BNF_FP_Basic
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imports BNF_Comp BNF_Ctr_Sugar
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keywords
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  "defaults"
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begin
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lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
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by auto
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lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
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by blast
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lemma unit_case_Unity: "(case u of () => f) = f"
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by (cases u) (hypsubst, rule unit.cases)
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lemma prod_case_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
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by simp
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lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
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by simp
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lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
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by clarify
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lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
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by auto
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lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
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unfolding o_def fun_eq_iff by simp
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lemma o_bij:
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  assumes gf: "g \<circ> f = id" and fg: "f \<circ> g = id"
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  shows "bij f"
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unfolding bij_def inj_on_def surj_def proof safe
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  fix a1 a2 assume "f a1 = f a2"
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  hence "g ( f a1) = g (f a2)" by simp
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  thus "a1 = a2" using gf unfolding fun_eq_iff by simp
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next
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  fix b
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  have "b = f (g b)"
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  using fg unfolding fun_eq_iff by simp
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  thus "EX a. b = f a" by blast
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qed
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lemma ssubst_mem: "\<lbrakk>t = s; s \<in> X\<rbrakk> \<Longrightarrow> t \<in> X" by simp
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lemma sum_case_step:
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"sum_case (sum_case f' g') g (Inl p) = sum_case f' g' p"
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"sum_case f (sum_case f' g') (Inr p) = sum_case f' g' p"
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by auto
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lemma one_pointE: "\<lbrakk>\<And>x. s = x \<Longrightarrow> P\<rbrakk> \<Longrightarrow> P"
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by simp
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lemma obj_one_pointE: "\<forall>x. s = x \<longrightarrow> P \<Longrightarrow> P"
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by blast
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lemma obj_sumE_f:
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"\<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"
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by (rule allI) (metis sumE)
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lemma obj_sumE: "\<lbrakk>\<forall>x. s = Inl x \<longrightarrow> P; \<forall>x. s = Inr x \<longrightarrow> P\<rbrakk> \<Longrightarrow> P"
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by (cases s) auto
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lemma sum_case_if:
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"sum_case f g (if p then Inl x else Inr y) = (if p then f x else g y)"
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by simp
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lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"
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by blast
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lemma UN_compreh_eq_eq:
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"\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
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"\<Union>{y. \<exists>x\<in>A. y = {x}} = A"
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by blast+
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lemma Inl_Inr_False: "(Inl x = Inr y) = False"
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by simp
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lemma prod_set_simps:
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"fsts (x, y) = {x}"
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"snds (x, y) = {y}"
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lemma sum_set_simps:
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"setl (Inl x) = {x}"
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"setl (Inr x) = {}"
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"setr (Inl x) = {}"
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"setr (Inr x) = {x}"
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unfolding sum_set_defs by simp+
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lemma prod_rel_simp:
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"prod_rel P Q (x, y) (x', y') \<longleftrightarrow> P x x' \<and> Q y y'"
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unfolding prod_rel_def by simp
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lemma sum_rel_simps:
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"sum_rel P Q (Inl x) (Inl x') \<longleftrightarrow> P x x'"
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"sum_rel P Q (Inr y) (Inr y') \<longleftrightarrow> Q y y'"
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"sum_rel P Q (Inl x) (Inr y') \<longleftrightarrow> False"
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"sum_rel P Q (Inr y) (Inl x') \<longleftrightarrow> False"
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unfolding sum_rel_def by simp+
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lemma spec2: "\<forall>x y. P x y \<Longrightarrow> P x y"
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by blast
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ML_file "Tools/bnf_fp_util.ML"
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ML_file "Tools/bnf_fp_def_sugar_tactics.ML"
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ML_file "Tools/bnf_fp_def_sugar.ML"
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end