(* Title: HOL/BNF/BNF_FP.thy
Author: Dmitriy Traytel, TU Muenchen
Author: Jasmin Blanchette, TU Muenchen
Copyright 2012
Basic fixed point operations on bounded natural functors.
*)
header {* Basic Fixed Point Operations on Bounded Natural Functors *}
theory BNF_FP
imports BNF_Comp BNF_Wrap
keywords
"defaults"
begin
lemma mp_conj: "(P \<longrightarrow> Q) \<and> R \<Longrightarrow> P \<Longrightarrow> R \<and> Q"
by auto
lemma eq_sym_Unity_conv: "(x = (() = ())) = x"
by blast
lemma unit_case_Unity: "(case u of () => f) = f"
by (cases u) (hypsubst, rule unit.cases)
lemma prod_case_Pair_iden: "(case p of (x, y) \<Rightarrow> (x, y)) = p"
by simp
lemma unit_all_impI: "(P () \<Longrightarrow> Q ()) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
by simp
lemma prod_all_impI: "(\<And>x y. P (x, y) \<Longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
by clarify
lemma prod_all_impI_step: "(\<And>x. \<forall>y. P (x, y) \<longrightarrow> Q (x, y)) \<Longrightarrow> \<forall>x. P x \<longrightarrow> Q x"
by auto
lemma all_unit_eq: "(\<And>x. PROP P x) \<equiv> PROP P ()"
by simp
lemma all_prod_eq: "(\<And>x. PROP P x) \<equiv> (\<And>a b. PROP P (a, b))"
by clarsimp
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
lemma pointfree_idE: "f \<circ> g = id \<Longrightarrow> f (g x) = x"
unfolding o_def fun_eq_iff by simp
lemma o_bij:
assumes gf: "g \<circ> f = id" and fg: "f \<circ> 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> s = f x \<longrightarrow> P"
by (cases x) 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 (rule allI) (rule obj_sumE_f')
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> s = f (Inr x) \<longrightarrow> P"
by (cases x) blast+
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 (rule allI) (rule obj_sum_step')
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
lemma sum_case_o_inj:
"sum_case f g \<circ> Inl = f"
"sum_case f g \<circ> Inr = g"
by auto
lemma ident_o_ident: "(\<lambda>x. x) \<circ> (\<lambda>x. x) = (\<lambda>x. x)"
by (rule o_def)
lemma mem_UN_compreh_eq: "(z : \<Union>{y. \<exists>x\<in>A. y = F x}) = (\<exists>x\<in>A. z : F x)"
by blast
lemma UN_compreh_eq_eq:
"\<Union>{y. \<exists>x\<in>A. y = {}} = {}"
"\<Union>{y. \<exists>x\<in>A. y = {x}} = A"
by blast+
lemma prod_set_simps:
"fsts (x, y) = {x}"
"snds (x, y) = {y}"
unfolding fsts_def snds_def by simp+
lemma sum_set_simps:
"setl (Inl x) = {x}"
"setl (Inr x) = {}"
"setr (Inl x) = {}"
"setr (Inr x) = {x}"
unfolding sum_set_defs by simp+
lemma prod_rel_simp:
"prod_rel P Q (x, y) (x', y') \<longleftrightarrow> P x x' \<and> Q y y'"
unfolding prod_rel_def by simp
lemma sum_rel_simps:
"sum_rel P Q (Inl x) (Inl x') \<longleftrightarrow> P x x'"
"sum_rel P Q (Inr y) (Inr y') \<longleftrightarrow> Q y y'"
"sum_rel P Q (Inl x) (Inr y') \<longleftrightarrow> False"
"sum_rel P Q (Inr y) (Inl x') \<longleftrightarrow> False"
unfolding sum_rel_def by simp+
ML_file "Tools/bnf_fp.ML"
ML_file "Tools/bnf_fp_def_sugar_tactics.ML"
ML_file "Tools/bnf_fp_def_sugar.ML"
end