Nominal/Nominal2_Base.thy
author Christian Urban <urbanc@in.tum.de>
Wed, 04 Apr 2012 06:19:38 +0100
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updated to Isabelle version April 1
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(*  Title:      Nominal2_Base
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    Authors:    Christian Urban, Brian Huffman, Cezary Kaliszyk
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    Basic definitions and lemma infrastructure for 
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    Nominal Isabelle. 
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*)
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theory Nominal2_Base
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imports Main 
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        "~~/src/HOL/Library/Infinite_Set"
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        "~~/src/HOL/Quotient_Examples/FSet"
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keywords
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  "atom_decl" "equivariance" :: thy_decl 
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uses ("nominal_basics.ML")
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     ("nominal_thmdecls.ML")
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     ("nominal_permeq.ML")
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     ("nominal_library.ML")
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     ("nominal_atoms.ML")
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     ("nominal_eqvt.ML")
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begin
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section {* Atoms and Sorts *}
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text {* A simple implementation for atom_sorts is strings. *}
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(* types atom_sort = string *)
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text {* To deal with Church-like binding we use trees of  
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  strings as sorts. *}
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datatype atom_sort = Sort "string" "atom_sort list"
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datatype atom = Atom atom_sort nat
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text {* Basic projection function. *}
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primrec
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  sort_of :: "atom \<Rightarrow> atom_sort"
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where
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  "sort_of (Atom s n) = s"
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primrec
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  nat_of :: "atom \<Rightarrow> nat"
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where
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  "nat_of (Atom s n) = n"
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text {* There are infinitely many atoms of each sort. *}
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lemma INFM_sort_of_eq: 
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  shows "INFM a. sort_of a = s"
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proof -
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  have "INFM i. sort_of (Atom s i) = s" by simp
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  moreover have "inj (Atom s)" by (simp add: inj_on_def)
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  ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
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qed
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lemma infinite_sort_of_eq:
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  shows "infinite {a. sort_of a = s}"
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  using INFM_sort_of_eq unfolding INFM_iff_infinite .
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lemma atom_infinite [simp]: 
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  shows "infinite (UNIV :: atom set)"
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  using subset_UNIV infinite_sort_of_eq
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  by (rule infinite_super)
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lemma obtain_atom:
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  fixes X :: "atom set"
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  assumes X: "finite X"
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  obtains a where "a \<notin> X" "sort_of a = s"
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proof -
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  from X have "MOST a. a \<notin> X"
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    unfolding MOST_iff_cofinite by simp
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  with INFM_sort_of_eq
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  have "INFM a. sort_of a = s \<and> a \<notin> X"
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    by (rule INFM_conjI)
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  then obtain a where "a \<notin> X" "sort_of a = s"
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    by (auto elim: INFM_E)
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  then show ?thesis ..
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qed
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lemma atom_components_eq_iff:
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  fixes a b :: atom
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  shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
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  by (induct a, induct b, simp)
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section {* Sort-Respecting Permutations *}
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typedef perm =
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  "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
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proof
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  show "id \<in> ?perm" by simp
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qed
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lemma permI:
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  assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a"
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  shows "f \<in> perm"
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  using assms unfolding perm_def MOST_iff_cofinite by simp
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lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f"
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  unfolding perm_def by simp
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lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}"
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  unfolding perm_def by simp
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lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a"
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  unfolding perm_def by simp
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lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x"
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  unfolding perm_def MOST_iff_cofinite by simp
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lemma perm_id: "id \<in> perm"
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  unfolding perm_def by simp
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lemma perm_comp:
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  assumes f: "f \<in> perm" and g: "g \<in> perm"
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  shows "(f \<circ> g) \<in> perm"
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apply (rule permI)
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apply (rule bij_comp)
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apply (rule perm_is_bij [OF g])
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apply (rule perm_is_bij [OF f])
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apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
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apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
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apply (simp)
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apply (simp add: perm_is_sort_respecting [OF f])
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apply (simp add: perm_is_sort_respecting [OF g])
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done
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lemma perm_inv:
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  assumes f: "f \<in> perm"
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  shows "(inv f) \<in> perm"
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apply (rule permI)
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apply (rule bij_imp_bij_inv)
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apply (rule perm_is_bij [OF f])
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apply (rule MOST_mono [OF perm_MOST [OF f]])
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apply (erule subst, rule inv_f_f)
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apply (rule bij_is_inj [OF perm_is_bij [OF f]])
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apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
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apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
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done
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lemma bij_Rep_perm: "bij (Rep_perm p)"
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  using Rep_perm [of p] unfolding perm_def by simp
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lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}"
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  using Rep_perm [of p] unfolding perm_def by simp
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lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
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  using Rep_perm [of p] unfolding perm_def by simp
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lemma Rep_perm_ext:
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  "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2"
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  by (simp add: fun_eq_iff Rep_perm_inject [symmetric])
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instance perm :: size ..
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subsection {* Permutations form a (multiplicative) group *}
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instantiation perm :: group_add
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begin
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definition
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  "0 = Abs_perm id"
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definition
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  "- p = Abs_perm (inv (Rep_perm p))"
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definition
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  "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
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definition
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  "(p1::perm) - p2 = p1 + - p2"
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lemma Rep_perm_0: "Rep_perm 0 = id"
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  unfolding zero_perm_def
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  by (simp add: Abs_perm_inverse perm_id)
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lemma Rep_perm_add:
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  "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
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  unfolding plus_perm_def
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  by (simp add: Abs_perm_inverse perm_comp Rep_perm)
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lemma Rep_perm_uminus:
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  "Rep_perm (- p) = inv (Rep_perm p)"
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  unfolding uminus_perm_def
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  by (simp add: Abs_perm_inverse perm_inv Rep_perm)
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instance
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apply default
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unfolding Rep_perm_inject [symmetric]
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unfolding minus_perm_def
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unfolding Rep_perm_add
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unfolding Rep_perm_uminus
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unfolding Rep_perm_0
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by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
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end
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section {* Implementation of swappings *}
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definition
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  swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
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where
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  "(a \<rightleftharpoons> b) =
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    Abs_perm (if sort_of a = sort_of b 
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              then (\<lambda>c. if a = c then b else if b = c then a else c) 
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              else id)"
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lemma Rep_perm_swap:
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  "Rep_perm (a \<rightleftharpoons> b) =
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    (if sort_of a = sort_of b 
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     then (\<lambda>c. if a = c then b else if b = c then a else c)
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     else id)"
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unfolding swap_def
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apply (rule Abs_perm_inverse)
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apply (rule permI)
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apply (auto simp add: bij_def inj_on_def surj_def)[1]
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apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
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apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
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apply (simp)
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apply (simp)
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done
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lemmas Rep_perm_simps =
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  Rep_perm_0
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  Rep_perm_add
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  Rep_perm_uminus
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  Rep_perm_swap
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lemma swap_different_sorts [simp]:
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  "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
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  by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
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lemma swap_cancel:
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  shows "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
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  and   "(a \<rightleftharpoons> b) + (b \<rightleftharpoons> a) = 0"
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  by (rule_tac [!] Rep_perm_ext) 
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     (simp_all add: Rep_perm_simps fun_eq_iff)
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lemma swap_self [simp]:
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  "(a \<rightleftharpoons> a) = 0"
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  by (rule Rep_perm_ext, simp add: Rep_perm_simps fun_eq_iff)
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lemma minus_swap [simp]:
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  "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
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  by (rule minus_unique [OF swap_cancel(1)])
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lemma swap_commute:
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  "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
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  by (rule Rep_perm_ext)
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     (simp add: Rep_perm_swap fun_eq_iff)
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lemma swap_triple:
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  assumes "a \<noteq> b" and "c \<noteq> b"
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  assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
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  shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
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  using assms
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  by (rule_tac Rep_perm_ext)
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     (auto simp add: Rep_perm_simps fun_eq_iff)
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section {* Permutation Types *}
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text {*
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  Infix syntax for @{text permute} has higher precedence than
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  addition, but lower than unary minus.
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*}
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class pt =
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  fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
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  assumes permute_zero [simp]: "0 \<bullet> x = x"
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  assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
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begin
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lemma permute_diff [simp]:
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  shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
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  unfolding diff_minus by simp
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lemma permute_minus_cancel [simp]:
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  shows "p \<bullet> - p \<bullet> x = x"
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  and   "- p \<bullet> p \<bullet> x = x"
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  unfolding permute_plus [symmetric] by simp_all
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lemma permute_swap_cancel [simp]:
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  shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
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  unfolding permute_plus [symmetric]
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  by (simp add: swap_cancel)
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lemma permute_swap_cancel2 [simp]:
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  shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x"
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  unfolding permute_plus [symmetric]
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  by (simp add: swap_commute)
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lemma inj_permute [simp]: 
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  shows "inj (permute p)"
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  by (rule inj_on_inverseI)
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     (rule permute_minus_cancel)
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lemma surj_permute [simp]: 
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  shows "surj (permute p)"
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  by (rule surjI, rule permute_minus_cancel)
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lemma bij_permute [simp]: 
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  shows "bij (permute p)"
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  by (rule bijI [OF inj_permute surj_permute])
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lemma inv_permute: 
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  shows "inv (permute p) = permute (- p)"
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  by (rule inv_equality) (simp_all)
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lemma permute_minus: 
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  shows "permute (- p) = inv (permute p)"
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  by (simp add: inv_permute)
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lemma permute_eq_iff [simp]: 
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  shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y"
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  by (rule inj_permute [THEN inj_eq])
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end
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subsection {* Permutations for atoms *}
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instantiation atom :: pt
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begin
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   326
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definition
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  "p \<bullet> a = (Rep_perm p) a"
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instance 
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apply(default)
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apply(simp_all add: permute_atom_def Rep_perm_simps)
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done
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end
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lemma sort_of_permute [simp]:
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  shows "sort_of (p \<bullet> a) = sort_of a"
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  unfolding permute_atom_def by (rule sort_of_Rep_perm)
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lemma swap_atom:
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  shows "(a \<rightleftharpoons> b) \<bullet> c =
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           (if sort_of a = sort_of b
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            then (if c = a then b else if c = b then a else c) else c)"
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  unfolding permute_atom_def
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  by (simp add: Rep_perm_swap)
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lemma swap_atom_simps [simp]:
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  "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
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  "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
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  "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
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  unfolding swap_atom by simp_all
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   353
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lemma perm_eq_iff:
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  fixes p q :: "perm"
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  shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
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  unfolding permute_atom_def
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  by (metis Rep_perm_ext ext)
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   360
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subsection {* Permutations for permutations *}
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instantiation perm :: pt
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   364
begin
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   365
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definition
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  "p \<bullet> q = p + q - p"
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   368
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instance
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apply default
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apply (simp add: permute_perm_def)
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apply (simp add: permute_perm_def diff_minus minus_add add_assoc)
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done
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end
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   376
1879
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   377
lemma permute_self: 
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  shows "p \<bullet> p = p"
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   379
  unfolding permute_perm_def 
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  by (simp add: diff_minus add_assoc)
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   381
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   382
lemma pemute_minus_self:
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   383
  shows "- p \<bullet> p = p"
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   384
  unfolding permute_perm_def 
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   385
  by (simp add: diff_minus add_assoc)
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   386
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   387
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   388
subsection {* Permutations for functions *}
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   389
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   390
instantiation "fun" :: (pt, pt) pt
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   391
begin
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   392
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   393
definition
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   394
  "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
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   395
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   396
instance
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   397
apply default
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   398
apply (simp add: permute_fun_def)
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   399
apply (simp add: permute_fun_def minus_add)
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   400
done
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   401
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   402
end
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   403
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   404
lemma permute_fun_app_eq:
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   405
  shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
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   406
  unfolding permute_fun_def by simp
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   407
2663
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   408
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   409
subsection {* Permutations for booleans *}
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   410
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   411
instantiation bool :: pt
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   412
begin
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   413
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   414
definition "p \<bullet> (b::bool) = b"
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   415
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   416
instance
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   417
apply(default) 
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apply(simp_all add: permute_bool_def)
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   419
done
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   420
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   421
end
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   422
1557
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   423
lemma permute_boolE:
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   424
  fixes P::"bool"
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   425
  shows "p \<bullet> P \<Longrightarrow> P"
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   426
  by (simp add: permute_bool_def)
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   427
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   428
lemma permute_boolI:
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   429
  fixes P::"bool"
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   430
  shows "P \<Longrightarrow> p \<bullet> P"
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   431
  by(simp add: permute_bool_def)
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   432
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   433
subsection {* Permutations for sets *}
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   434
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   435
instantiation "set" :: (pt) pt
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   436
begin
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   437
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   438
definition
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   439
  "p \<bullet> X = {p \<bullet> x | x. x \<in> X}" 
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   440
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   441
instance
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   442
apply default
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   443
apply (auto simp add: permute_set_def)
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   444
done
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   445
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   446
end
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   447
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   448
lemma permute_set_eq:
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   449
  shows "p \<bullet> X = {x. - p \<bullet> x \<in> X}"
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   450
unfolding permute_set_def
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   451
by (auto) (metis permute_minus_cancel(1))
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   452
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   453
lemma permute_set_eq_image:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   454
  shows "p \<bullet> X = permute p ` X"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   455
  unfolding permute_set_def by auto
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   456
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   457
lemma permute_set_eq_vimage:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   458
  shows "p \<bullet> X = permute (- p) -` X"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   459
  unfolding permute_set_eq vimage_def
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   460
  by simp
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   461
  
2588
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
   462
lemma permute_finite [simp]:
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
   463
  shows "finite (p \<bullet> X) = finite X"
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   464
  unfolding permute_set_eq_vimage
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   465
  using bij_permute by (rule finite_vimage_iff)
2588
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
   466
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   467
lemma swap_set_not_in:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   468
  assumes a: "a \<notin> S" "b \<notin> S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   469
  shows "(a \<rightleftharpoons> b) \<bullet> S = S"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   470
  unfolding permute_set_def
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   471
  using a by (auto simp add: swap_atom)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   472
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   473
lemma swap_set_in:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   474
  assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   475
  shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   476
  unfolding permute_set_def
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   477
  using a by (auto simp add: swap_atom)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   478
2669
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   479
lemma swap_set_in_eq:
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   480
  assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   481
  shows "(a \<rightleftharpoons> b) \<bullet> S = (S - {a}) \<union> {b}"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   482
  unfolding permute_set_def
2669
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   483
  using a by (auto simp add: swap_atom)
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   484
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   485
lemma swap_set_both_in:
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   486
  assumes a: "a \<in> S" "b \<in> S"
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   487
  shows "(a \<rightleftharpoons> b) \<bullet> S = S"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   488
  unfolding permute_set_def
2669
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   489
  using a by (auto simp add: swap_atom)
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
   490
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   491
lemma mem_permute_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   492
  shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   493
  unfolding permute_set_def
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   494
  by auto
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   495
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   496
lemma empty_eqvt:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   497
  shows "p \<bullet> {} = {}"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   498
  unfolding permute_set_def
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   499
  by (simp)
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   500
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   501
lemma insert_eqvt:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   502
  shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   503
  unfolding permute_set_eq_image image_insert ..
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
   504
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   505
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   506
subsection {* Permutations for @{typ unit} *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   507
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   508
instantiation unit :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   509
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   510
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   511
definition "p \<bullet> (u::unit) = u"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   512
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   513
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   514
by (default) (simp_all add: permute_unit_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   515
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   516
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   517
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   518
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   519
subsection {* Permutations for products *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   520
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
   521
instantiation prod :: (pt, pt) pt
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   522
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   523
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   524
primrec 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   525
  permute_prod 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   526
where
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   527
  Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   528
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   529
instance
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   530
by default auto
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   531
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   532
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   533
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   534
subsection {* Permutations for sums *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   535
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
   536
instantiation sum :: (pt, pt) pt
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   537
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   538
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   539
primrec 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   540
  permute_sum 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   541
where
2982
4a00077c008f completed the eqvt-proofs for functions; they are stored under the name function_name.eqvt and added to the eqvt-list
Christian Urban <urbanc@in.tum.de>
parents: 2972
diff changeset
   542
  Inl_eqvt: "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
4a00077c008f completed the eqvt-proofs for functions; they are stored under the name function_name.eqvt and added to the eqvt-list
Christian Urban <urbanc@in.tum.de>
parents: 2972
diff changeset
   543
| Inr_eqvt: "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   544
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   545
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   546
by (default) (case_tac [!] x, simp_all)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   547
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   548
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   549
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   550
subsection {* Permutations for @{typ "'a list"} *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   551
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   552
instantiation list :: (pt) pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   553
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   554
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   555
primrec 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   556
  permute_list 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   557
where
2982
4a00077c008f completed the eqvt-proofs for functions; they are stored under the name function_name.eqvt and added to the eqvt-list
Christian Urban <urbanc@in.tum.de>
parents: 2972
diff changeset
   558
  Nil_eqvt:  "p \<bullet> [] = []"
4a00077c008f completed the eqvt-proofs for functions; they are stored under the name function_name.eqvt and added to the eqvt-list
Christian Urban <urbanc@in.tum.de>
parents: 2972
diff changeset
   559
| Cons_eqvt: "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   560
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   561
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   562
by (default) (induct_tac [!] x, simp_all)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   563
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   564
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   565
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   566
lemma set_eqvt:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   567
  shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   568
  by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   569
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   570
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   571
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   572
subsection {* Permutations for @{typ "'a option"} *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   573
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   574
instantiation option :: (pt) pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   575
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   576
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   577
primrec 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   578
  permute_option 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   579
where
2982
4a00077c008f completed the eqvt-proofs for functions; they are stored under the name function_name.eqvt and added to the eqvt-list
Christian Urban <urbanc@in.tum.de>
parents: 2972
diff changeset
   580
  None_eqvt: "p \<bullet> None = None"
4a00077c008f completed the eqvt-proofs for functions; they are stored under the name function_name.eqvt and added to the eqvt-list
Christian Urban <urbanc@in.tum.de>
parents: 2972
diff changeset
   581
| Some_eqvt: "p \<bullet> (Some x) = Some (p \<bullet> x)"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   582
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   583
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   584
by (default) (induct_tac [!] x, simp_all)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   585
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   586
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   587
3121
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   588
subsection {* Permutations for @{typ "'a multiset"} *}
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   589
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   590
instantiation multiset :: (pt) pt
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   591
begin
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   592
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   593
definition
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   594
  "p \<bullet> M = {# p \<bullet> x. x :# M #}"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   595
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   596
instance 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   597
proof
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   598
  fix M :: "'a multiset" and p q :: "perm"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   599
  show "0 \<bullet> M = M" 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   600
    unfolding permute_multiset_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   601
    by (induct_tac M) (simp_all)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   602
  show "(p + q) \<bullet> M = p \<bullet> q \<bullet> M" 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   603
    unfolding permute_multiset_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   604
    by (induct_tac M) (simp_all)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   605
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   606
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   607
end
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   608
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   609
lemma permute_multiset [simp]:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   610
  fixes M N::"('a::pt) multiset"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   611
  shows "(p \<bullet> {#}) = ({#} ::('a::pt) multiset)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   612
  and   "(p \<bullet> {# x #}) = {# p \<bullet> x #}"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   613
  and   "(p \<bullet> (M + N)) = (p \<bullet> M) + (p \<bullet> N)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   614
  unfolding permute_multiset_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   615
  by (simp_all)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   616
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   617
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   618
subsection {* Permutations for @{typ "'a fset"} *}
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   619
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   620
lemma permute_fset_rsp[quot_respect]:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   621
  shows "(op = ===> list_eq ===> list_eq) permute permute"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   622
  unfolding fun_rel_def
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   623
  by (simp add: set_eqvt[symmetric])
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   624
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   625
instantiation fset :: (pt) pt
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   626
begin
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   627
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   628
quotient_definition
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   629
  "permute_fset :: perm \<Rightarrow> 'a fset \<Rightarrow> 'a fset"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   630
is
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   631
  "permute :: perm \<Rightarrow> 'a list \<Rightarrow> 'a list"
3152
da59c94bed7e updated to Isabelle version April 1
Christian Urban <urbanc@in.tum.de>
parents: 3147
diff changeset
   632
  by (simp add: set_eqvt[symmetric])
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   633
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   634
instance 
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   635
proof
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   636
  fix x :: "'a fset" and p q :: "perm"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   637
  show "0 \<bullet> x = x" by (descending) (simp)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   638
  show "(p + q) \<bullet> x = p \<bullet> q \<bullet> x" by (descending) (simp)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   639
qed
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   640
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   641
end
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   642
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   643
lemma permute_fset [simp]:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   644
  fixes S::"('a::pt) fset"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   645
  shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   646
  and   "(p \<bullet> insert_fset x S) = insert_fset (p \<bullet> x) (p \<bullet> S)"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   647
  by (lifting permute_list.simps)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   648
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   649
lemma fset_eqvt: 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   650
  shows "p \<bullet> (fset S) = fset (p \<bullet> S)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   651
  by (lifting set_eqvt)
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   652
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   653
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   654
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   655
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   656
instantiation char :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   657
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   658
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   659
definition "p \<bullet> (c::char) = c"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   660
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   661
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   662
by (default) (simp_all add: permute_char_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   663
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   664
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   665
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   666
instantiation nat :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   667
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   668
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   669
definition "p \<bullet> (n::nat) = n"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   670
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   671
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   672
by (default) (simp_all add: permute_nat_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   673
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   674
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   675
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   676
instantiation int :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   677
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   678
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   679
definition "p \<bullet> (i::int) = i"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   680
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   681
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   682
by (default) (simp_all add: permute_int_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   683
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   684
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   685
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   686
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   687
section {* Pure types *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   688
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   689
text {* Pure types will have always empty support. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   690
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   691
class pure = pt +
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   692
  assumes permute_pure: "p \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   693
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   694
text {* Types @{typ unit} and @{typ bool} are pure. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   695
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   696
instance unit :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   697
proof qed (rule permute_unit_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   698
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   699
instance bool :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   700
proof qed (rule permute_bool_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   701
2635
64b4cb2c2bf8 simple cases for string rule inductions
Christian Urban <urbanc@in.tum.de>
parents: 2632
diff changeset
   702
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   703
text {* Other type constructors preserve purity. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   704
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   705
instance "fun" :: (pure, pure) pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   706
by default (simp add: permute_fun_def permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   707
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   708
instance set :: (pure) pure
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   709
by default (simp add: permute_set_def permute_pure)
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   710
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
   711
instance prod :: (pure, pure) pure
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   712
by default (induct_tac x, simp add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   713
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
   714
instance sum :: (pure, pure) pure
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   715
by default (induct_tac x, simp_all add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   716
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   717
instance list :: (pure) pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   718
by default (induct_tac x, simp_all add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   719
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   720
instance option :: (pure) pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   721
by default (induct_tac x, simp_all add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   722
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   723
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   724
subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   725
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   726
instance char :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   727
proof qed (rule permute_char_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   728
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   729
instance nat :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   730
proof qed (rule permute_nat_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   731
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   732
instance int :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   733
proof qed (rule permute_int_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   734
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   735
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   736
section {* Infrastructure for Equivariance and Perm_simp *}
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   737
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   738
subsection {* Basic functions about permutations *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   739
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   740
use "nominal_basics.ML"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   741
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   742
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   743
subsection {* Eqvt infrastructure *}
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   744
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   745
text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_raw} *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   746
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   747
use "nominal_thmdecls.ML"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   748
setup "Nominal_ThmDecls.setup"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   749
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   750
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   751
lemmas [eqvt] =
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   752
  (* pt types *)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   753
  permute_prod.simps 
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   754
  permute_list.simps 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   755
  permute_option.simps 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   756
  permute_sum.simps
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   757
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   758
  (* sets *)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   759
  empty_eqvt insert_eqvt set_eqvt 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   760
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   761
  (* fsets *)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   762
  permute_fset fset_eqvt
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   763
3121
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   764
  (* multisets *)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   765
  permute_multiset
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   766
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   767
subsection {* perm_simp infrastructure *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   768
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   769
definition
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   770
  "unpermute p = permute (- p)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   771
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   772
lemma eqvt_apply:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   773
  fixes f :: "'a::pt \<Rightarrow> 'b::pt" 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   774
  and x :: "'a::pt"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   775
  shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   776
  unfolding permute_fun_def by simp
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   777
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   778
lemma eqvt_lambda:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   779
  fixes f :: "'a::pt \<Rightarrow> 'b::pt"
2753
445518561867 eqvt_lambda without eta-expansion
Christian Urban <urbanc@in.tum.de>
parents: 2743
diff changeset
   780
  shows "p \<bullet> f \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   781
  unfolding permute_fun_def unpermute_def by simp
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   782
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   783
lemma eqvt_bound:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   784
  shows "p \<bullet> unpermute p x \<equiv> x"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   785
  unfolding unpermute_def by simp
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   786
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   787
text {* provides perm_simp methods *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   788
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   789
use "nominal_permeq.ML"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   790
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   791
method_setup perm_simp =
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   792
 {* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_simp_meth *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   793
 {* pushes permutations inside. *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   794
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   795
method_setup perm_strict_simp =
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   796
 {* Nominal_Permeq.args_parser >> Nominal_Permeq.perm_strict_simp_meth *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   797
 {* pushes permutations inside, raises an error if it cannot solve all permutations. *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   798
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   799
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   800
subsubsection {* Equivariance for permutations and swapping *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   801
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   802
lemma permute_eqvt:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   803
  shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   804
  unfolding permute_perm_def by simp
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   805
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   806
(* the normal version of this lemma would cause loops *)
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   807
lemma permute_eqvt_raw [eqvt_raw]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   808
  shows "p \<bullet> permute \<equiv> permute"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   809
apply(simp add: fun_eq_iff permute_fun_def)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   810
apply(subst permute_eqvt)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   811
apply(simp)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   812
done
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   813
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   814
lemma zero_perm_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   815
  shows "p \<bullet> (0::perm) = 0"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   816
  unfolding permute_perm_def by simp
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   817
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   818
lemma add_perm_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   819
  fixes p p1 p2 :: perm
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   820
  shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   821
  unfolding permute_perm_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   822
  by (simp add: perm_eq_iff)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   823
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   824
lemma swap_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   825
  shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   826
  unfolding permute_perm_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   827
  by (auto simp add: swap_atom perm_eq_iff)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   828
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   829
lemma uminus_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   830
  fixes p q::"perm"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   831
  shows "p \<bullet> (- q) = - (p \<bullet> q)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   832
  unfolding permute_perm_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   833
  by (simp add: diff_minus minus_add add_assoc)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   834
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   835
subsubsection {* Equivariance of Logical Operators *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   836
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   837
lemma eq_eqvt [eqvt]:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   838
  shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   839
  unfolding permute_eq_iff permute_bool_def ..
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   840
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   841
lemma Not_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   842
  shows "p \<bullet> (\<not> A) \<longleftrightarrow> \<not> (p \<bullet> A)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   843
  by (simp add: permute_bool_def)
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   844
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   845
lemma conj_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   846
  shows "p \<bullet> (A \<and> B) \<longleftrightarrow> (p \<bullet> A) \<and> (p \<bullet> B)"
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   847
  by (simp add: permute_bool_def)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   848
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   849
lemma imp_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   850
  shows "p \<bullet> (A \<longrightarrow> B) \<longleftrightarrow> (p \<bullet> A) \<longrightarrow> (p \<bullet> B)"
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   851
  by (simp add: permute_bool_def)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   852
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   853
declare imp_eqvt[folded induct_implies_def, eqvt]
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   854
2743
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   855
lemma all_eqvt [eqvt]:
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   856
  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   857
  unfolding All_def
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   858
  by (perm_simp) (rule refl)
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   859
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   860
declare all_eqvt[folded induct_forall_def, eqvt]
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   861
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   862
lemma ex_eqvt [eqvt]:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   863
  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
2743
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   864
  unfolding Ex_def
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
   865
  by (perm_simp) (rule refl)
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   866
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   867
lemma ex1_eqvt [eqvt]:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   868
  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   869
  unfolding Ex1_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   870
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   871
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   872
lemma if_eqvt [eqvt]:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   873
  shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   874
  by (simp add: permute_fun_def permute_bool_def)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   875
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   876
lemma True_eqvt [eqvt]:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   877
  shows "p \<bullet> True = True"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   878
  unfolding permute_bool_def ..
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   879
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   880
lemma False_eqvt [eqvt]:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   881
  shows "p \<bullet> False = False"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   882
  unfolding permute_bool_def ..
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   883
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   884
lemma disj_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   885
  shows "p \<bullet> (A \<or> B) \<longleftrightarrow> (p \<bullet> A) \<or> (p \<bullet> B)"
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   886
  by (simp add: permute_bool_def)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   887
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   888
lemma all_eqvt2:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   889
  shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   890
  by (perm_simp add: permute_minus_cancel) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   891
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   892
lemma ex_eqvt2:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   893
  shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   894
  by (perm_simp add: permute_minus_cancel) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   895
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   896
lemma ex1_eqvt2:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   897
  shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   898
  by (perm_simp add: permute_minus_cancel) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   899
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   900
lemma the_eqvt:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   901
  assumes unique: "\<exists>!x. P x"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   902
  shows "(p \<bullet> (THE x. P x)) = (THE x. (p \<bullet> P) x)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   903
  apply(rule the1_equality [symmetric])
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   904
  apply(rule_tac p="-p" in permute_boolE)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   905
  apply(perm_simp add: permute_minus_cancel)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   906
  apply(rule unique)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   907
  apply(rule_tac p="-p" in permute_boolE)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   908
  apply(perm_simp add: permute_minus_cancel)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   909
  apply(rule theI'[OF unique])
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   910
  done
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   911
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   912
lemma the_eqvt2:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   913
  assumes unique: "\<exists>!x. P x"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   914
  shows "(p \<bullet> (THE x. P x)) = (THE x. p \<bullet> P (- p \<bullet> x))"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   915
  apply(rule the1_equality [symmetric])
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   916
  apply(simp add: ex1_eqvt2[symmetric])
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   917
  apply(simp add: permute_bool_def unique)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   918
  apply(simp add: permute_bool_def)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   919
  apply(rule theI'[OF unique])
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   920
  done
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   921
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   922
subsubsection {* Equivariance of Set operators *}
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   923
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   924
lemma mem_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   925
  shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   926
  unfolding permute_bool_def permute_set_def
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   927
  by (auto)
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   928
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   929
lemma Collect_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   930
  shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   931
  unfolding permute_set_eq permute_fun_def
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   932
  by (auto simp add: permute_bool_def)
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   933
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   934
lemma inter_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   935
  shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   936
  unfolding Int_def
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   937
  by (perm_simp) (rule refl)
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   938
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   939
lemma Bex_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   940
  shows "p \<bullet> (\<exists>x \<in> S. P x) = (\<exists>x \<in> (p \<bullet> S). (p \<bullet> P) x)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   941
  unfolding Bex_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   942
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   943
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   944
lemma Ball_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   945
  shows "p \<bullet> (\<forall>x \<in> S. P x) = (\<forall>x \<in> (p \<bullet> S). (p \<bullet> P) x)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   946
  unfolding Ball_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   947
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   948
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   949
lemma image_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   950
  shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   951
  unfolding image_def
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   952
  by (perm_simp) (rule refl)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   953
3050
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   954
lemma Image_eqvt [eqvt]:
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   955
  shows "p \<bullet> (R `` A) = (p \<bullet> R) `` (p \<bullet> A)"
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   956
  unfolding Image_def
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   957
  by (perm_simp) (rule refl)
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   958
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   959
lemma UNIV_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   960
  shows "p \<bullet> UNIV = UNIV"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   961
  unfolding UNIV_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   962
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   963
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   964
lemma union_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   965
  shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   966
  unfolding Un_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   967
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   968
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   969
lemma Diff_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   970
  fixes A B :: "'a::pt set"
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   971
  shows "p \<bullet> (A - B) = (p \<bullet> A) - (p \<bullet> B)"
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   972
  unfolding set_diff_eq
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   973
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   974
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   975
lemma Compl_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   976
  fixes A :: "'a::pt set"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   977
  shows "p \<bullet> (- A) = - (p \<bullet> A)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   978
  unfolding Compl_eq_Diff_UNIV
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   979
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   980
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   981
lemma subset_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   982
  shows "p \<bullet> (S \<subseteq> T) \<longleftrightarrow> (p \<bullet> S) \<subseteq> (p \<bullet> T)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   983
  unfolding subset_eq
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   984
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   985
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   986
lemma psubset_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   987
  shows "p \<bullet> (S \<subset> T) \<longleftrightarrow> (p \<bullet> S) \<subset> (p \<bullet> T)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   988
  unfolding psubset_eq
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   989
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   990
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   991
lemma vimage_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   992
  shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   993
  unfolding vimage_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   994
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   995
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   996
lemma Union_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   997
  shows "p \<bullet> (\<Union> S) = \<Union> (p \<bullet> S)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   998
  unfolding Union_eq 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
   999
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1000
2777
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1001
lemma Inter_eqvt [eqvt]:
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1002
  shows "p \<bullet> (\<Inter> S) = \<Inter> (p \<bullet> S)"
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1003
  unfolding Inter_eq 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1004
  by (perm_simp) (rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1005
3147
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1006
lemma foldr_eqvt[eqvt]:
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1007
  "p \<bullet> foldr a b c = foldr (p \<bullet> a) (p \<bullet> b) (p \<bullet> c)"
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1008
  apply (induct b)
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1009
  apply simp_all
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1010
  apply (perm_simp)
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1011
  apply simp
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1012
  done
d24e70483051 Clean the proof of Aux
Cezary Kaliszyk <cezarykaliszyk@gmail.com>
parents: 3134
diff changeset
  1013
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1014
(* FIXME: eqvt attribute *)
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1015
lemma Sigma_eqvt:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1016
  shows "(p \<bullet> (X \<times> Y)) = (p \<bullet> X) \<times> (p \<bullet> Y)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1017
unfolding Sigma_def
3026
b037ae269f50 updated to Isabelle 19 Sept
Christian Urban <urbanc@in.tum.de>
parents: 2987
diff changeset
  1018
unfolding SUP_def
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1019
by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1020
2777
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1021
text {* 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1022
  In order to prove that lfp is equivariant we need two
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1023
  auxiliary classes which specify that (op <=) and
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1024
  Inf are equivariant. Instances for bool and fun are 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1025
  given.
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1026
*}
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1027
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1028
class le_eqvt =  order + 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1029
  assumes le_eqvt [eqvt]: "p \<bullet> (x \<le> y) = ((p \<bullet> x) \<le> (p \<bullet> (y::('a::{pt, order}))))"
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1030
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1031
class inf_eqvt = complete_lattice +
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1032
  assumes inf_eqvt [eqvt]: "p \<bullet> (Inf X) = Inf (p \<bullet> (X::('a::{pt, Inf}) set))"
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1033
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1034
instantiation bool :: le_eqvt
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1035
begin
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1036
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1037
instance 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1038
apply(default)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1039
unfolding le_bool_def
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1040
apply(perm_simp)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1041
apply(rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1042
done
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1043
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1044
end
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1045
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1046
instantiation "fun" :: (pt, le_eqvt) le_eqvt
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1047
begin
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1048
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1049
instance 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1050
apply(default)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1051
unfolding le_fun_def
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1052
apply(perm_simp)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1053
apply(rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1054
done 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1055
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1056
end
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1057
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1058
instantiation bool :: inf_eqvt
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1059
begin
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1060
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1061
instance 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1062
apply(default)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1063
unfolding Inf_bool_def
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1064
apply(perm_simp)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1065
apply(rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1066
done
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1067
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1068
end
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1069
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1070
instantiation "fun" :: (pt, inf_eqvt) inf_eqvt
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1071
begin
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1072
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1073
instance 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1074
apply(default)
2987
27aab7a105eb updated for new Isabelle (11. Aug.)
Christian Urban <urbanc@in.tum.de>
parents: 2982
diff changeset
  1075
unfolding Inf_fun_def INF_def
2777
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1076
apply(perm_simp)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1077
apply(rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1078
done 
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1079
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1080
end
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1081
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1082
lemma lfp_eqvt [eqvt]:
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1083
  fixes F::"('a \<Rightarrow> 'b) \<Rightarrow> ('a::pt \<Rightarrow> 'b::{inf_eqvt, le_eqvt})"
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1084
  shows "p \<bullet> (lfp F) = lfp (p \<bullet> F)"
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1085
unfolding lfp_def
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1086
by (perm_simp) (rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1087
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1088
lemma finite_eqvt [eqvt]:
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1089
  shows "p \<bullet> finite A = finite (p \<bullet> A)"
2777
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1090
unfolding finite_def
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1091
by (perm_simp) (rule refl)
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1092
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1093
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1094
subsubsection {* Equivariance for product operations *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1095
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1096
lemma fst_eqvt [eqvt]:
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1097
  shows "p \<bullet> (fst x) = fst (p \<bullet> x)"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1098
  by (cases x) simp
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1099
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1100
lemma snd_eqvt [eqvt]:
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1101
  shows "p \<bullet> (snd x) = snd (p \<bullet> x)"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1102
  by (cases x) simp
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1103
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1104
lemma split_eqvt [eqvt]: 
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1105
  shows "p \<bullet> (split P x) = split (p \<bullet> P) (p \<bullet> x)"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1106
  unfolding split_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1107
  by (perm_simp) (rule refl)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1108
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1109
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1110
subsubsection {* Equivariance for list operations *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1111
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1112
lemma append_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1113
  shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1114
  by (induct xs) auto
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1115
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1116
lemma rev_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1117
  shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1118
  by (induct xs) (simp_all add: append_eqvt)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1119
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1120
lemma map_eqvt [eqvt]: 
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1121
  shows "p \<bullet> (map f xs) = map (p \<bullet> f) (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1122
  by (induct xs) (simp_all, simp only: permute_fun_app_eq)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1123
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1124
lemma removeAll_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1125
  shows "p \<bullet> (removeAll x xs) = removeAll (p \<bullet> x) (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1126
  by (induct xs) (auto)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1127
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1128
lemma filter_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1129
  shows "p \<bullet> (filter f xs) = filter (p \<bullet> f) (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1130
apply(induct xs)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1131
apply(simp)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1132
apply(simp only: filter.simps permute_list.simps if_eqvt)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1133
apply(simp only: permute_fun_app_eq)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1134
done
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1135
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1136
lemma distinct_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1137
  shows "p \<bullet> (distinct xs) = distinct (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1138
apply(induct xs)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1139
apply(simp add: permute_bool_def)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1140
apply(simp add: conj_eqvt Not_eqvt mem_eqvt set_eqvt)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1141
done
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1142
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1143
lemma length_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1144
  shows "p \<bullet> (length xs) = length (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1145
by (induct xs) (simp_all add: permute_pure)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1146
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1147
2972
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1148
subsubsection {* Equivariance for @{typ "'a option"} *}
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1149
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1150
lemma option_map_eqvt[eqvt]:
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1151
  shows "p \<bullet> (Option.map f x) = Option.map (p \<bullet> f) (p \<bullet> x)"
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1152
  by (cases x) (simp_all, simp add: permute_fun_app_eq)
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1153
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1154
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1155
subsubsection {* Equivariance for @{typ "'a fset"} *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1156
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1157
lemma in_fset_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1158
  shows "(p \<bullet> (x |\<in>| S)) = ((p \<bullet> x) |\<in>| (p \<bullet> S))"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1159
unfolding in_fset
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1160
by (perm_simp) (simp)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1161
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1162
lemma union_fset_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1163
  shows "(p \<bullet> (S |\<union>| T)) = ((p \<bullet> S) |\<union>| (p \<bullet> T))"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1164
  by (induct S) (simp_all)
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1165
3101
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1166
lemma inter_fset_eqvt [eqvt]:
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1167
  shows "(p \<bullet> (S |\<inter>| T)) = ((p \<bullet> S) |\<inter>| (p \<bullet> T))"
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1168
  apply(descending)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1169
  unfolding list_eq_def inter_list_def
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1170
  apply(perm_simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1171
  apply(simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1172
  done
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1173
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1174
lemma subset_fset_eqvt [eqvt]:
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1175
  shows "(p \<bullet> (S |\<subseteq>| T)) = ((p \<bullet> S) |\<subseteq>| (p \<bullet> T))"
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1176
  apply(descending)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1177
  unfolding sub_list_def
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1178
  apply(perm_simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1179
  apply(simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1180
  done
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1181
  
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1182
lemma map_fset_eqvt [eqvt]: 
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1183
  shows "p \<bullet> (map_fset f S) = map_fset (p \<bullet> f) (p \<bullet> S)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1184
  by (lifting map_eqvt)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1185
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1186
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1187
section {* Supp, Freshness and Supports *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1188
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1189
context pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1190
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1191
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1192
definition
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1193
  supp :: "'a \<Rightarrow> atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1194
where
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1195
  "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1196
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1197
definition
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1198
  fresh :: "atom \<Rightarrow> 'a \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1199
where   
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1200
  "a \<sharp> x \<equiv> a \<notin> supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1201
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1202
end
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1203
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1204
lemma supp_conv_fresh: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1205
  shows "supp x = {a. \<not> a \<sharp> x}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1206
  unfolding fresh_def by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1207
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1208
lemma swap_rel_trans:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1209
  assumes "sort_of a = sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1210
  assumes "sort_of b = sort_of c"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1211
  assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1212
  assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1213
  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1214
proof (cases)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1215
  assume "a = b \<or> c = b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1216
  with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1217
next
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1218
  assume *: "\<not> (a = b \<or> c = b)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1219
  have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1220
    using assms by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1221
  also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1222
    using assms * by (simp add: swap_triple)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1223
  finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1224
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1225
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1226
lemma swap_fresh_fresh:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1227
  assumes a: "a \<sharp> x" 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1228
  and     b: "b \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1229
  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1230
proof (cases)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1231
  assume asm: "sort_of a = sort_of b" 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1232
  have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}" 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1233
    using a b unfolding fresh_def supp_def by simp_all
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1234
  then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1235
  then obtain c 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1236
    where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1237
    by (rule obtain_atom) (auto)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1238
  then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1239
next
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1240
  assume "sort_of a \<noteq> sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1241
  then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1242
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1243
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1244
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1245
subsection {* supp and fresh are equivariant *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1246
2760
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1247
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1248
lemma supp_eqvt [eqvt]:
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1249
  shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1250
  unfolding supp_def
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1251
  by (perm_simp)
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1252
     (simp only: permute_eqvt[symmetric])
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1253
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1254
lemma fresh_eqvt [eqvt]:
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1255
  shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1256
  unfolding fresh_def
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1257
  by (perm_simp) (rule refl)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1258
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1259
lemma fresh_permute_iff:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1260
  shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
2760
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1261
  by (simp only: fresh_eqvt[symmetric] permute_bool_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1262
2683
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1263
lemma fresh_permute_left:
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1264
  shows "a \<sharp> p \<bullet> x \<longleftrightarrow> - p \<bullet> a \<sharp> x"
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1265
proof
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1266
  assume "a \<sharp> p \<bullet> x"
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1267
  then have "- p \<bullet> a \<sharp> - p \<bullet> p \<bullet> x" by (simp only: fresh_permute_iff)
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1268
  then show "- p \<bullet> a \<sharp> x" by simp
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1269
next
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1270
  assume "- p \<bullet> a \<sharp> x"
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1271
  then have "p \<bullet> - p \<bullet> a \<sharp> p \<bullet> x" by (simp only: fresh_permute_iff)
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1272
  then show "a \<sharp> p \<bullet> x" by simp
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1273
qed
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1274
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1275
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1276
section {* supports *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1277
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1278
definition
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1279
  supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1280
where  
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1281
  "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1282
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1283
lemma supp_is_subset:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1284
  fixes S :: "atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1285
  and   x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1286
  assumes a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1287
  and     a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1288
  shows "(supp x) \<subseteq> S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1289
proof (rule ccontr)
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1290
  assume "\<not> (supp x \<subseteq> S)"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1291
  then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1292
  from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding supports_def by auto
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1293
  then have "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1294
  with a2 have "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}" by (simp add: finite_subset)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1295
  then have "a \<notin> (supp x)" unfolding supp_def by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1296
  with b1 show False by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1297
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1298
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1299
lemma supports_finite:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1300
  fixes S :: "atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1301
  and   x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1302
  assumes a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1303
  and     a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1304
  shows "finite (supp x)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1305
proof -
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1306
  have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1307
  then show "finite (supp x)" using a2 by (simp add: finite_subset)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1308
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1309
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1310
lemma supp_supports:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1311
  fixes x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1312
  shows "(supp x) supports x"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1313
unfolding supports_def
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1314
proof (intro strip)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1315
  fix a b
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1316
  assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1317
  then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1318
  then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1319
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1320
2955
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1321
lemma supports_fresh:
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1322
  fixes x :: "'a::pt"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1323
  assumes a1: "S supports x"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1324
  and     a2: "finite S"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1325
  and     a3: "a \<notin> S"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1326
  shows "a \<sharp> x"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1327
unfolding fresh_def
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1328
proof -
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1329
  have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1330
  then show "a \<notin> (supp x)" using a3 by auto
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1331
qed
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1332
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1333
lemma supp_is_least_supports:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1334
  fixes S :: "atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1335
  and   x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1336
  assumes  a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1337
  and      a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1338
  and      a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1339
  shows "(supp x) = S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1340
proof (rule equalityI)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1341
  show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1342
  with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1343
  have "(supp x) supports x" by (rule supp_supports)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1344
  with fin a3 show "S \<subseteq> supp x" by blast
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1345
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1346
2955
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1347
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1348
lemma subsetCI: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1349
  shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1350
  by auto
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1351
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1352
lemma finite_supp_unique:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1353
  assumes a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1354
  assumes a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1355
  assumes a3: "\<And>a b. \<lbrakk>a \<in> S; b \<notin> S; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1356
  shows "(supp x) = S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1357
  using a1 a2
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1358
proof (rule supp_is_least_supports)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1359
  fix S'
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1360
  assume "finite S'" and "S' supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1361
  show "S \<subseteq> S'"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1362
  proof (rule subsetCI)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1363
    fix a
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1364
    assume "a \<in> S" and "a \<notin> S'"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1365
    have "finite (S \<union> S')"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1366
      using `finite S` `finite S'` by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1367
    then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1368
      by (rule obtain_atom)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1369
    then have "b \<notin> S" and "b \<notin> S'"  and "sort_of a = sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1370
      by simp_all
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1371
    then have "(a \<rightleftharpoons> b) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1372
      using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1373
    moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1374
      using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1375
      by (rule a3)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1376
    ultimately show "False" by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1377
  qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1378
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1379
2475
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1380
section {* Support w.r.t. relations *}
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1381
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1382
text {* 
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1383
  This definition is used for unquotient types, where
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1384
  alpha-equivalence does not coincide with equality.
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1385
*}
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1386
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1387
definition 
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1388
  "supp_rel R x = {a. infinite {b. \<not>(R ((a \<rightleftharpoons> b) \<bullet> x) x)}}"
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1389
486d4647bb37 supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
parents: 2470
diff changeset
  1390
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1391
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1392
section {* Finitely-supported types *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1393
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1394
class fs = pt +
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1395
  assumes finite_supp: "finite (supp x)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1396
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1397
lemma pure_supp: 
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1398
  fixes x::"'a::pure"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1399
  shows "supp x = {}"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1400
  unfolding supp_def by (simp add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1401
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1402
lemma pure_fresh:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1403
  fixes x::"'a::pure"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1404
  shows "a \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1405
  unfolding fresh_def by (simp add: pure_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1406
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1407
instance pure < fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1408
by default (simp add: pure_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1409
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1410
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1411
subsection  {* Type @{typ atom} is finitely-supported. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1412
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1413
lemma supp_atom:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1414
  shows "supp a = {a}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1415
apply (rule finite_supp_unique)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1416
apply (clarsimp simp add: supports_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1417
apply simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1418
apply simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1419
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1420
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1421
lemma fresh_atom: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1422
  shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1423
  unfolding fresh_def supp_atom by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1424
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1425
instance atom :: fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1426
by default (simp add: supp_atom)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1427
1933
9eab1dfc14d2 moved lemmas from FSet.thy to do with atom to Nominal2_Base, and to do with 'a::at set to Nominal2_Atoms; moved Nominal2_Eqvt.thy one up to be loaded before Nominal2_Atoms
Christian Urban <urbanc@in.tum.de>
parents: 1932
diff changeset
  1428
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1429
section {* Type @{typ perm} is finitely-supported. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1430
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1431
lemma perm_swap_eq:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1432
  shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1433
unfolding permute_perm_def
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1434
by (metis add_diff_cancel minus_perm_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1435
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1436
lemma supports_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1437
  shows "{a. p \<bullet> a \<noteq> a} supports p"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1438
  unfolding supports_def
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1439
  unfolding perm_swap_eq
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1440
  by (simp add: swap_eqvt)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1441
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1442
lemma finite_perm_lemma: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1443
  shows "finite {a::atom. p \<bullet> a \<noteq> a}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1444
  using finite_Rep_perm [of p]
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1445
  unfolding permute_atom_def .
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1446
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1447
lemma supp_perm:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1448
  shows "supp p = {a. p \<bullet> a \<noteq> a}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1449
apply (rule finite_supp_unique)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1450
apply (rule supports_perm)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1451
apply (rule finite_perm_lemma)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1452
apply (simp add: perm_swap_eq swap_eqvt)
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1453
apply (auto simp add: perm_eq_iff swap_atom)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1454
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1455
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1456
lemma fresh_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1457
  shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1458
  unfolding fresh_def 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1459
  by (simp add: supp_perm)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1460
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1461
lemma supp_swap:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1462
  shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1463
  by (auto simp add: supp_perm swap_atom)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1464
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1465
lemma fresh_zero_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1466
  shows "a \<sharp> (0::perm)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1467
  unfolding fresh_perm by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1468
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1469
lemma supp_zero_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1470
  shows "supp (0::perm) = {}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1471
  unfolding supp_perm by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1472
1087
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1473
lemma fresh_plus_perm:
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1474
  fixes p q::perm
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1475
  assumes "a \<sharp> p" "a \<sharp> q"
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1476
  shows "a \<sharp> (p + q)"
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1477
  using assms
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1478
  unfolding fresh_def
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1479
  by (auto simp add: supp_perm)
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1480
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1481
lemma supp_plus_perm:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1482
  fixes p q::perm
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1483
  shows "supp (p + q) \<subseteq> supp p \<union> supp q"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1484
  by (auto simp add: supp_perm)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1485
1087
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1486
lemma fresh_minus_perm:
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1487
  fixes p::perm
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1488
  shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1489
  unfolding fresh_def
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1490
  unfolding supp_perm
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1491
  apply(simp)
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1492
  apply(metis permute_minus_cancel)
1087
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1493
  done
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1494
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1495
lemma supp_minus_perm:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1496
  fixes p::perm
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1497
  shows "supp (- p) = supp p"
1087
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1498
  unfolding supp_conv_fresh
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1499
  by (simp add: fresh_minus_perm)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1500
1305
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1501
lemma plus_perm_eq:
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1502
  fixes p q::"perm"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1503
  assumes asm: "supp p \<inter> supp q = {}"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1504
  shows "p + q = q + p"
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1505
unfolding perm_eq_iff
1305
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1506
proof
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1507
  fix a::"atom"
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1508
  show "(p + q) \<bullet> a = (q + p) \<bullet> a"
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1509
  proof -
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1510
    { assume "a \<notin> supp p" "a \<notin> supp q"
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1511
      then have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1512
	by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1513
    }
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1514
    moreover
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1515
    { assume a: "a \<in> supp p" "a \<notin> supp q"
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1516
      then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1517
      then have "p \<bullet> a \<notin> supp q" using asm by auto
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1518
      with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1519
	by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1520
    }
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1521
    moreover
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1522
    { assume a: "a \<notin> supp p" "a \<in> supp q"
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1523
      then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1524
      then have "q \<bullet> a \<notin> supp p" using asm by auto 
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1525
      with a have "(p + q) \<bullet> a = (q + p) \<bullet> a" 
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1526
	by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1527
    }
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1528
    ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a" 
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1529
      using asm by blast
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1530
  qed
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1531
qed
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1532
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1533
lemma supp_plus_perm_eq:
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1534
  fixes p q::perm
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1535
  assumes asm: "supp p \<inter> supp q = {}"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1536
  shows "supp (p + q) = supp p \<union> supp q"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1537
proof -
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1538
  { fix a::"atom"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1539
    assume "a \<in> supp p"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1540
    then have "a \<notin> supp q" using asm by auto
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1541
    then have "a \<in> supp (p + q)" using `a \<in> supp p` 
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1542
      by (simp add: supp_perm)
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1543
  }
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1544
  moreover
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1545
  { fix a::"atom"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1546
    assume "a \<in> supp q"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1547
    then have "a \<notin> supp p" using asm by auto
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1548
    then have "a \<in> supp (q + p)" using `a \<in> supp q` 
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1549
      by (simp add: supp_perm)
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1550
    then have "a \<in> supp (p + q)" using asm plus_perm_eq
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1551
      by metis
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1552
  }
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1553
  ultimately have "supp p \<union> supp q \<subseteq> supp (p + q)"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1554
    by blast
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1555
  then show "supp (p + q) = supp p \<union> supp q" using supp_plus_perm
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1556
    by blast
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1557
qed
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1558
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1559
instance perm :: fs
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1560
by default (simp add: supp_perm finite_perm_lemma)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1561
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1562
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1563
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1564
section {* Finite Support instances for other types *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1565
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1566
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1567
subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1568
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1569
lemma supp_Pair: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1570
  shows "supp (x, y) = supp x \<union> supp y"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1571
  by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1572
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1573
lemma fresh_Pair: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1574
  shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1575
  by (simp add: fresh_def supp_Pair)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1576
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1577
lemma supp_Unit:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1578
  shows "supp () = {}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1579
  by (simp add: supp_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1580
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1581
lemma fresh_Unit:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1582
  shows "a \<sharp> ()"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1583
  by (simp add: fresh_def supp_Unit)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1584
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
  1585
instance prod :: (fs, fs) fs
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1586
apply default
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1587
apply (case_tac x)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1588
apply (simp add: supp_Pair finite_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1589
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1590
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1591
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1592
subsection {* Type @{typ "'a + 'b"} is finitely supported *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1593
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1594
lemma supp_Inl: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1595
  shows "supp (Inl x) = supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1596
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1597
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1598
lemma supp_Inr: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1599
  shows "supp (Inr x) = supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1600
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1601
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1602
lemma fresh_Inl: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1603
  shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1604
  by (simp add: fresh_def supp_Inl)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1605
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1606
lemma fresh_Inr: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1607
  shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1608
  by (simp add: fresh_def supp_Inr)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1609
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
  1610
instance sum :: (fs, fs) fs
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1611
apply default
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1612
apply (case_tac x)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1613
apply (simp_all add: supp_Inl supp_Inr finite_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1614
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1615
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1616
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1617
subsection {* Type @{typ "'a option"} is finitely supported *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1618
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1619
lemma supp_None: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1620
  shows "supp None = {}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1621
by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1622
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1623
lemma supp_Some: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1624
  shows "supp (Some x) = supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1625
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1626
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1627
lemma fresh_None: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1628
  shows "a \<sharp> None"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1629
  by (simp add: fresh_def supp_None)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1630
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1631
lemma fresh_Some: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1632
  shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1633
  by (simp add: fresh_def supp_Some)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1634
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1635
instance option :: (fs) fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1636
apply default
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1637
apply (induct_tac x)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1638
apply (simp_all add: supp_None supp_Some finite_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1639
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1640
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1641
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1642
subsubsection {* Type @{typ "'a list"} is finitely supported *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1643
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1644
lemma supp_Nil: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1645
  shows "supp [] = {}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1646
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1647
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1648
lemma fresh_Nil: 
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1649
  shows "a \<sharp> []"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1650
  by (simp add: fresh_def supp_Nil)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1651
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1652
lemma supp_Cons: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1653
  shows "supp (x # xs) = supp x \<union> supp xs"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1654
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1655
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1656
lemma fresh_Cons: 
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1657
  shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1658
  by (simp add: fresh_def supp_Cons)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1659
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1660
lemma supp_append:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1661
  shows "supp (xs @ ys) = supp xs \<union> supp ys"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1662
  by (induct xs) (auto simp add: supp_Nil supp_Cons)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1663
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1664
lemma fresh_append:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1665
  shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1666
  by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1667
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1668
lemma supp_rev:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1669
  shows "supp (rev xs) = supp xs"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1670
  by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1671
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1672
lemma fresh_rev:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1673
  shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1674
  by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1675
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1676
lemma supp_removeAll:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1677
  fixes x::"atom"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1678
  shows "supp (removeAll x xs) = supp xs - {x}"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1679
  by (induct xs)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1680
     (auto simp add: supp_Nil supp_Cons supp_atom)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1681
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1682
lemma supp_of_atom_list:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1683
  fixes as::"atom list"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1684
  shows "supp as = set as"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1685
by (induct as)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1686
   (simp_all add: supp_Nil supp_Cons supp_atom)
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1687
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1688
instance list :: (fs) fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1689
apply default
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1690
apply (induct_tac x)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1691
apply (simp_all add: supp_Nil supp_Cons finite_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1692
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1693
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1694
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1695
section {* Support and Freshness for Applications *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1696
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1697
lemma fresh_conv_MOST: 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1698
  shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1699
  unfolding fresh_def supp_def 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1700
  unfolding MOST_iff_cofinite by simp
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1701
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1702
lemma fresh_fun_app:
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1703
  assumes "a \<sharp> f" and "a \<sharp> x" 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1704
  shows "a \<sharp> f x"
2003
b53e98bfb298 added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
parents: 1973
diff changeset
  1705
  using assms
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1706
  unfolding fresh_conv_MOST
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1707
  unfolding permute_fun_app_eq
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1708
  by (elim MOST_rev_mp) (simp)
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1709
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1710
lemma supp_fun_app:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1711
  shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1712
  using fresh_fun_app
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1713
  unfolding fresh_def
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1714
  by auto
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1715
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1716
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1717
subsection {* Equivariance Predicate @{text eqvt} and @{text eqvt_at}*}
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1718
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1719
definition
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1720
  "eqvt f \<equiv> \<forall>p. p \<bullet> f = f"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1721
2868
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1722
lemma eqvt_boolI:
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1723
  fixes f::"bool"
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1724
  shows "eqvt f"
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1725
unfolding eqvt_def by (simp add: permute_bool_def)
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1726
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1727
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1728
text {* equivariance of a function at a given argument *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1729
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1730
definition
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1731
 "eqvt_at f x \<equiv> \<forall>p. p \<bullet> (f x) = f (p \<bullet> x)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1732
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1733
lemma eqvtI:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1734
  shows "(\<And>p. p \<bullet> f \<equiv> f) \<Longrightarrow> eqvt f"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1735
unfolding eqvt_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1736
by simp
2003
b53e98bfb298 added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
parents: 1973
diff changeset
  1737
2955
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1738
lemma eqvt_at_perm:
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1739
  assumes "eqvt_at f x"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1740
  shows "eqvt_at f (q \<bullet> x)"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1741
proof -
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1742
  { fix p::"perm"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1743
    have "p \<bullet> (f (q \<bullet> x)) = p \<bullet> q \<bullet> (f x)"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1744
      using assms by (simp add: eqvt_at_def)
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1745
    also have "\<dots> = (p + q) \<bullet> (f x)" by simp
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1746
    also have "\<dots> = f ((p + q) \<bullet> x)"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1747
      using assms by (simp add: eqvt_at_def)
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1748
    finally have "p \<bullet> (f (q \<bullet> x)) = f (p \<bullet> q \<bullet> x)" by simp } 
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1749
  then show "eqvt_at f (q \<bullet> x)" unfolding eqvt_at_def
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1750
    by simp
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1751
qed
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1752
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1753
lemma supp_fun_eqvt:
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1754
  assumes a: "eqvt f"
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1755
  shows "supp f = {}"
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1756
  using a
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1757
  unfolding eqvt_def
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1758
  unfolding supp_def 
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1759
  by simp
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1760
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1761
lemma fresh_fun_eqvt_app:
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1762
  assumes a: "eqvt f"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1763
  shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1764
proof -
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1765
  from a have "supp f = {}" by (simp add: supp_fun_eqvt)
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1766
  then show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1767
    unfolding fresh_def
2003
b53e98bfb298 added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
parents: 1973
diff changeset
  1768
    using supp_fun_app by auto
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1769
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1770
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1771
lemma supp_fun_app_eqvt:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1772
  assumes a: "eqvt f"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1773
  shows "supp (f x) \<subseteq> supp x"
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1774
  using fresh_fun_eqvt_app[OF a]
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1775
  unfolding fresh_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1776
  by auto
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1777
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1778
lemma supp_eqvt_at:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1779
  assumes asm: "eqvt_at f x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1780
  and     fin: "finite (supp x)"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1781
  shows "supp (f x) \<subseteq> supp x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1782
apply(rule supp_is_subset)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1783
unfolding supports_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1784
unfolding fresh_def[symmetric]
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1785
using asm
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1786
apply(simp add: eqvt_at_def)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1787
apply(simp add: swap_fresh_fresh)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1788
apply(rule fin)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1789
done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1790
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1791
lemma finite_supp_eqvt_at:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1792
  assumes asm: "eqvt_at f x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1793
  and     fin: "finite (supp x)"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1794
  shows "finite (supp (f x))"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1795
apply(rule finite_subset)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1796
apply(rule supp_eqvt_at[OF asm fin])
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1797
apply(rule fin)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1798
done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1799
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1800
lemma fresh_eqvt_at:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1801
  assumes asm: "eqvt_at f x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1802
  and     fin: "finite (supp x)"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1803
  and     fresh: "a \<sharp> x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1804
  shows "a \<sharp> f x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1805
using fresh
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1806
unfolding fresh_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1807
using supp_eqvt_at[OF asm fin]
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1808
by auto
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1809
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1810
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1811
subsection {* helper functions for nominal_functions *}
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1812
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1813
lemma THE_defaultI2:
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1814
  assumes "\<exists>!x. P x" "\<And>x. P x \<Longrightarrow> Q x"
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1815
  shows "Q (THE_default d P)"
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1816
by (iprover intro: assms THE_defaultI')
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1817
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1818
lemma the_default_eqvt:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1819
  assumes unique: "\<exists>!x. P x"
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1820
  shows "(p \<bullet> (THE_default d P)) = (THE_default (p \<bullet> d) (p \<bullet> P))"
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1821
  apply(rule THE_default1_equality [symmetric])
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1822
  apply(rule_tac p="-p" in permute_boolE)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1823
  apply(simp add: ex1_eqvt)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1824
  apply(rule unique)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1825
  apply(rule_tac p="-p" in permute_boolE)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1826
  apply(rule subst[OF permute_fun_app_eq])
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1827
  apply(simp)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1828
  apply(rule THE_defaultI'[OF unique])
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1829
  done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1830
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1831
lemma fundef_ex1_eqvt:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1832
  fixes x::"'a::pt"
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1833
  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (G x))"
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1834
  assumes eqvt: "eqvt G"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1835
  assumes ex1: "\<exists>!y. G x y"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1836
  shows "(p \<bullet> (f x)) = f (p \<bullet> x)"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1837
  apply(simp only: f_def)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1838
  apply(subst the_default_eqvt)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1839
  apply(rule ex1)
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1840
  apply(rule THE_default1_equality [symmetric])
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1841
  apply(rule_tac p="-p" in permute_boolE)
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1842
  apply(perm_simp add: permute_minus_cancel)
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1843
  using eqvt[simplified eqvt_def]
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1844
  apply(simp)
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1845
  apply(rule ex1)
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1846
  apply(rule THE_defaultI2) 
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1847
  apply(rule_tac p="-p" in permute_boolE)
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1848
  apply(perm_simp add: permute_minus_cancel)
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1849
  apply(rule ex1)
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1850
  apply(perm_simp)
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1851
  using eqvt[simplified eqvt_def]
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1852
  apply(simp)
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1853
  done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1854
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1855
lemma fundef_ex1_eqvt_at:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1856
  fixes x::"'a::pt"
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1857
  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (G x))"
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1858
  assumes eqvt: "eqvt G"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1859
  assumes ex1: "\<exists>!y. G x y"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1860
  shows "eqvt_at f x"
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1861
  unfolding eqvt_at_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1862
  using assms
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1863
  by (auto intro: fundef_ex1_eqvt)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1864
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1865
lemma fundef_ex1_prop:
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1866
  fixes x::"'a::pt"
2848
da7e6655cd4c fixed the problem when giving a complex default-term; the fundef lemmas in Nominal_Base were not general enough
Christian Urban <urbanc@in.tum.de>
parents: 2847
diff changeset
  1867
  assumes f_def: "f == (\<lambda>x::'a. THE_default (d x) (G x))"
2820
77e1d9f2925e slightly stronger property in fundef_ex_prop
Christian Urban <urbanc@in.tum.de>
parents: 2818
diff changeset
  1868
  assumes P_all: "\<And>x y. G x y \<Longrightarrow> P x y"
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1869
  assumes ex1: "\<exists>!y. G x y"
2820
77e1d9f2925e slightly stronger property in fundef_ex_prop
Christian Urban <urbanc@in.tum.de>
parents: 2818
diff changeset
  1870
  shows "P x (f x)"
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1871
  unfolding f_def
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1872
  using ex1
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1873
  apply(erule_tac ex1E)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1874
  apply(rule THE_defaultI2)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1875
  apply(blast)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1876
  apply(rule P_all)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1877
  apply(assumption)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1878
  done
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1879
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1880
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1881
section {* Support of Finite Sets of Finitely Supported Elements *}
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1882
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1883
text {* support and freshness for atom sets *}
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1884
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1885
lemma supp_finite_atom_set:
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1886
  fixes S::"atom set"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1887
  assumes "finite S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1888
  shows "supp S = S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1889
  apply(rule finite_supp_unique)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1890
  apply(simp add: supports_def)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1891
  apply(simp add: swap_set_not_in)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1892
  apply(rule assms)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1893
  apply(simp add: swap_set_in)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1894
done
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1895
2742
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1896
lemma supp_cofinite_atom_set:
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1897
  fixes S::"atom set"
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1898
  assumes "finite (UNIV - S)"
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1899
  shows "supp S = (UNIV - S)"
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1900
  apply(rule finite_supp_unique)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1901
  apply(simp add: supports_def)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1902
  apply(simp add: swap_set_both_in)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1903
  apply(rule assms)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1904
  apply(subst swap_commute)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1905
  apply(simp add: swap_set_in)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1906
done
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1907
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1908
lemma fresh_finite_atom_set:
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1909
  fixes S::"atom set"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1910
  assumes "finite S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1911
  shows "a \<sharp> S \<longleftrightarrow> a \<notin> S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1912
  unfolding fresh_def
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1913
  by (simp add: supp_finite_atom_set[OF assms])
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1914
2679
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1915
lemma fresh_minus_atom_set:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1916
  fixes S::"atom set"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1917
  assumes "finite S"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1918
  shows "a \<sharp> S - T \<longleftrightarrow> (a \<notin> T \<longrightarrow> a \<sharp> S)"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1919
  unfolding fresh_def
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1920
  by (auto simp add: supp_finite_atom_set assms)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1921
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1922
lemma Union_supports_set:
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1923
  shows "(\<Union>x \<in> S. supp x) supports S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1924
proof -
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1925
  { fix a b
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1926
    have "\<forall>x \<in> S. (a \<rightleftharpoons> b) \<bullet> x = x \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> S = S"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
  1927
      unfolding permute_set_def by force
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1928
  }
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1929
  then show "(\<Union>x \<in> S. supp x) supports S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1930
    unfolding supports_def 
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1931
    by (simp add: fresh_def[symmetric] swap_fresh_fresh)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1932
qed
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1933
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1934
lemma Union_of_finite_supp_sets:
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1935
  fixes S::"('a::fs set)"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1936
  assumes fin: "finite S"   
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1937
  shows "finite (\<Union>x\<in>S. supp x)"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1938
  using fin by (induct) (auto simp add: finite_supp)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1939
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1940
lemma Union_included_in_supp:
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1941
  fixes S::"('a::fs set)"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1942
  assumes fin: "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1943
  shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1944
proof -
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1945
  have eqvt: "eqvt (\<lambda>S. \<Union> supp ` S)" 
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1946
    unfolding eqvt_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1947
    by (perm_simp) (simp)
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1948
  have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)"
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1949
    by (rule supp_finite_atom_set[symmetric]) (rule Union_of_finite_supp_sets[OF fin])
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1950
  also have "\<dots> = supp ((\<lambda>S. \<Union> supp ` S) S)" by simp
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1951
  also have "\<dots> \<subseteq> supp S" using eqvt
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1952
    by (rule supp_fun_app_eqvt)
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1953
  finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1954
qed
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1955
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1956
lemma supp_of_finite_sets:
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1957
  fixes S::"('a::fs set)"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1958
  assumes fin: "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1959
  shows "(supp S) = (\<Union>x\<in>S. supp x)"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1960
apply(rule subset_antisym)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1961
apply(rule supp_is_subset)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1962
apply(rule Union_supports_set)
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1963
apply(rule Union_of_finite_supp_sets[OF fin])
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1964
apply(rule Union_included_in_supp[OF fin])
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1965
done
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1966
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1967
lemma finite_sets_supp:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1968
  fixes S::"('a::fs set)"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1969
  assumes "finite S"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1970
  shows "finite (supp S)"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1971
using assms
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1972
by (simp only: supp_of_finite_sets Union_of_finite_supp_sets)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1973
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1974
lemma supp_of_finite_union:
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1975
  fixes S T::"('a::fs) set"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1976
  assumes fin1: "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1977
  and     fin2: "finite T"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1978
  shows "supp (S \<union> T) = supp S \<union> supp T"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1979
  using fin1 fin2
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1980
  by (simp add: supp_of_finite_sets)
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1981
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1982
lemma supp_of_finite_insert:
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1983
  fixes S::"('a::fs) set"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1984
  assumes fin:  "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1985
  shows "supp (insert x S) = supp x \<union> supp S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1986
  using fin
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1987
  by (simp add: supp_of_finite_sets)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1988
2588
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1989
lemma fresh_finite_insert:
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1990
  fixes S::"('a::fs) set"
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1991
  assumes fin:  "finite S"
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1992
  shows "a \<sharp> (insert x S) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1993
  using fin unfolding fresh_def
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1994
  by (simp add: supp_of_finite_insert)
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1995
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1996
lemma supp_set_empty:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1997
  shows "supp {} = {}"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1998
  unfolding supp_def
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1999
  by (simp add: empty_eqvt)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2000
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2001
lemma fresh_set_empty:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2002
  shows "a \<sharp> {}"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2003
  by (simp add: fresh_def supp_set_empty)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2004
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2005
lemma supp_set:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2006
  fixes xs :: "('a::fs) list"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2007
  shows "supp (set xs) = supp xs"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2008
apply(induct xs)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2009
apply(simp add: supp_set_empty supp_Nil)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2010
apply(simp add: supp_Cons supp_of_finite_insert)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2011
done
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2012
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2013
lemma fresh_set:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2014
  fixes xs :: "('a::fs) list"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2015
  shows "a \<sharp> (set xs) \<longleftrightarrow> a \<sharp> xs"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2016
unfolding fresh_def
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2017
by (simp add: supp_set)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2018
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2019
3121
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2020
subsection {* Type @{typ "'a multiset"} is finitely supported *}
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2021
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2022
lemma set_of_eqvt[eqvt]:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2023
  shows "p \<bullet> (set_of M) = set_of (p \<bullet> M)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2024
by (induct M) (simp_all add: insert_eqvt empty_eqvt)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2025
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2026
lemma supp_set_of:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2027
  shows "supp (set_of M) \<subseteq> supp M"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2028
  apply (rule supp_fun_app_eqvt)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2029
  unfolding eqvt_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2030
  apply(perm_simp)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2031
  apply(simp)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2032
  done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2033
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2034
lemma Union_finite_multiset:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2035
  fixes M::"'a::fs multiset"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2036
  shows "finite (\<Union>{supp x | x. x \<in># M})"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2037
proof - 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2038
  have "finite (\<Union>(supp ` {x. x \<in># M}))"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2039
    by (induct M) (simp_all add: Collect_imp_eq Collect_neg_eq finite_supp)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2040
  then show "finite (\<Union>{supp x | x. x \<in># M})"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2041
    by (simp only: image_Collect)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2042
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2043
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2044
lemma Union_supports_multiset:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2045
  shows "\<Union>{supp x | x. x :# M} supports M"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2046
proof -
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2047
  have sw: "\<And>a b. ((\<And>x. x :# M \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x) \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> M = M)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2048
    unfolding permute_multiset_def 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2049
    apply(induct M)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2050
    apply(simp_all)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2051
    done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2052
  show "(\<Union>{supp x | x. x :# M}) supports M"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2053
    unfolding supports_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2054
    apply(clarify)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2055
    apply(rule sw)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2056
    apply(rule swap_fresh_fresh)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2057
    apply(simp_all only: fresh_def)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2058
    apply(auto)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2059
    apply(metis neq0_conv)+
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2060
    done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2061
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2062
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2063
lemma Union_included_multiset:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2064
  fixes M::"('a::fs multiset)" 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2065
  shows "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2066
proof -
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2067
  have "(\<Union>{supp x | x. x \<in># M}) = (\<Union>{supp x | x. x \<in> set_of M})" by simp
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2068
  also have "... \<subseteq> (\<Union>x \<in> set_of M. supp x)" by auto
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2069
  also have "... = supp (set_of M)" by (simp add: subst supp_of_finite_sets)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2070
  also have " ... \<subseteq> supp M" by (rule supp_set_of)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2071
  finally show "(\<Union>{supp x | x. x \<in># M}) \<subseteq> supp M" .
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2072
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2073
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2074
lemma supp_of_multisets:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2075
  fixes M::"('a::fs multiset)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2076
  shows "(supp M) = (\<Union>{supp x | x. x :# M})"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2077
apply(rule subset_antisym)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2078
apply(rule supp_is_subset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2079
apply(rule Union_supports_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2080
apply(rule Union_finite_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2081
apply(rule Union_included_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2082
done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2083
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2084
lemma multisets_supp_finite:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2085
  fixes M::"('a::fs multiset)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2086
  shows "finite (supp M)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2087
by (simp only: supp_of_multisets Union_finite_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2088
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2089
lemma supp_of_multiset_union:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2090
  fixes M N::"('a::fs) multiset"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2091
  shows "supp (M + N) = supp M \<union> supp N"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2092
  by (auto simp add: supp_of_multisets)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2093
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2094
lemma supp_empty_mset [simp]:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2095
  shows "supp {#} = {}"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2096
  unfolding supp_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2097
  by simp
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2098
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2099
instance multiset :: (fs) fs
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2100
  apply (default)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2101
  apply (rule multisets_supp_finite)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2102
  done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2103
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2104
subsection {* Type @{typ "'a fset"} is finitely supported *}
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2105
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2106
lemma supp_fset [simp]:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2107
  shows "supp (fset S) = supp S"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2108
  unfolding supp_def
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2109
  by (simp add: fset_eqvt fset_cong)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2110
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2111
lemma supp_empty_fset [simp]:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2112
  shows "supp {||} = {}"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2113
  unfolding supp_def
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2114
  by simp
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2115
2641
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2116
lemma fresh_empty_fset:
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2117
  shows "a \<sharp> {||}"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2118
unfolding fresh_def
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2119
by (simp)
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2120
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2121
lemma supp_insert_fset [simp]:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2122
  fixes x::"'a::fs"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2123
  and   S::"'a fset"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2124
  shows "supp (insert_fset x S) = supp x \<union> supp S"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2125
  apply(subst supp_fset[symmetric])
2587
78623a0f294b tuned proof to reduce number of warnings
Christian Urban <urbanc@in.tum.de>
parents: 2586
diff changeset
  2126
  apply(simp add: supp_of_finite_insert)
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2127
  done
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2128
2641
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2129
lemma fresh_insert_fset:
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2130
  fixes x::"'a::fs"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2131
  and   S::"'a fset"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2132
  shows "a \<sharp> insert_fset x S \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> S"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2133
  unfolding fresh_def
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2134
  by (simp)
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2135
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2136
lemma fset_finite_supp:
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2137
  fixes S::"('a::fs) fset"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2138
  shows "finite (supp S)"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2139
  by (induct S) (simp_all add: finite_supp)
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2140
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2141
lemma supp_union_fset:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2142
  fixes S T::"'a::fs fset"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2143
  shows "supp (S |\<union>| T) = supp S \<union> supp T"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2144
by (induct S) (auto)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2145
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2146
lemma fresh_union_fset:
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2147
  fixes S T::"'a::fs fset"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2148
  shows "a \<sharp> S |\<union>| T \<longleftrightarrow> a \<sharp> S \<and> a \<sharp> T"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2149
unfolding fresh_def
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2150
by (simp add: supp_union_fset)
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2151
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2152
instance fset :: (fs) fs
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2153
  apply (default)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2154
  apply (rule fset_finite_supp)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2155
  done
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2156
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2157
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2158
section {* Freshness and Fresh-Star *}
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2159
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2160
lemma fresh_Unit_elim: 
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2161
  shows "(a \<sharp> () \<Longrightarrow> PROP C) \<equiv> PROP C"
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2162
  by (simp add: fresh_Unit)
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2163
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2164
lemma fresh_Pair_elim: 
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2165
  shows "(a \<sharp> (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> a \<sharp> y \<Longrightarrow> PROP C)"
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2166
  by rule (simp_all add: fresh_Pair)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2167
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2168
(* this rule needs to be added before the fresh_prodD is *)
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2169
(* added to the simplifier with mksimps                  *) 
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2170
lemma [simp]:
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2171
  shows "a \<sharp> x1 \<Longrightarrow> a \<sharp> x2 \<Longrightarrow> a \<sharp> (x1, x2)"
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2172
  by (simp add: fresh_Pair)
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2173
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2174
lemma fresh_PairD:
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2175
  shows "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> x"
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2176
  and   "a \<sharp> (x, y) \<Longrightarrow> a \<sharp> y"
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2177
  by (simp_all add: fresh_Pair)
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2178
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2179
declaration {* fn _ =>
3051
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2180
let
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2181
  val mksimps_pairs = (@{const_name Nominal2_Base.fresh}, @{thms fresh_PairD}) :: mksimps_pairs
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2182
in
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2183
  Simplifier.map_ss (fn ss => Simplifier.set_mksimps (mksimps mksimps_pairs) ss)
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2184
end
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2185
*}
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2186
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2187
text {* The fresh-star generalisation of fresh is used in strong
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2188
  induction principles. *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2189
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2190
definition 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2191
  fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2192
where 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2193
  "as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2194
2507
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2195
lemma fresh_star_supp_conv:
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2196
  shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2197
by (auto simp add: fresh_star_def fresh_def)
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2198
2675
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2199
lemma fresh_star_perm_set_conv:
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2200
  fixes p::"perm"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2201
  assumes fresh: "as \<sharp>* p"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2202
  and     fin: "finite as"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2203
  shows "supp p \<sharp>* as"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2204
apply(rule fresh_star_supp_conv)
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2205
apply(simp add: supp_finite_atom_set fin fresh)
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2206
done
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2207
2679
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2208
lemma fresh_star_atom_set_conv:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2209
  assumes fresh: "as \<sharp>* bs"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2210
  and     fin: "finite as" "finite bs"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2211
  shows "bs \<sharp>* as"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2212
using fresh
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2213
unfolding fresh_star_def fresh_def
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2214
by (auto simp add: supp_finite_atom_set fin)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2215
2730
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2216
lemma atom_fresh_star_disjoint:
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2217
  assumes fin: "finite bs" 
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2218
  shows "as \<sharp>* bs \<longleftrightarrow> (as \<inter> bs = {})"
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2219
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2220
unfolding fresh_star_def fresh_def
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2221
by (auto simp add: supp_finite_atom_set fin)
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2222
2675
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2223
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2224
lemma fresh_star_Pair:
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2225
  shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)" 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2226
  by (auto simp add: fresh_star_def fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2227
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2228
lemma fresh_star_list:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2229
  shows "as \<sharp>* (xs @ ys) \<longleftrightarrow> as \<sharp>* xs \<and> as \<sharp>* ys"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2230
  and   "as \<sharp>* (x # xs) \<longleftrightarrow> as \<sharp>* x \<and> as \<sharp>* xs"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2231
  and   "as \<sharp>* []"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2232
by (auto simp add: fresh_star_def fresh_Nil fresh_Cons fresh_append)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2233
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2234
lemma fresh_star_set:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2235
  fixes xs::"('a::fs) list"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2236
  shows "as \<sharp>* set xs \<longleftrightarrow> as \<sharp>* xs"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2237
unfolding fresh_star_def
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2238
by (simp add: fresh_set)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2239
2611
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2240
lemma fresh_star_singleton:
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2241
  fixes a::"atom"
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2242
  shows "as \<sharp>* {a} \<longleftrightarrow> as \<sharp>* a"
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2243
  by (simp add: fresh_star_def fresh_finite_insert fresh_set_empty)
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2244
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2245
lemma fresh_star_fset:
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2246
  fixes xs::"('a::fs) list"
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2247
  shows "as \<sharp>* fset S \<longleftrightarrow> as \<sharp>* S"
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2248
by (simp add: fresh_star_def fresh_def) 
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2249
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2250
lemma fresh_star_Un:
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2251
  shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2252
  by (auto simp add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2253
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2254
lemma fresh_star_insert:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2255
  shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2256
  by (auto simp add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2257
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2258
lemma fresh_star_Un_elim:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2259
  "((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2260
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2261
  apply(rule)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2262
  apply(erule meta_mp)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2263
  apply(auto)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2264
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2265
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2266
lemma fresh_star_insert_elim:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2267
  "(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2268
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2269
  by rule (simp_all add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2270
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2271
lemma fresh_star_empty_elim:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2272
  "({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2273
  by (simp add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2274
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2275
lemma fresh_star_Unit_elim: 
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2276
  shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2277
  by (simp add: fresh_star_def fresh_Unit) 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2278
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2279
lemma fresh_star_Pair_elim: 
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2280
  shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2281
  by (rule, simp_all add: fresh_star_Pair)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2282
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2283
lemma fresh_star_zero:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2284
  shows "as \<sharp>* (0::perm)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2285
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2286
  by (simp add: fresh_zero_perm)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2287
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2288
lemma fresh_star_plus:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2289
  fixes p q::perm
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2290
  shows "\<lbrakk>a \<sharp>* p;  a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2291
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2292
  by (simp add: fresh_plus_perm)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2293
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2294
lemma fresh_star_permute_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2295
  shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2296
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2297
  by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2298
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2299
lemma fresh_star_eqvt [eqvt]:
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  2300
  shows "p \<bullet> (as \<sharp>* x) \<longleftrightarrow> (p \<bullet> as) \<sharp>* (p \<bullet> x)"
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2301
unfolding fresh_star_def
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2302
by (perm_simp) (rule refl)
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2303
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2304
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2305
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2306
section {* Induction principle for permutations *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2307
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2308
lemma smaller_supp:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2309
  assumes a: "a \<in> supp p"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2310
  shows "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2311
proof -
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2312
  have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subseteq> supp p"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2313
    unfolding supp_perm by (auto simp add: swap_atom)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2314
  moreover
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2315
  have "a \<notin> supp ((p \<bullet> a \<rightleftharpoons> a) + p)" by (simp add: supp_perm)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2316
  then have "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<noteq> supp p" using a by auto
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2317
  ultimately 
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2318
  show "supp ((p \<bullet> a \<rightleftharpoons> a) + p) \<subset> supp p" by auto
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2319
qed
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2320
  
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2321
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2322
lemma perm_struct_induct[consumes 1, case_names zero swap]:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2323
  assumes S: "supp p \<subseteq> S"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2324
  and zero: "P 0"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2325
  and swap: "\<And>p a b. \<lbrakk>P p; supp p \<subseteq> S; a \<in> S; b \<in> S; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2326
  shows "P p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2327
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2328
  have "finite (supp p)" by (simp add: finite_supp)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2329
  then show "P p" using S
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2330
  proof(induct A\<equiv>"supp p" arbitrary: p rule: finite_psubset_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2331
    case (psubset p)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2332
    then have ih: "\<And>q. supp q \<subset> supp p \<Longrightarrow> P q" by auto
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2333
    have as: "supp p \<subseteq> S" by fact
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2334
    { assume "supp p = {}"
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  2335
      then have "p = 0" by (simp add: supp_perm perm_eq_iff)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2336
      then have "P p" using zero by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2337
    }
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2338
    moreover
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2339
    { assume "supp p \<noteq> {}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2340
      then obtain a where a0: "a \<in> supp p" by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2341
      then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2342
        using as by (auto simp add: supp_atom supp_perm swap_atom)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2343
      let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2344
      have a2: "supp ?q \<subset> supp p" using a0 smaller_supp by simp
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2345
      then have "P ?q" using ih by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2346
      moreover
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2347
      have "supp ?q \<subseteq> S" using as a2 by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2348
      ultimately  have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2349
      moreover 
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  2350
      have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: perm_eq_iff)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2351
      ultimately have "P p" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2352
    }
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2353
    ultimately show "P p" by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2354
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2355
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2356
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2357
lemma perm_simple_struct_induct[case_names zero swap]:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2358
  assumes zero: "P 0"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2359
  and     swap: "\<And>p a b. \<lbrakk>P p; a \<noteq> b; sort_of a = sort_of b\<rbrakk> \<Longrightarrow> P ((a \<rightleftharpoons> b) + p)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2360
  shows "P p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2361
by (rule_tac S="supp p" in perm_struct_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2362
   (auto intro: zero swap)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2363
2669
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2364
lemma perm_struct_induct2[consumes 1, case_names zero swap plus]:
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2365
  assumes S: "supp p \<subseteq> S"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2366
  assumes zero: "P 0"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2367
  assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b; a \<in> S; b \<in> S\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2368
  assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2369
  shows "P p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2370
using S
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2371
by (induct p rule: perm_struct_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2372
   (auto intro: zero plus swap simp add: supp_swap)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2373
2669
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2374
lemma perm_simple_struct_induct2[case_names zero swap plus]:
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2375
  assumes zero: "P 0"
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2376
  assumes swap: "\<And>a b. \<lbrakk>sort_of a = sort_of b; a \<noteq> b\<rbrakk> \<Longrightarrow> P (a \<rightleftharpoons> b)"
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2377
  assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2\<rbrakk> \<Longrightarrow> P (p1 + p2)"
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2378
  shows "P p"
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2379
by (rule_tac S="supp p" in perm_struct_induct2)
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2380
   (auto intro: zero swap plus)
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2381
2679
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2382
lemma supp_perm_singleton:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2383
  fixes p::"perm"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2384
  shows "supp p \<subseteq> {b} \<longleftrightarrow> p = 0"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2385
proof -
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2386
  { assume "supp p \<subseteq> {b}"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2387
    then have "p = 0"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2388
      by (induct p rule: perm_struct_induct) (simp_all)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2389
  }
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2390
  then show "supp p \<subseteq> {b} \<longleftrightarrow> p = 0" by (auto simp add: supp_zero_perm)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2391
qed
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2392
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2393
lemma supp_perm_pair:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2394
  fixes p::"perm"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2395
  shows "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2396
proof -
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2397
  { assume "supp p \<subseteq> {a, b}"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2398
    then have "p = 0 \<or> p = (b \<rightleftharpoons> a)"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2399
      apply (induct p rule: perm_struct_induct) 
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2400
      apply (auto simp add: swap_cancel supp_zero_perm supp_swap)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2401
      apply (simp add: swap_commute)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2402
      done
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2403
  }
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2404
  then show "supp p \<subseteq> {a, b} \<longleftrightarrow> p = 0 \<or> p = (b \<rightleftharpoons> a)" 
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2405
    by (auto simp add: supp_zero_perm supp_swap split: if_splits)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2406
qed
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2407
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2408
lemma supp_perm_eq:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2409
  assumes "(supp x) \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2410
  shows "p \<bullet> x = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2411
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2412
  from assms have "supp p \<subseteq> {a. a \<sharp> x}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2413
    unfolding supp_perm fresh_star_def fresh_def by auto
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2414
  then show "p \<bullet> x = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2415
  proof (induct p rule: perm_struct_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2416
    case zero
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2417
    show "0 \<bullet> x = x" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2418
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2419
    case (swap p a b)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2420
    then have "a \<sharp> x" "b \<sharp> x" "p \<bullet> x = x" by simp_all
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2421
    then show "((a \<rightleftharpoons> b) + p) \<bullet> x = x" by (simp add: swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2422
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2423
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2424
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2425
text {* same lemma as above, but proved with a different induction principle *}
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2426
lemma supp_perm_eq_test:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2427
  assumes "(supp x) \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2428
  shows "p \<bullet> x = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2429
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2430
  from assms have "supp p \<subseteq> {a. a \<sharp> x}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2431
    unfolding supp_perm fresh_star_def fresh_def by auto
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2432
  then show "p \<bullet> x = x"
2669
1d1772a89026 the function translating lambda terms to locally nameless lambda terms; still needs a stronger abs_eq_iff lemma...at the moment only proved for restrictions
Christian Urban <urbanc@in.tum.de>
parents: 2668
diff changeset
  2433
  proof (induct p rule: perm_struct_induct2)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2434
    case zero
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2435
    show "0 \<bullet> x = x" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2436
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2437
    case (swap a b)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2438
    then have "a \<sharp> x" "b \<sharp> x" by simp_all
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2439
    then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2440
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2441
    case (plus p1 p2)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2442
    have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2443
    then show "(p1 + p2) \<bullet> x = x" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2444
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2445
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2446
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2447
lemma perm_supp_eq:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2448
  assumes a: "(supp p) \<sharp>* x"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2449
  shows "p \<bullet> x = x"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2450
proof -
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2451
  from assms have "supp p \<subseteq> {a. a \<sharp> x}"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2452
    unfolding supp_perm fresh_star_def fresh_def by auto
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2453
  then show "p \<bullet> x = x"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2454
  proof (induct p rule: perm_struct_induct2)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2455
    case zero
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2456
    show "0 \<bullet> x = x" by simp
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2457
  next
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2458
    case (swap a b)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2459
    then have "a \<sharp> x" "b \<sharp> x" by simp_all
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2460
    then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2461
  next
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2462
    case (plus p1 p2)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2463
    have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2464
    then show "(p1 + p2) \<bullet> x = x" by simp
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2465
  qed
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2466
qed
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2467
3065
51ef8a3cb6ef updated to lates changes in the datatype package
Christian Urban <urbanc@in.tum.de>
parents: 3051
diff changeset
  2468
51ef8a3cb6ef updated to lates changes in the datatype package
Christian Urban <urbanc@in.tum.de>
parents: 3051
diff changeset
  2469
51ef8a3cb6ef updated to lates changes in the datatype package
Christian Urban <urbanc@in.tum.de>
parents: 3051
diff changeset
  2470
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2471
lemma supp_perm_perm_eq:
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2472
  assumes a: "\<forall>a \<in> supp x. p \<bullet> a = q \<bullet> a"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2473
  shows "p \<bullet> x = q \<bullet> x"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2474
proof -
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2475
  from a have "\<forall>a \<in> supp x. (-q + p) \<bullet> a = a" by simp
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2476
  then have "\<forall>a \<in> supp x. a \<notin> supp (-q + p)" 
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2477
    unfolding supp_perm by simp
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2478
  then have "supp x \<sharp>* (-q + p)"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2479
    unfolding fresh_star_def fresh_def by simp
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2480
  then have "(-q + p) \<bullet> x = x" by (simp only: supp_perm_eq)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2481
  then show "p \<bullet> x = q \<bullet> x"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2482
    by (metis permute_minus_cancel permute_plus)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2483
qed
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2484
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2485
text {* disagreement set *}
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2486
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2487
definition
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2488
  dset :: "perm \<Rightarrow> perm \<Rightarrow> atom set"
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2489
where
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2490
  "dset p q = {a::atom. p \<bullet> a \<noteq> q \<bullet> a}"
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2491
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2492
lemma ds_fresh:
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2493
  assumes "dset p q \<sharp>* x"
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2494
  shows "p \<bullet> x = q \<bullet> x"
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2495
using assms
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2496
unfolding dset_def fresh_star_def fresh_def
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2497
by (auto intro: supp_perm_perm_eq)
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2498
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2499
lemma atom_set_perm_eq:
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2500
  assumes a: "as \<sharp>* p"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2501
  shows "p \<bullet> as = as"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2502
proof -
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2503
  from a have "supp p \<subseteq> {a. a \<notin> as}"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2504
    unfolding supp_perm fresh_star_def fresh_def by auto
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2505
  then show "p \<bullet> as = as"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2506
  proof (induct p rule: perm_struct_induct)
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2507
    case zero
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2508
    show "0 \<bullet> as = as" by simp
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2509
  next
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2510
    case (swap p a b)
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2511
    then have "a \<notin> as" "b \<notin> as" "p \<bullet> as = as" by simp_all
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2512
    then show "((a \<rightleftharpoons> b) + p) \<bullet> as = as" by (simp add: swap_set_not_in)
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2513
  qed
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2514
qed
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2515
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2516
section {* Avoiding of atom sets *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2517
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2518
text {* 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2519
  For every set of atoms, there is another set of atoms
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2520
  avoiding a finitely supported c and there is a permutation
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2521
  which 'translates' between both sets.
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2522
*}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2523
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2524
lemma at_set_avoiding_aux:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2525
  fixes Xs::"atom set"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2526
  and   As::"atom set"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2527
  assumes b: "Xs \<subseteq> As"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2528
  and     c: "finite As"
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2529
  shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) = (Xs \<union> (p \<bullet> Xs))"
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2530
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2531
  from b c have "finite Xs" by (rule finite_subset)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2532
  then show ?thesis using b
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2533
  proof (induct rule: finite_subset_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2534
    case empty
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2535
    have "0 \<bullet> {} \<inter> As = {}" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2536
    moreover
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2537
    have "supp (0::perm) = {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2538
    ultimately show ?case by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2539
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2540
    case (insert x Xs)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2541
    then obtain p where
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2542
      p1: "(p \<bullet> Xs) \<inter> As = {}" and 
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2543
      p2: "supp p = (Xs \<union> (p \<bullet> Xs))" by blast
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2544
    from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2545
    with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2546
    hence px: "p \<bullet> x = x" unfolding supp_perm by simp
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2547
    have "finite (As \<union> p \<bullet> Xs \<union> supp p)"
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2548
      using `finite As` `finite Xs`
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2549
      by (simp add: permute_set_eq_image finite_supp)
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2550
    then obtain y where "y \<notin> (As \<union> p \<bullet> Xs \<union> supp p)" "sort_of y = sort_of x"
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2551
      by (rule obtain_atom)
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2552
    hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "y \<notin> supp p" "sort_of y = sort_of x"
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2553
      by simp_all
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2554
    hence py: "p \<bullet> y = y" "x \<noteq> y" using `x \<in> As`
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2555
      by (auto simp add: supp_perm)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2556
    let ?q = "(x \<rightleftharpoons> y) + p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2557
    have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2558
      unfolding insert_eqvt
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2559
      using `p \<bullet> x = x` `sort_of y = sort_of x`
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2560
      using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2561
      by (simp add: swap_atom swap_set_not_in)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2562
    have "?q \<bullet> insert x Xs \<inter> As = {}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2563
      using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2564
      unfolding q by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2565
    moreover
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2566
    have "supp (x \<rightleftharpoons> y) \<inter> supp p = {}" using px py `sort_of y = sort_of x`
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2567
      unfolding supp_swap by (simp add: supp_perm)
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2568
    then have "supp ?q = (supp (x \<rightleftharpoons> y) \<union> supp p)" 
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2569
      by (simp add: supp_plus_perm_eq)
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2570
    then have "supp ?q = insert x Xs \<union> ?q \<bullet> insert x Xs"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2571
      using p2 `sort_of y = sort_of x` `x \<noteq> y` unfolding q supp_swap
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2572
      by auto
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2573
    ultimately show ?case by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2574
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2575
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2576
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2577
lemma at_set_avoiding:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2578
  assumes a: "finite Xs"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2579
  and     b: "finite (supp c)"
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2580
  obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) = (Xs \<union> (p \<bullet> Xs))"
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2581
  using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2582
  unfolding fresh_star_def fresh_def by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2583
2589
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2584
lemma at_set_avoiding1:
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2585
  assumes "finite xs"
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2586
  and     "finite (supp c)"
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2587
  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c"
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2588
using assms
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2589
apply(erule_tac c="c" in at_set_avoiding)
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2590
apply(auto)
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2591
done
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2592
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2593
lemma at_set_avoiding2:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2594
  assumes "finite xs"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2595
  and     "finite (supp c)" "finite (supp x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2596
  and     "xs \<sharp>* x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2597
  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2598
using assms
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2599
apply(erule_tac c="(c, x)" in at_set_avoiding)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2600
apply(simp add: supp_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2601
apply(rule_tac x="p" in exI)
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2602
apply(simp add: fresh_star_Pair)
2507
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2603
apply(rule fresh_star_supp_conv)
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2604
apply(auto simp add: fresh_star_def)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2605
done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2606
2573
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2607
lemma at_set_avoiding3:
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2608
  assumes "finite xs"
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2609
  and     "finite (supp c)" "finite (supp x)"
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2610
  and     "xs \<sharp>* x"
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2611
  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p \<and> supp p = xs \<union> (p \<bullet> xs)"
2586
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2612
using assms
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2613
apply(erule_tac c="(c, x)" in at_set_avoiding)
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2614
apply(simp add: supp_Pair)
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2615
apply(rule_tac x="p" in exI)
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2616
apply(simp add: fresh_star_Pair)
2586
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2617
apply(rule fresh_star_supp_conv)
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2618
apply(auto simp add: fresh_star_def)
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2619
done
2573
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2620
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2621
lemma at_set_avoiding2_atom:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2622
  assumes "finite (supp c)" "finite (supp x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2623
  and     b: "a \<sharp> x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2624
  shows "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2625
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2626
  have a: "{a} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2627
  obtain p where p1: "(p \<bullet> {a}) \<sharp>* c" and p2: "supp x \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2628
    using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2629
  have c: "(p \<bullet> a) \<sharp> c" using p1
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2630
    unfolding fresh_star_def Ball_def 
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
  2631
    by(erule_tac x="p \<bullet> a" in allE) (simp add: permute_set_def)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2632
  hence "p \<bullet> a \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2633
  then show "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p" by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2634
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2635
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2636
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2637
section {* Renaming permutations *}
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2638
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2639
lemma set_renaming_perm:
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2640
  assumes b: "finite bs"
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2641
  shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2642
using b
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2643
proof (induct)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2644
  case empty
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2645
  have "(\<forall>b \<in> {}. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> {} \<union> p \<bullet> {}"
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
  2646
    by (simp add: permute_set_def supp_perm)
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2647
  then show "\<exists>q. (\<forall>b \<in> {}. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> {} \<union> p \<bullet> {}" by blast
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2648
next
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2649
  case (insert a bs)
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2650
  then have " \<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> p \<bullet> bs" by simp 
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2651
  then obtain q where *: "\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> bs \<union> p \<bullet> bs"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2652
    by (metis empty_subsetI insert(3) supp_swap) 
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2653
  { assume 1: "q \<bullet> a = p \<bullet> a"
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2654
    have "\<forall>b \<in> (insert a bs). q \<bullet> b = p \<bullet> b" using 1 * by simp
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2655
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2656
    have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2657
      using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2658
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2659
    have "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2660
  }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2661
  moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2662
  { assume 2: "q \<bullet> a \<noteq> p \<bullet> a"
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2663
    def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q"
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2664
    have "\<forall>b \<in> insert a bs. q' \<bullet> b = p \<bullet> b" using 2 * `a \<notin> bs` unfolding q'_def
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2665
      by (auto simp add: swap_atom)
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2666
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2667
    { have "{q \<bullet> a, p \<bullet> a} \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2668
	using ** 
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2669
	apply (auto simp add: supp_perm insert_eqvt)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2670
	apply (subgoal_tac "q \<bullet> a \<in> bs \<union> p \<bullet> bs")
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2671
	apply(auto)[1]
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2672
	apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}")
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2673
	apply(blast)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2674
	apply(simp)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2675
	done
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2676
      then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by (simp add: supp_swap)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2677
      moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2678
      have "supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2679
	using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2680
      ultimately 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2681
      have "supp q' \<subseteq> insert a bs \<union> p \<bullet> insert a bs" 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2682
        unfolding q'_def using supp_plus_perm by blast
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2683
    }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2684
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2685
    have "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs" by blast
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2686
  }
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2687
  ultimately show "\<exists>q. (\<forall>b \<in> insert a bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> insert a bs \<union> p \<bullet> insert a bs"
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2688
    by blast
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2689
qed
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2690
2672
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2691
lemma set_renaming_perm2:
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2692
  shows "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)"
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2693
proof -
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2694
  have "finite (bs \<inter> supp p)" by (simp add: finite_supp)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2695
  then obtain q 
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2696
    where *: "\<forall>b \<in> bs \<inter> supp p. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> (bs \<inter> supp p) \<union> (p \<bullet> (bs \<inter> supp p))"
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2697
    using set_renaming_perm by blast
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2698
  from ** have "supp q \<subseteq> bs \<union> (p \<bullet> bs)" by (auto simp add: inter_eqvt)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2699
  moreover
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2700
  have "\<forall>b \<in> bs - supp p. q \<bullet> b = p \<bullet> b" 
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2701
    apply(auto)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2702
    apply(subgoal_tac "b \<notin> supp q")
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2703
    apply(simp add: fresh_def[symmetric])
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2704
    apply(simp add: fresh_perm)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2705
    apply(clarify)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2706
    apply(rotate_tac 2)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2707
    apply(drule subsetD[OF **])
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2708
    apply(simp add: inter_eqvt supp_eqvt permute_self)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2709
    done
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2710
  ultimately have "(\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" using * by auto
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2711
  then show "\<exists>q. (\<forall>b \<in> bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> bs \<union> (p \<bullet> bs)" by blast
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2712
qed
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2713
    
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2714
lemma list_renaming_perm:
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2715
  shows "\<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> (p \<bullet> set bs)"
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2716
proof (induct bs)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2717
  case (Cons a bs)
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2718
  then have " \<exists>q. (\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set bs \<union> p \<bullet> (set bs)"  by simp
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2719
  then obtain q where *: "\<forall>b \<in> set bs. q \<bullet> b = p \<bullet> b" and **: "supp q \<subseteq> set bs \<union> p \<bullet> (set bs)"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2720
    by (blast)
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2721
  { assume 1: "a \<in> set bs"
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2722
    have "q \<bullet> a = p \<bullet> a" using * 1 by (induct bs) (auto)
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2723
    then have "\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b" using * by simp 
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2724
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2725
    have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2726
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2727
    have "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2728
  }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2729
  moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2730
  { assume 2: "a \<notin> set bs"
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2731
    def q' \<equiv> "((q \<bullet> a) \<rightleftharpoons> (p \<bullet> a)) + q"
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2732
    have "\<forall>b \<in> set (a # bs). q' \<bullet> b = p \<bullet> b" 
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2733
      unfolding q'_def using 2 * `a \<notin> set bs` by (auto simp add: swap_atom)
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2734
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2735
    { have "{q \<bullet> a, p \<bullet> a} \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2736
	using **
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2737
	apply (auto simp add: supp_perm insert_eqvt)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2738
	apply (subgoal_tac "q \<bullet> a \<in> set bs \<union> p \<bullet> set bs")
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2739
	apply(auto)[1]
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2740
	apply(subgoal_tac "q \<bullet> a \<in> {a. q \<bullet> a \<noteq> a}")
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2741
	apply(blast)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2742
	apply(simp)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2743
	done
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2744
      then have "supp (q \<bullet> a \<rightleftharpoons> p \<bullet> a) \<subseteq> set (a # bs) \<union> p \<bullet> set (a # bs)" by (simp add: supp_swap)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2745
      moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2746
      have "supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2747
	using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2748
      ultimately 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2749
      have "supp q' \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2750
        unfolding q'_def using supp_plus_perm by blast
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2751
    }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2752
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2753
    have "\<exists>q. (\<forall>b \<in> set (a # bs).  q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))" by blast
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2754
  }
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2755
  ultimately show "\<exists>q. (\<forall>b \<in> set (a # bs). q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set (a # bs) \<union> p \<bullet> (set (a # bs))"
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2756
    by blast
2771
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2757
next
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2758
 case Nil
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2759
  have "(\<forall>b \<in> set []. 0 \<bullet> b = p \<bullet> b) \<and> supp (0::perm) \<subseteq> set [] \<union> p \<bullet> set []" 
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2760
    by (simp add: supp_zero_perm)
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2761
  then show "\<exists>q. (\<forall>b \<in> set []. q \<bullet> b = p \<bullet> b) \<and> supp q \<subseteq> set [] \<union> p \<bullet> (set [])" by blast
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2762
qed
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2763
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2764
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2765
section {* Concrete Atoms Types *}
1962
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2766
1972
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2767
text {*
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2768
  Class @{text at_base} allows types containing multiple sorts of atoms.
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2769
  Class @{text at} only allows types with a single sort.
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2770
*}
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2771
1962
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2772
class at_base = pt +
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2773
  fixes atom :: "'a \<Rightarrow> atom"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2774
  assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2775
  assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2776
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2777
declare atom_eqvt[eqvt]
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2778
1962
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2779
class at = at_base +
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2780
  assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2781
2900
d66430c7c4f1 an alternative FCB for Abs_lst1; seems simpler but not as simple as I thought; not sure whether it generalises to multiple binders.
Christian Urban <urbanc@in.tum.de>
parents: 2891
diff changeset
  2782
lemma sort_ineq [simp]:
d66430c7c4f1 an alternative FCB for Abs_lst1; seems simpler but not as simple as I thought; not sure whether it generalises to multiple binders.
Christian Urban <urbanc@in.tum.de>
parents: 2891
diff changeset
  2783
  assumes "sort_of (atom a) \<noteq> sort_of (atom b)"
d66430c7c4f1 an alternative FCB for Abs_lst1; seems simpler but not as simple as I thought; not sure whether it generalises to multiple binders.
Christian Urban <urbanc@in.tum.de>
parents: 2891
diff changeset
  2784
  shows "atom a \<noteq> atom b"
d66430c7c4f1 an alternative FCB for Abs_lst1; seems simpler but not as simple as I thought; not sure whether it generalises to multiple binders.
Christian Urban <urbanc@in.tum.de>
parents: 2891
diff changeset
  2785
using assms by metis
d66430c7c4f1 an alternative FCB for Abs_lst1; seems simpler but not as simple as I thought; not sure whether it generalises to multiple binders.
Christian Urban <urbanc@in.tum.de>
parents: 2891
diff changeset
  2786
1962
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2787
lemma supp_at_base: 
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2788
  fixes a::"'a::at_base"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2789
  shows "supp a = {atom a}"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2790
  by (simp add: supp_atom [symmetric] supp_def atom_eqvt)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2791
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2792
lemma fresh_at_base: 
2891
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2793
  shows  "sort_of a \<noteq> sort_of (atom b) \<Longrightarrow> a \<sharp> b"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2794
  and "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2795
  unfolding fresh_def 
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2796
  apply(simp_all add: supp_at_base)
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2797
  apply(metis)
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2798
  done
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  2799
  
2609
666ffc8a92a9 freshness theorem in strong exhausts; (temporarily includes a cheat_tac to make all tests go through)
Christian Urban <urbanc@in.tum.de>
parents: 2599
diff changeset
  2800
lemma fresh_atom_at_base: 
666ffc8a92a9 freshness theorem in strong exhausts; (temporarily includes a cheat_tac to make all tests go through)
Christian Urban <urbanc@in.tum.de>
parents: 2599
diff changeset
  2801
  fixes b::"'a::at_base"
666ffc8a92a9 freshness theorem in strong exhausts; (temporarily includes a cheat_tac to make all tests go through)
Christian Urban <urbanc@in.tum.de>
parents: 2599
diff changeset
  2802
  shows "a \<sharp> atom b \<longleftrightarrow> a \<sharp> b"
666ffc8a92a9 freshness theorem in strong exhausts; (temporarily includes a cheat_tac to make all tests go through)
Christian Urban <urbanc@in.tum.de>
parents: 2599
diff changeset
  2803
  by (simp add: fresh_def supp_at_base supp_atom)
666ffc8a92a9 freshness theorem in strong exhausts; (temporarily includes a cheat_tac to make all tests go through)
Christian Urban <urbanc@in.tum.de>
parents: 2599
diff changeset
  2804
2611
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2805
lemma fresh_star_atom_at_base: 
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2806
  fixes b::"'a::at_base"
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2807
  shows "as \<sharp>* atom b \<longleftrightarrow> as \<sharp>* b"
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2808
  by (simp add: fresh_star_def fresh_atom_at_base)
2609
666ffc8a92a9 freshness theorem in strong exhausts; (temporarily includes a cheat_tac to make all tests go through)
Christian Urban <urbanc@in.tum.de>
parents: 2599
diff changeset
  2809
1962
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2810
instance at_base < fs
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2811
proof qed (simp add: supp_at_base)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2812
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2813
lemma at_base_infinite [simp]:
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2814
  shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2815
proof
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2816
  obtain a :: 'a where "True" by auto
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2817
  assume "finite ?U"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2818
  hence "finite (atom ` ?U)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2819
    by (rule finite_imageI)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2820
  then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2821
    by (rule obtain_atom)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2822
  from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2823
    unfolding atom_eqvt [symmetric]
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2824
    by (simp add: swap_atom)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2825
  hence "b \<in> atom ` ?U" by simp
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2826
  with b(1) show "False" by simp
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2827
qed
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2828
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2829
lemma swap_at_base_simps [simp]:
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2830
  fixes x y::"'a::at_base"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2831
  shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2832
  and   "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2833
  and   "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2834
  unfolding atom_eq_iff [symmetric]
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2835
  unfolding atom_eqvt [symmetric]
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2836
  by simp_all
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2837
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2838
lemma obtain_at_base:
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2839
  assumes X: "finite X"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2840
  obtains a::"'a::at_base" where "atom a \<notin> X"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2841
proof -
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2842
  have "inj (atom :: 'a \<Rightarrow> atom)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2843
    by (simp add: inj_on_def)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2844
  with X have "finite (atom -` X :: 'a set)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2845
    by (rule finite_vimageI)
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2846
  with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2847
    by auto
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2848
  then obtain a :: 'a where "atom a \<notin> X"
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2849
    by auto
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2850
  thus ?thesis ..
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2851
qed
84a13d1e2511 moved mk_atom into the library; that meant that concrete atom classes need to be in Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1941
diff changeset
  2852
2685
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2853
lemma obtain_fresh':
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2854
  assumes fin: "finite (supp x)"
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2855
  obtains a::"'a::at_base" where "atom a \<sharp> x"
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2856
using obtain_at_base[where X="supp x"]
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2857
by (auto simp add: fresh_def fin)
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2858
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2859
lemma obtain_fresh:
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2860
  fixes x::"'b::fs"
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2861
  obtains a::"'a::at_base" where "atom a \<sharp> x"
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2862
  by (rule obtain_fresh') (auto simp add: finite_supp)
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2863
1973
fc5ce7f22b74 use the more general type-class at_base
Christian Urban <urbanc@in.tum.de>
parents: 1972
diff changeset
  2864
lemma supp_finite_set_at_base:
1971
8daf6ff5e11a simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1962
diff changeset
  2865
  assumes a: "finite S"
8daf6ff5e11a simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1962
diff changeset
  2866
  shows "supp S = atom ` S"
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2867
apply(simp add: supp_of_finite_sets[OF a])
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  2868
apply(simp add: supp_at_base)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  2869
apply(auto)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  2870
done
1971
8daf6ff5e11a simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1962
diff changeset
  2871
2743
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2872
(* FIXME 
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2873
lemma supp_cofinite_set_at_base:
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2874
  assumes a: "finite (UNIV - S)"
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2875
  shows "supp S = atom ` (UNIV - S)"
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2876
apply(rule finite_supp_unique)
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2877
*)
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2878
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2879
lemma fresh_finite_set_at_base:
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2880
  fixes a::"'a::at_base"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2881
  assumes a: "finite S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2882
  shows "atom a \<sharp> S \<longleftrightarrow> a \<notin> S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2883
  unfolding fresh_def
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2884
  apply(simp add: supp_finite_set_at_base[OF a])
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2885
  apply(subst inj_image_mem_iff)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2886
  apply(simp add: inj_on_def)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2887
  apply(simp)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2888
  done
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2889
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2890
lemma fresh_at_base_permute_iff [simp]:
2683
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2891
  fixes a::"'a::at_base"
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2892
  shows "atom (p \<bullet> a) \<sharp> p \<bullet> x \<longleftrightarrow> atom a \<sharp> x"
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2893
  unfolding atom_eqvt[symmetric]
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2894
  by (simp add: fresh_permute_iff)
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2895
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2896
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2897
section {* Infrastructure for concrete atom types *}
1971
8daf6ff5e11a simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
parents: 1962
diff changeset
  2898
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2899
definition
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2900
  flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2901
where
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2902
  "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2903
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2904
lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2905
  unfolding flip_def by (rule swap_self)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2906
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2907
lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2908
  unfolding flip_def by (rule swap_commute)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2909
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2910
lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2911
  unfolding flip_def by (rule minus_swap)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2912
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2913
lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2914
  unfolding flip_def by (rule swap_cancel)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2915
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2916
lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2917
  unfolding permute_plus [symmetric] add_flip_cancel by simp
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2918
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2919
lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2920
  by (simp add: flip_commute)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2921
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2922
lemma flip_eqvt [eqvt]: 
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2923
  fixes a b c::"'a::at_base"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2924
  shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2925
  unfolding flip_def
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2926
  by (simp add: swap_eqvt atom_eqvt)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2927
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2928
lemma flip_at_base_simps [simp]:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2929
  shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2930
  and   "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2931
  and   "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2932
  and   "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2933
  unfolding flip_def
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2934
  unfolding atom_eq_iff [symmetric]
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2935
  unfolding atom_eqvt [symmetric]
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2936
  by simp_all
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2937
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2938
text {* the following two lemmas do not hold for at_base, 
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2939
  only for single sort atoms from at *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2940
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2941
lemma permute_flip_at:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2942
  fixes a b c::"'a::at"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2943
  shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2944
  unfolding flip_def
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2945
  apply (rule atom_eq_iff [THEN iffD1])
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2946
  apply (subst atom_eqvt [symmetric])
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2947
  apply (simp add: swap_atom)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2948
  done
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2949
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2950
lemma flip_at_simps [simp]:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2951
  fixes a b::"'a::at"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2952
  shows "(a \<leftrightarrow> b) \<bullet> a = b" 
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2953
  and   "(a \<leftrightarrow> b) \<bullet> b = a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2954
  unfolding permute_flip_at by simp_all
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2955
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2956
lemma flip_fresh_fresh:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2957
  fixes a b::"'a::at_base"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2958
  assumes "atom a \<sharp> x" "atom b \<sharp> x"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2959
  shows "(a \<leftrightarrow> b) \<bullet> x = x"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2960
using assms
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2961
by (simp add: flip_def swap_fresh_fresh)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2962
2683
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2963
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2964
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2965
subsection {* Syntax for coercing at-elements to the atom-type *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2966
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2967
syntax
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2968
  "_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2969
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2970
translations
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2971
  "_atom_constrain a t" => "CONST atom (_constrain a t)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2972
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2973
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2974
subsection {* A lemma for proving instances of class @{text at}. *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2975
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2976
setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2977
setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2978
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2979
text {*
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2980
  New atom types are defined as subtypes of @{typ atom}.
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2981
*}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2982
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2983
lemma exists_eq_simple_sort: 
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2984
  shows "\<exists>a. a \<in> {a. sort_of a = s}"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2985
  by (rule_tac x="Atom s 0" in exI, simp)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2986
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2987
lemma exists_eq_sort: 
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2988
  shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2989
  by (rule_tac x="Atom (sort_fun x) y" in exI, simp)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2990
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2991
lemma at_base_class:
2847
Christian Urban <urbanc@in.tum.de>
parents: 2820
diff changeset
  2992
  fixes sort_fun :: "'b \<Rightarrow> atom_sort"
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2993
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2994
  assumes type: "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2995
  assumes atom_def: "\<And>a. atom a = Rep a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2996
  assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2997
  shows "OFCLASS('a, at_base_class)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2998
proof
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2999
  interpret type_definition Rep Abs "{a. sort_of a \<in> range sort_fun}" by (rule type)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3000
  have sort_of_Rep: "\<And>a. sort_of (Rep a) \<in> range sort_fun" using Rep by simp
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3001
  fix a b :: 'a and p p1 p2 :: perm
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3002
  show "0 \<bullet> a = a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3003
    unfolding permute_def by (simp add: Rep_inverse)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3004
  show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3005
    unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3006
  show "atom a = atom b \<longleftrightarrow> a = b"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3007
    unfolding atom_def by (simp add: Rep_inject)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3008
  show "p \<bullet> atom a = atom (p \<bullet> a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3009
    unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3010
qed
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3011
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3012
(*
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3013
lemma at_class:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3014
  fixes s :: atom_sort
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3015
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3016
  assumes type: "type_definition Rep Abs {a. sort_of a \<in> range (\<lambda>x::unit. s)}"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3017
  assumes atom_def: "\<And>a. atom a = Rep a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3018
  assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3019
  shows "OFCLASS('a, at_class)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3020
proof
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3021
  interpret type_definition Rep Abs "{a. sort_of a \<in> range (\<lambda>x::unit. s)}" by (rule type)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3022
  have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3023
  fix a b :: 'a and p p1 p2 :: perm
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3024
  show "0 \<bullet> a = a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3025
    unfolding permute_def by (simp add: Rep_inverse)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3026
  show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3027
    unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3028
  show "sort_of (atom a) = sort_of (atom b)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3029
    unfolding atom_def by (simp add: sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3030
  show "atom a = atom b \<longleftrightarrow> a = b"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3031
    unfolding atom_def by (simp add: Rep_inject)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3032
  show "p \<bullet> atom a = atom (p \<bullet> a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3033
    unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3034
qed
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3035
*)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3036
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3037
lemma at_class:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3038
  fixes s :: atom_sort
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3039
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3040
  assumes type: "type_definition Rep Abs {a. sort_of a = s}"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3041
  assumes atom_def: "\<And>a. atom a = Rep a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3042
  assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3043
  shows "OFCLASS('a, at_class)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3044
proof
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3045
  interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3046
  have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3047
  fix a b :: 'a and p p1 p2 :: perm
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3048
  show "0 \<bullet> a = a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3049
    unfolding permute_def by (simp add: Rep_inverse)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3050
  show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3051
    unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3052
  show "sort_of (atom a) = sort_of (atom b)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3053
    unfolding atom_def by (simp add: sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3054
  show "atom a = atom b \<longleftrightarrow> a = b"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3055
    unfolding atom_def by (simp add: Rep_inject)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3056
  show "p \<bullet> atom a = atom (p \<bullet> a)"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3057
    unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3058
qed
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3059
2891
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3060
lemma at_class_sort:
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3061
  fixes s :: atom_sort
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3062
  fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3063
  fixes a::"'a"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3064
  assumes type: "type_definition Rep Abs {a. sort_of a = s}"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3065
  assumes atom_def: "\<And>a. atom a = Rep a"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3066
  shows "sort_of (atom a) = s"
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3067
  using atom_def type
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3068
  unfolding type_definition_def by simp
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3069
304dfe6cc83a the simplifier can simplify "sort (atom a)" if a is a concrete atom type declared with atom_decl
Christian Urban <urbanc@in.tum.de>
parents: 2868
diff changeset
  3070
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3071
setup {* Sign.add_const_constraint
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3072
  (@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3073
setup {* Sign.add_const_constraint
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3074
  (@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3075
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3076
section {* The freshness lemma according to Andy Pitts *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3077
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3078
lemma freshness_lemma:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3079
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3080
  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3081
  shows  "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3082
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3083
  from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3084
    by (auto simp add: fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3085
  show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3086
  proof (intro exI allI impI)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3087
    fix a :: 'a
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3088
    assume a3: "atom a \<sharp> h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3089
    show "h a = h b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3090
    proof (cases "a = b")
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3091
      assume "a = b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3092
      thus "h a = h b" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3093
    next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3094
      assume "a \<noteq> b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3095
      hence "atom a \<sharp> b" by (simp add: fresh_at_base)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3096
      with a3 have "atom a \<sharp> h b" 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3097
        by (rule fresh_fun_app)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3098
      with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3099
        by (rule swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3100
      from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3101
        by (rule swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3102
      from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3103
      also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3104
        by (rule permute_fun_app_eq)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3105
      also have "\<dots> = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3106
        using d2 by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3107
      finally show "h a = h b"  by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3108
    qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3109
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3110
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3111
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3112
lemma freshness_lemma_unique:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3113
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3114
  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3115
  shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3116
proof (rule ex_ex1I)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3117
  from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3118
    by (rule freshness_lemma)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3119
next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3120
  fix x y
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3121
  assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3122
  assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3123
  from a x y show "x = y"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3124
    by (auto simp add: fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3125
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3126
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3127
text {* packaging the freshness lemma into a function *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3128
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3129
definition
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3130
  fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3131
where
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3132
  "fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3133
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3134
lemma fresh_fun_apply:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3135
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3136
  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3137
  assumes b: "atom a \<sharp> h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3138
  shows "fresh_fun h = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3139
unfolding fresh_fun_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3140
proof (rule the_equality)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3141
  show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3142
  proof (intro strip)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3143
    fix a':: 'a
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3144
    assume c: "atom a' \<sharp> h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3145
    from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3146
    with b c show "h a' = h a" by auto
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3147
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3148
next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3149
  fix fr :: 'b
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3150
  assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3151
  with b show "fr = h a" by auto
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3152
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3153
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3154
lemma fresh_fun_apply':
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3155
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3156
  assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3157
  shows "fresh_fun h = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3158
  apply (rule fresh_fun_apply)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3159
  apply (auto simp add: fresh_Pair intro: a)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3160
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3161
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3162
lemma fresh_fun_eqvt:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3163
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3164
  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3165
  shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3166
  using a
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3167
  apply (clarsimp simp add: fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3168
  apply (subst fresh_fun_apply', assumption+)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3169
  apply (drule fresh_permute_iff [where p=p, THEN iffD2])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3170
  apply (drule fresh_permute_iff [where p=p, THEN iffD2])
2683
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  3171
  apply (simp only: atom_eqvt permute_fun_app_eq [where f=h])
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3172
  apply (erule (1) fresh_fun_apply' [symmetric])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3173
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3174
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3175
lemma fresh_fun_supports:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3176
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3177
  assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3178
  shows "(supp h) supports (fresh_fun h)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3179
  apply (simp add: supports_def fresh_def [symmetric])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3180
  apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3181
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3182
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3183
notation fresh_fun (binder "FRESH " 10)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3184
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3185
lemma FRESH_f_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3186
  fixes P :: "'a::at \<Rightarrow> 'b::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3187
  fixes f :: "'b \<Rightarrow> 'c::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3188
  assumes P: "finite (supp P)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3189
  shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3190
proof -
2685
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  3191
  obtain a::'a where "atom a \<sharp> P" using P by (rule obtain_fresh')
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3192
  show "(FRESH x. f (P x)) = f (FRESH x. P x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3193
    apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3194
    apply (cut_tac `atom a \<sharp> P`)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3195
    apply (simp add: fresh_conv_MOST)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3196
    apply (elim MOST_rev_mp, rule MOST_I, clarify)
2479
a9b6a00b1ba0 updated to Isabelle Sept 16
Christian Urban <urbanc@in.tum.de>
parents: 2475
diff changeset
  3197
    apply (simp add: permute_fun_def permute_pure fun_eq_iff)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3198
    apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3199
    apply (rule refl)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3200
    done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3201
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3202
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3203
lemma FRESH_binop_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3204
  fixes P :: "'a::at \<Rightarrow> 'b::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3205
  fixes Q :: "'a::at \<Rightarrow> 'c::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3206
  fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3207
  assumes P: "finite (supp P)" 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3208
  and     Q: "finite (supp Q)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3209
  shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3210
proof -
2685
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  3211
  from assms have "finite (supp (P, Q))" by (simp add: supp_Pair)
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  3212
  then obtain a::'a where "atom a \<sharp> (P, Q)" by (rule obtain_fresh') 
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  3213
  then have "atom a \<sharp> P" and "atom a \<sharp> Q" by (simp_all add: fresh_Pair)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3214
  show ?thesis
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3215
    apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3216
    apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3217
    apply (simp add: fresh_conv_MOST)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3218
    apply (elim MOST_rev_mp, rule MOST_I, clarify)
2479
a9b6a00b1ba0 updated to Isabelle Sept 16
Christian Urban <urbanc@in.tum.de>
parents: 2475
diff changeset
  3219
    apply (simp add: permute_fun_def permute_pure fun_eq_iff)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3220
    apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3221
    apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3222
    apply (rule refl)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3223
    done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3224
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3225
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3226
lemma FRESH_conj_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3227
  fixes P Q :: "'a::at \<Rightarrow> bool"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3228
  assumes P: "finite (supp P)" and Q: "finite (supp Q)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3229
  shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3230
using P Q by (rule FRESH_binop_iff)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3231
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3232
lemma FRESH_disj_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3233
  fixes P Q :: "'a::at \<Rightarrow> bool"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3234
  assumes P: "finite (supp P)" and Q: "finite (supp Q)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3235
  shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3236
using P Q by (rule FRESH_binop_iff)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3237
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3238
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3239
section {* Library functions for the nominal infrastructure *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3240
1833
2050b5723c04 added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
parents: 1774
diff changeset
  3241
use "nominal_library.ML"
2050b5723c04 added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
parents: 1774
diff changeset
  3242
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  3243
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3244
section {* Automation for creating concrete atom types *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3245
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3246
text {* at the moment only single-sort concrete atoms are supported *}
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3247
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3248
use "nominal_atoms.ML"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3249
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3250
2733
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3251
section {* automatic equivariance procedure for inductive definitions *}
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3252
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3253
use "nominal_eqvt.ML"
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3254
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3255
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  3256
2948
b0b2adafb6d2 made the tests go through again
Christian Urban <urbanc@in.tum.de>
parents: 2908
diff changeset
  3257
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  3258
end