Nominal/Nominal2_Base.thy
author Christian Urban <urbanc@in.tum.de>
Sat, 17 Mar 2012 05:13:59 +0000
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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"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   632
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   633
instance 
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   634
proof
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   635
  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
   636
  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
   637
  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
   638
qed
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   639
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   640
end
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   641
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
   642
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
   643
  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
   644
  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
   645
  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
   646
  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
   647
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
   648
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
   649
  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
   650
  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
   651
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   652
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   653
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
   654
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   655
instantiation char :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   656
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   657
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   658
definition "p \<bullet> (c::char) = c"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   659
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   660
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   661
by (default) (simp_all add: permute_char_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   662
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   663
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   664
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   665
instantiation nat :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   666
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   667
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   668
definition "p \<bullet> (n::nat) = n"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   669
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   670
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   671
by (default) (simp_all add: permute_nat_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   672
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   673
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   674
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   675
instantiation int :: pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   676
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   677
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   678
definition "p \<bullet> (i::int) = i"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   679
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   680
instance 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
   681
by (default) (simp_all add: permute_int_def)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   682
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   683
end
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   684
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
section {* Pure types *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   687
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   688
text {* Pure types will have always empty support. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   689
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   690
class pure = pt +
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   691
  assumes permute_pure: "p \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   692
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   693
text {* Types @{typ unit} and @{typ bool} are pure. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   694
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   695
instance unit :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   696
proof qed (rule permute_unit_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   697
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   698
instance bool :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   699
proof qed (rule permute_bool_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   700
2635
64b4cb2c2bf8 simple cases for string rule inductions
Christian Urban <urbanc@in.tum.de>
parents: 2632
diff changeset
   701
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   702
text {* Other type constructors preserve purity. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   703
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   704
instance "fun" :: (pure, pure) pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   705
by default (simp add: permute_fun_def permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   706
3104
f7c4b8e6918b updated to explicit set type constructor (post Isabelle 3rd January)
Christian Urban <urbanc@in.tum.de>
parents: 3101
diff changeset
   707
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
   708
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
   709
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
   710
instance prod :: (pure, pure) pure
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   711
by default (induct_tac x, simp add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   712
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
   713
instance sum :: (pure, pure) pure
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   714
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
   715
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   716
instance list :: (pure) pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   717
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
   718
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   719
instance option :: (pure) pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   720
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
   721
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
subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   724
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   725
instance char :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   726
proof qed (rule permute_char_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   727
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   728
instance nat :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   729
proof qed (rule permute_nat_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   730
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   731
instance int :: pure
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   732
proof qed (rule permute_int_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
   733
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
   734
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   735
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
   736
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
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
   738
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
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
   740
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
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   742
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
   743
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
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
   745
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
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
   747
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
   748
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
lemmas [eqvt] =
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   751
  (* pt types *)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   752
  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
   753
  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
   754
  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
   755
  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
   756
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
  (* 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
   758
  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
   759
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
  (* 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
   761
  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
   762
3121
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   763
  (* multisets *)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
   764
  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
   765
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
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
   767
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
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
   769
  "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
   770
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
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
   772
  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
   773
  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
   774
  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
   775
  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
   776
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
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
   778
  fixes f :: "'a::pt \<Rightarrow> 'b::pt"
2753
445518561867 eqvt_lambda without eta-expansion
Christian Urban <urbanc@in.tum.de>
parents: 2743
diff changeset
   779
  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
   780
  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
   781
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
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
   783
  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
   784
  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
   785
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
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
   787
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
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
   789
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
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
   791
 {* 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
   792
 {* 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
   793
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
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
   795
 {* 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
   796
 {* 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
   797
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   798
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   799
subsubsection {* Equivariance for permutations and swapping *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   800
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   801
lemma permute_eqvt:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   802
  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
   803
  unfolding permute_perm_def by simp
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   804
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
   805
(* 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
   806
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
   807
  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
   808
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
   809
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
   810
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
   811
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
   812
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   813
lemma zero_perm_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   814
  shows "p \<bullet> (0::perm) = 0"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   815
  unfolding permute_perm_def by simp
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   816
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   817
lemma add_perm_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   818
  fixes p p1 p2 :: perm
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   819
  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
   820
  unfolding permute_perm_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   821
  by (simp add: perm_eq_iff)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   822
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   823
lemma swap_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   824
  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
   825
  unfolding permute_perm_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   826
  by (auto simp add: swap_atom perm_eq_iff)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   827
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   828
lemma uminus_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   829
  fixes p q::"perm"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   830
  shows "p \<bullet> (- q) = - (p \<bullet> q)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   831
  unfolding permute_perm_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   832
  by (simp add: diff_minus minus_add add_assoc)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   833
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
   834
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
   835
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
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
   837
  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
   838
  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
   839
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
lemma Not_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   841
  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
   842
  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
   843
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
lemma conj_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   845
  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
   846
  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
   847
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
lemma imp_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   849
  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
   850
  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
   851
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
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
   853
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
   854
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
   855
  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
   856
  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
   857
  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
   858
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
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
   860
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
   861
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
   862
  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
   863
  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
   864
  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
   865
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
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
   867
  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
   868
  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
   869
  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
   870
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
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
   872
  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
   873
  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
   874
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
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
   876
  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
   877
  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
   878
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
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
   880
  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
   881
  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
   882
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
lemma disj_eqvt [eqvt]:
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   884
  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
   885
  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
   886
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
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
   888
  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
   889
  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
   890
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
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
   892
  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
   893
  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
   894
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
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
   896
  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
   897
  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
   898
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
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
   900
  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
   901
  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
   902
  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
   903
  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
   904
  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
   905
  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
   906
  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
   907
  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
   908
  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
   909
  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
   910
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
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
   912
  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
   913
  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
   914
  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
   915
  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
   916
  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
   917
  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
   918
  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
   919
  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
   920
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   921
subsubsection {* Equivariance of Set operators *}
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   922
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   923
lemma mem_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   924
  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
   925
  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
   926
  by (auto)
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   927
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   928
lemma Collect_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   929
  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
   930
  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
   931
  by (auto simp add: permute_bool_def)
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   932
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   933
lemma inter_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   934
  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
   935
  unfolding Int_def
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   936
  by (perm_simp) (rule refl)
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   937
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   938
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
   939
  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
   940
  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
   941
  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
   942
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   943
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
   944
  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
   945
  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
   946
  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
   947
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   948
lemma image_eqvt [eqvt]:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   949
  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
   950
  unfolding image_def
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   951
  by (perm_simp) (rule refl)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
   952
3050
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   953
lemma Image_eqvt [eqvt]:
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   954
  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
   955
  unfolding Image_def
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   956
  by (perm_simp) (rule refl)
7519ebb41145 added eqvt-lemma for Image
Christian Urban <urbanc@in.tum.de>
parents: 3026
diff changeset
   957
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   958
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
   959
  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
   960
  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
   961
  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
   962
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   963
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
   964
  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
   965
  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
   966
  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
   967
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   968
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
   969
  fixes A B :: "'a::pt set"
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   970
  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
   971
  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
   972
  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
   973
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   974
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
   975
  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
   976
  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
   977
  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
   978
  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
   979
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   980
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
   981
  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
   982
  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
   983
  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
   984
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   985
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
   986
  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
   987
  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
   988
  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
   989
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   990
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
   991
  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
   992
  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
   993
  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
   994
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
   995
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
   996
  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
   997
  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
   998
  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
   999
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
  1000
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
  1001
  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
  1002
  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
  1003
  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
  1004
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1005
(* 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
  1006
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
  1007
  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
  1008
unfolding Sigma_def
3026
b037ae269f50 updated to Isabelle 19 Sept
Christian Urban <urbanc@in.tum.de>
parents: 2987
diff changeset
  1009
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
  1010
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
  1011
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
  1012
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
  1013
  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
  1014
  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
  1015
  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
  1016
  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
  1017
*}
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1018
75a95431cd8b proved that lfp is equivariant (that simplifies equivariance proofs of inductively defined predicates)
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1019
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
  1020
  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
  1021
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
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
  1023
  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
  1024
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
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
  1026
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
  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
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
  1029
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
  1030
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
  1031
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
  1032
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
  1033
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
  1034
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
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
  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
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
  1038
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
  1039
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
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
  1041
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
  1042
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
  1043
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
  1044
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
  1045
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
  1046
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
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
  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
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
  1050
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
  1051
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
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
  1053
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
  1054
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
  1055
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
  1056
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
  1057
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
  1058
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
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
  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
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
  1062
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
  1063
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
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
  1065
apply(default)
2987
27aab7a105eb updated for new Isabelle (11. Aug.)
Christian Urban <urbanc@in.tum.de>
parents: 2982
diff changeset
  1066
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
  1067
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
  1068
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
  1069
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
  1070
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
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
  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
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
  1074
  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
  1075
  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
  1076
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
  1077
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
  1078
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1079
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
  1080
  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
  1081
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
  1082
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
  1083
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1084
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1085
subsubsection {* Equivariance for product operations *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1086
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1087
lemma fst_eqvt [eqvt]:
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1088
  shows "p \<bullet> (fst x) = fst (p \<bullet> x)"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1089
  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
  1090
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1091
lemma snd_eqvt [eqvt]:
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1092
  shows "p \<bullet> (snd x) = snd (p \<bullet> x)"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1093
  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
  1094
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1095
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
  1096
  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
  1097
  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
  1098
  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
  1099
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1100
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1101
subsubsection {* Equivariance for list operations *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1102
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1103
lemma append_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1104
  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
  1105
  by (induct xs) auto
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1106
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1107
lemma rev_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1108
  shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1109
  by (induct xs) (simp_all add: append_eqvt)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1110
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1111
lemma map_eqvt [eqvt]: 
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1112
  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
  1113
  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
  1114
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1115
lemma removeAll_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1116
  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
  1117
  by (induct xs) (auto)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1118
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1119
lemma filter_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1120
  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
  1121
apply(induct xs)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1122
apply(simp)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1123
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
  1124
apply(simp only: permute_fun_app_eq)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1125
done
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1126
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1127
lemma distinct_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1128
  shows "p \<bullet> (distinct xs) = distinct (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1129
apply(induct xs)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1130
apply(simp add: permute_bool_def)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1131
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
  1132
done
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1133
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1134
lemma length_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1135
  shows "p \<bullet> (length xs) = length (p \<bullet> xs)"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1136
by (induct xs) (simp_all add: permute_pure)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1137
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1138
2972
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1139
subsubsection {* Equivariance for @{typ "'a option"} *}
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1140
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1141
lemma option_map_eqvt[eqvt]:
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1142
  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
  1143
  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
  1144
84afb941df53 moved eqvt for Option.map
Christian Urban <urbanc@in.tum.de>
parents: 2955
diff changeset
  1145
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1146
subsubsection {* Equivariance for @{typ "'a fset"} *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1147
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1148
lemma in_fset_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1149
  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
  1150
unfolding in_fset
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1151
by (perm_simp) (simp)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1152
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1153
lemma union_fset_eqvt [eqvt]:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1154
  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
  1155
  by (induct S) (simp_all)
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1156
3101
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1157
lemma inter_fset_eqvt [eqvt]:
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1158
  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
  1159
  apply(descending)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1160
  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
  1161
  apply(perm_simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1162
  apply(simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1163
  done
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1164
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1165
lemma subset_fset_eqvt [eqvt]:
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1166
  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
  1167
  apply(descending)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1168
  unfolding sub_list_def
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1169
  apply(perm_simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1170
  apply(simp)
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1171
  done
09acd7e116e8 added two eqvt lemmas for fset-operators
Christian Urban <urbanc@in.tum.de>
parents: 3065
diff changeset
  1172
  
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1173
lemma map_fset_eqvt [eqvt]: 
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1174
  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
  1175
  by (lifting map_eqvt)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1176
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1177
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1178
section {* Supp, Freshness and Supports *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1179
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1180
context pt
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1181
begin
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1182
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1183
definition
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1184
  supp :: "'a \<Rightarrow> atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1185
where
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1186
  "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
  1187
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1188
definition
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1189
  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
  1190
where   
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1191
  "a \<sharp> x \<equiv> a \<notin> supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1192
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1193
end
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1194
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1195
lemma supp_conv_fresh: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1196
  shows "supp x = {a. \<not> a \<sharp> x}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1197
  unfolding fresh_def by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1198
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1199
lemma swap_rel_trans:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1200
  assumes "sort_of a = sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1201
  assumes "sort_of b = sort_of c"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1202
  assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1203
  assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1204
  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1205
proof (cases)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1206
  assume "a = b \<or> c = b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1207
  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
  1208
next
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1209
  assume *: "\<not> (a = b \<or> c = b)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1210
  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
  1211
    using assms by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1212
  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
  1213
    using assms * by (simp add: swap_triple)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1214
  finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1215
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1216
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1217
lemma swap_fresh_fresh:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1218
  assumes a: "a \<sharp> x" 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1219
  and     b: "b \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1220
  shows "(a \<rightleftharpoons> b) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1221
proof (cases)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1222
  assume asm: "sort_of a = sort_of b" 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1223
  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
  1224
    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
  1225
  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
  1226
  then obtain c 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1227
    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
  1228
    by (rule obtain_atom) (auto)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1229
  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
  1230
next
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1231
  assume "sort_of a \<noteq> sort_of b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1232
  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
  1233
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1234
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1235
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1236
subsection {* supp and fresh are equivariant *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1237
2760
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1238
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1239
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
  1240
  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
  1241
  unfolding supp_def
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1242
  by (perm_simp)
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1243
     (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
  1244
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1245
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
  1246
  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
  1247
  unfolding fresh_def
8f833ebc4b58 eqvt of supp and fresh is proved using equivariance infrastructure
Christian Urban <urbanc@in.tum.de>
parents: 2753
diff changeset
  1248
  by (perm_simp) (rule refl)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1249
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1250
lemma fresh_permute_iff:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1251
  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
  1252
  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
  1253
2683
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1254
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
  1255
  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
  1256
proof
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1257
  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
  1258
  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
  1259
  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
  1260
next
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1261
  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
  1262
  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
  1263
  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
  1264
qed
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1265
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  1266
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1267
section {* supports *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1268
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1269
definition
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1270
  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
  1271
where  
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1272
  "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
  1273
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1274
lemma supp_is_subset:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1275
  fixes S :: "atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1276
  and   x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1277
  assumes a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1278
  and     a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1279
  shows "(supp x) \<subseteq> S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1280
proof (rule ccontr)
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1281
  assume "\<not> (supp x \<subseteq> S)"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1282
  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
  1283
  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
  1284
  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
  1285
  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
  1286
  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
  1287
  with b1 show False by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1288
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1289
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1290
lemma supports_finite:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1291
  fixes S :: "atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1292
  and   x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1293
  assumes a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1294
  and     a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1295
  shows "finite (supp x)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1296
proof -
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1297
  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
  1298
  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
  1299
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1300
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1301
lemma supp_supports:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1302
  fixes x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1303
  shows "(supp x) supports x"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1304
unfolding supports_def
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1305
proof (intro strip)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1306
  fix a b
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1307
  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
  1308
  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
  1309
  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
  1310
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1311
2955
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1312
lemma supports_fresh:
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1313
  fixes x :: "'a::pt"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1314
  assumes a1: "S supports x"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1315
  and     a2: "finite S"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1316
  and     a3: "a \<notin> S"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1317
  shows "a \<sharp> x"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1318
unfolding fresh_def
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1319
proof -
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1320
  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
  1321
  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
  1322
qed
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1323
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1324
lemma supp_is_least_supports:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1325
  fixes S :: "atom set"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1326
  and   x :: "'a::pt"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1327
  assumes  a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1328
  and      a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1329
  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
  1330
  shows "(supp x) = S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1331
proof (rule equalityI)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1332
  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
  1333
  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
  1334
  have "(supp x) supports x" by (rule supp_supports)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1335
  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
  1336
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1337
2955
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1338
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1339
lemma subsetCI: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1340
  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
  1341
  by auto
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1342
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1343
lemma finite_supp_unique:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1344
  assumes a1: "S supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1345
  assumes a2: "finite S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1346
  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
  1347
  shows "(supp x) = S"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1348
  using a1 a2
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1349
proof (rule supp_is_least_supports)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1350
  fix S'
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1351
  assume "finite S'" and "S' supports x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1352
  show "S \<subseteq> S'"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1353
  proof (rule subsetCI)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1354
    fix a
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1355
    assume "a \<in> S" and "a \<notin> S'"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1356
    have "finite (S \<union> S')"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1357
      using `finite S` `finite S'` by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1358
    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
  1359
      by (rule obtain_atom)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1360
    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
  1361
      by simp_all
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1362
    then have "(a \<rightleftharpoons> b) \<bullet> x = x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1363
      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
  1364
    moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1365
      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
  1366
      by (rule a3)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1367
    ultimately show "False" by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1368
  qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1369
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1370
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
  1371
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
  1372
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
  1373
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
  1374
  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
  1375
  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
  1376
*}
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
  1377
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
  1378
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
  1379
  "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
  1380
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
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1382
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1383
section {* Finitely-supported types *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1384
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1385
class fs = pt +
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1386
  assumes finite_supp: "finite (supp x)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1387
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1388
lemma pure_supp: 
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1389
  fixes x::"'a::pure"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1390
  shows "supp x = {}"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1391
  unfolding supp_def by (simp add: permute_pure)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1392
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1393
lemma pure_fresh:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1394
  fixes x::"'a::pure"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1395
  shows "a \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1396
  unfolding fresh_def by (simp add: pure_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1397
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1398
instance pure < fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1399
by default (simp add: pure_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1400
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
subsection  {* Type @{typ atom} is finitely-supported. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1403
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1404
lemma supp_atom:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1405
  shows "supp a = {a}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1406
apply (rule finite_supp_unique)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1407
apply (clarsimp simp add: supports_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1408
apply simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1409
apply simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1410
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1411
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1412
lemma fresh_atom: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1413
  shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1414
  unfolding fresh_def supp_atom by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1415
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1416
instance atom :: fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1417
by default (simp add: supp_atom)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1418
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
  1419
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1420
section {* Type @{typ perm} is finitely-supported. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1421
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1422
lemma perm_swap_eq:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1423
  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
  1424
unfolding permute_perm_def
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1425
by (metis add_diff_cancel minus_perm_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1426
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1427
lemma supports_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1428
  shows "{a. p \<bullet> a \<noteq> a} supports p"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1429
  unfolding supports_def
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1430
  unfolding perm_swap_eq
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1431
  by (simp add: swap_eqvt)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1432
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1433
lemma finite_perm_lemma: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1434
  shows "finite {a::atom. p \<bullet> a \<noteq> a}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1435
  using finite_Rep_perm [of p]
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1436
  unfolding permute_atom_def .
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1437
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1438
lemma supp_perm:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1439
  shows "supp p = {a. p \<bullet> a \<noteq> a}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1440
apply (rule finite_supp_unique)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1441
apply (rule supports_perm)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1442
apply (rule finite_perm_lemma)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1443
apply (simp add: perm_swap_eq swap_eqvt)
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1444
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
  1445
done
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 fresh_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1448
  shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1449
  unfolding fresh_def 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1450
  by (simp add: supp_perm)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1451
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1452
lemma supp_swap:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1453
  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
  1454
  by (auto simp add: supp_perm swap_atom)
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_zero_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1457
  shows "a \<sharp> (0::perm)"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1458
  unfolding fresh_perm by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1459
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1460
lemma supp_zero_perm: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1461
  shows "supp (0::perm) = {}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1462
  unfolding supp_perm by simp
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1463
1087
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1464
lemma fresh_plus_perm:
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1465
  fixes p q::perm
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1466
  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
  1467
  shows "a \<sharp> (p + q)"
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1468
  using assms
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1469
  unfolding fresh_def
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1470
  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
  1471
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1472
lemma supp_plus_perm:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1473
  fixes p q::perm
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1474
  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
  1475
  by (auto simp add: supp_perm)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1476
1087
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1477
lemma fresh_minus_perm:
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1478
  fixes p::perm
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1479
  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
  1480
  unfolding fresh_def
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1481
  unfolding supp_perm
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1482
  apply(simp)
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1483
  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
  1484
  done
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1485
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1486
lemma supp_minus_perm:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1487
  fixes p::perm
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1488
  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
  1489
  unfolding supp_conv_fresh
bb7f4457091a moved some lemmas to Nominal; updated all files
Christian Urban <urbanc@in.tum.de>
parents: 1062
diff changeset
  1490
  by (simp add: fresh_minus_perm)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1491
1305
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1492
lemma plus_perm_eq:
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1493
  fixes p q::"perm"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1494
  assumes asm: "supp p \<inter> supp q = {}"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1495
  shows "p + q = q + p"
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1496
unfolding perm_eq_iff
1305
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1497
proof
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1498
  fix a::"atom"
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1499
  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
  1500
  proof -
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1501
    { 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
  1502
      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
  1503
	by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1504
    }
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1505
    moreover
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1506
    { 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
  1507
      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
  1508
      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
  1509
      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
  1510
	by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1511
    }
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1512
    moreover
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1513
    { 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
  1514
      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
  1515
      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
  1516
      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
  1517
	by (simp add: supp_perm)
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1518
    }
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1519
    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
  1520
      using asm by blast
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1521
  qed
61319a9af976 updated (added lemma about commuting permutations)
Christian Urban <urbanc@in.tum.de>
parents: 1258
diff changeset
  1522
qed
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1523
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1524
lemma supp_plus_perm_eq:
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1525
  fixes p q::perm
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1526
  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
  1527
  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
  1528
proof -
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1529
  { fix a::"atom"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1530
    assume "a \<in> supp p"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1531
    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
  1532
    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
  1533
      by (simp add: supp_perm)
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1534
  }
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1535
  moreover
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1536
  { fix a::"atom"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1537
    assume "a \<in> supp q"
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1538
    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
  1539
    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
  1540
      by (simp add: supp_perm)
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 asm plus_perm_eq
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1542
      by metis
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
  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
  1545
    by blast
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1546
  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
  1547
    by blast
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1548
qed
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1549
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1550
instance perm :: fs
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1551
by default (simp add: supp_perm finite_perm_lemma)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1552
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1553
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  1554
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1555
section {* Finite Support instances for other types *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1556
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1557
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1558
subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1559
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1560
lemma supp_Pair: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1561
  shows "supp (x, y) = supp x \<union> supp y"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1562
  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
  1563
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1564
lemma fresh_Pair: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1565
  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
  1566
  by (simp add: fresh_def supp_Pair)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1567
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1568
lemma supp_Unit:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1569
  shows "supp () = {}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1570
  by (simp add: supp_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1571
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1572
lemma fresh_Unit:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1573
  shows "a \<sharp> ()"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1574
  by (simp add: fresh_def supp_Unit)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1575
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
  1576
instance prod :: (fs, fs) fs
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1577
apply default
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1578
apply (case_tac x)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1579
apply (simp add: supp_Pair finite_supp)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1580
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1581
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1582
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1583
subsection {* Type @{typ "'a + 'b"} is finitely supported *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1584
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1585
lemma supp_Inl: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1586
  shows "supp (Inl x) = supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1587
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1588
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1589
lemma supp_Inr: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1590
  shows "supp (Inr x) = supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1591
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1592
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1593
lemma fresh_Inl: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1594
  shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1595
  by (simp add: fresh_def supp_Inl)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1596
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1597
lemma fresh_Inr: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1598
  shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1599
  by (simp add: fresh_def supp_Inr)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1600
2378
2f13fe48c877 updated to new Isabelle; made FSet more "quiet"
Christian Urban <urbanc@in.tum.de>
parents: 2310
diff changeset
  1601
instance sum :: (fs, fs) fs
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1602
apply default
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1603
apply (case_tac x)
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1604
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
  1605
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1606
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1607
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1608
subsection {* Type @{typ "'a option"} is finitely supported *}
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1609
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1610
lemma supp_None: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1611
  shows "supp None = {}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1612
by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1613
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1614
lemma supp_Some: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1615
  shows "supp (Some x) = supp x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1616
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1617
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1618
lemma fresh_None: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1619
  shows "a \<sharp> None"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1620
  by (simp add: fresh_def supp_None)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1621
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1622
lemma fresh_Some: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1623
  shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1624
  by (simp add: fresh_def supp_Some)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1625
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1626
instance option :: (fs) fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1627
apply default
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1628
apply (induct_tac x)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1629
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
  1630
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1631
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1632
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1633
subsubsection {* Type @{typ "'a list"} is finitely supported *}
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
lemma supp_Nil: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1636
  shows "supp [] = {}"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1637
  by (simp add: supp_def)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1638
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1639
lemma fresh_Nil: 
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1640
  shows "a \<sharp> []"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1641
  by (simp add: fresh_def supp_Nil)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1642
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1643
lemma supp_Cons: 
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1644
  shows "supp (x # xs) = supp x \<union> supp xs"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1645
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
  1646
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1647
lemma fresh_Cons: 
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1648
  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
  1649
  by (simp add: fresh_def supp_Cons)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1650
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1651
lemma supp_append:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1652
  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
  1653
  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
  1654
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1655
lemma fresh_append:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1656
  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
  1657
  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
  1658
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
  1659
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
  1660
  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
  1661
  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
  1662
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1663
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
  1664
  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
  1665
  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
  1666
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1667
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
  1668
  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
  1669
  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
  1670
  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
  1671
     (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
  1672
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1673
lemma supp_of_atom_list:
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1674
  fixes as::"atom list"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1675
  shows "supp as = set as"
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1676
by (induct as)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1677
   (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
  1678
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1679
instance list :: (fs) fs
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1680
apply default
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1681
apply (induct_tac x)
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1682
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
  1683
done
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1684
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1685
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  1686
section {* Support and Freshness for Applications *}
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1687
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1688
lemma fresh_conv_MOST: 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1689
  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
  1690
  unfolding fresh_def supp_def 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1691
  unfolding MOST_iff_cofinite by simp
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1692
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1693
lemma fresh_fun_app:
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1694
  assumes "a \<sharp> f" and "a \<sharp> x" 
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1695
  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
  1696
  using assms
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1697
  unfolding fresh_conv_MOST
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1698
  unfolding permute_fun_app_eq
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  1699
  by (elim MOST_rev_mp) (simp)
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1700
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1701
lemma supp_fun_app:
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1702
  shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
1879
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1703
  using fresh_fun_app
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1704
  unfolding fresh_def
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1705
  by auto
869d1183e082 tuned proofs
Christian Urban <urbanc@in.tum.de>
parents: 1833
diff changeset
  1706
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  1707
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1708
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
  1709
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1710
definition
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1711
  "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
  1712
2868
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1713
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
  1714
  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
  1715
  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
  1716
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
  1717
2b8e387d2dfc got rid of the boolean flag in the raw_equivariance function
Christian Urban <urbanc@in.tum.de>
parents: 2849
diff changeset
  1718
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1719
text {* equivariance of a function at a given argument *}
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1720
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1721
definition
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1722
 "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
  1723
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1724
lemma eqvtI:
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1725
  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
  1726
unfolding eqvt_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1727
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
  1728
2955
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1729
lemma eqvt_at_perm:
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1730
  assumes "eqvt_at f x"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1731
  shows "eqvt_at f (q \<bullet> x)"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1732
proof -
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1733
  { fix p::"perm"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1734
    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
  1735
      using assms by (simp add: eqvt_at_def)
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1736
    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
  1737
    also have "\<dots> = f ((p + q) \<bullet> x)"
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1738
      using assms by (simp add: eqvt_at_def)
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1739
    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
  1740
  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
  1741
    by simp
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1742
qed
4049a2651dd9 more on the NBE function
Christian Urban <urbanc@in.tum.de>
parents: 2948
diff changeset
  1743
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1744
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
  1745
  assumes a: "eqvt f"
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1746
  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
  1747
  using a
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1748
  unfolding eqvt_def
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1749
  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
  1750
  by simp
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1751
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1752
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
  1753
  assumes a: "eqvt f"
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1754
  shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1755
proof -
1941
Christian Urban <urbanc@in.tum.de>
parents: 1933
diff changeset
  1756
  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
  1757
  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
  1758
    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
  1759
    using supp_fun_app by auto
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1760
qed
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  1761
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
  1762
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
  1763
  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
  1764
  shows "supp (f x) \<subseteq> supp x"
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1765
  using fresh_fun_eqvt_app[OF a]
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1766
  unfolding fresh_def
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1767
  by auto
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1768
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1769
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
  1770
  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
  1771
  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
  1772
  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
  1773
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
  1774
unfolding supports_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1775
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
  1776
using asm
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1777
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
  1778
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
  1779
apply(rule fin)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1780
done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1781
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1782
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
  1783
  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
  1784
  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
  1785
  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
  1786
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
  1787
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
  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 fresh_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
  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
  1795
  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
  1796
using fresh
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1797
unfolding fresh_def
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1798
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
  1799
by auto
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1800
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
  1801
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  1802
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
  1803
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1804
lemma THE_defaultI2:
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1805
  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
  1806
  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
  1807
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
  1808
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1809
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
  1810
  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
  1811
  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
  1812
  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
  1813
  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
  1814
  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
  1815
  apply(rule unique)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1816
  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
  1817
  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
  1818
  apply(simp)
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1819
  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
  1820
  done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1821
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1822
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
  1823
  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
  1824
  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
  1825
  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
  1826
  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
  1827
  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
  1828
  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
  1829
  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
  1830
  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
  1831
  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
  1832
  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
  1833
  apply(perm_simp add: permute_minus_cancel)
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1834
  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
  1835
  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
  1836
  apply(rule ex1)
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1837
  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
  1838
  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
  1839
  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
  1840
  apply(rule ex1)
2849
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1841
  apply(perm_simp)
31c338d562fd tuned some proofs
Christian Urban <urbanc@in.tum.de>
parents: 2848
diff changeset
  1842
  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
  1843
  apply(simp)
2663
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1844
  done
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1845
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1846
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
  1847
  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
  1848
  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
  1849
  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
  1850
  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
  1851
  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
  1852
  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
  1853
  using assms
54aade5d0fe6 moved high level code from LamTest into the main libraries.
Christian Urban <urbanc@in.tum.de>
parents: 2659
diff changeset
  1854
  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
  1855
2818
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1856
lemma fundef_ex1_prop:
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1857
  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
  1858
  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
  1859
  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
  1860
  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
  1861
  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
  1862
  unfolding f_def
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1863
  using ex1
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1864
  apply(erule_tac ex1E)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1865
  apply(rule THE_defaultI2)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1866
  apply(blast)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1867
  apply(rule P_all)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1868
  apply(assumption)
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1869
  done
8fe80e9f796d added a more general lemma fro fundef_ex1
Christian Urban <urbanc@in.tum.de>
parents: 2810
diff changeset
  1870
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  1871
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1872
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
  1873
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1874
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
  1875
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1876
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
  1877
  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
  1878
  assumes "finite S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1879
  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
  1880
  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
  1881
  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
  1882
  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
  1883
  apply(rule assms)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1884
  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
  1885
done
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1886
2742
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1887
lemma supp_cofinite_atom_set:
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1888
  fixes S::"atom set"
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1889
  assumes "finite (UNIV - S)"
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1890
  shows "supp S = (UNIV - S)"
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1891
  apply(rule finite_supp_unique)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1892
  apply(simp add: supports_def)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1893
  apply(simp add: swap_set_both_in)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1894
  apply(rule assms)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1895
  apply(subst swap_commute)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1896
  apply(simp add: swap_set_in)
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1897
done
f1192e3474e0 more on the paper
Christian Urban <urbanc@in.tum.de>
parents: 2735
diff changeset
  1898
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1899
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
  1900
  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
  1901
  assumes "finite S"
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1902
  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
  1903
  unfolding fresh_def
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  1904
  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
  1905
2679
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1906
lemma fresh_minus_atom_set:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1907
  fixes S::"atom set"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1908
  assumes "finite S"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1909
  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
  1910
  unfolding fresh_def
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  1911
  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
  1912
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1913
lemma Union_supports_set:
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1914
  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
  1915
proof -
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1916
  { fix a b
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1917
    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
  1918
      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
  1919
  }
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1920
  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
  1921
    unfolding supports_def 
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1922
    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
  1923
qed
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1924
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1925
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
  1926
  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
  1927
  assumes fin: "finite S"   
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1928
  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
  1929
  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
  1930
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1931
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
  1932
  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
  1933
  assumes fin: "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1934
  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
  1935
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
  1936
  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
  1937
    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
  1938
    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
  1939
  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
  1940
    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
  1941
  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
  1942
  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
  1943
    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
  1944
  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
  1945
qed
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1946
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1947
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
  1948
  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
  1949
  assumes fin: "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1950
  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
  1951
apply(rule subset_antisym)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1952
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
  1953
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
  1954
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
  1955
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
  1956
done
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1957
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1958
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
  1959
  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
  1960
  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
  1961
  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
  1962
using assms
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1963
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
  1964
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1965
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
  1966
  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
  1967
  assumes fin1: "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1968
  and     fin2: "finite T"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1969
  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
  1970
  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
  1971
  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
  1972
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  1973
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
  1974
  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
  1975
  assumes fin:  "finite S"
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  1976
  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
  1977
  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
  1978
  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
  1979
2588
8f5420681039 completed the strong exhausts rules for Foo2 using general lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2587
diff changeset
  1980
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
  1981
  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
  1982
  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
  1983
  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
  1984
  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
  1985
  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
  1986
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1987
lemma supp_set_empty:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1988
  shows "supp {} = {}"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1989
  unfolding supp_def
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1990
  by (simp add: empty_eqvt)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1991
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1992
lemma fresh_set_empty:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1993
  shows "a \<sharp> {}"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1994
  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
  1995
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1996
lemma supp_set:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1997
  fixes xs :: "('a::fs) list"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1998
  shows "supp (set xs) = supp xs"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  1999
apply(induct xs)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2000
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
  2001
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
  2002
done
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2003
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2004
lemma fresh_set:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2005
  fixes xs :: "('a::fs) list"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2006
  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
  2007
unfolding fresh_def
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2008
by (simp add: supp_set)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2009
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2010
3121
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2011
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
  2012
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2013
lemma set_of_eqvt[eqvt]:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2014
  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
  2015
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
  2016
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2017
lemma supp_set_of:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2018
  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
  2019
  apply (rule supp_fun_app_eqvt)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2020
  unfolding eqvt_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2021
  apply(perm_simp)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2022
  apply(simp)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2023
  done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2024
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2025
lemma Union_finite_multiset:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2026
  fixes M::"'a::fs multiset"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2027
  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
  2028
proof - 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2029
  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
  2030
    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
  2031
  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
  2032
    by (simp only: image_Collect)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2033
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2034
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2035
lemma Union_supports_multiset:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2036
  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
  2037
proof -
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2038
  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
  2039
    unfolding permute_multiset_def 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2040
    apply(induct M)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2041
    apply(simp_all)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2042
    done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2043
  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
  2044
    unfolding supports_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2045
    apply(clarify)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2046
    apply(rule sw)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2047
    apply(rule swap_fresh_fresh)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2048
    apply(simp_all only: fresh_def)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2049
    apply(auto)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2050
    apply(metis neq0_conv)+
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
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2053
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2054
lemma Union_included_multiset:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2055
  fixes M::"('a::fs multiset)" 
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2056
  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
  2057
proof -
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2058
  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
  2059
  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
  2060
  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
  2061
  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
  2062
  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
  2063
qed
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2064
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2065
lemma supp_of_multisets:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2066
  fixes M::"('a::fs multiset)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2067
  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
  2068
apply(rule subset_antisym)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2069
apply(rule supp_is_subset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2070
apply(rule Union_supports_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2071
apply(rule Union_finite_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2072
apply(rule Union_included_multiset)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2073
done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2074
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2075
lemma multisets_supp_finite:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2076
  fixes M::"('a::fs multiset)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2077
  shows "finite (supp M)"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2078
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
  2079
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2080
lemma supp_of_multiset_union:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2081
  fixes M N::"('a::fs) multiset"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2082
  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
  2083
  by (auto simp add: supp_of_multisets)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2084
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2085
lemma supp_empty_mset [simp]:
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2086
  shows "supp {#} = {}"
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2087
  unfolding supp_def
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2088
  by simp
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2089
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2090
instance multiset :: (fs) fs
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2091
  apply (default)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2092
  apply (rule multisets_supp_finite)
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2093
  done
878de0084b62 added fs and pt for multisets
Christian Urban <urbanc@in.tum.de>
parents: 3104
diff changeset
  2094
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2095
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
  2096
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2097
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
  2098
  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
  2099
  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
  2100
  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
  2101
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2102
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
  2103
  shows "supp {||} = {}"
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2104
  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
  2105
  by simp
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2106
2641
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2107
lemma fresh_empty_fset:
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2108
  shows "a \<sharp> {||}"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2109
unfolding fresh_def
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2110
by (simp)
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2111
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2112
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
  2113
  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
  2114
  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
  2115
  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
  2116
  apply(subst supp_fset[symmetric])
2587
78623a0f294b tuned proof to reduce number of warnings
Christian Urban <urbanc@in.tum.de>
parents: 2586
diff changeset
  2117
  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
  2118
  done
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2119
2641
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2120
lemma fresh_insert_fset:
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2121
  fixes x::"'a::fs"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2122
  and   S::"'a fset"
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2123
  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
  2124
  unfolding fresh_def
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2125
  by (simp)
592d17e26e09 some further lemmas for fsets
Christian Urban <urbanc@in.tum.de>
parents: 2635
diff changeset
  2126
2565
6bf332360510 moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
parents: 2560
diff changeset
  2127
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
  2128
  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
  2129
  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
  2130
  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
  2131
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
  2132
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
  2133
  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
  2134
  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
  2135
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
  2136
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  2137
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
  2138
  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
  2139
  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
  2140
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
  2141
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
  2142
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2143
instance fset :: (fs) fs
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2144
  apply (default)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2145
  apply (rule fset_finite_supp)
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2146
  done
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2147
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
  2148
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2149
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
  2150
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2151
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
  2152
  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
  2153
  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
  2154
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2155
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
  2156
  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
  2157
  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
  2158
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2159
(* 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
  2160
(* 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
  2161
lemma [simp]:
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2162
  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
  2163
  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
  2164
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2165
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
  2166
  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
  2167
  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
  2168
  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
  2169
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2170
declaration {* fn _ =>
3051
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2171
let
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2172
  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
  2173
in
a06de111c70e updated to Isabelle 26 Nov
Christian Urban <urbanc@in.tum.de>
parents: 3050
diff changeset
  2174
  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
  2175
end
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2176
*}
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2177
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2178
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
  2179
  induction principles. *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2180
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2181
definition 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2182
  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
  2183
where 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2184
  "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
  2185
2507
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2186
lemma fresh_star_supp_conv:
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2187
  shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2188
by (auto simp add: fresh_star_def fresh_def)
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2189
2675
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2190
lemma fresh_star_perm_set_conv:
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2191
  fixes p::"perm"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2192
  assumes fresh: "as \<sharp>* p"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2193
  and     fin: "finite as"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2194
  shows "supp p \<sharp>* as"
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2195
apply(rule fresh_star_supp_conv)
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2196
apply(simp add: supp_finite_atom_set fin fresh)
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2197
done
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2198
2679
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2199
lemma fresh_star_atom_set_conv:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2200
  assumes fresh: "as \<sharp>* bs"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2201
  and     fin: "finite as" "finite bs"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2202
  shows "bs \<sharp>* as"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2203
using fresh
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2204
unfolding fresh_star_def fresh_def
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2205
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
  2206
2730
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2207
lemma atom_fresh_star_disjoint:
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2208
  assumes fin: "finite bs" 
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2209
  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
  2210
eebc24b9cf39 added a lemma about fresh_star and Abs
Christian Urban <urbanc@in.tum.de>
parents: 2708
diff changeset
  2211
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
  2212
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
  2213
2675
68ccf847507d defined properly substitution
Christian Urban <urbanc@in.tum.de>
parents: 2672
diff changeset
  2214
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2215
lemma fresh_star_Pair:
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2216
  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
  2217
  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
  2218
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2219
lemma fresh_star_list:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2220
  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
  2221
  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
  2222
  and   "as \<sharp>* []"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2223
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
  2224
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2225
lemma fresh_star_set:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2226
  fixes xs::"('a::fs) list"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2227
  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
  2228
unfolding fresh_star_def
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2229
by (simp add: fresh_set)
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2230
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
  2231
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
  2232
  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
  2233
  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
  2234
  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
  2235
3d101f2f817c simple cases for strong inducts done; infrastructure for the difficult ones is there
Christian Urban <urbanc@in.tum.de>
parents: 2609
diff changeset
  2236
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
  2237
  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
  2238
  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
  2239
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
  2240
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2241
lemma fresh_star_Un:
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2242
  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
  2243
  by (auto simp add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2244
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2245
lemma fresh_star_insert:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2246
  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
  2247
  by (auto simp add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2248
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2249
lemma fresh_star_Un_elim:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2250
  "((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
  2251
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2252
  apply(rule)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2253
  apply(erule meta_mp)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2254
  apply(auto)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2255
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2256
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2257
lemma fresh_star_insert_elim:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2258
  "(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
  2259
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2260
  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
  2261
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2262
lemma fresh_star_empty_elim:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2263
  "({} \<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
  2264
  by (simp add: fresh_star_def)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2265
2632
e8732350a29f added small example for strong inductions; functions still need a sorry
Christian Urban <urbanc@in.tum.de>
parents: 2614
diff changeset
  2266
lemma fresh_star_Unit_elim: 
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2267
  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
  2268
  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
  2269
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2270
lemma fresh_star_Pair_elim: 
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2271
  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
  2272
  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
  2273
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2274
lemma fresh_star_zero:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2275
  shows "as \<sharp>* (0::perm)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2276
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2277
  by (simp add: fresh_zero_perm)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2278
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2279
lemma fresh_star_plus:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2280
  fixes p q::perm
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2281
  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
  2282
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2283
  by (simp add: fresh_plus_perm)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2284
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2285
lemma fresh_star_permute_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2286
  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
  2287
  unfolding fresh_star_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2288
  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
  2289
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
  2290
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
  2291
  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
  2292
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
  2293
by (perm_simp) (rule refl)
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2294
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2295
2735
d97e04126a3d a bit more tuning
Christian Urban <urbanc@in.tum.de>
parents: 2733
diff changeset
  2296
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2297
section {* Induction principle for permutations *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2298
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2299
lemma smaller_supp:
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2300
  assumes a: "a \<in> supp p"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2301
  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
  2302
proof -
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2303
  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
  2304
    unfolding supp_perm by (auto simp add: swap_atom)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2305
  moreover
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2306
  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
  2307
  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
  2308
  ultimately 
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2309
  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
  2310
qed
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2311
  
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2312
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2313
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
  2314
  assumes S: "supp p \<subseteq> S"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2315
  and zero: "P 0"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2316
  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
  2317
  shows "P p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2318
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2319
  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
  2320
  then show "P p" using S
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2321
  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
  2322
    case (psubset p)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2323
    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
  2324
    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
  2325
    { assume "supp p = {}"
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  2326
      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
  2327
      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
  2328
    }
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2329
    moreover
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2330
    { assume "supp p \<noteq> {}"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2331
      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
  2332
      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
  2333
        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
  2334
      let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2335
      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
  2336
      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
  2337
      moreover
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2338
      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
  2339
      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
  2340
      moreover 
2732
9abc4a70540c some slight polishing
Christian Urban <urbanc@in.tum.de>
parents: 2730
diff changeset
  2341
      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
  2342
      ultimately have "P p" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2343
    }
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2344
    ultimately show "P p" by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2345
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2346
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2347
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2348
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
  2349
  assumes zero: "P 0"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2350
  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
  2351
  shows "P p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2352
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
  2353
   (auto intro: zero swap)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2354
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
  2355
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
  2356
  assumes S: "supp p \<subseteq> S"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2357
  assumes zero: "P 0"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2358
  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
  2359
  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
  2360
  shows "P p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2361
using S
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2362
by (induct p rule: perm_struct_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2363
   (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
  2364
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
  2365
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
  2366
  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
  2367
  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
  2368
  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
  2369
  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
  2370
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
  2371
   (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
  2372
2679
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2373
lemma supp_perm_singleton:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2374
  fixes p::"perm"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2375
  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
  2376
proof -
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2377
  { assume "supp p \<subseteq> {b}"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2378
    then have "p = 0"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2379
      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
  2380
  }
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2381
  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
  2382
qed
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2383
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2384
lemma supp_perm_pair:
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2385
  fixes p::"perm"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2386
  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
  2387
proof -
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2388
  { assume "supp p \<subseteq> {a, b}"
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2389
    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
  2390
      apply (induct p rule: perm_struct_induct) 
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2391
      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
  2392
      apply (simp add: swap_commute)
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2393
      done
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2394
  }
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2395
  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
  2396
    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
  2397
qed
e003e5e36bae added Minimal file to test things
Christian Urban <urbanc@in.tum.de>
parents: 2675
diff changeset
  2398
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2399
lemma supp_perm_eq:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2400
  assumes "(supp x) \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2401
  shows "p \<bullet> x = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2402
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2403
  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
  2404
    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
  2405
  then show "p \<bullet> x = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2406
  proof (induct p rule: perm_struct_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2407
    case zero
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2408
    show "0 \<bullet> x = x" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2409
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2410
    case (swap p a b)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2411
    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
  2412
    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
  2413
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2414
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2415
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2416
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
  2417
lemma supp_perm_eq_test:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2418
  assumes "(supp x) \<sharp>* p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2419
  shows "p \<bullet> x = x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2420
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2421
  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
  2422
    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
  2423
  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
  2424
  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
  2425
    case zero
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2426
    show "0 \<bullet> x = x" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2427
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2428
    case (swap a b)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2429
    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
  2430
    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
  2431
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2432
    case (plus p1 p2)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2433
    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
  2434
    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
  2435
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2436
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2437
2591
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2438
lemma perm_supp_eq:
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2439
  assumes a: "(supp p) \<sharp>* x"
35c570891a3a isarfied some of the high-level proofs
Christian Urban <urbanc@in.tum.de>
parents: 2589
diff changeset
  2440
  shows "p \<bullet> x = x"
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2441
proof -
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2442
  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
  2443
    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
  2444
  then show "p \<bullet> x = x"
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2445
  proof (induct p rule: perm_struct_induct2)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2446
    case zero
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2447
    show "0 \<bullet> x = x" by simp
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2448
  next
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2449
    case (swap a b)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2450
    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
  2451
    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
  2452
  next
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2453
    case (plus p1 p2)
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2454
    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
  2455
    then show "(p1 + p2) \<bullet> x = x" by simp
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2456
  qed
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2457
qed
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2458
3065
51ef8a3cb6ef updated to lates changes in the datatype package
Christian Urban <urbanc@in.tum.de>
parents: 3051
diff changeset
  2459
51ef8a3cb6ef updated to lates changes in the datatype package
Christian Urban <urbanc@in.tum.de>
parents: 3051
diff changeset
  2460
51ef8a3cb6ef updated to lates changes in the datatype package
Christian Urban <urbanc@in.tum.de>
parents: 3051
diff changeset
  2461
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2462
lemma supp_perm_perm_eq:
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2463
  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
  2464
  shows "p \<bullet> x = q \<bullet> x"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2465
proof -
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2466
  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
  2467
  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
  2468
    unfolding supp_perm by simp
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2469
  then have "supp x \<sharp>* (-q + p)"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2470
    unfolding fresh_star_def fresh_def by simp
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2471
  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
  2472
  then show "p \<bullet> x = q \<bullet> x"
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2473
    by (metis permute_minus_cancel permute_plus)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2474
qed
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2475
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2476
text {* disagreement set *}
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2477
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2478
definition
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2479
  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
  2480
where
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2481
  "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
  2482
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2483
lemma ds_fresh:
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2484
  assumes "dset p q \<sharp>* x"
2907
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2485
  shows "p \<bullet> x = q \<bullet> x"
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2486
using assms
2908
ad426ba60606 renamed ds to dset (disagreement set)
Christian Urban <urbanc@in.tum.de>
parents: 2907
diff changeset
  2487
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
  2488
by (auto intro: supp_perm_perm_eq)
9096338a7985 added small lemma about disagreement set
Christian Urban <urbanc@in.tum.de>
parents: 2900
diff changeset
  2489
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2490
lemma atom_set_perm_eq:
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2491
  assumes a: "as \<sharp>* p"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2492
  shows "p \<bullet> as = as"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2493
proof -
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2494
  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
  2495
    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
  2496
  then show "p \<bullet> as = as"
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2497
  proof (induct p rule: perm_struct_induct)
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2498
    case zero
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2499
    show "0 \<bullet> as = as" by simp
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2500
  next
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2501
    case (swap p a b)
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2502
    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
  2503
    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
  2504
  qed
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2505
qed
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2506
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2507
section {* Avoiding of atom sets *}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2508
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2509
text {* 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2510
  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
  2511
  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
  2512
  which 'translates' between both sets.
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2513
*}
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2514
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2515
lemma at_set_avoiding_aux:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2516
  fixes Xs::"atom set"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2517
  and   As::"atom set"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2518
  assumes b: "Xs \<subseteq> As"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2519
  and     c: "finite As"
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2520
  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
  2521
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2522
  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
  2523
  then show ?thesis using b
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2524
  proof (induct rule: finite_subset_induct)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2525
    case empty
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2526
    have "0 \<bullet> {} \<inter> As = {}" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2527
    moreover
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2528
    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
  2529
    ultimately show ?case by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2530
  next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2531
    case (insert x Xs)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2532
    then obtain p where
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2533
      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
  2534
      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
  2535
    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
  2536
    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
  2537
    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
  2538
    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
  2539
      using `finite As` `finite Xs`
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2540
      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
  2541
    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
  2542
      by (rule obtain_atom)
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2543
    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
  2544
      by simp_all
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2545
    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
  2546
      by (auto simp add: supp_perm)
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2547
    let ?q = "(x \<rightleftharpoons> y) + p"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2548
    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
  2549
      unfolding insert_eqvt
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2550
      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
  2551
      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
  2552
      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
  2553
    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
  2554
      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
  2555
      unfolding q by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2556
    moreover
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2557
    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
  2558
      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
  2559
    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
  2560
      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
  2561
    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
  2562
      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
  2563
      by auto
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2564
    ultimately show ?case by blast
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2565
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2566
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2567
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2568
lemma at_set_avoiding:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2569
  assumes a: "finite Xs"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2570
  and     b: "finite (supp c)"
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2571
  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
  2572
  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
  2573
  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
  2574
2589
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2575
lemma at_set_avoiding1:
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2576
  assumes "finite xs"
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2577
  and     "finite (supp c)"
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2578
  shows "\<exists>p. (p \<bullet> xs) \<sharp>* c"
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2579
using assms
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2580
apply(erule_tac c="c" in at_set_avoiding)
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2581
apply(auto)
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2582
done
9781db0e2196 completed proofs in Foo2
Christian Urban <urbanc@in.tum.de>
parents: 2588
diff changeset
  2583
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2584
lemma at_set_avoiding2:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2585
  assumes "finite xs"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2586
  and     "finite (supp c)" "finite (supp x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2587
  and     "xs \<sharp>* x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2588
  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
  2589
using assms
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2590
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
  2591
apply(simp add: supp_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2592
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
  2593
apply(simp add: fresh_star_Pair)
2507
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2594
apply(rule fresh_star_supp_conv)
f5621efe5a20 changed to llncs
Christian Urban <urbanc@in.tum.de>
parents: 2479
diff changeset
  2595
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
  2596
done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2597
2573
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2598
lemma at_set_avoiding3:
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2599
  assumes "finite xs"
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2600
  and     "finite (supp c)" "finite (supp x)"
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2601
  and     "xs \<sharp>* x"
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2602
  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
  2603
using assms
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2604
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
  2605
apply(simp add: supp_Pair)
3ebc7ecfb0dd disabled the Foo examples, because of heavy work
Christian Urban <urbanc@in.tum.de>
parents: 2573
diff changeset
  2606
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
  2607
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
  2608
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
  2609
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
  2610
done
2573
6c131c089ce2 added example Foo2.thy
Christian Urban <urbanc@in.tum.de>
parents: 2568
diff changeset
  2611
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2612
lemma at_set_avoiding2_atom:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2613
  assumes "finite (supp c)" "finite (supp x)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2614
  and     b: "a \<sharp> x"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2615
  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
  2616
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2617
  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
  2618
  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
  2619
    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
  2620
  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
  2621
    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
  2622
    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
  2623
  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
  2624
  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
  2625
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2626
2614
0d7a1703fe28 a stronger statement for at_set_avoiding
Christian Urban <urbanc@in.tum.de>
parents: 2611
diff changeset
  2627
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2628
section {* Renaming permutations *}
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2629
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2630
lemma set_renaming_perm:
2659
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2631
  assumes b: "finite bs"
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2632
  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
  2633
using b
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2634
proof (induct)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2635
  case empty
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2636
  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
  2637
    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
  2638
  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
  2639
next
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2640
  case (insert a bs)
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2641
  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
  2642
  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
  2643
    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
  2644
  { 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
  2645
    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
  2646
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2647
    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
  2648
      using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2649
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2650
    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
  2651
  }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2652
  moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2653
  { 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
  2654
    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
  2655
    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
  2656
      by (auto simp add: swap_atom)
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2657
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2658
    { 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
  2659
	using ** 
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2660
	apply (auto simp add: supp_perm insert_eqvt)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2661
	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
  2662
	apply(auto)[1]
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2663
	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
  2664
	apply(blast)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2665
	apply(simp)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2666
	done
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2667
      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
  2668
      moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2669
      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
  2670
	using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2671
      ultimately 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2672
      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
  2673
        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
  2674
    }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2675
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2676
    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
  2677
  }
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2678
  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
  2679
    by blast
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2680
qed
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2681
2672
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2682
lemma set_renaming_perm2:
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2683
  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
  2684
proof -
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2685
  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
  2686
  then obtain q 
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2687
    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
  2688
    using set_renaming_perm by blast
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2689
  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
  2690
  moreover
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2691
  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
  2692
    apply(auto)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2693
    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
  2694
    apply(simp add: fresh_def[symmetric])
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2695
    apply(simp add: fresh_perm)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2696
    apply(clarify)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2697
    apply(rotate_tac 2)
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2698
    apply(drule subsetD[OF **])
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2699
    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
  2700
    done
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2701
  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
  2702
  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
  2703
qed
7e7662890477 removed finiteness assumption from set_rename_perm
Christian Urban <urbanc@in.tum.de>
parents: 2669
diff changeset
  2704
    
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2705
lemma list_renaming_perm:
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2706
  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
  2707
proof (induct bs)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2708
  case (Cons a bs)
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2709
  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
  2710
  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
  2711
    by (blast)
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2712
  { assume 1: "a \<in> set bs"
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2713
    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
  2714
    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
  2715
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2716
    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
  2717
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2718
    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
  2719
  }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2720
  moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2721
  { assume 2: "a \<notin> set bs"
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2722
    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
  2723
    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
  2724
      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
  2725
    moreover 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2726
    { 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
  2727
	using **
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2728
	apply (auto simp add: supp_perm insert_eqvt)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2729
	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
  2730
	apply(auto)[1]
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2731
	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
  2732
	apply(blast)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2733
	apply(simp)
619ecb57db38 strengthened renaming lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2657
diff changeset
  2734
	done
2599
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2735
      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
  2736
      moreover
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2737
      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
  2738
	using ** by (auto simp add: insert_eqvt)
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2739
      ultimately 
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2740
      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
  2741
        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
  2742
    }
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2743
    ultimately 
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2744
    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
  2745
  }
2668
92c001d93225 modified the renaming_perm lemmas
Christian Urban <urbanc@in.tum.de>
parents: 2663
diff changeset
  2746
  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
  2747
    by blast
2771
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2748
next
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2749
 case Nil
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2750
  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
  2751
    by (simp add: supp_zero_perm)
66ef2a2c64fb more to the pearl paper
Christian Urban <urbanc@in.tum.de>
parents: 2760
diff changeset
  2752
  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
  2753
qed
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2754
d6fe94028a5d moved general theorems into the libraries
Christian Urban <urbanc@in.tum.de>
parents: 2591
diff changeset
  2755
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  2756
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
  2757
1972
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2758
text {*
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2759
  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
  2760
  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
  2761
*}
40db835442a0 deleted left-over code
Christian Urban <urbanc@in.tum.de>
parents: 1971
diff changeset
  2762
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
  2763
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
  2764
  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
  2765
  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
  2766
  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
  2767
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
  2768
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
  2769
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
  2770
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
  2771
  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
  2772
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
  2773
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
  2774
  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
  2775
  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
  2776
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
  2777
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
  2778
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
  2779
  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
  2780
  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
  2781
  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
  2782
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
  2783
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
  2784
  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
  2785
  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
  2786
  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
  2787
  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
  2788
  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
  2789
  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
  2790
  
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
  2791
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
  2792
  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
  2793
  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
  2794
  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
  2795
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
  2796
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
  2797
  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
  2798
  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
  2799
  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
  2800
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
  2801
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
  2802
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
  2803
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
  2804
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
  2805
  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
  2806
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
  2807
  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
  2808
  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
  2809
  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
  2810
    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
  2811
  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
  2812
    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
  2813
  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
  2814
    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
  2815
    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
  2816
  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
  2817
  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
  2818
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
  2819
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
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
  2821
  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
  2822
  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
  2823
  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
  2824
  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
  2825
  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
  2826
  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
  2827
  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
  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 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
  2830
  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
  2831
  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
  2832
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
  2833
  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
  2834
    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
  2835
  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
  2836
    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
  2837
  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
  2838
    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
  2839
  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
  2840
    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
  2841
  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
  2842
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
  2843
2685
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2844
lemma obtain_fresh':
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2845
  assumes fin: "finite (supp x)"
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2846
  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
  2847
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
  2848
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
  2849
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2850
lemma obtain_fresh:
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2851
  fixes x::"'b::fs"
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  2852
  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
  2853
  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
  2854
1973
fc5ce7f22b74 use the more general type-class at_base
Christian Urban <urbanc@in.tum.de>
parents: 1972
diff changeset
  2855
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
  2856
  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
  2857
  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
  2858
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
  2859
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
  2860
apply(auto)
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  2861
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
  2862
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
  2863
(* 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
  2864
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
  2865
  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
  2866
  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
  2867
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
  2868
*)
7085ab735de7 equivariance for All and Ex can be proved in terms of their definition
Christian Urban <urbanc@in.tum.de>
parents: 2742
diff changeset
  2869
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2870
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
  2871
  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
  2872
  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
  2873
  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
  2874
  unfolding fresh_def
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2875
  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
  2876
  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
  2877
  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
  2878
  apply(simp)
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2879
  done
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2880
2776
8e0f0b2b51dd more on pearl-paper
Christian Urban <urbanc@in.tum.de>
parents: 2771
diff changeset
  2881
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
  2882
  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
  2883
  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
  2884
  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
  2885
  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
  2886
2657
1ea9c059fc0f a few lemmas about freshness for at and at_base
Christian Urban <urbanc@in.tum.de>
parents: 2641
diff changeset
  2887
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2888
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
  2889
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2890
definition
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2891
  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
  2892
where
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2893
  "(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
  2894
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2895
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
  2896
  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
  2897
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2898
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
  2899
  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
  2900
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2901
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
  2902
  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
  2903
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2904
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
  2905
  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
  2906
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2907
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
  2908
  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
  2909
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2910
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
  2911
  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
  2912
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
  2913
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
  2914
  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
  2915
  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
  2916
  unfolding flip_def
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2917
  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
  2918
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2919
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
  2920
  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
  2921
  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
  2922
  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
  2923
  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
  2924
  unfolding flip_def
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2925
  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
  2926
  unfolding atom_eqvt [symmetric]
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2927
  by simp_all
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2928
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2929
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
  2930
  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
  2931
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2932
lemma permute_flip_at:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2933
  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
  2934
  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
  2935
  unfolding flip_def
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2936
  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
  2937
  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
  2938
  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
  2939
  done
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 flip_at_simps [simp]:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2942
  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
  2943
  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
  2944
  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
  2945
  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
  2946
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2947
lemma flip_fresh_fresh:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2948
  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
  2949
  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
  2950
  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
  2951
using assms
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2952
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
  2953
2683
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2954
42c0d011a177 ported some of the old proofs to serve as testcases
Christian Urban <urbanc@in.tum.de>
parents: 2679
diff changeset
  2955
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2956
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
  2957
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2958
syntax
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2959
  "_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
  2960
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2961
translations
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2962
  "_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
  2963
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2964
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2965
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
  2966
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2967
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
  2968
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
  2969
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2970
text {*
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2971
  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
  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
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
  2975
  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
  2976
  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
  2977
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2978
lemma exists_eq_sort: 
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2979
  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
  2980
  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
  2981
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2982
lemma at_base_class:
2847
Christian Urban <urbanc@in.tum.de>
parents: 2820
diff changeset
  2983
  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
  2984
  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
  2985
  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
  2986
  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
  2987
  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
  2988
  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
  2989
proof
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  2990
  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
  2991
  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
  2992
  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
  2993
  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
  2994
    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
  2995
  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
  2996
    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
  2997
  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
  2998
    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
  2999
  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
  3000
    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
  3001
qed
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3002
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3003
(*
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3004
lemma at_class:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3005
  fixes s :: atom_sort
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3006
  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
  3007
  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
  3008
  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
  3009
  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
  3010
  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
  3011
proof
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3012
  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
  3013
  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
  3014
  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
  3015
  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
  3016
    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
  3017
  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
  3018
    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
  3019
  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
  3020
    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
  3021
  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
  3022
    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
  3023
  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
  3024
    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
  3025
qed
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3026
*)
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3027
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3028
lemma at_class:
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3029
  fixes s :: atom_sort
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3030
  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
  3031
  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
  3032
  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
  3033
  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
  3034
  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
  3035
proof
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3036
  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
  3037
  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
  3038
  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
  3039
  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
  3040
    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
  3041
  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
  3042
    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
  3043
  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
  3044
    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
  3045
  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
  3046
    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
  3047
  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
  3048
    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
  3049
qed
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3050
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
  3051
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
  3052
  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
  3053
  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
  3054
  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
  3055
  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
  3056
  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
  3057
  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
  3058
  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
  3059
  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
  3060
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
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3062
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
  3063
  (@{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
  3064
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
  3065
  (@{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
  3066
2470
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3067
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
  3068
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3069
lemma freshness_lemma:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3070
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3071
  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
  3072
  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
  3073
proof -
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3074
  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
  3075
    by (auto simp add: fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3076
  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
  3077
  proof (intro exI allI impI)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3078
    fix a :: 'a
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3079
    assume a3: "atom a \<sharp> h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3080
    show "h a = h b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3081
    proof (cases "a = b")
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3082
      assume "a = b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3083
      thus "h a = h b" by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3084
    next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3085
      assume "a \<noteq> b"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3086
      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
  3087
      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
  3088
        by (rule fresh_fun_app)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3089
      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
  3090
        by (rule swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3091
      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
  3092
        by (rule swap_fresh_fresh)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3093
      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
  3094
      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
  3095
        by (rule permute_fun_app_eq)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3096
      also have "\<dots> = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3097
        using d2 by simp
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3098
      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
  3099
    qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3100
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3101
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3102
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3103
lemma freshness_lemma_unique:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3104
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3105
  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
  3106
  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
  3107
proof (rule ex_ex1I)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3108
  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
  3109
    by (rule freshness_lemma)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3110
next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3111
  fix x y
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3112
  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
  3113
  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
  3114
  from a x y show "x = y"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3115
    by (auto simp add: fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3116
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3117
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3118
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
  3119
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3120
definition
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3121
  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
  3122
where
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3123
  "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
  3124
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3125
lemma fresh_fun_apply:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3126
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3127
  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
  3128
  assumes b: "atom a \<sharp> h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3129
  shows "fresh_fun h = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3130
unfolding fresh_fun_def
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3131
proof (rule the_equality)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3132
  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
  3133
  proof (intro strip)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3134
    fix a':: 'a
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3135
    assume c: "atom a' \<sharp> h"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3136
    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
  3137
    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
  3138
  qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3139
next
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3140
  fix fr :: 'b
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3141
  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
  3142
  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
  3143
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3144
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3145
lemma fresh_fun_apply':
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3146
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3147
  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
  3148
  shows "fresh_fun h = h a"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3149
  apply (rule fresh_fun_apply)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3150
  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
  3151
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3152
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3153
lemma fresh_fun_eqvt:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3154
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3155
  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
  3156
  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
  3157
  using a
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3158
  apply (clarsimp simp add: fresh_Pair)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3159
  apply (subst fresh_fun_apply', assumption+)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3160
  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
  3161
  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
  3162
  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
  3163
  apply (erule (1) fresh_fun_apply' [symmetric])
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3164
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3165
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3166
lemma fresh_fun_supports:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3167
  fixes h :: "'a::at \<Rightarrow> 'b::pt"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3168
  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
  3169
  shows "(supp h) supports (fresh_fun h)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3170
  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
  3171
  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
  3172
  done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3173
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3174
notation fresh_fun (binder "FRESH " 10)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3175
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3176
lemma FRESH_f_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3177
  fixes P :: "'a::at \<Rightarrow> 'b::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3178
  fixes f :: "'b \<Rightarrow> 'c::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3179
  assumes P: "finite (supp P)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3180
  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
  3181
proof -
2685
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  3182
  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
  3183
  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
  3184
    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
  3185
    apply (cut_tac `atom a \<sharp> P`)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3186
    apply (simp add: fresh_conv_MOST)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3187
    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
  3188
    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
  3189
    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
  3190
    apply (rule refl)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3191
    done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3192
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3193
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3194
lemma FRESH_binop_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3195
  fixes P :: "'a::at \<Rightarrow> 'b::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3196
  fixes Q :: "'a::at \<Rightarrow> 'c::pure"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3197
  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
  3198
  assumes P: "finite (supp P)" 
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3199
  and     Q: "finite (supp Q)"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3200
  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
  3201
proof -
2685
1df873b63cb2 added obtain_fresh lemma; tuned Lambda.thy
Christian Urban <urbanc@in.tum.de>
parents: 2683
diff changeset
  3202
  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
  3203
  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
  3204
  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
  3205
  show ?thesis
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3206
    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
  3207
    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
  3208
    apply (simp add: fresh_conv_MOST)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3209
    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
  3210
    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
  3211
    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
  3212
    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
  3213
    apply (rule refl)
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3214
    done
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3215
qed
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3216
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3217
lemma FRESH_conj_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3218
  fixes P Q :: "'a::at \<Rightarrow> bool"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3219
  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
  3220
  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
  3221
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
  3222
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3223
lemma FRESH_disj_iff:
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3224
  fixes P Q :: "'a::at \<Rightarrow> bool"
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3225
  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
  3226
  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
  3227
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
  3228
bdb1eab47161 moved everything out of Nominal_Supp
Christian Urban <urbanc@in.tum.de>
parents: 2467
diff changeset
  3229
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3230
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
  3231
1833
2050b5723c04 added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
parents: 1774
diff changeset
  3232
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
  3233
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  3234
2467
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3235
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
  3236
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3237
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
  3238
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3239
use "nominal_atoms.ML"
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3240
67b3933c3190 got rid of Nominal_Atoms (folded into Nominal2_Base)
Christian Urban <urbanc@in.tum.de>
parents: 2466
diff changeset
  3241
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
  3242
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
  3243
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3244
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
  3245
5f6fefdbf055 split the library into a basics file; merged Nominal_Eqvt into Nominal_Base
Christian Urban <urbanc@in.tum.de>
parents: 2732
diff changeset
  3246
2466
47c840599a6b cleaned a bit various thy-files in Nominal-General
Christian Urban <urbanc@in.tum.de>
parents: 2378
diff changeset
  3247
2948
b0b2adafb6d2 made the tests go through again
Christian Urban <urbanc@in.tum.de>
parents: 2908
diff changeset
  3248
1062
dfea9e739231 rollback of the test
Christian Urban <urbanc@in.tum.de>
parents: 1061
diff changeset
  3249
end