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(* Title: Nominal2_Base
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Authors: Brian Huffman, Christian Urban
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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 Infinite_Set
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"~~/src/HOL/Quotient_Examples/FSet"
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added a library for basic nominal functions; separated nominal_eqvt file
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uses ("nominal_library.ML")
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("nominal_atoms.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 i) = 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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1930
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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 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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"(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
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by (rule Rep_perm_ext)
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(simp 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])
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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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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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lemma expand_perm_eq:
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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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subsection {* Permutations for permutations *}
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instantiation perm :: pt
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begin
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definition
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"p \<bullet> q = p + q - p"
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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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lemma permute_self:
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shows "p \<bullet> p = p"
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unfolding permute_perm_def
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by (simp add: diff_minus add_assoc)
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lemma permute_eqvt:
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shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
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unfolding permute_perm_def by simp
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lemma zero_perm_eqvt:
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shows "p \<bullet> (0::perm) = 0"
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unfolding permute_perm_def by simp
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lemma add_perm_eqvt:
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fixes p p1 p2 :: perm
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shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
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unfolding permute_perm_def
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by (simp add: expand_perm_eq)
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lemma swap_eqvt:
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shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
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unfolding permute_perm_def
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by (auto simp add: swap_atom expand_perm_eq)
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lemma uminus_eqvt:
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fixes p q::"perm"
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shows "p \<bullet> (- q) = - (p \<bullet> q)"
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unfolding permute_perm_def
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by (simp add: diff_minus minus_add add_assoc)
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subsection {* Permutations for functions *}
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instantiation "fun" :: (pt, pt) pt
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begin
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definition
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"p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
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instance
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apply default
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apply (simp add: permute_fun_def)
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apply (simp add: permute_fun_def minus_add)
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done
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end
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lemma permute_fun_app_eq:
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shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
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unfolding permute_fun_def by simp
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subsection {* Permutations for booleans *}
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instantiation bool :: pt
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begin
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definition "p \<bullet> (b::bool) = b"
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instance
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apply(default)
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apply(simp_all add: permute_bool_def)
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done
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end
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lemma Not_eqvt:
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shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
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by (simp add: permute_bool_def)
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lemma conj_eqvt:
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shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"
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by (simp add: permute_bool_def)
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lemma imp_eqvt:
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shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"
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by (simp add: permute_bool_def)
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lemma ex_eqvt:
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shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
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unfolding permute_fun_def permute_bool_def
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by (auto, rule_tac x="p \<bullet> x" in exI, simp)
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lemma all_eqvt:
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shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
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unfolding permute_fun_def permute_bool_def
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by (auto, drule_tac x="p \<bullet> x" in spec, simp)
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
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lemma permute_boolE:
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diff
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fixes P::"bool"
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
changeset
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shows "p \<bullet> P \<Longrightarrow> P"
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
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by (simp add: permute_bool_def)
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
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lemma permute_boolI:
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
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fixes P::"bool"
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
Christian Urban <urbanc@in.tum.de>
diff
changeset
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shows "P \<Longrightarrow> p \<bullet> P"
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moved infinite_Un into mainstream Isabelle; moved permute_boolI/E lemmas
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diff
changeset
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by(simp add: permute_bool_def)
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subsection {* Permutations for sets *}
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lemma permute_set_eq:
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fixes x::"'a::pt"
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and p::"perm"
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shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
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unfolding permute_fun_def
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unfolding permute_bool_def
+ − 472
apply(auto simp add: mem_def)
1062
+ − 473
apply(rule_tac x="- p \<bullet> x" in exI)
+ − 474
apply(simp)
+ − 475
done
+ − 476
+ − 477
lemma permute_set_eq_image:
+ − 478
shows "p \<bullet> X = permute p ` X"
1879
+ − 479
unfolding permute_set_eq by auto
1062
+ − 480
+ − 481
lemma permute_set_eq_vimage:
+ − 482
shows "p \<bullet> X = permute (- p) -` X"
1879
+ − 483
unfolding permute_fun_def permute_bool_def
+ − 484
unfolding vimage_def Collect_def mem_def ..
1062
+ − 485
+ − 486
lemma swap_set_not_in:
+ − 487
assumes a: "a \<notin> S" "b \<notin> S"
+ − 488
shows "(a \<rightleftharpoons> b) \<bullet> S = S"
1879
+ − 489
unfolding permute_set_eq
+ − 490
using a by (auto simp add: swap_atom)
1062
+ − 491
+ − 492
lemma swap_set_in:
+ − 493
assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
+ − 494
shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
1879
+ − 495
unfolding permute_set_eq
+ − 496
using a by (auto simp add: swap_atom)
1062
+ − 497
2470
+ − 498
lemma mem_permute_iff:
+ − 499
shows "(p \<bullet> x) \<in> (p \<bullet> X) \<longleftrightarrow> x \<in> X"
+ − 500
unfolding mem_def permute_fun_def permute_bool_def
+ − 501
by simp
+ − 502
+ − 503
lemma mem_eqvt:
+ − 504
shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
+ − 505
unfolding mem_def
+ − 506
by (simp add: permute_fun_app_eq)
+ − 507
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 508
lemma empty_eqvt:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 509
shows "p \<bullet> {} = {}"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 510
unfolding empty_def Collect_def
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 511
by (simp add: permute_fun_def permute_bool_def)
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 512
2470
+ − 513
lemma insert_eqvt:
+ − 514
shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
+ − 515
unfolding permute_set_eq_image image_insert ..
+ − 516
+ − 517
1062
+ − 518
subsection {* Permutations for units *}
+ − 519
+ − 520
instantiation unit :: pt
+ − 521
begin
+ − 522
+ − 523
definition "p \<bullet> (u::unit) = u"
+ − 524
1879
+ − 525
instance
+ − 526
by (default) (simp_all add: permute_unit_def)
1062
+ − 527
+ − 528
end
+ − 529
+ − 530
+ − 531
subsection {* Permutations for products *}
+ − 532
2378
+ − 533
instantiation prod :: (pt, pt) pt
1062
+ − 534
begin
+ − 535
+ − 536
primrec
+ − 537
permute_prod
+ − 538
where
+ − 539
Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
+ − 540
+ − 541
instance
+ − 542
by default auto
+ − 543
+ − 544
end
+ − 545
+ − 546
subsection {* Permutations for sums *}
+ − 547
2378
+ − 548
instantiation sum :: (pt, pt) pt
1062
+ − 549
begin
+ − 550
+ − 551
primrec
+ − 552
permute_sum
+ − 553
where
+ − 554
"p \<bullet> (Inl x) = Inl (p \<bullet> x)"
+ − 555
| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
+ − 556
1879
+ − 557
instance
+ − 558
by (default) (case_tac [!] x, simp_all)
1062
+ − 559
+ − 560
end
+ − 561
+ − 562
subsection {* Permutations for lists *}
+ − 563
+ − 564
instantiation list :: (pt) pt
+ − 565
begin
+ − 566
+ − 567
primrec
+ − 568
permute_list
+ − 569
where
+ − 570
"p \<bullet> [] = []"
+ − 571
| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
+ − 572
1879
+ − 573
instance
+ − 574
by (default) (induct_tac [!] x, simp_all)
1062
+ − 575
+ − 576
end
+ − 577
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 578
lemma set_eqvt:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 579
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>
diff
changeset
+ − 580
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>
diff
changeset
+ − 581
1062
+ − 582
subsection {* Permutations for options *}
+ − 583
+ − 584
instantiation option :: (pt) pt
+ − 585
begin
+ − 586
+ − 587
primrec
+ − 588
permute_option
+ − 589
where
+ − 590
"p \<bullet> None = None"
+ − 591
| "p \<bullet> (Some x) = Some (p \<bullet> x)"
+ − 592
1879
+ − 593
instance
+ − 594
by (default) (induct_tac [!] x, simp_all)
1062
+ − 595
+ − 596
end
+ − 597
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 598
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 599
subsection {* Permutations for fsets *}
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 600
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 601
lemma permute_fset_rsp[quot_respect]:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 602
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>
diff
changeset
+ − 603
unfolding fun_rel_def
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 604
by (simp add: set_eqvt[symmetric])
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 605
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 606
instantiation fset :: (pt) pt
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 607
begin
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 608
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 609
quotient_definition
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 610
"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>
diff
changeset
+ − 611
is
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 612
"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>
diff
changeset
+ − 613
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 614
instance
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 615
proof
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 616
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>
diff
changeset
+ − 617
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>
diff
changeset
+ − 618
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>
diff
changeset
+ − 619
qed
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 620
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 621
end
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 622
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 623
lemma permute_fset [simp]:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 624
fixes S::"('a::pt) fset"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 625
shows "(p \<bullet> {||}) = ({||} ::('a::pt) fset)"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 626
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>
diff
changeset
+ − 627
by (lifting permute_list.simps)
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 628
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 629
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 630
1062
+ − 631
subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
+ − 632
+ − 633
instantiation char :: pt
+ − 634
begin
+ − 635
+ − 636
definition "p \<bullet> (c::char) = c"
+ − 637
1879
+ − 638
instance
+ − 639
by (default) (simp_all add: permute_char_def)
1062
+ − 640
+ − 641
end
+ − 642
+ − 643
instantiation nat :: pt
+ − 644
begin
+ − 645
+ − 646
definition "p \<bullet> (n::nat) = n"
+ − 647
1879
+ − 648
instance
+ − 649
by (default) (simp_all add: permute_nat_def)
1062
+ − 650
+ − 651
end
+ − 652
+ − 653
instantiation int :: pt
+ − 654
begin
+ − 655
+ − 656
definition "p \<bullet> (i::int) = i"
+ − 657
1879
+ − 658
instance
+ − 659
by (default) (simp_all add: permute_int_def)
1062
+ − 660
+ − 661
end
+ − 662
+ − 663
+ − 664
section {* Pure types *}
+ − 665
+ − 666
text {* Pure types will have always empty support. *}
+ − 667
+ − 668
class pure = pt +
+ − 669
assumes permute_pure: "p \<bullet> x = x"
+ − 670
+ − 671
text {* Types @{typ unit} and @{typ bool} are pure. *}
+ − 672
+ − 673
instance unit :: pure
+ − 674
proof qed (rule permute_unit_def)
+ − 675
+ − 676
instance bool :: pure
+ − 677
proof qed (rule permute_bool_def)
+ − 678
+ − 679
text {* Other type constructors preserve purity. *}
+ − 680
+ − 681
instance "fun" :: (pure, pure) pure
+ − 682
by default (simp add: permute_fun_def permute_pure)
+ − 683
2378
+ − 684
instance prod :: (pure, pure) pure
1062
+ − 685
by default (induct_tac x, simp add: permute_pure)
+ − 686
2378
+ − 687
instance sum :: (pure, pure) pure
1062
+ − 688
by default (induct_tac x, simp_all add: permute_pure)
+ − 689
+ − 690
instance list :: (pure) pure
+ − 691
by default (induct_tac x, simp_all add: permute_pure)
+ − 692
+ − 693
instance option :: (pure) pure
+ − 694
by default (induct_tac x, simp_all add: permute_pure)
+ − 695
+ − 696
+ − 697
subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
+ − 698
+ − 699
instance char :: pure
+ − 700
proof qed (rule permute_char_def)
+ − 701
+ − 702
instance nat :: pure
+ − 703
proof qed (rule permute_nat_def)
+ − 704
+ − 705
instance int :: pure
+ − 706
proof qed (rule permute_int_def)
+ − 707
+ − 708
+ − 709
subsection {* Supp, Freshness and Supports *}
+ − 710
+ − 711
context pt
+ − 712
begin
+ − 713
+ − 714
definition
+ − 715
supp :: "'a \<Rightarrow> atom set"
+ − 716
where
+ − 717
"supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
+ − 718
+ − 719
end
+ − 720
+ − 721
definition
+ − 722
fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
+ − 723
where
+ − 724
"a \<sharp> x \<equiv> a \<notin> supp x"
+ − 725
+ − 726
lemma supp_conv_fresh:
+ − 727
shows "supp x = {a. \<not> a \<sharp> x}"
+ − 728
unfolding fresh_def by simp
+ − 729
+ − 730
lemma swap_rel_trans:
+ − 731
assumes "sort_of a = sort_of b"
+ − 732
assumes "sort_of b = sort_of c"
+ − 733
assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
+ − 734
assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
+ − 735
shows "(a \<rightleftharpoons> b) \<bullet> x = x"
+ − 736
proof (cases)
+ − 737
assume "a = b \<or> c = b"
+ − 738
with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
+ − 739
next
+ − 740
assume *: "\<not> (a = b \<or> c = b)"
+ − 741
have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
+ − 742
using assms by simp
+ − 743
also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
+ − 744
using assms * by (simp add: swap_triple)
+ − 745
finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
+ − 746
qed
+ − 747
+ − 748
lemma swap_fresh_fresh:
+ − 749
assumes a: "a \<sharp> x"
+ − 750
and b: "b \<sharp> x"
+ − 751
shows "(a \<rightleftharpoons> b) \<bullet> x = x"
+ − 752
proof (cases)
+ − 753
assume asm: "sort_of a = sort_of b"
+ − 754
have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}"
+ − 755
using a b unfolding fresh_def supp_def by simp_all
+ − 756
then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
+ − 757
then obtain c
+ − 758
where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
+ − 759
by (rule obtain_atom) (auto)
+ − 760
then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
+ − 761
next
+ − 762
assume "sort_of a \<noteq> sort_of b"
+ − 763
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
+ − 764
qed
+ − 765
+ − 766
+ − 767
subsection {* supp and fresh are equivariant *}
+ − 768
+ − 769
lemma finite_Collect_bij:
+ − 770
assumes a: "bij f"
+ − 771
shows "finite {x. P (f x)} = finite {x. P x}"
+ − 772
by (metis a finite_vimage_iff vimage_Collect_eq)
+ − 773
+ − 774
lemma fresh_permute_iff:
+ − 775
shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
+ − 776
proof -
+ − 777
have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
+ − 778
unfolding fresh_def supp_def by simp
+ − 779
also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
1879
+ − 780
using bij_permute by (rule finite_Collect_bij[symmetric])
1062
+ − 781
also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}"
+ − 782
by (simp only: permute_eqvt [of p] swap_eqvt)
+ − 783
also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
+ − 784
by (simp only: permute_eq_iff)
+ − 785
also have "\<dots> \<longleftrightarrow> a \<sharp> x"
+ − 786
unfolding fresh_def supp_def by simp
1879
+ − 787
finally show "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x" .
1062
+ − 788
qed
+ − 789
+ − 790
lemma fresh_eqvt:
+ − 791
shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
1879
+ − 792
unfolding permute_bool_def
+ − 793
by (simp add: fresh_permute_iff)
1062
+ − 794
+ − 795
lemma supp_eqvt:
+ − 796
fixes p :: "perm"
+ − 797
and x :: "'a::pt"
+ − 798
shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
+ − 799
unfolding supp_conv_fresh
1879
+ − 800
unfolding Collect_def
+ − 801
unfolding permute_fun_def
1062
+ − 802
by (simp add: Not_eqvt fresh_eqvt)
+ − 803
+ − 804
subsection {* supports *}
+ − 805
+ − 806
definition
+ − 807
supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
+ − 808
where
+ − 809
"S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
+ − 810
+ − 811
lemma supp_is_subset:
+ − 812
fixes S :: "atom set"
+ − 813
and x :: "'a::pt"
+ − 814
assumes a1: "S supports x"
+ − 815
and a2: "finite S"
+ − 816
shows "(supp x) \<subseteq> S"
+ − 817
proof (rule ccontr)
1879
+ − 818
assume "\<not> (supp x \<subseteq> S)"
1062
+ − 819
then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
1879
+ − 820
from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" unfolding supports_def by auto
+ − 821
then have "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
1062
+ − 822
with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
+ − 823
then have "a \<notin> (supp x)" unfolding supp_def by simp
+ − 824
with b1 show False by simp
+ − 825
qed
+ − 826
+ − 827
lemma supports_finite:
+ − 828
fixes S :: "atom set"
+ − 829
and x :: "'a::pt"
+ − 830
assumes a1: "S supports x"
+ − 831
and a2: "finite S"
+ − 832
shows "finite (supp x)"
+ − 833
proof -
+ − 834
have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
+ − 835
then show "finite (supp x)" using a2 by (simp add: finite_subset)
+ − 836
qed
+ − 837
+ − 838
lemma supp_supports:
+ − 839
fixes x :: "'a::pt"
+ − 840
shows "(supp x) supports x"
1879
+ − 841
unfolding supports_def
+ − 842
proof (intro strip)
1062
+ − 843
fix a b
+ − 844
assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
+ − 845
then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
1879
+ − 846
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
1062
+ − 847
qed
+ − 848
+ − 849
lemma supp_is_least_supports:
+ − 850
fixes S :: "atom set"
+ − 851
and x :: "'a::pt"
+ − 852
assumes a1: "S supports x"
+ − 853
and a2: "finite S"
+ − 854
and a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
+ − 855
shows "(supp x) = S"
+ − 856
proof (rule equalityI)
+ − 857
show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
+ − 858
with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
+ − 859
have "(supp x) supports x" by (rule supp_supports)
+ − 860
with fin a3 show "S \<subseteq> supp x" by blast
+ − 861
qed
+ − 862
+ − 863
lemma subsetCI:
+ − 864
shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
+ − 865
by auto
+ − 866
+ − 867
lemma finite_supp_unique:
+ − 868
assumes a1: "S supports x"
+ − 869
assumes a2: "finite S"
+ − 870
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"
+ − 871
shows "(supp x) = S"
+ − 872
using a1 a2
+ − 873
proof (rule supp_is_least_supports)
+ − 874
fix S'
+ − 875
assume "finite S'" and "S' supports x"
+ − 876
show "S \<subseteq> S'"
+ − 877
proof (rule subsetCI)
+ − 878
fix a
+ − 879
assume "a \<in> S" and "a \<notin> S'"
+ − 880
have "finite (S \<union> S')"
+ − 881
using `finite S` `finite S'` by simp
+ − 882
then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
+ − 883
by (rule obtain_atom)
+ − 884
then have "b \<notin> S" and "b \<notin> S'" and "sort_of a = sort_of b"
+ − 885
by simp_all
+ − 886
then have "(a \<rightleftharpoons> b) \<bullet> x = x"
+ − 887
using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
+ − 888
moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
+ − 889
using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
+ − 890
by (rule a3)
+ − 891
ultimately show "False" by simp
+ − 892
qed
+ − 893
qed
+ − 894
2475
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 895
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>
diff
changeset
+ − 896
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 897
text {*
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 898
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>
diff
changeset
+ − 899
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>
diff
changeset
+ − 900
*}
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 901
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 902
definition
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 903
"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>
diff
changeset
+ − 904
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 905
486d4647bb37
supp-proofs work except for CoreHaskell and Modules (induct is probably not finding the correct instance)
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 906
1062
+ − 907
section {* Finitely-supported types *}
+ − 908
+ − 909
class fs = pt +
+ − 910
assumes finite_supp: "finite (supp x)"
+ − 911
+ − 912
lemma pure_supp:
+ − 913
shows "supp (x::'a::pure) = {}"
+ − 914
unfolding supp_def by (simp add: permute_pure)
+ − 915
+ − 916
lemma pure_fresh:
+ − 917
fixes x::"'a::pure"
+ − 918
shows "a \<sharp> x"
+ − 919
unfolding fresh_def by (simp add: pure_supp)
+ − 920
+ − 921
instance pure < fs
+ − 922
by default (simp add: pure_supp)
+ − 923
+ − 924
+ − 925
subsection {* Type @{typ atom} is finitely-supported. *}
+ − 926
+ − 927
lemma supp_atom:
+ − 928
shows "supp a = {a}"
+ − 929
apply (rule finite_supp_unique)
+ − 930
apply (clarsimp simp add: supports_def)
+ − 931
apply simp
+ − 932
apply simp
+ − 933
done
+ − 934
+ − 935
lemma fresh_atom:
+ − 936
shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
+ − 937
unfolding fresh_def supp_atom by simp
+ − 938
+ − 939
instance atom :: fs
+ − 940
by default (simp add: supp_atom)
+ − 941
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>
diff
changeset
+ − 942
section {* Support for finite sets of atoms *}
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>
diff
changeset
+ − 943
2466
+ − 944
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>
diff
changeset
+ − 945
lemma supp_finite_atom_set:
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>
diff
changeset
+ − 946
fixes S::"atom set"
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>
diff
changeset
+ − 947
assumes "finite S"
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>
diff
changeset
+ − 948
shows "supp S = S"
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>
diff
changeset
+ − 949
apply(rule finite_supp_unique)
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>
diff
changeset
+ − 950
apply(simp add: supports_def)
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>
diff
changeset
+ − 951
apply(simp add: swap_set_not_in)
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>
diff
changeset
+ − 952
apply(rule assms)
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>
diff
changeset
+ − 953
apply(simp add: swap_set_in)
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>
diff
changeset
+ − 954
done
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>
diff
changeset
+ − 955
1062
+ − 956
section {* Type @{typ perm} is finitely-supported. *}
+ − 957
+ − 958
lemma perm_swap_eq:
+ − 959
shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
+ − 960
unfolding permute_perm_def
+ − 961
by (metis add_diff_cancel minus_perm_def)
+ − 962
+ − 963
lemma supports_perm:
+ − 964
shows "{a. p \<bullet> a \<noteq> a} supports p"
+ − 965
unfolding supports_def
1879
+ − 966
unfolding perm_swap_eq
+ − 967
by (simp add: swap_eqvt)
1062
+ − 968
+ − 969
lemma finite_perm_lemma:
+ − 970
shows "finite {a::atom. p \<bullet> a \<noteq> a}"
+ − 971
using finite_Rep_perm [of p]
+ − 972
unfolding permute_atom_def .
+ − 973
+ − 974
lemma supp_perm:
+ − 975
shows "supp p = {a. p \<bullet> a \<noteq> a}"
+ − 976
apply (rule finite_supp_unique)
+ − 977
apply (rule supports_perm)
+ − 978
apply (rule finite_perm_lemma)
+ − 979
apply (simp add: perm_swap_eq swap_eqvt)
+ − 980
apply (auto simp add: expand_perm_eq swap_atom)
+ − 981
done
+ − 982
+ − 983
lemma fresh_perm:
+ − 984
shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
1879
+ − 985
unfolding fresh_def
+ − 986
by (simp add: supp_perm)
1062
+ − 987
+ − 988
lemma supp_swap:
+ − 989
shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
+ − 990
by (auto simp add: supp_perm swap_atom)
+ − 991
+ − 992
lemma fresh_zero_perm:
+ − 993
shows "a \<sharp> (0::perm)"
+ − 994
unfolding fresh_perm by simp
+ − 995
+ − 996
lemma supp_zero_perm:
+ − 997
shows "supp (0::perm) = {}"
+ − 998
unfolding supp_perm by simp
+ − 999
1087
+ − 1000
lemma fresh_plus_perm:
+ − 1001
fixes p q::perm
+ − 1002
assumes "a \<sharp> p" "a \<sharp> q"
+ − 1003
shows "a \<sharp> (p + q)"
+ − 1004
using assms
+ − 1005
unfolding fresh_def
+ − 1006
by (auto simp add: supp_perm)
+ − 1007
1062
+ − 1008
lemma supp_plus_perm:
+ − 1009
fixes p q::perm
+ − 1010
shows "supp (p + q) \<subseteq> supp p \<union> supp q"
+ − 1011
by (auto simp add: supp_perm)
+ − 1012
1087
+ − 1013
lemma fresh_minus_perm:
+ − 1014
fixes p::perm
+ − 1015
shows "a \<sharp> (- p) \<longleftrightarrow> a \<sharp> p"
+ − 1016
unfolding fresh_def
1879
+ − 1017
unfolding supp_perm
+ − 1018
apply(simp)
+ − 1019
apply(metis permute_minus_cancel)
1087
+ − 1020
done
+ − 1021
1062
+ − 1022
lemma supp_minus_perm:
+ − 1023
fixes p::perm
+ − 1024
shows "supp (- p) = supp p"
1087
+ − 1025
unfolding supp_conv_fresh
+ − 1026
by (simp add: fresh_minus_perm)
1062
+ − 1027
+ − 1028
instance perm :: fs
+ − 1029
by default (simp add: supp_perm finite_perm_lemma)
+ − 1030
1305
+ − 1031
lemma plus_perm_eq:
+ − 1032
fixes p q::"perm"
1879
+ − 1033
assumes asm: "supp p \<inter> supp q = {}"
1305
+ − 1034
shows "p + q = q + p"
+ − 1035
unfolding expand_perm_eq
+ − 1036
proof
+ − 1037
fix a::"atom"
+ − 1038
show "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1039
proof -
+ − 1040
{ assume "a \<notin> supp p" "a \<notin> supp q"
+ − 1041
then have "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1042
by (simp add: supp_perm)
+ − 1043
}
+ − 1044
moreover
+ − 1045
{ assume a: "a \<in> supp p" "a \<notin> supp q"
+ − 1046
then have "p \<bullet> a \<in> supp p" by (simp add: supp_perm)
+ − 1047
then have "p \<bullet> a \<notin> supp q" using asm by auto
+ − 1048
with a have "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1049
by (simp add: supp_perm)
+ − 1050
}
+ − 1051
moreover
+ − 1052
{ assume a: "a \<notin> supp p" "a \<in> supp q"
+ − 1053
then have "q \<bullet> a \<in> supp q" by (simp add: supp_perm)
+ − 1054
then have "q \<bullet> a \<notin> supp p" using asm by auto
+ − 1055
with a have "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1056
by (simp add: supp_perm)
+ − 1057
}
+ − 1058
ultimately show "(p + q) \<bullet> a = (q + p) \<bullet> a"
+ − 1059
using asm by blast
+ − 1060
qed
+ − 1061
qed
1062
+ − 1062
+ − 1063
section {* Finite Support instances for other types *}
+ − 1064
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1065
1062
+ − 1066
subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
+ − 1067
+ − 1068
lemma supp_Pair:
+ − 1069
shows "supp (x, y) = supp x \<union> supp y"
+ − 1070
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+ − 1071
+ − 1072
lemma fresh_Pair:
+ − 1073
shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
+ − 1074
by (simp add: fresh_def supp_Pair)
+ − 1075
2470
+ − 1076
lemma supp_Unit:
+ − 1077
shows "supp () = {}"
+ − 1078
by (simp add: supp_def)
+ − 1079
+ − 1080
lemma fresh_Unit:
+ − 1081
shows "a \<sharp> ()"
+ − 1082
by (simp add: fresh_def supp_Unit)
+ − 1083
2378
+ − 1084
instance prod :: (fs, fs) fs
1062
+ − 1085
apply default
+ − 1086
apply (induct_tac x)
+ − 1087
apply (simp add: supp_Pair finite_supp)
+ − 1088
done
+ − 1089
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1090
1062
+ − 1091
subsection {* Type @{typ "'a + 'b"} is finitely supported *}
+ − 1092
+ − 1093
lemma supp_Inl:
+ − 1094
shows "supp (Inl x) = supp x"
+ − 1095
by (simp add: supp_def)
+ − 1096
+ − 1097
lemma supp_Inr:
+ − 1098
shows "supp (Inr x) = supp x"
+ − 1099
by (simp add: supp_def)
+ − 1100
+ − 1101
lemma fresh_Inl:
+ − 1102
shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
+ − 1103
by (simp add: fresh_def supp_Inl)
+ − 1104
+ − 1105
lemma fresh_Inr:
+ − 1106
shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
+ − 1107
by (simp add: fresh_def supp_Inr)
+ − 1108
2378
+ − 1109
instance sum :: (fs, fs) fs
1062
+ − 1110
apply default
+ − 1111
apply (induct_tac x)
+ − 1112
apply (simp_all add: supp_Inl supp_Inr finite_supp)
+ − 1113
done
+ − 1114
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1115
1062
+ − 1116
subsection {* Type @{typ "'a option"} is finitely supported *}
+ − 1117
+ − 1118
lemma supp_None:
+ − 1119
shows "supp None = {}"
+ − 1120
by (simp add: supp_def)
+ − 1121
+ − 1122
lemma supp_Some:
+ − 1123
shows "supp (Some x) = supp x"
+ − 1124
by (simp add: supp_def)
+ − 1125
+ − 1126
lemma fresh_None:
+ − 1127
shows "a \<sharp> None"
+ − 1128
by (simp add: fresh_def supp_None)
+ − 1129
+ − 1130
lemma fresh_Some:
+ − 1131
shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
+ − 1132
by (simp add: fresh_def supp_Some)
+ − 1133
+ − 1134
instance option :: (fs) fs
+ − 1135
apply default
+ − 1136
apply (induct_tac x)
+ − 1137
apply (simp_all add: supp_None supp_Some finite_supp)
+ − 1138
done
+ − 1139
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1140
1062
+ − 1141
subsubsection {* Type @{typ "'a list"} is finitely supported *}
+ − 1142
+ − 1143
lemma supp_Nil:
+ − 1144
shows "supp [] = {}"
+ − 1145
by (simp add: supp_def)
+ − 1146
+ − 1147
lemma supp_Cons:
+ − 1148
shows "supp (x # xs) = supp x \<union> supp xs"
+ − 1149
by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
+ − 1150
+ − 1151
lemma fresh_Nil:
+ − 1152
shows "a \<sharp> []"
+ − 1153
by (simp add: fresh_def supp_Nil)
+ − 1154
+ − 1155
lemma fresh_Cons:
+ − 1156
shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
+ − 1157
by (simp add: fresh_def supp_Cons)
+ − 1158
+ − 1159
instance list :: (fs) fs
+ − 1160
apply default
+ − 1161
apply (induct_tac x)
+ − 1162
apply (simp_all add: supp_Nil supp_Cons finite_supp)
+ − 1163
done
+ − 1164
2466
+ − 1165
2470
+ − 1166
section {* Support and Freshness for Applications *}
1062
+ − 1167
1879
+ − 1168
lemma fresh_conv_MOST:
+ − 1169
shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
+ − 1170
unfolding fresh_def supp_def
+ − 1171
unfolding MOST_iff_cofinite by simp
+ − 1172
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1173
lemma supp_subset_fresh:
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1174
assumes a: "\<And>a. a \<sharp> x \<Longrightarrow> a \<sharp> y"
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1175
shows "supp y \<subseteq> supp x"
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1176
using a
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1177
unfolding fresh_def
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1178
by blast
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1179
1879
+ − 1180
lemma fresh_fun_app:
+ − 1181
assumes "a \<sharp> f" and "a \<sharp> x"
+ − 1182
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>
diff
changeset
+ − 1183
using assms
1879
+ − 1184
unfolding fresh_conv_MOST
+ − 1185
unfolding permute_fun_app_eq
+ − 1186
by (elim MOST_rev_mp, simp)
+ − 1187
1062
+ − 1188
lemma supp_fun_app:
+ − 1189
shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
1879
+ − 1190
using fresh_fun_app
+ − 1191
unfolding fresh_def
+ − 1192
by auto
+ − 1193
2470
+ − 1194
text {* Support of Equivariant Functions *}
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1195
1941
+ − 1196
lemma supp_fun_eqvt:
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1197
assumes a: "\<And>p. p \<bullet> f = f"
1941
+ − 1198
shows "supp f = {}"
+ − 1199
unfolding supp_def
+ − 1200
using a by simp
+ − 1201
1062
+ − 1202
lemma fresh_fun_eqvt_app:
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1203
assumes a: "\<And>p. p \<bullet> f = f"
1062
+ − 1204
shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
+ − 1205
proof -
1941
+ − 1206
from a have "supp f = {}" by (simp add: supp_fun_eqvt)
1879
+ − 1207
then show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
1062
+ − 1208
unfolding fresh_def
2003
b53e98bfb298
added lemmas establishing the support of finite sets of finitely supported elements
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1209
using supp_fun_app by auto
1062
+ − 1210
qed
+ − 1211
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>
diff
changeset
+ − 1212
2466
+ − 1213
section {* Support of Finite Sets of Finitely Supported Elements *}
+ − 1214
+ − 1215
lemma Union_fresh:
+ − 1216
shows "a \<sharp> S \<Longrightarrow> a \<sharp> (\<Union>x \<in> S. supp x)"
+ − 1217
unfolding Union_image_eq[symmetric]
+ − 1218
apply(rule_tac f="\<lambda>S. \<Union> supp ` S" in fresh_fun_eqvt_app)
+ − 1219
apply(simp add: permute_fun_def UNION_def Collect_def Bex_def ex_eqvt mem_def conj_eqvt)
+ − 1220
apply(subst permute_fun_app_eq)
+ − 1221
back
+ − 1222
apply(simp add: supp_eqvt)
+ − 1223
apply(assumption)
+ − 1224
done
+ − 1225
+ − 1226
lemma Union_supports_set:
+ − 1227
shows "(\<Union>x \<in> S. supp x) supports S"
+ − 1228
proof -
+ − 1229
{ fix a b
+ − 1230
have "\<forall>x \<in> S. (a \<rightleftharpoons> b) \<bullet> x = x \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> S = S"
+ − 1231
unfolding permute_set_eq by force
+ − 1232
}
+ − 1233
then show "(\<Union>x \<in> S. supp x) supports S"
+ − 1234
unfolding supports_def
+ − 1235
by (simp add: fresh_def[symmetric] swap_fresh_fresh)
+ − 1236
qed
+ − 1237
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1238
lemma Union_of_finite_supp_sets:
2466
+ − 1239
fixes S::"('a::fs set)"
+ − 1240
assumes fin: "finite S"
+ − 1241
shows "finite (\<Union>x\<in>S. supp x)"
+ − 1242
using fin by (induct) (auto simp add: finite_supp)
+ − 1243
+ − 1244
lemma Union_included_in_supp:
+ − 1245
fixes S::"('a::fs set)"
+ − 1246
assumes fin: "finite S"
+ − 1247
shows "(\<Union>x\<in>S. supp x) \<subseteq> supp S"
+ − 1248
proof -
+ − 1249
have "(\<Union>x\<in>S. supp x) = supp (\<Union>x\<in>S. supp x)"
+ − 1250
by (rule supp_finite_atom_set[symmetric])
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1251
(rule Union_of_finite_supp_sets[OF fin])
2466
+ − 1252
also have "\<dots> \<subseteq> supp S"
+ − 1253
by (rule supp_subset_fresh)
+ − 1254
(simp add: Union_fresh)
+ − 1255
finally show "(\<Union>x\<in>S. supp x) \<subseteq> supp S" .
+ − 1256
qed
+ − 1257
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1258
lemma supp_of_finite_sets:
2466
+ − 1259
fixes S::"('a::fs set)"
+ − 1260
assumes fin: "finite S"
+ − 1261
shows "(supp S) = (\<Union>x\<in>S. supp x)"
+ − 1262
apply(rule subset_antisym)
+ − 1263
apply(rule supp_is_subset)
+ − 1264
apply(rule Union_supports_set)
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1265
apply(rule Union_of_finite_supp_sets[OF fin])
2466
+ − 1266
apply(rule Union_included_in_supp[OF fin])
+ − 1267
done
+ − 1268
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1269
lemma finite_sets_supp:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1270
fixes S::"('a::fs set)"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1271
assumes "finite S"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1272
shows "finite (supp S)"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1273
using assms
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1274
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>
diff
changeset
+ − 1275
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1276
lemma supp_of_finite_union:
2466
+ − 1277
fixes S T::"('a::fs) set"
+ − 1278
assumes fin1: "finite S"
+ − 1279
and fin2: "finite T"
+ − 1280
shows "supp (S \<union> T) = supp S \<union> supp T"
+ − 1281
using fin1 fin2
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1282
by (simp add: supp_of_finite_sets)
2466
+ − 1283
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1284
lemma supp_of_finite_insert:
2466
+ − 1285
fixes S::"('a::fs) set"
+ − 1286
assumes fin: "finite S"
+ − 1287
shows "supp (insert x S) = supp x \<union> supp S"
+ − 1288
using fin
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1289
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>
diff
changeset
+ − 1290
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1291
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1292
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>
diff
changeset
+ − 1293
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1294
lemma fset_eqvt:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1295
shows "p \<bullet> (fset S) = fset (p \<bullet> S)"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1296
by (lifting set_eqvt)
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1297
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1298
lemma supp_fset [simp]:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1299
shows "supp (fset S) = supp S"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1300
unfolding supp_def
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1301
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>
diff
changeset
+ − 1302
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1303
lemma supp_empty_fset [simp]:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1304
shows "supp {||} = {}"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1305
unfolding supp_def
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1306
by simp
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1307
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1308
lemma supp_insert_fset [simp]:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1309
fixes x::"'a::fs"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1310
and S::"'a fset"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1311
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>
diff
changeset
+ − 1312
apply(subst supp_fset[symmetric])
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1313
apply(simp add: supp_fset supp_of_finite_insert)
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1314
done
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1315
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1316
lemma fset_finite_supp:
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1317
fixes S::"('a::fs) fset"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1318
shows "finite (supp S)"
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1319
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>
diff
changeset
+ − 1320
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1321
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1322
instance fset :: (fs) fs
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1323
apply (default)
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1324
apply (rule fset_finite_supp)
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1325
done
2466
+ − 1326
+ − 1327
2470
+ − 1328
section {* Fresh-Star *}
+ − 1329
+ − 1330
+ − 1331
text {* The fresh-star generalisation of fresh is used in strong
+ − 1332
induction principles. *}
+ − 1333
+ − 1334
definition
+ − 1335
fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
+ − 1336
where
+ − 1337
"as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
+ − 1338
2507
+ − 1339
lemma fresh_star_supp_conv:
+ − 1340
shows "supp x \<sharp>* y \<Longrightarrow> supp y \<sharp>* x"
+ − 1341
by (auto simp add: fresh_star_def fresh_def)
+ − 1342
2470
+ − 1343
lemma fresh_star_prod:
+ − 1344
fixes as::"atom set"
+ − 1345
shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
+ − 1346
by (auto simp add: fresh_star_def fresh_Pair)
+ − 1347
+ − 1348
lemma fresh_star_union:
+ − 1349
shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
+ − 1350
by (auto simp add: fresh_star_def)
+ − 1351
+ − 1352
lemma fresh_star_insert:
+ − 1353
shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
+ − 1354
by (auto simp add: fresh_star_def)
+ − 1355
+ − 1356
lemma fresh_star_Un_elim:
+ − 1357
"((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
+ − 1358
unfolding fresh_star_def
+ − 1359
apply(rule)
+ − 1360
apply(erule meta_mp)
+ − 1361
apply(auto)
+ − 1362
done
+ − 1363
+ − 1364
lemma fresh_star_insert_elim:
+ − 1365
"(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
+ − 1366
unfolding fresh_star_def
+ − 1367
by rule (simp_all add: fresh_star_def)
+ − 1368
+ − 1369
lemma fresh_star_empty_elim:
+ − 1370
"({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
+ − 1371
by (simp add: fresh_star_def)
+ − 1372
+ − 1373
lemma fresh_star_unit_elim:
+ − 1374
shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
+ − 1375
by (simp add: fresh_star_def fresh_Unit)
+ − 1376
+ − 1377
lemma fresh_star_prod_elim:
+ − 1378
shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
+ − 1379
by (rule, simp_all add: fresh_star_prod)
+ − 1380
+ − 1381
lemma fresh_star_zero:
+ − 1382
shows "as \<sharp>* (0::perm)"
+ − 1383
unfolding fresh_star_def
+ − 1384
by (simp add: fresh_zero_perm)
+ − 1385
+ − 1386
lemma fresh_star_plus:
+ − 1387
fixes p q::perm
+ − 1388
shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
+ − 1389
unfolding fresh_star_def
+ − 1390
by (simp add: fresh_plus_perm)
+ − 1391
+ − 1392
lemma fresh_star_permute_iff:
+ − 1393
shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
+ − 1394
unfolding fresh_star_def
+ − 1395
by (metis mem_permute_iff permute_minus_cancel(1) fresh_permute_iff)
+ − 1396
+ − 1397
lemma fresh_star_eqvt:
+ − 1398
shows "(p \<bullet> (as \<sharp>* x)) = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
+ − 1399
unfolding fresh_star_def
+ − 1400
unfolding Ball_def
+ − 1401
apply(simp add: all_eqvt)
+ − 1402
apply(subst permute_fun_def)
+ − 1403
apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)
+ − 1404
done
+ − 1405
+ − 1406
+ − 1407
section {* Induction principle for permutations *}
+ − 1408
+ − 1409
+ − 1410
lemma perm_struct_induct[consumes 1, case_names zero swap]:
+ − 1411
assumes S: "supp p \<subseteq> S"
+ − 1412
and zero: "P 0"
+ − 1413
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)"
+ − 1414
shows "P p"
+ − 1415
proof -
+ − 1416
have "finite (supp p)" by (simp add: finite_supp)
+ − 1417
then show "P p" using S
+ − 1418
proof(induct A\<equiv>"supp p" arbitrary: p rule: finite_psubset_induct)
+ − 1419
case (psubset p)
+ − 1420
then have ih: "\<And>q. supp q \<subset> supp p \<Longrightarrow> P q" by auto
+ − 1421
have as: "supp p \<subseteq> S" by fact
+ − 1422
{ assume "supp p = {}"
+ − 1423
then have "p = 0" by (simp add: supp_perm expand_perm_eq)
+ − 1424
then have "P p" using zero by simp
+ − 1425
}
+ − 1426
moreover
+ − 1427
{ assume "supp p \<noteq> {}"
+ − 1428
then obtain a where a0: "a \<in> supp p" by blast
+ − 1429
then have a1: "p \<bullet> a \<in> S" "a \<in> S" "sort_of (p \<bullet> a) = sort_of a" "p \<bullet> a \<noteq> a"
+ − 1430
using as by (auto simp add: supp_atom supp_perm swap_atom)
+ − 1431
let ?q = "(p \<bullet> a \<rightleftharpoons> a) + p"
+ − 1432
have a2: "supp ?q \<subseteq> supp p" unfolding supp_perm by (auto simp add: swap_atom)
+ − 1433
moreover
+ − 1434
have "a \<notin> supp ?q" by (simp add: supp_perm)
+ − 1435
then have "supp ?q \<noteq> supp p" using a0 by auto
+ − 1436
ultimately have "supp ?q \<subset> supp p" using a2 by auto
+ − 1437
then have "P ?q" using ih by simp
+ − 1438
moreover
+ − 1439
have "supp ?q \<subseteq> S" using as a2 by simp
+ − 1440
ultimately have "P ((p \<bullet> a \<rightleftharpoons> a) + ?q)" using as a1 swap by simp
+ − 1441
moreover
+ − 1442
have "p = (p \<bullet> a \<rightleftharpoons> a) + ?q" by (simp add: expand_perm_eq)
+ − 1443
ultimately have "P p" by simp
+ − 1444
}
+ − 1445
ultimately show "P p" by blast
+ − 1446
qed
+ − 1447
qed
+ − 1448
+ − 1449
lemma perm_simple_struct_induct[case_names zero swap]:
+ − 1450
assumes zero: "P 0"
+ − 1451
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)"
+ − 1452
shows "P p"
+ − 1453
by (rule_tac S="supp p" in perm_struct_induct)
+ − 1454
(auto intro: zero swap)
+ − 1455
+ − 1456
lemma perm_subset_induct[consumes 1, case_names zero swap plus]:
+ − 1457
assumes S: "supp p \<subseteq> S"
+ − 1458
assumes zero: "P 0"
+ − 1459
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)"
+ − 1460
assumes plus: "\<And>p1 p2. \<lbrakk>P p1; P p2; supp p1 \<subseteq> S; supp p2 \<subseteq> S\<rbrakk> \<Longrightarrow> P (p1 + p2)"
+ − 1461
shows "P p"
+ − 1462
using S
+ − 1463
by (induct p rule: perm_struct_induct)
+ − 1464
(auto intro: zero plus swap simp add: supp_swap)
+ − 1465
+ − 1466
lemma supp_perm_eq:
+ − 1467
assumes "(supp x) \<sharp>* p"
+ − 1468
shows "p \<bullet> x = x"
+ − 1469
proof -
+ − 1470
from assms have "supp p \<subseteq> {a. a \<sharp> x}"
+ − 1471
unfolding supp_perm fresh_star_def fresh_def by auto
+ − 1472
then show "p \<bullet> x = x"
+ − 1473
proof (induct p rule: perm_struct_induct)
+ − 1474
case zero
+ − 1475
show "0 \<bullet> x = x" by simp
+ − 1476
next
+ − 1477
case (swap p a b)
+ − 1478
then have "a \<sharp> x" "b \<sharp> x" "p \<bullet> x = x" by simp_all
+ − 1479
then show "((a \<rightleftharpoons> b) + p) \<bullet> x = x" by (simp add: swap_fresh_fresh)
+ − 1480
qed
+ − 1481
qed
+ − 1482
+ − 1483
lemma supp_perm_eq_test:
+ − 1484
assumes "(supp x) \<sharp>* p"
+ − 1485
shows "p \<bullet> x = x"
+ − 1486
proof -
+ − 1487
from assms have "supp p \<subseteq> {a. a \<sharp> x}"
+ − 1488
unfolding supp_perm fresh_star_def fresh_def by auto
+ − 1489
then show "p \<bullet> x = x"
+ − 1490
proof (induct p rule: perm_subset_induct)
+ − 1491
case zero
+ − 1492
show "0 \<bullet> x = x" by simp
+ − 1493
next
+ − 1494
case (swap a b)
+ − 1495
then have "a \<sharp> x" "b \<sharp> x" by simp_all
+ − 1496
then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (simp add: swap_fresh_fresh)
+ − 1497
next
+ − 1498
case (plus p1 p2)
+ − 1499
have "p1 \<bullet> x = x" "p2 \<bullet> x = x" by fact+
+ − 1500
then show "(p1 + p2) \<bullet> x = x" by simp
+ − 1501
qed
+ − 1502
qed
+ − 1503
+ − 1504
+ − 1505
section {* Avoiding of atom sets *}
+ − 1506
+ − 1507
text {*
+ − 1508
For every set of atoms, there is another set of atoms
+ − 1509
avoiding a finitely supported c and there is a permutation
+ − 1510
which 'translates' between both sets.
+ − 1511
*}
+ − 1512
+ − 1513
lemma at_set_avoiding_aux:
+ − 1514
fixes Xs::"atom set"
+ − 1515
and As::"atom set"
+ − 1516
assumes b: "Xs \<subseteq> As"
+ − 1517
and c: "finite As"
+ − 1518
shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+ − 1519
proof -
+ − 1520
from b c have "finite Xs" by (rule finite_subset)
+ − 1521
then show ?thesis using b
+ − 1522
proof (induct rule: finite_subset_induct)
+ − 1523
case empty
+ − 1524
have "0 \<bullet> {} \<inter> As = {}" by simp
+ − 1525
moreover
+ − 1526
have "supp (0::perm) \<subseteq> {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
+ − 1527
ultimately show ?case by blast
+ − 1528
next
+ − 1529
case (insert x Xs)
+ − 1530
then obtain p where
+ − 1531
p1: "(p \<bullet> Xs) \<inter> As = {}" and
+ − 1532
p2: "supp p \<subseteq> (Xs \<union> (p \<bullet> Xs))" by blast
+ − 1533
from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
+ − 1534
with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
+ − 1535
hence px: "p \<bullet> x = x" unfolding supp_perm by simp
+ − 1536
have "finite (As \<union> p \<bullet> Xs)"
+ − 1537
using `finite As` `finite Xs`
+ − 1538
by (simp add: permute_set_eq_image)
+ − 1539
then obtain y where "y \<notin> (As \<union> p \<bullet> Xs)" "sort_of y = sort_of x"
+ − 1540
by (rule obtain_atom)
+ − 1541
hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "sort_of y = sort_of x"
+ − 1542
by simp_all
+ − 1543
let ?q = "(x \<rightleftharpoons> y) + p"
+ − 1544
have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
+ − 1545
unfolding insert_eqvt
+ − 1546
using `p \<bullet> x = x` `sort_of y = sort_of x`
+ − 1547
using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
+ − 1548
by (simp add: swap_atom swap_set_not_in)
+ − 1549
have "?q \<bullet> insert x Xs \<inter> As = {}"
+ − 1550
using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
+ − 1551
unfolding q by simp
+ − 1552
moreover
+ − 1553
have "supp ?q \<subseteq> insert x Xs \<union> ?q \<bullet> insert x Xs"
+ − 1554
using p2 unfolding q
+ − 1555
by (intro subset_trans [OF supp_plus_perm])
+ − 1556
(auto simp add: supp_swap)
+ − 1557
ultimately show ?case by blast
+ − 1558
qed
+ − 1559
qed
+ − 1560
+ − 1561
lemma at_set_avoiding:
+ − 1562
assumes a: "finite Xs"
+ − 1563
and b: "finite (supp c)"
+ − 1564
obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+ − 1565
using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
+ − 1566
unfolding fresh_star_def fresh_def by blast
+ − 1567
+ − 1568
lemma at_set_avoiding2:
+ − 1569
assumes "finite xs"
+ − 1570
and "finite (supp c)" "finite (supp x)"
+ − 1571
and "xs \<sharp>* x"
+ − 1572
shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
+ − 1573
using assms
+ − 1574
apply(erule_tac c="(c, x)" in at_set_avoiding)
+ − 1575
apply(simp add: supp_Pair)
+ − 1576
apply(rule_tac x="p" in exI)
+ − 1577
apply(simp add: fresh_star_prod)
2507
+ − 1578
apply(rule fresh_star_supp_conv)
+ − 1579
apply(auto simp add: fresh_star_def)
2470
+ − 1580
done
+ − 1581
+ − 1582
lemma at_set_avoiding2_atom:
+ − 1583
assumes "finite (supp c)" "finite (supp x)"
+ − 1584
and b: "a \<sharp> x"
+ − 1585
shows "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p"
+ − 1586
proof -
+ − 1587
have a: "{a} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
+ − 1588
obtain p where p1: "(p \<bullet> {a}) \<sharp>* c" and p2: "supp x \<sharp>* p"
+ − 1589
using at_set_avoiding2[of "{a}" "c" "x"] assms a by blast
+ − 1590
have c: "(p \<bullet> a) \<sharp> c" using p1
+ − 1591
unfolding fresh_star_def Ball_def
+ − 1592
by(erule_tac x="p \<bullet> a" in allE) (simp add: permute_set_eq)
+ − 1593
hence "p \<bullet> a \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
+ − 1594
then show "\<exists>p. (p \<bullet> a) \<sharp> c \<and> supp x \<sharp>* p" by blast
+ − 1595
qed
+ − 1596
+ − 1597
+ − 1598
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>
diff
changeset
+ − 1599
1972
+ − 1600
text {*
+ − 1601
Class @{text at_base} allows types containing multiple sorts of atoms.
+ − 1602
Class @{text at} only allows types with a single sort.
+ − 1603
*}
+ − 1604
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>
diff
changeset
+ − 1605
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>
diff
changeset
+ − 1606
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>
diff
changeset
+ − 1607
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>
diff
changeset
+ − 1608
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>
diff
changeset
+ − 1609
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>
diff
changeset
+ − 1610
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>
diff
changeset
+ − 1611
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>
diff
changeset
+ − 1612
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>
diff
changeset
+ − 1613
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>
diff
changeset
+ − 1614
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>
diff
changeset
+ − 1615
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>
diff
changeset
+ − 1616
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>
diff
changeset
+ − 1617
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>
diff
changeset
+ − 1618
lemma fresh_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>
diff
changeset
+ − 1619
shows "a \<sharp> b \<longleftrightarrow> a \<noteq> 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>
diff
changeset
+ − 1620
unfolding fresh_def by (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>
diff
changeset
+ − 1621
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>
diff
changeset
+ − 1622
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>
diff
changeset
+ − 1623
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>
diff
changeset
+ − 1624
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>
diff
changeset
+ − 1625
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>
diff
changeset
+ − 1626
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>
diff
changeset
+ − 1627
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>
diff
changeset
+ − 1628
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>
diff
changeset
+ − 1629
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>
diff
changeset
+ − 1630
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>
diff
changeset
+ − 1631
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>
diff
changeset
+ − 1632
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>
diff
changeset
+ − 1633
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>
diff
changeset
+ − 1634
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>
diff
changeset
+ − 1635
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>
diff
changeset
+ − 1636
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>
diff
changeset
+ − 1637
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>
diff
changeset
+ − 1638
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>
diff
changeset
+ − 1639
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>
diff
changeset
+ − 1640
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>
diff
changeset
+ − 1641
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>
diff
changeset
+ − 1642
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>
diff
changeset
+ − 1643
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>
diff
changeset
+ − 1644
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>
diff
changeset
+ − 1645
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>
diff
changeset
+ − 1646
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>
diff
changeset
+ − 1647
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>
diff
changeset
+ − 1648
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>
diff
changeset
+ − 1649
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>
diff
changeset
+ − 1650
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>
diff
changeset
+ − 1651
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>
diff
changeset
+ − 1652
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>
diff
changeset
+ − 1653
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>
diff
changeset
+ − 1654
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>
diff
changeset
+ − 1655
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>
diff
changeset
+ − 1656
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>
diff
changeset
+ − 1657
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>
diff
changeset
+ − 1658
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>
diff
changeset
+ − 1659
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>
diff
changeset
+ − 1660
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>
diff
changeset
+ − 1661
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>
diff
changeset
+ − 1662
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>
diff
changeset
+ − 1663
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>
diff
changeset
+ − 1664
1973
+ − 1665
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>
diff
changeset
+ − 1666
assumes a: "finite S"
8daf6ff5e11a
simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1667
shows "supp S = atom ` S"
2565
6bf332360510
moved most material fron Nominal2_FSet into the Nominal_Base theory
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1668
apply(simp add: supp_of_finite_sets[OF a])
2466
+ − 1669
apply(simp add: supp_at_base)
+ − 1670
apply(auto)
+ − 1671
done
1971
8daf6ff5e11a
simpliied and moved the remaining lemmas about the atom-function to Nominal2_Base
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 1672
2467
+ − 1673
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>
diff
changeset
+ − 1674
2467
+ − 1675
section {* A swapping operation for concrete atoms *}
+ − 1676
+ − 1677
definition
+ − 1678
flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
+ − 1679
where
+ − 1680
"(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
+ − 1681
+ − 1682
lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
+ − 1683
unfolding flip_def by (rule swap_self)
+ − 1684
+ − 1685
lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
+ − 1686
unfolding flip_def by (rule swap_commute)
+ − 1687
+ − 1688
lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
+ − 1689
unfolding flip_def by (rule minus_swap)
+ − 1690
+ − 1691
lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
+ − 1692
unfolding flip_def by (rule swap_cancel)
+ − 1693
+ − 1694
lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
+ − 1695
unfolding permute_plus [symmetric] add_flip_cancel by simp
+ − 1696
+ − 1697
lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
+ − 1698
by (simp add: flip_commute)
+ − 1699
+ − 1700
lemma flip_eqvt:
+ − 1701
fixes a b c::"'a::at_base"
+ − 1702
shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
+ − 1703
unfolding flip_def
+ − 1704
by (simp add: swap_eqvt atom_eqvt)
+ − 1705
+ − 1706
lemma flip_at_base_simps [simp]:
+ − 1707
shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
+ − 1708
and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
+ − 1709
and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
+ − 1710
and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
+ − 1711
unfolding flip_def
+ − 1712
unfolding atom_eq_iff [symmetric]
+ − 1713
unfolding atom_eqvt [symmetric]
+ − 1714
by simp_all
+ − 1715
+ − 1716
text {* the following two lemmas do not hold for at_base,
+ − 1717
only for single sort atoms from at *}
+ − 1718
+ − 1719
lemma permute_flip_at:
+ − 1720
fixes a b c::"'a::at"
+ − 1721
shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
+ − 1722
unfolding flip_def
+ − 1723
apply (rule atom_eq_iff [THEN iffD1])
+ − 1724
apply (subst atom_eqvt [symmetric])
+ − 1725
apply (simp add: swap_atom)
+ − 1726
done
+ − 1727
+ − 1728
lemma flip_at_simps [simp]:
+ − 1729
fixes a b::"'a::at"
+ − 1730
shows "(a \<leftrightarrow> b) \<bullet> a = b"
+ − 1731
and "(a \<leftrightarrow> b) \<bullet> b = a"
+ − 1732
unfolding permute_flip_at by simp_all
+ − 1733
+ − 1734
lemma flip_fresh_fresh:
+ − 1735
fixes a b::"'a::at_base"
+ − 1736
assumes "atom a \<sharp> x" "atom b \<sharp> x"
+ − 1737
shows "(a \<leftrightarrow> b) \<bullet> x = x"
+ − 1738
using assms
+ − 1739
by (simp add: flip_def swap_fresh_fresh)
+ − 1740
+ − 1741
subsection {* Syntax for coercing at-elements to the atom-type *}
+ − 1742
+ − 1743
syntax
+ − 1744
"_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)
+ − 1745
+ − 1746
translations
+ − 1747
"_atom_constrain a t" => "CONST atom (_constrain a t)"
+ − 1748
+ − 1749
+ − 1750
subsection {* A lemma for proving instances of class @{text at}. *}
+ − 1751
+ − 1752
setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}
+ − 1753
setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}
+ − 1754
+ − 1755
text {*
+ − 1756
New atom types are defined as subtypes of @{typ atom}.
+ − 1757
*}
+ − 1758
+ − 1759
lemma exists_eq_simple_sort:
+ − 1760
shows "\<exists>a. a \<in> {a. sort_of a = s}"
+ − 1761
by (rule_tac x="Atom s 0" in exI, simp)
+ − 1762
+ − 1763
lemma exists_eq_sort:
+ − 1764
shows "\<exists>a. a \<in> {a. sort_of a \<in> range sort_fun}"
+ − 1765
by (rule_tac x="Atom (sort_fun x) y" in exI, simp)
+ − 1766
+ − 1767
lemma at_base_class:
+ − 1768
fixes sort_fun :: "'b \<Rightarrow>atom_sort"
+ − 1769
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
+ − 1770
assumes type: "type_definition Rep Abs {a. sort_of a \<in> range sort_fun}"
+ − 1771
assumes atom_def: "\<And>a. atom a = Rep a"
+ − 1772
assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
+ − 1773
shows "OFCLASS('a, at_base_class)"
+ − 1774
proof
+ − 1775
interpret type_definition Rep Abs "{a. sort_of a \<in> range sort_fun}" by (rule type)
+ − 1776
have sort_of_Rep: "\<And>a. sort_of (Rep a) \<in> range sort_fun" using Rep by simp
+ − 1777
fix a b :: 'a and p p1 p2 :: perm
+ − 1778
show "0 \<bullet> a = a"
+ − 1779
unfolding permute_def by (simp add: Rep_inverse)
+ − 1780
show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
+ − 1781
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1782
show "atom a = atom b \<longleftrightarrow> a = b"
+ − 1783
unfolding atom_def by (simp add: Rep_inject)
+ − 1784
show "p \<bullet> atom a = atom (p \<bullet> a)"
+ − 1785
unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1786
qed
+ − 1787
+ − 1788
(*
+ − 1789
lemma at_class:
+ − 1790
fixes s :: atom_sort
+ − 1791
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
+ − 1792
assumes type: "type_definition Rep Abs {a. sort_of a \<in> range (\<lambda>x::unit. s)}"
+ − 1793
assumes atom_def: "\<And>a. atom a = Rep a"
+ − 1794
assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
+ − 1795
shows "OFCLASS('a, at_class)"
+ − 1796
proof
+ − 1797
interpret type_definition Rep Abs "{a. sort_of a \<in> range (\<lambda>x::unit. s)}" by (rule type)
+ − 1798
have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
+ − 1799
fix a b :: 'a and p p1 p2 :: perm
+ − 1800
show "0 \<bullet> a = a"
+ − 1801
unfolding permute_def by (simp add: Rep_inverse)
+ − 1802
show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
+ − 1803
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1804
show "sort_of (atom a) = sort_of (atom b)"
+ − 1805
unfolding atom_def by (simp add: sort_of_Rep)
+ − 1806
show "atom a = atom b \<longleftrightarrow> a = b"
+ − 1807
unfolding atom_def by (simp add: Rep_inject)
+ − 1808
show "p \<bullet> atom a = atom (p \<bullet> a)"
+ − 1809
unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1810
qed
+ − 1811
*)
+ − 1812
+ − 1813
lemma at_class:
+ − 1814
fixes s :: atom_sort
+ − 1815
fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
+ − 1816
assumes type: "type_definition Rep Abs {a. sort_of a = s}"
+ − 1817
assumes atom_def: "\<And>a. atom a = Rep a"
+ − 1818
assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
+ − 1819
shows "OFCLASS('a, at_class)"
+ − 1820
proof
+ − 1821
interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type)
+ − 1822
have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by (simp add: image_def)
+ − 1823
fix a b :: 'a and p p1 p2 :: perm
+ − 1824
show "0 \<bullet> a = a"
+ − 1825
unfolding permute_def by (simp add: Rep_inverse)
+ − 1826
show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
+ − 1827
unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1828
show "sort_of (atom a) = sort_of (atom b)"
+ − 1829
unfolding atom_def by (simp add: sort_of_Rep)
+ − 1830
show "atom a = atom b \<longleftrightarrow> a = b"
+ − 1831
unfolding atom_def by (simp add: Rep_inject)
+ − 1832
show "p \<bullet> atom a = atom (p \<bullet> a)"
+ − 1833
unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
+ − 1834
qed
+ − 1835
+ − 1836
setup {* Sign.add_const_constraint
+ − 1837
(@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}
+ − 1838
setup {* Sign.add_const_constraint
+ − 1839
(@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}
+ − 1840
+ − 1841
2470
+ − 1842
+ − 1843
section {* The freshness lemma according to Andy Pitts *}
+ − 1844
+ − 1845
lemma freshness_lemma:
+ − 1846
fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1847
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1848
shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1849
proof -
+ − 1850
from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
+ − 1851
by (auto simp add: fresh_Pair)
+ − 1852
show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1853
proof (intro exI allI impI)
+ − 1854
fix a :: 'a
+ − 1855
assume a3: "atom a \<sharp> h"
+ − 1856
show "h a = h b"
+ − 1857
proof (cases "a = b")
+ − 1858
assume "a = b"
+ − 1859
thus "h a = h b" by simp
+ − 1860
next
+ − 1861
assume "a \<noteq> b"
+ − 1862
hence "atom a \<sharp> b" by (simp add: fresh_at_base)
+ − 1863
with a3 have "atom a \<sharp> h b"
+ − 1864
by (rule fresh_fun_app)
+ − 1865
with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
+ − 1866
by (rule swap_fresh_fresh)
+ − 1867
from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
+ − 1868
by (rule swap_fresh_fresh)
+ − 1869
from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
+ − 1870
also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
+ − 1871
by (rule permute_fun_app_eq)
+ − 1872
also have "\<dots> = h a"
+ − 1873
using d2 by simp
+ − 1874
finally show "h a = h b" by simp
+ − 1875
qed
+ − 1876
qed
+ − 1877
qed
+ − 1878
+ − 1879
lemma freshness_lemma_unique:
+ − 1880
fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1881
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1882
shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1883
proof (rule ex_ex1I)
+ − 1884
from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1885
by (rule freshness_lemma)
+ − 1886
next
+ − 1887
fix x y
+ − 1888
assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ − 1889
assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
+ − 1890
from a x y show "x = y"
+ − 1891
by (auto simp add: fresh_Pair)
+ − 1892
qed
+ − 1893
+ − 1894
text {* packaging the freshness lemma into a function *}
+ − 1895
+ − 1896
definition
+ − 1897
fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
+ − 1898
where
+ − 1899
"fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
+ − 1900
+ − 1901
lemma fresh_fun_apply:
+ − 1902
fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1903
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1904
assumes b: "atom a \<sharp> h"
+ − 1905
shows "fresh_fun h = h a"
+ − 1906
unfolding fresh_fun_def
+ − 1907
proof (rule the_equality)
+ − 1908
show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
+ − 1909
proof (intro strip)
+ − 1910
fix a':: 'a
+ − 1911
assume c: "atom a' \<sharp> h"
+ − 1912
from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
+ − 1913
with b c show "h a' = h a" by auto
+ − 1914
qed
+ − 1915
next
+ − 1916
fix fr :: 'b
+ − 1917
assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
+ − 1918
with b show "fr = h a" by auto
+ − 1919
qed
+ − 1920
+ − 1921
lemma fresh_fun_apply':
+ − 1922
fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1923
assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
+ − 1924
shows "fresh_fun h = h a"
+ − 1925
apply (rule fresh_fun_apply)
+ − 1926
apply (auto simp add: fresh_Pair intro: a)
+ − 1927
done
+ − 1928
+ − 1929
lemma fresh_fun_eqvt:
+ − 1930
fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1931
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1932
shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
+ − 1933
using a
+ − 1934
apply (clarsimp simp add: fresh_Pair)
+ − 1935
apply (subst fresh_fun_apply', assumption+)
+ − 1936
apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+ − 1937
apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+ − 1938
apply (simp add: atom_eqvt permute_fun_app_eq [where f=h])
+ − 1939
apply (erule (1) fresh_fun_apply' [symmetric])
+ − 1940
done
+ − 1941
+ − 1942
lemma fresh_fun_supports:
+ − 1943
fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ − 1944
assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ − 1945
shows "(supp h) supports (fresh_fun h)"
+ − 1946
apply (simp add: supports_def fresh_def [symmetric])
+ − 1947
apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
+ − 1948
done
+ − 1949
+ − 1950
notation fresh_fun (binder "FRESH " 10)
+ − 1951
+ − 1952
lemma FRESH_f_iff:
+ − 1953
fixes P :: "'a::at \<Rightarrow> 'b::pure"
+ − 1954
fixes f :: "'b \<Rightarrow> 'c::pure"
+ − 1955
assumes P: "finite (supp P)"
+ − 1956
shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
+ − 1957
proof -
+ − 1958
obtain a::'a where "atom a \<notin> supp P"
+ − 1959
using P by (rule obtain_at_base)
+ − 1960
hence "atom a \<sharp> P"
+ − 1961
by (simp add: fresh_def)
+ − 1962
show "(FRESH x. f (P x)) = f (FRESH x. P x)"
+ − 1963
apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
+ − 1964
apply (cut_tac `atom a \<sharp> P`)
+ − 1965
apply (simp add: fresh_conv_MOST)
+ − 1966
apply (elim MOST_rev_mp, rule MOST_I, clarify)
2479
+ − 1967
apply (simp add: permute_fun_def permute_pure fun_eq_iff)
2470
+ − 1968
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+ − 1969
apply (rule refl)
+ − 1970
done
+ − 1971
qed
+ − 1972
+ − 1973
lemma FRESH_binop_iff:
+ − 1974
fixes P :: "'a::at \<Rightarrow> 'b::pure"
+ − 1975
fixes Q :: "'a::at \<Rightarrow> 'c::pure"
+ − 1976
fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
+ − 1977
assumes P: "finite (supp P)"
+ − 1978
and Q: "finite (supp Q)"
+ − 1979
shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
+ − 1980
proof -
+ − 1981
from assms have "finite (supp P \<union> supp Q)" by simp
+ − 1982
then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)"
+ − 1983
by (rule obtain_at_base)
+ − 1984
hence "atom a \<sharp> P" and "atom a \<sharp> Q"
+ − 1985
by (simp_all add: fresh_def)
+ − 1986
show ?thesis
+ − 1987
apply (subst fresh_fun_apply' [where a=a, OF _ pure_fresh])
+ − 1988
apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
+ − 1989
apply (simp add: fresh_conv_MOST)
+ − 1990
apply (elim MOST_rev_mp, rule MOST_I, clarify)
2479
+ − 1991
apply (simp add: permute_fun_def permute_pure fun_eq_iff)
2470
+ − 1992
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+ − 1993
apply (subst fresh_fun_apply' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
+ − 1994
apply (rule refl)
+ − 1995
done
+ − 1996
qed
+ − 1997
+ − 1998
lemma FRESH_conj_iff:
+ − 1999
fixes P Q :: "'a::at \<Rightarrow> bool"
+ − 2000
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+ − 2001
shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
+ − 2002
using P Q by (rule FRESH_binop_iff)
+ − 2003
+ − 2004
lemma FRESH_disj_iff:
+ − 2005
fixes P Q :: "'a::at \<Rightarrow> bool"
+ − 2006
assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+ − 2007
shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
+ − 2008
using P Q by (rule FRESH_binop_iff)
+ − 2009
+ − 2010
2467
+ − 2011
section {* Library functions for the nominal infrastructure *}
+ − 2012
1833
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 2013
use "nominal_library.ML"
2050b5723c04
added a library for basic nominal functions; separated nominal_eqvt file
Christian Urban <urbanc@in.tum.de>
diff
changeset
+ − 2014
2466
+ − 2015
2467
+ − 2016
section {* Automation for creating concrete atom types *}
+ − 2017
+ − 2018
text {* at the moment only single-sort concrete atoms are supported *}
+ − 2019
+ − 2020
use "nominal_atoms.ML"
+ − 2021
+ − 2022
2466
+ − 2023
1062
+ − 2024
end