--- /dev/null Thu Jan 01 00:00:00 1970 +0000
+++ b/Nominal-General/Nominal2_Supp.thy Sun Apr 04 21:39:28 2010 +0200
@@ -0,0 +1,501 @@
+(* Title: Nominal2_Supp
+ Authors: Brian Huffman, Christian Urban
+
+ Supplementary Lemmas and Definitions for
+ Nominal Isabelle.
+*)
+theory Nominal2_Supp
+imports Nominal2_Base Nominal2_Eqvt Nominal2_Atoms
+begin
+
+
+section {* Fresh-Star *}
+
+text {* The fresh-star generalisation of fresh is used in strong
+ induction principles. *}
+
+definition
+ fresh_star :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp>* _" [80,80] 80)
+where
+ "as \<sharp>* x \<equiv> \<forall>a \<in> as. a \<sharp> x"
+
+lemma fresh_star_prod:
+ fixes as::"atom set"
+ shows "as \<sharp>* (x, y) = (as \<sharp>* x \<and> as \<sharp>* y)"
+ by (auto simp add: fresh_star_def fresh_Pair)
+
+lemma fresh_star_union:
+ shows "(as \<union> bs) \<sharp>* x = (as \<sharp>* x \<and> bs \<sharp>* x)"
+ by (auto simp add: fresh_star_def)
+
+lemma fresh_star_insert:
+ shows "(insert a as) \<sharp>* x = (a \<sharp> x \<and> as \<sharp>* x)"
+ by (auto simp add: fresh_star_def)
+
+lemma fresh_star_Un_elim:
+ "((as \<union> bs) \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (as \<sharp>* x \<Longrightarrow> bs \<sharp>* x \<Longrightarrow> PROP C)"
+ unfolding fresh_star_def
+ apply(rule)
+ apply(erule meta_mp)
+ apply(auto)
+ done
+
+lemma fresh_star_insert_elim:
+ "(insert a as \<sharp>* x \<Longrightarrow> PROP C) \<equiv> (a \<sharp> x \<Longrightarrow> as \<sharp>* x \<Longrightarrow> PROP C)"
+ unfolding fresh_star_def
+ by rule (simp_all add: fresh_star_def)
+
+lemma fresh_star_empty_elim:
+ "({} \<sharp>* x \<Longrightarrow> PROP C) \<equiv> PROP C"
+ by (simp add: fresh_star_def)
+
+lemma fresh_star_unit_elim:
+ shows "(a \<sharp>* () \<Longrightarrow> PROP C) \<equiv> PROP C"
+ by (simp add: fresh_star_def fresh_unit)
+
+lemma fresh_star_prod_elim:
+ shows "(a \<sharp>* (x, y) \<Longrightarrow> PROP C) \<equiv> (a \<sharp>* x \<Longrightarrow> a \<sharp>* y \<Longrightarrow> PROP C)"
+ by (rule, simp_all add: fresh_star_prod)
+
+lemma fresh_star_plus:
+ fixes p q::perm
+ shows "\<lbrakk>a \<sharp>* p; a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"
+ unfolding fresh_star_def
+ by (simp add: fresh_plus_perm)
+
+lemma fresh_star_permute_iff:
+ shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"
+ unfolding fresh_star_def
+ by (metis mem_permute_iff permute_minus_cancel fresh_permute_iff)
+
+lemma fresh_star_eqvt:
+ shows "(p \<bullet> (as \<sharp>* x)) = (p \<bullet> as) \<sharp>* (p \<bullet> x)"
+unfolding fresh_star_def
+unfolding Ball_def
+apply(simp add: all_eqvt)
+apply(subst permute_fun_def)
+apply(simp add: imp_eqvt fresh_eqvt mem_eqvt)
+done
+
+section {* Avoiding of atom sets *}
+
+text {*
+ For every set of atoms, there is another set of atoms
+ avoiding a finitely supported c and there is a permutation
+ which 'translates' between both sets.
+*}
+
+lemma at_set_avoiding_aux:
+ fixes Xs::"atom set"
+ and As::"atom set"
+ assumes b: "Xs \<subseteq> As"
+ and c: "finite As"
+ shows "\<exists>p. (p \<bullet> Xs) \<inter> As = {} \<and> (supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+proof -
+ from b c have "finite Xs" by (rule finite_subset)
+ then show ?thesis using b
+ proof (induct rule: finite_subset_induct)
+ case empty
+ have "0 \<bullet> {} \<inter> As = {}" by simp
+ moreover
+ have "supp (0::perm) \<subseteq> {} \<union> 0 \<bullet> {}" by (simp add: supp_zero_perm)
+ ultimately show ?case by blast
+ next
+ case (insert x Xs)
+ then obtain p where
+ p1: "(p \<bullet> Xs) \<inter> As = {}" and
+ p2: "supp p \<subseteq> (Xs \<union> (p \<bullet> Xs))" by blast
+ from `x \<in> As` p1 have "x \<notin> p \<bullet> Xs" by fast
+ with `x \<notin> Xs` p2 have "x \<notin> supp p" by fast
+ hence px: "p \<bullet> x = x" unfolding supp_perm by simp
+ have "finite (As \<union> p \<bullet> Xs)"
+ using `finite As` `finite Xs`
+ by (simp add: permute_set_eq_image)
+ then obtain y where "y \<notin> (As \<union> p \<bullet> Xs)" "sort_of y = sort_of x"
+ by (rule obtain_atom)
+ hence y: "y \<notin> As" "y \<notin> p \<bullet> Xs" "sort_of y = sort_of x"
+ by simp_all
+ let ?q = "(x \<rightleftharpoons> y) + p"
+ have q: "?q \<bullet> insert x Xs = insert y (p \<bullet> Xs)"
+ unfolding insert_eqvt
+ using `p \<bullet> x = x` `sort_of y = sort_of x`
+ using `x \<notin> p \<bullet> Xs` `y \<notin> p \<bullet> Xs`
+ by (simp add: swap_atom swap_set_not_in)
+ have "?q \<bullet> insert x Xs \<inter> As = {}"
+ using `y \<notin> As` `p \<bullet> Xs \<inter> As = {}`
+ unfolding q by simp
+ moreover
+ have "supp ?q \<subseteq> insert x Xs \<union> ?q \<bullet> insert x Xs"
+ using p2 unfolding q
+ apply (intro subset_trans [OF supp_plus_perm])
+ apply (auto simp add: supp_swap)
+ done
+ ultimately show ?case by blast
+ qed
+qed
+
+lemma at_set_avoiding:
+ assumes a: "finite Xs"
+ and b: "finite (supp c)"
+ obtains p::"perm" where "(p \<bullet> Xs)\<sharp>*c" and "(supp p) \<subseteq> (Xs \<union> (p \<bullet> Xs))"
+ using a b at_set_avoiding_aux [where Xs="Xs" and As="Xs \<union> supp c"]
+ unfolding fresh_star_def fresh_def by blast
+
+
+section {* The freshness lemma according to Andrew Pitts *}
+
+lemma fresh_conv_MOST:
+ shows "a \<sharp> x \<longleftrightarrow> (MOST b. (a \<rightleftharpoons> b) \<bullet> x = x)"
+ unfolding fresh_def supp_def MOST_iff_cofinite by simp
+
+lemma fresh_apply:
+ assumes "a \<sharp> f" and "a \<sharp> x"
+ shows "a \<sharp> f x"
+ using assms unfolding fresh_conv_MOST
+ unfolding permute_fun_app_eq [where f=f]
+ by (elim MOST_rev_mp, simp)
+
+lemma freshness_lemma:
+ fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ shows "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+proof -
+ from a obtain b where a1: "atom b \<sharp> h" and a2: "atom b \<sharp> h b"
+ by (auto simp add: fresh_Pair)
+ show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ proof (intro exI allI impI)
+ fix a :: 'a
+ assume a3: "atom a \<sharp> h"
+ show "h a = h b"
+ proof (cases "a = b")
+ assume "a = b"
+ thus "h a = h b" by simp
+ next
+ assume "a \<noteq> b"
+ hence "atom a \<sharp> b" by (simp add: fresh_at_base)
+ with a3 have "atom a \<sharp> h b" by (rule fresh_apply)
+ with a2 have d1: "(atom b \<rightleftharpoons> atom a) \<bullet> (h b) = (h b)"
+ by (rule swap_fresh_fresh)
+ from a1 a3 have d2: "(atom b \<rightleftharpoons> atom a) \<bullet> h = h"
+ by (rule swap_fresh_fresh)
+ from d1 have "h b = (atom b \<rightleftharpoons> atom a) \<bullet> (h b)" by simp
+ also have "\<dots> = ((atom b \<rightleftharpoons> atom a) \<bullet> h) ((atom b \<rightleftharpoons> atom a) \<bullet> b)"
+ by (rule permute_fun_app_eq)
+ also have "\<dots> = h a"
+ using d2 by simp
+ finally show "h a = h b" by simp
+ qed
+ qed
+qed
+
+lemma freshness_lemma_unique:
+ fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ shows "\<exists>!x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+proof (rule ex_ex1I)
+ from a show "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ by (rule freshness_lemma)
+next
+ fix x y
+ assume x: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = x"
+ assume y: "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = y"
+ from a x y show "x = y"
+ by (auto simp add: fresh_Pair)
+qed
+
+text {* packaging the freshness lemma into a function *}
+
+definition
+ fresh_fun :: "('a::at \<Rightarrow> 'b::pt) \<Rightarrow> 'b"
+where
+ "fresh_fun h = (THE x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x)"
+
+lemma fresh_fun_app:
+ fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ assumes b: "atom a \<sharp> h"
+ shows "fresh_fun h = h a"
+unfolding fresh_fun_def
+proof (rule the_equality)
+ show "\<forall>a'. atom a' \<sharp> h \<longrightarrow> h a' = h a"
+ proof (intro strip)
+ fix a':: 'a
+ assume c: "atom a' \<sharp> h"
+ from a have "\<exists>x. \<forall>a. atom a \<sharp> h \<longrightarrow> h a = x" by (rule freshness_lemma)
+ with b c show "h a' = h a" by auto
+ qed
+next
+ fix fr :: 'b
+ assume "\<forall>a. atom a \<sharp> h \<longrightarrow> h a = fr"
+ with b show "fr = h a" by auto
+qed
+
+lemma fresh_fun_app':
+ fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ assumes a: "atom a \<sharp> h" "atom a \<sharp> h a"
+ shows "fresh_fun h = h a"
+ apply (rule fresh_fun_app)
+ apply (auto simp add: fresh_Pair intro: a)
+ done
+
+lemma fresh_fun_eqvt:
+ fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ shows "p \<bullet> (fresh_fun h) = fresh_fun (p \<bullet> h)"
+ using a
+ apply (clarsimp simp add: fresh_Pair)
+ apply (subst fresh_fun_app', assumption+)
+ apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+ apply (drule fresh_permute_iff [where p=p, THEN iffD2])
+ apply (simp add: atom_eqvt permute_fun_app_eq [where f=h])
+ apply (erule (1) fresh_fun_app' [symmetric])
+ done
+
+lemma fresh_fun_supports:
+ fixes h :: "'a::at \<Rightarrow> 'b::pt"
+ assumes a: "\<exists>a. atom a \<sharp> (h, h a)"
+ shows "(supp h) supports (fresh_fun h)"
+ apply (simp add: supports_def fresh_def [symmetric])
+ apply (simp add: fresh_fun_eqvt [OF a] swap_fresh_fresh)
+ done
+
+notation fresh_fun (binder "FRESH " 10)
+
+lemma FRESH_f_iff:
+ fixes P :: "'a::at \<Rightarrow> 'b::pure"
+ fixes f :: "'b \<Rightarrow> 'c::pure"
+ assumes P: "finite (supp P)"
+ shows "(FRESH x. f (P x)) = f (FRESH x. P x)"
+proof -
+ obtain a::'a where "atom a \<notin> supp P"
+ using P by (rule obtain_at_base)
+ hence "atom a \<sharp> P"
+ by (simp add: fresh_def)
+ show "(FRESH x. f (P x)) = f (FRESH x. P x)"
+ apply (subst fresh_fun_app' [where a=a, OF _ pure_fresh])
+ apply (cut_tac `atom a \<sharp> P`)
+ apply (simp add: fresh_conv_MOST)
+ apply (elim MOST_rev_mp, rule MOST_I, clarify)
+ apply (simp add: permute_fun_def permute_pure expand_fun_eq)
+ apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+ apply (rule refl)
+ done
+qed
+
+lemma FRESH_binop_iff:
+ fixes P :: "'a::at \<Rightarrow> 'b::pure"
+ fixes Q :: "'a::at \<Rightarrow> 'c::pure"
+ fixes binop :: "'b \<Rightarrow> 'c \<Rightarrow> 'd::pure"
+ assumes P: "finite (supp P)"
+ and Q: "finite (supp Q)"
+ shows "(FRESH x. binop (P x) (Q x)) = binop (FRESH x. P x) (FRESH x. Q x)"
+proof -
+ from assms have "finite (supp P \<union> supp Q)" by simp
+ then obtain a::'a where "atom a \<notin> (supp P \<union> supp Q)"
+ by (rule obtain_at_base)
+ hence "atom a \<sharp> P" and "atom a \<sharp> Q"
+ by (simp_all add: fresh_def)
+ show ?thesis
+ apply (subst fresh_fun_app' [where a=a, OF _ pure_fresh])
+ apply (cut_tac `atom a \<sharp> P` `atom a \<sharp> Q`)
+ apply (simp add: fresh_conv_MOST)
+ apply (elim MOST_rev_mp, rule MOST_I, clarify)
+ apply (simp add: permute_fun_def permute_pure expand_fun_eq)
+ apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> P` pure_fresh])
+ apply (subst fresh_fun_app' [where a=a, OF `atom a \<sharp> Q` pure_fresh])
+ apply (rule refl)
+ done
+qed
+
+lemma FRESH_conj_iff:
+ fixes P Q :: "'a::at \<Rightarrow> bool"
+ assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+ shows "(FRESH x. P x \<and> Q x) \<longleftrightarrow> (FRESH x. P x) \<and> (FRESH x. Q x)"
+using P Q by (rule FRESH_binop_iff)
+
+lemma FRESH_disj_iff:
+ fixes P Q :: "'a::at \<Rightarrow> bool"
+ assumes P: "finite (supp P)" and Q: "finite (supp Q)"
+ shows "(FRESH x. P x \<or> Q x) \<longleftrightarrow> (FRESH x. P x) \<or> (FRESH x. Q x)"
+using P Q by (rule FRESH_binop_iff)
+
+
+section {* An example of a function without finite support *}
+
+primrec
+ nat_of :: "atom \<Rightarrow> nat"
+where
+ "nat_of (Atom s n) = n"
+
+lemma atom_eq_iff:
+ fixes a b :: atom
+ shows "a = b \<longleftrightarrow> sort_of a = sort_of b \<and> nat_of a = nat_of b"
+ by (induct a, induct b, simp)
+
+lemma not_fresh_nat_of:
+ shows "\<not> a \<sharp> nat_of"
+unfolding fresh_def supp_def
+proof (clarsimp)
+ assume "finite {b. (a \<rightleftharpoons> b) \<bullet> nat_of \<noteq> nat_of}"
+ hence "finite ({a} \<union> {b. (a \<rightleftharpoons> b) \<bullet> nat_of \<noteq> nat_of})"
+ by simp
+ then obtain b where
+ b1: "b \<noteq> a" and
+ b2: "sort_of b = sort_of a" and
+ b3: "(a \<rightleftharpoons> b) \<bullet> nat_of = nat_of"
+ by (rule obtain_atom) auto
+ have "nat_of a = (a \<rightleftharpoons> b) \<bullet> (nat_of a)" by (simp add: permute_nat_def)
+ also have "\<dots> = ((a \<rightleftharpoons> b) \<bullet> nat_of) ((a \<rightleftharpoons> b) \<bullet> a)" by (simp add: permute_fun_app_eq)
+ also have "\<dots> = nat_of ((a \<rightleftharpoons> b) \<bullet> a)" using b3 by simp
+ also have "\<dots> = nat_of b" using b2 by simp
+ finally have "nat_of a = nat_of b" by simp
+ with b2 have "a = b" by (simp add: atom_eq_iff)
+ with b1 show "False" by simp
+qed
+
+lemma supp_nat_of:
+ shows "supp nat_of = UNIV"
+ using not_fresh_nat_of [unfolded fresh_def] by auto
+
+
+section {* Support for sets of atoms *}
+
+lemma supp_finite_atom_set:
+ fixes S::"atom set"
+ assumes "finite S"
+ shows "supp S = S"
+ apply(rule finite_supp_unique)
+ apply(simp add: supports_def)
+ apply(simp add: swap_set_not_in)
+ apply(rule assms)
+ apply(simp add: swap_set_in)
+done
+
+
+section {* transpositions of permutations *}
+
+fun
+ add_perm
+where
+ "add_perm [] = 0"
+| "add_perm ((a, b) # xs) = (a \<rightleftharpoons> b) + add_perm xs"
+
+lemma add_perm_append:
+ shows "add_perm (xs @ ys) = add_perm xs + add_perm ys"
+by (induct xs arbitrary: ys)
+ (auto simp add: add_assoc)
+
+lemma perm_list_exists:
+ fixes p::perm
+ shows "\<exists>xs. p = add_perm xs \<and> supp xs \<subseteq> supp p"
+apply(induct p taking: "\<lambda>p::perm. card (supp p)" rule: measure_induct)
+apply(rename_tac p)
+apply(case_tac "supp p = {}")
+apply(simp)
+apply(simp add: supp_perm)
+apply(rule_tac x="[]" in exI)
+apply(simp add: supp_Nil)
+apply(simp add: expand_perm_eq)
+apply(subgoal_tac "\<exists>x. x \<in> supp p")
+defer
+apply(auto)[1]
+apply(erule exE)
+apply(drule_tac x="p + (((- p) \<bullet> x) \<rightleftharpoons> x)" in spec)
+apply(drule mp)
+defer
+apply(erule exE)
+apply(rule_tac x="xs @ [((- p) \<bullet> x, x)]" in exI)
+apply(simp add: add_perm_append)
+apply(erule conjE)
+apply(drule sym)
+apply(simp)
+apply(simp add: expand_perm_eq)
+apply(simp add: supp_append)
+apply(simp add: supp_perm supp_Cons supp_Nil supp_Pair supp_atom)
+apply(rule conjI)
+prefer 2
+apply(auto)[1]
+apply (metis permute_atom_def_raw permute_minus_cancel(2))
+defer
+apply(rule psubset_card_mono)
+apply(simp add: finite_supp)
+apply(rule psubsetI)
+defer
+apply(subgoal_tac "x \<notin> supp (p + (- p \<bullet> x \<rightleftharpoons> x))")
+apply(blast)
+apply(simp add: supp_perm)
+defer
+apply(auto simp add: supp_perm)[1]
+apply(case_tac "x = xa")
+apply(simp)
+apply(case_tac "((- p) \<bullet> x) = xa")
+apply(simp)
+apply(case_tac "sort_of xa = sort_of x")
+apply(simp)
+apply(auto)[1]
+apply(simp)
+apply(simp)
+apply(subgoal_tac "{a. p \<bullet> (- p \<bullet> x \<rightleftharpoons> x) \<bullet> a \<noteq> a} \<subseteq> {a. p \<bullet> a \<noteq> a}")
+apply(blast)
+apply(auto simp add: supp_perm)[1]
+apply(case_tac "x = xa")
+apply(simp)
+apply(case_tac "((- p) \<bullet> x) = xa")
+apply(simp)
+apply(case_tac "sort_of xa = sort_of x")
+apply(simp)
+apply(auto)[1]
+apply(simp)
+apply(simp)
+done
+
+section {* Lemma about support and permutations *}
+
+lemma supp_perm_eq:
+ assumes a: "(supp x) \<sharp>* p"
+ shows "p \<bullet> x = x"
+proof -
+ obtain xs where eq: "p = add_perm xs" and sub: "supp xs \<subseteq> supp p"
+ using perm_list_exists by blast
+ from a have "\<forall>a \<in> supp p. a \<sharp> x"
+ by (auto simp add: fresh_star_def fresh_def supp_perm)
+ with eq sub have "\<forall>a \<in> supp xs. a \<sharp> x" by auto
+ then have "add_perm xs \<bullet> x = x"
+ by (induct xs rule: add_perm.induct)
+ (simp_all add: supp_Cons supp_Pair supp_atom swap_fresh_fresh)
+ then show "p \<bullet> x = x" using eq by simp
+qed
+
+section {* at_set_avoiding2 *}
+
+lemma at_set_avoiding2:
+ assumes "finite xs"
+ and "finite (supp c)" "finite (supp x)"
+ and "xs \<sharp>* x"
+ shows "\<exists>p. (p \<bullet> xs) \<sharp>* c \<and> supp x \<sharp>* p"
+using assms
+apply(erule_tac c="(c, x)" in at_set_avoiding)
+apply(simp add: supp_Pair)
+apply(rule_tac x="p" in exI)
+apply(simp add: fresh_star_prod)
+apply(subgoal_tac "\<forall>a \<in> supp p. a \<sharp> x")
+apply(auto simp add: fresh_star_def fresh_def supp_perm)[1]
+apply(auto simp add: fresh_star_def fresh_def)
+done
+
+lemma at_set_avoiding2_atom:
+ assumes "finite (supp c)" "finite (supp x)"
+ and b: "xa \<sharp> x"
+ shows "\<exists>p. (p \<bullet> xa) \<sharp> c \<and> supp x \<sharp>* p"
+proof -
+ have a: "{xa} \<sharp>* x" unfolding fresh_star_def by (simp add: b)
+ obtain p where p1: "(p \<bullet> {xa}) \<sharp>* c" and p2: "supp x \<sharp>* p"
+ using at_set_avoiding2[of "{xa}" "c" "x"] assms a by blast
+ have c: "(p \<bullet> xa) \<sharp> c" using p1
+ unfolding fresh_star_def Ball_def
+ by (erule_tac x="p \<bullet> xa" in allE) (simp add: eqvts)
+ hence "p \<bullet> xa \<sharp> c \<and> supp x \<sharp>* p" using p2 by blast
+ then show ?thesis by blast
+qed
+
+end