--- a/Quot/Nominal/Nominal2_Supp.thy Thu Feb 04 14:55:52 2010 +0100
+++ b/Quot/Nominal/Nominal2_Supp.thy Thu Feb 04 15:16:34 2010 +0100
@@ -1,375 +1,1 @@
-(* 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
- "xs \<sharp>* c \<equiv> \<forall>x \<in> xs. x \<sharp> c"
-
-lemma fresh_star_prod:
- fixes xs::"atom set"
- shows "xs \<sharp>* (a, b) = (xs \<sharp>* a \<and> xs \<sharp>* b)"
- by (auto simp add: fresh_star_def fresh_Pair)
-
-lemma fresh_star_union:
- shows "(xs \<union> ys) \<sharp>* c = (xs \<sharp>* c \<and> ys \<sharp>* c)"
- by (auto simp add: fresh_star_def)
-
-lemma fresh_star_insert:
- shows "(insert x ys) \<sharp>* c = (x \<sharp> c \<and> ys \<sharp>* c)"
- by (auto simp add: fresh_star_def)
-
-lemma fresh_star_Un_elim:
- "((S \<union> T) \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (S \<sharp>* c \<Longrightarrow> T \<sharp>* c \<Longrightarrow> PROP C)"
- unfolding fresh_star_def
- apply(rule)
- apply(erule meta_mp)
- apply(auto)
- done
-
-lemma fresh_star_insert_elim:
- "(insert x S \<sharp>* c \<Longrightarrow> PROP C) \<equiv> (x \<sharp> c \<Longrightarrow> S \<sharp>* c \<Longrightarrow> PROP C)"
- unfolding fresh_star_def
- by rule (simp_all add: fresh_star_def)
-
-lemma fresh_star_empty_elim:
- "({} \<sharp>* c \<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)
-
-
-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)
- 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
-
-
-(*
-lemma supp_infinite:
- fixes S::"atom set"
- assumes asm: "finite (UNIV - S)"
- shows "(supp S) = (UNIV - S)"
-apply(rule finite_supp_unique)
-apply(auto simp add: supports_def permute_set_eq swap_atom)[1]
-apply(rule asm)
-apply(auto simp add: permute_set_eq swap_atom)[1]
-done
-
-lemma supp_infinite_coinfinite:
- fixes S::"atom set"
- assumes asm1: "infinite S"
- and asm2: "infinite (UNIV-S)"
- shows "(supp S) = (UNIV::atom set)"
-*)
-
-
-end
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