--- a/Quot/Nominal/Nominal2_Atoms.thy Thu Feb 04 14:55:52 2010 +0100
+++ b/Quot/Nominal/Nominal2_Atoms.thy Thu Feb 04 15:16:34 2010 +0100
@@ -1,221 +1,1 @@
-(* Title: Nominal2_Atoms
- Authors: Brian Huffman, Christian Urban
-
- Definitions for concrete atom types.
-*)
-theory Nominal2_Atoms
-imports Nominal2_Base
-uses ("atom_decl.ML")
-begin
-
-section {* Concrete atom types *}
-
-text {*
- Class @{text at_base} allows types containing multiple sorts of atoms.
- Class @{text at} only allows types with a single sort.
-*}
-
-class at_base = pt +
- fixes atom :: "'a \<Rightarrow> atom"
- assumes atom_eq_iff [simp]: "atom a = atom b \<longleftrightarrow> a = b"
- assumes atom_eqvt: "p \<bullet> (atom a) = atom (p \<bullet> a)"
-
-class at = at_base +
- assumes sort_of_atom_eq [simp]: "sort_of (atom a) = sort_of (atom b)"
-
-instance at < at_base ..
-
-lemma supp_at_base:
- fixes a::"'a::at_base"
- shows "supp a = {atom a}"
- by (simp add: supp_atom [symmetric] supp_def atom_eqvt)
-
-lemma fresh_at:
- shows "a \<sharp> b \<longleftrightarrow> a \<noteq> atom b"
- unfolding fresh_def by (simp add: supp_at_base)
-
-instance at_base < fs
-proof qed (simp add: supp_at_base)
-
-
-lemma at_base_infinite [simp]:
- shows "infinite (UNIV :: 'a::at_base set)" (is "infinite ?U")
-proof
- obtain a :: 'a where "True" by auto
- assume "finite ?U"
- hence "finite (atom ` ?U)"
- by (rule finite_imageI)
- then obtain b where b: "b \<notin> atom ` ?U" "sort_of b = sort_of (atom a)"
- by (rule obtain_atom)
- from b(2) have "b = atom ((atom a \<rightleftharpoons> b) \<bullet> a)"
- unfolding atom_eqvt [symmetric]
- by (simp add: swap_atom)
- hence "b \<in> atom ` ?U" by simp
- with b(1) show "False" by simp
-qed
-
-lemma swap_at_base_simps [simp]:
- fixes x y::"'a::at_base"
- shows "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> x = y"
- and "sort_of (atom x) = sort_of (atom y) \<Longrightarrow> (atom x \<rightleftharpoons> atom y) \<bullet> y = x"
- and "atom x \<noteq> a \<Longrightarrow> atom x \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x"
- unfolding atom_eq_iff [symmetric]
- unfolding atom_eqvt [symmetric]
- by simp_all
-
-lemma obtain_at_base:
- assumes X: "finite X"
- obtains a::"'a::at_base" where "atom a \<notin> X"
-proof -
- have "inj (atom :: 'a \<Rightarrow> atom)"
- by (simp add: inj_on_def)
- with X have "finite (atom -` X :: 'a set)"
- by (rule finite_vimageI)
- with at_base_infinite have "atom -` X \<noteq> (UNIV :: 'a set)"
- by auto
- then obtain a :: 'a where "atom a \<notin> X"
- by auto
- thus ?thesis ..
-qed
-
-
-section {* A swapping operation for concrete atoms *}
-
-definition
- flip :: "'a::at_base \<Rightarrow> 'a \<Rightarrow> perm" ("'(_ \<leftrightarrow> _')")
-where
- "(a \<leftrightarrow> b) = (atom a \<rightleftharpoons> atom b)"
-
-lemma flip_self [simp]: "(a \<leftrightarrow> a) = 0"
- unfolding flip_def by (rule swap_self)
-
-lemma flip_commute: "(a \<leftrightarrow> b) = (b \<leftrightarrow> a)"
- unfolding flip_def by (rule swap_commute)
-
-lemma minus_flip [simp]: "- (a \<leftrightarrow> b) = (a \<leftrightarrow> b)"
- unfolding flip_def by (rule minus_swap)
-
-lemma add_flip_cancel: "(a \<leftrightarrow> b) + (a \<leftrightarrow> b) = 0"
- unfolding flip_def by (rule swap_cancel)
-
-lemma permute_flip_cancel [simp]: "(a \<leftrightarrow> b) \<bullet> (a \<leftrightarrow> b) \<bullet> x = x"
- unfolding permute_plus [symmetric] add_flip_cancel by simp
-
-lemma permute_flip_cancel2 [simp]: "(a \<leftrightarrow> b) \<bullet> (b \<leftrightarrow> a) \<bullet> x = x"
- by (simp add: flip_commute)
-
-lemma flip_eqvt:
- fixes a b c::"'a::at_base"
- shows "p \<bullet> (a \<leftrightarrow> b) = (p \<bullet> a \<leftrightarrow> p \<bullet> b)"
- unfolding flip_def
- by (simp add: swap_eqvt atom_eqvt)
-
-lemma flip_at_base_simps [simp]:
- shows "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> a = b"
- and "sort_of (atom a) = sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> b = a"
- and "\<lbrakk>a \<noteq> c; b \<noteq> c\<rbrakk> \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> c = c"
- and "sort_of (atom a) \<noteq> sort_of (atom b) \<Longrightarrow> (a \<leftrightarrow> b) \<bullet> x = x"
- unfolding flip_def
- unfolding atom_eq_iff [symmetric]
- unfolding atom_eqvt [symmetric]
- by simp_all
-
-text {* the following two lemmas do not hold for at_base,
- only for single sort atoms from at *}
-
-lemma permute_flip_at:
- fixes a b c::"'a::at"
- shows "(a \<leftrightarrow> b) \<bullet> c = (if c = a then b else if c = b then a else c)"
- unfolding flip_def
- apply (rule atom_eq_iff [THEN iffD1])
- apply (subst atom_eqvt [symmetric])
- apply (simp add: swap_atom)
- done
-
-lemma flip_at_simps [simp]:
- fixes a b::"'a::at"
- shows "(a \<leftrightarrow> b) \<bullet> a = b"
- and "(a \<leftrightarrow> b) \<bullet> b = a"
- unfolding permute_flip_at by simp_all
-
-
-subsection {* Syntax for coercing at-elements to the atom-type *}
-
-syntax
- "_atom_constrain" :: "logic \<Rightarrow> type \<Rightarrow> logic" ("_:::_" [4, 0] 3)
-
-translations
- "_atom_constrain a t" => "atom (_constrain a t)"
-
-
-subsection {* A lemma for proving instances of class @{text at}. *}
-
-setup {* Sign.add_const_constraint (@{const_name "permute"}, NONE) *}
-setup {* Sign.add_const_constraint (@{const_name "atom"}, NONE) *}
-
-text {*
- New atom types are defined as subtypes of @{typ atom}.
-*}
-
-lemma exists_eq_sort:
- shows "\<exists>a. a \<in> {a. sort_of a = s}"
- by (rule_tac x="Atom s 0" in exI, simp)
-
-lemma at_base_class:
- fixes s :: atom_sort
- fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
- assumes type: "type_definition Rep Abs {a. P (sort_of a)}"
- assumes atom_def: "\<And>a. atom a = Rep a"
- assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
- shows "OFCLASS('a, at_base_class)"
-proof
- interpret type_definition Rep Abs "{a. P (sort_of a)}" by (rule type)
- have sort_of_Rep: "\<And>a. P (sort_of (Rep a))" using Rep by simp
- fix a b :: 'a and p p1 p2 :: perm
- show "0 \<bullet> a = a"
- unfolding permute_def by (simp add: Rep_inverse)
- show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
- unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
- show "atom a = atom b \<longleftrightarrow> a = b"
- unfolding atom_def by (simp add: Rep_inject)
- show "p \<bullet> atom a = atom (p \<bullet> a)"
- unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
-qed
-
-lemma at_class:
- fixes s :: atom_sort
- fixes Rep :: "'a \<Rightarrow> atom" and Abs :: "atom \<Rightarrow> 'a"
- assumes type: "type_definition Rep Abs {a. sort_of a = s}"
- assumes atom_def: "\<And>a. atom a = Rep a"
- assumes permute_def: "\<And>p a. p \<bullet> a = Abs (p \<bullet> Rep a)"
- shows "OFCLASS('a, at_class)"
-proof
- interpret type_definition Rep Abs "{a. sort_of a = s}" by (rule type)
- have sort_of_Rep: "\<And>a. sort_of (Rep a) = s" using Rep by simp
- fix a b :: 'a and p p1 p2 :: perm
- show "0 \<bullet> a = a"
- unfolding permute_def by (simp add: Rep_inverse)
- show "(p1 + p2) \<bullet> a = p1 \<bullet> p2 \<bullet> a"
- unfolding permute_def by (simp add: Abs_inverse sort_of_Rep)
- show "sort_of (atom a) = sort_of (atom b)"
- unfolding atom_def by (simp add: sort_of_Rep)
- show "atom a = atom b \<longleftrightarrow> a = b"
- unfolding atom_def by (simp add: Rep_inject)
- show "p \<bullet> atom a = atom (p \<bullet> a)"
- unfolding permute_def atom_def by (simp add: Abs_inverse sort_of_Rep)
-qed
-
-setup {* Sign.add_const_constraint
- (@{const_name "permute"}, SOME @{typ "perm \<Rightarrow> 'a::pt \<Rightarrow> 'a"}) *}
-setup {* Sign.add_const_constraint
- (@{const_name "atom"}, SOME @{typ "'a::at_base \<Rightarrow> atom"}) *}
-
-
-section {* Automation for creating concrete atom types *}
-
-text {* at the moment only single-sort concrete atoms are supported *}
-
-use "atom_decl.ML"
-
-
-end
+/home/cu200/Isabelle/nominal-huffman/Nominal2_Atoms.thy
\ No newline at end of file
--- a/Quot/Nominal/Nominal2_Base.thy Thu Feb 04 14:55:52 2010 +0100
+++ b/Quot/Nominal/Nominal2_Base.thy Thu Feb 04 15:16:34 2010 +0100
@@ -1,1007 +1,1 @@
-(* Title: Nominal2_Base
- Authors: Brian Huffman, Christian Urban
-
- Basic definitions and lemma infrastructure for
- Nominal Isabelle.
-*)
-theory Nominal2_Base
-imports Main Infinite_Set
-begin
-
-section {* Atoms and Sorts *}
-
-text {* A simple implementation for atom_sorts is strings. *}
-(* types atom_sort = string *)
-
-text {* To deal with Church-like binding we use trees of
- strings as sorts. *}
-
-datatype atom_sort = Sort "string" "atom_sort list"
-
-datatype atom = Atom atom_sort nat
-
-
-text {* Basic projection function. *}
-
-primrec
- sort_of :: "atom \<Rightarrow> atom_sort"
-where
- "sort_of (Atom s i) = s"
-
-
-text {* There are infinitely many atoms of each sort. *}
-lemma INFM_sort_of_eq:
- shows "INFM a. sort_of a = s"
-proof -
- have "INFM i. sort_of (Atom s i) = s" by simp
- moreover have "inj (Atom s)" by (simp add: inj_on_def)
- ultimately show "INFM a. sort_of a = s" by (rule INFM_inj)
-qed
-
-lemma infinite_sort_of_eq:
- shows "infinite {a. sort_of a = s}"
- using INFM_sort_of_eq unfolding INFM_iff_infinite .
-
-lemma atom_infinite [simp]:
- shows "infinite (UNIV :: atom set)"
- using subset_UNIV infinite_sort_of_eq
- by (rule infinite_super)
-
-lemma obtain_atom:
- fixes X :: "atom set"
- assumes X: "finite X"
- obtains a where "a \<notin> X" "sort_of a = s"
-proof -
- from X have "MOST a. a \<notin> X"
- unfolding MOST_iff_cofinite by simp
- with INFM_sort_of_eq
- have "INFM a. sort_of a = s \<and> a \<notin> X"
- by (rule INFM_conjI)
- then obtain a where "a \<notin> X" "sort_of a = s"
- by (auto elim: INFM_E)
- then show ?thesis ..
-qed
-
-section {* Sort-Respecting Permutations *}
-
-typedef perm =
- "{f. bij f \<and> finite {a. f a \<noteq> a} \<and> (\<forall>a. sort_of (f a) = sort_of a)}"
-proof
- show "id \<in> ?perm" by simp
-qed
-
-lemma permI:
- assumes "bij f" and "MOST x. f x = x" and "\<And>a. sort_of (f a) = sort_of a"
- shows "f \<in> perm"
- using assms unfolding perm_def MOST_iff_cofinite by simp
-
-lemma perm_is_bij: "f \<in> perm \<Longrightarrow> bij f"
- unfolding perm_def by simp
-
-lemma perm_is_finite: "f \<in> perm \<Longrightarrow> finite {a. f a \<noteq> a}"
- unfolding perm_def by simp
-
-lemma perm_is_sort_respecting: "f \<in> perm \<Longrightarrow> sort_of (f a) = sort_of a"
- unfolding perm_def by simp
-
-lemma perm_MOST: "f \<in> perm \<Longrightarrow> MOST x. f x = x"
- unfolding perm_def MOST_iff_cofinite by simp
-
-lemma perm_id: "id \<in> perm"
- unfolding perm_def by simp
-
-lemma perm_comp:
- assumes f: "f \<in> perm" and g: "g \<in> perm"
- shows "(f \<circ> g) \<in> perm"
-apply (rule permI)
-apply (rule bij_comp)
-apply (rule perm_is_bij [OF g])
-apply (rule perm_is_bij [OF f])
-apply (rule MOST_rev_mp [OF perm_MOST [OF g]])
-apply (rule MOST_rev_mp [OF perm_MOST [OF f]])
-apply (simp)
-apply (simp add: perm_is_sort_respecting [OF f])
-apply (simp add: perm_is_sort_respecting [OF g])
-done
-
-lemma perm_inv:
- assumes f: "f \<in> perm"
- shows "(inv f) \<in> perm"
-apply (rule permI)
-apply (rule bij_imp_bij_inv)
-apply (rule perm_is_bij [OF f])
-apply (rule MOST_mono [OF perm_MOST [OF f]])
-apply (erule subst, rule inv_f_f)
-apply (rule bij_is_inj [OF perm_is_bij [OF f]])
-apply (rule perm_is_sort_respecting [OF f, THEN sym, THEN trans])
-apply (simp add: surj_f_inv_f [OF bij_is_surj [OF perm_is_bij [OF f]]])
-done
-
-lemma bij_Rep_perm: "bij (Rep_perm p)"
- using Rep_perm [of p] unfolding perm_def by simp
-
-lemma finite_Rep_perm: "finite {a. Rep_perm p a \<noteq> a}"
- using Rep_perm [of p] unfolding perm_def by simp
-
-lemma sort_of_Rep_perm: "sort_of (Rep_perm p a) = sort_of a"
- using Rep_perm [of p] unfolding perm_def by simp
-
-lemma Rep_perm_ext:
- "Rep_perm p1 = Rep_perm p2 \<Longrightarrow> p1 = p2"
- by (simp add: expand_fun_eq Rep_perm_inject [symmetric])
-
-
-subsection {* Permutations form a group *}
-
-instantiation perm :: group_add
-begin
-
-definition
- "0 = Abs_perm id"
-
-definition
- "- p = Abs_perm (inv (Rep_perm p))"
-
-definition
- "p + q = Abs_perm (Rep_perm p \<circ> Rep_perm q)"
-
-definition
- "(p1::perm) - p2 = p1 + - p2"
-
-lemma Rep_perm_0: "Rep_perm 0 = id"
- unfolding zero_perm_def
- by (simp add: Abs_perm_inverse perm_id)
-
-lemma Rep_perm_add:
- "Rep_perm (p1 + p2) = Rep_perm p1 \<circ> Rep_perm p2"
- unfolding plus_perm_def
- by (simp add: Abs_perm_inverse perm_comp Rep_perm)
-
-lemma Rep_perm_uminus:
- "Rep_perm (- p) = inv (Rep_perm p)"
- unfolding uminus_perm_def
- by (simp add: Abs_perm_inverse perm_inv Rep_perm)
-
-instance
-apply default
-unfolding Rep_perm_inject [symmetric]
-unfolding minus_perm_def
-unfolding Rep_perm_add
-unfolding Rep_perm_uminus
-unfolding Rep_perm_0
-by (simp_all add: o_assoc inv_o_cancel [OF bij_is_inj [OF bij_Rep_perm]])
-
-end
-
-
-section {* Implementation of swappings *}
-
-definition
- swap :: "atom \<Rightarrow> atom \<Rightarrow> perm" ("'(_ \<rightleftharpoons> _')")
-where
- "(a \<rightleftharpoons> b) =
- Abs_perm (if sort_of a = sort_of b
- then (\<lambda>c. if a = c then b else if b = c then a else c)
- else id)"
-
-lemma Rep_perm_swap:
- "Rep_perm (a \<rightleftharpoons> b) =
- (if sort_of a = sort_of b
- then (\<lambda>c. if a = c then b else if b = c then a else c)
- else id)"
-unfolding swap_def
-apply (rule Abs_perm_inverse)
-apply (rule permI)
-apply (auto simp add: bij_def inj_on_def surj_def)[1]
-apply (rule MOST_rev_mp [OF MOST_neq(1) [of a]])
-apply (rule MOST_rev_mp [OF MOST_neq(1) [of b]])
-apply (simp)
-apply (simp)
-done
-
-lemmas Rep_perm_simps =
- Rep_perm_0
- Rep_perm_add
- Rep_perm_uminus
- Rep_perm_swap
-
-lemma swap_different_sorts [simp]:
- "sort_of a \<noteq> sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) = 0"
- by (rule Rep_perm_ext) (simp add: Rep_perm_simps)
-
-lemma swap_cancel:
- "(a \<rightleftharpoons> b) + (a \<rightleftharpoons> b) = 0"
-by (rule Rep_perm_ext)
- (simp add: Rep_perm_simps expand_fun_eq)
-
-lemma swap_self [simp]:
- "(a \<rightleftharpoons> a) = 0"
- by (rule Rep_perm_ext, simp add: Rep_perm_simps expand_fun_eq)
-
-lemma minus_swap [simp]:
- "- (a \<rightleftharpoons> b) = (a \<rightleftharpoons> b)"
- by (rule minus_unique [OF swap_cancel])
-
-lemma swap_commute:
- "(a \<rightleftharpoons> b) = (b \<rightleftharpoons> a)"
- by (rule Rep_perm_ext)
- (simp add: Rep_perm_swap expand_fun_eq)
-
-lemma swap_triple:
- assumes "a \<noteq> b" and "c \<noteq> b"
- assumes "sort_of a = sort_of b" "sort_of b = sort_of c"
- shows "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
- using assms
- by (rule_tac Rep_perm_ext)
- (auto simp add: Rep_perm_simps expand_fun_eq)
-
-
-section {* Permutation Types *}
-
-text {*
- Infix syntax for @{text permute} has higher precedence than
- addition, but lower than unary minus.
-*}
-
-class pt =
- fixes permute :: "perm \<Rightarrow> 'a \<Rightarrow> 'a" ("_ \<bullet> _" [76, 75] 75)
- assumes permute_zero [simp]: "0 \<bullet> x = x"
- assumes permute_plus [simp]: "(p + q) \<bullet> x = p \<bullet> (q \<bullet> x)"
-begin
-
-lemma permute_diff [simp]:
- shows "(p - q) \<bullet> x = p \<bullet> - q \<bullet> x"
- unfolding diff_minus by simp
-
-lemma permute_minus_cancel [simp]:
- shows "p \<bullet> - p \<bullet> x = x"
- and "- p \<bullet> p \<bullet> x = x"
- unfolding permute_plus [symmetric] by simp_all
-
-lemma permute_swap_cancel [simp]:
- shows "(a \<rightleftharpoons> b) \<bullet> (a \<rightleftharpoons> b) \<bullet> x = x"
- unfolding permute_plus [symmetric]
- by (simp add: swap_cancel)
-
-lemma permute_swap_cancel2 [simp]:
- shows "(a \<rightleftharpoons> b) \<bullet> (b \<rightleftharpoons> a) \<bullet> x = x"
- unfolding permute_plus [symmetric]
- by (simp add: swap_commute)
-
-lemma inj_permute [simp]:
- shows "inj (permute p)"
- by (rule inj_on_inverseI)
- (rule permute_minus_cancel)
-
-lemma surj_permute [simp]:
- shows "surj (permute p)"
- by (rule surjI, rule permute_minus_cancel)
-
-lemma bij_permute [simp]:
- shows "bij (permute p)"
- by (rule bijI [OF inj_permute surj_permute])
-
-lemma inv_permute:
- shows "inv (permute p) = permute (- p)"
- by (rule inv_equality) (simp_all)
-
-lemma permute_minus:
- shows "permute (- p) = inv (permute p)"
- by (simp add: inv_permute)
-
-lemma permute_eq_iff [simp]:
- shows "p \<bullet> x = p \<bullet> y \<longleftrightarrow> x = y"
- by (rule inj_permute [THEN inj_eq])
-
-end
-
-subsection {* Permutations for atoms *}
-
-instantiation atom :: pt
-begin
-
-definition
- "p \<bullet> a = Rep_perm p a"
-
-instance
-apply(default)
-apply(simp_all add: permute_atom_def Rep_perm_simps)
-done
-
-end
-
-lemma sort_of_permute [simp]:
- shows "sort_of (p \<bullet> a) = sort_of a"
- unfolding permute_atom_def by (rule sort_of_Rep_perm)
-
-lemma swap_atom:
- shows "(a \<rightleftharpoons> b) \<bullet> c =
- (if sort_of a = sort_of b
- then (if c = a then b else if c = b then a else c) else c)"
- unfolding permute_atom_def
- by (simp add: Rep_perm_swap)
-
-lemma swap_atom_simps [simp]:
- "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> a = b"
- "sort_of a = sort_of b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> b = a"
- "c \<noteq> a \<Longrightarrow> c \<noteq> b \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> c = c"
- unfolding swap_atom by simp_all
-
-lemma expand_perm_eq:
- fixes p q :: "perm"
- shows "p = q \<longleftrightarrow> (\<forall>a::atom. p \<bullet> a = q \<bullet> a)"
- unfolding permute_atom_def
- by (metis Rep_perm_ext ext)
-
-
-subsection {* Permutations for permutations *}
-
-instantiation perm :: pt
-begin
-
-definition
- "p \<bullet> q = p + q - p"
-
-instance
-apply default
-apply (simp add: permute_perm_def)
-apply (simp add: permute_perm_def diff_minus minus_add add_assoc)
-done
-
-end
-
-lemma permute_self: "p \<bullet> p = p"
-unfolding permute_perm_def by (simp add: diff_minus add_assoc)
-
-lemma permute_eqvt:
- shows "p \<bullet> (q \<bullet> x) = (p \<bullet> q) \<bullet> (p \<bullet> x)"
- unfolding permute_perm_def by simp
-
-lemma zero_perm_eqvt:
- shows "p \<bullet> (0::perm) = 0"
- unfolding permute_perm_def by simp
-
-lemma add_perm_eqvt:
- fixes p p1 p2 :: perm
- shows "p \<bullet> (p1 + p2) = p \<bullet> p1 + p \<bullet> p2"
- unfolding permute_perm_def
- by (simp add: expand_perm_eq)
-
-lemma swap_eqvt:
- shows "p \<bullet> (a \<rightleftharpoons> b) = (p \<bullet> a \<rightleftharpoons> p \<bullet> b)"
- unfolding permute_perm_def
- by (auto simp add: swap_atom expand_perm_eq)
-
-
-subsection {* Permutations for functions *}
-
-instantiation "fun" :: (pt, pt) pt
-begin
-
-definition
- "p \<bullet> f = (\<lambda>x. p \<bullet> (f (- p \<bullet> x)))"
-
-instance
-apply default
-apply (simp add: permute_fun_def)
-apply (simp add: permute_fun_def minus_add)
-done
-
-end
-
-lemma permute_fun_app_eq:
- shows "p \<bullet> (f x) = (p \<bullet> f) (p \<bullet> x)"
-unfolding permute_fun_def by simp
-
-
-subsection {* Permutations for booleans *}
-
-instantiation bool :: pt
-begin
-
-definition "p \<bullet> (b::bool) = b"
-
-instance
-apply(default)
-apply(simp_all add: permute_bool_def)
-done
-
-end
-
-lemma Not_eqvt:
- shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
-by (simp add: permute_bool_def)
-
-
-subsection {* Permutations for sets *}
-
-lemma permute_set_eq:
- fixes x::"'a::pt"
- and p::"perm"
- shows "(p \<bullet> X) = {p \<bullet> x | x. x \<in> X}"
- apply(auto simp add: permute_fun_def permute_bool_def mem_def)
- apply(rule_tac x="- p \<bullet> x" in exI)
- apply(simp)
- done
-
-lemma permute_set_eq_image:
- shows "p \<bullet> X = permute p ` X"
-unfolding permute_set_eq by auto
-
-lemma permute_set_eq_vimage:
- shows "p \<bullet> X = permute (- p) -` X"
-unfolding permute_fun_def permute_bool_def
-unfolding vimage_def Collect_def mem_def ..
-
-lemma swap_set_not_in:
- assumes a: "a \<notin> S" "b \<notin> S"
- shows "(a \<rightleftharpoons> b) \<bullet> S = S"
- using a by (auto simp add: permute_set_eq swap_atom)
-
-lemma swap_set_in:
- assumes a: "a \<in> S" "b \<notin> S" "sort_of a = sort_of b"
- shows "(a \<rightleftharpoons> b) \<bullet> S \<noteq> S"
- using a by (auto simp add: permute_set_eq swap_atom)
-
-
-subsection {* Permutations for units *}
-
-instantiation unit :: pt
-begin
-
-definition "p \<bullet> (u::unit) = u"
-
-instance proof
-qed (simp_all add: permute_unit_def)
-
-end
-
-
-subsection {* Permutations for products *}
-
-instantiation "*" :: (pt, pt) pt
-begin
-
-primrec
- permute_prod
-where
- Pair_eqvt: "p \<bullet> (x, y) = (p \<bullet> x, p \<bullet> y)"
-
-instance
-by default auto
-
-end
-
-subsection {* Permutations for sums *}
-
-instantiation "+" :: (pt, pt) pt
-begin
-
-primrec
- permute_sum
-where
- "p \<bullet> (Inl x) = Inl (p \<bullet> x)"
-| "p \<bullet> (Inr y) = Inr (p \<bullet> y)"
-
-instance proof
-qed (case_tac [!] x, simp_all)
-
-end
-
-subsection {* Permutations for lists *}
-
-instantiation list :: (pt) pt
-begin
-
-primrec
- permute_list
-where
- "p \<bullet> [] = []"
-| "p \<bullet> (x # xs) = p \<bullet> x # p \<bullet> xs"
-
-instance proof
-qed (induct_tac [!] x, simp_all)
-
-end
-
-subsection {* Permutations for options *}
-
-instantiation option :: (pt) pt
-begin
-
-primrec
- permute_option
-where
- "p \<bullet> None = None"
-| "p \<bullet> (Some x) = Some (p \<bullet> x)"
-
-instance proof
-qed (induct_tac [!] x, simp_all)
-
-end
-
-subsection {* Permutations for @{typ char}, @{typ nat}, and @{typ int} *}
-
-instantiation char :: pt
-begin
-
-definition "p \<bullet> (c::char) = c"
-
-instance proof
-qed (simp_all add: permute_char_def)
-
-end
-
-instantiation nat :: pt
-begin
-
-definition "p \<bullet> (n::nat) = n"
-
-instance proof
-qed (simp_all add: permute_nat_def)
-
-end
-
-instantiation int :: pt
-begin
-
-definition "p \<bullet> (i::int) = i"
-
-instance proof
-qed (simp_all add: permute_int_def)
-
-end
-
-
-section {* Pure types *}
-
-text {* Pure types will have always empty support. *}
-
-class pure = pt +
- assumes permute_pure: "p \<bullet> x = x"
-
-text {* Types @{typ unit} and @{typ bool} are pure. *}
-
-instance unit :: pure
-proof qed (rule permute_unit_def)
-
-instance bool :: pure
-proof qed (rule permute_bool_def)
-
-text {* Other type constructors preserve purity. *}
-
-instance "fun" :: (pure, pure) pure
-by default (simp add: permute_fun_def permute_pure)
-
-instance "*" :: (pure, pure) pure
-by default (induct_tac x, simp add: permute_pure)
-
-instance "+" :: (pure, pure) pure
-by default (induct_tac x, simp_all add: permute_pure)
-
-instance list :: (pure) pure
-by default (induct_tac x, simp_all add: permute_pure)
-
-instance option :: (pure) pure
-by default (induct_tac x, simp_all add: permute_pure)
-
-
-subsection {* Types @{typ char}, @{typ nat}, and @{typ int} *}
-
-instance char :: pure
-proof qed (rule permute_char_def)
-
-instance nat :: pure
-proof qed (rule permute_nat_def)
-
-instance int :: pure
-proof qed (rule permute_int_def)
-
-
-subsection {* Supp, Freshness and Supports *}
-
-context pt
-begin
-
-definition
- supp :: "'a \<Rightarrow> atom set"
-where
- "supp x = {a. infinite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}}"
-
-end
-
-definition
- fresh :: "atom \<Rightarrow> 'a::pt \<Rightarrow> bool" ("_ \<sharp> _" [55, 55] 55)
-where
- "a \<sharp> x \<equiv> a \<notin> supp x"
-
-lemma supp_conv_fresh:
- shows "supp x = {a. \<not> a \<sharp> x}"
- unfolding fresh_def by simp
-
-lemma swap_rel_trans:
- assumes "sort_of a = sort_of b"
- assumes "sort_of b = sort_of c"
- assumes "(a \<rightleftharpoons> c) \<bullet> x = x"
- assumes "(b \<rightleftharpoons> c) \<bullet> x = x"
- shows "(a \<rightleftharpoons> b) \<bullet> x = x"
-proof (cases)
- assume "a = b \<or> c = b"
- with assms show "(a \<rightleftharpoons> b) \<bullet> x = x" by auto
-next
- assume *: "\<not> (a = b \<or> c = b)"
- have "((a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c)) \<bullet> x = x"
- using assms by simp
- also have "(a \<rightleftharpoons> c) + (b \<rightleftharpoons> c) + (a \<rightleftharpoons> c) = (a \<rightleftharpoons> b)"
- using assms * by (simp add: swap_triple)
- finally show "(a \<rightleftharpoons> b) \<bullet> x = x" .
-qed
-
-lemma swap_fresh_fresh:
- assumes a: "a \<sharp> x"
- and b: "b \<sharp> x"
- shows "(a \<rightleftharpoons> b) \<bullet> x = x"
-proof (cases)
- assume asm: "sort_of a = sort_of b"
- have "finite {c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x}" "finite {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x}"
- using a b unfolding fresh_def supp_def by simp_all
- then have "finite ({c. (a \<rightleftharpoons> c) \<bullet> x \<noteq> x} \<union> {c. (b \<rightleftharpoons> c) \<bullet> x \<noteq> x})" by simp
- then obtain c
- where "(a \<rightleftharpoons> c) \<bullet> x = x" "(b \<rightleftharpoons> c) \<bullet> x = x" "sort_of c = sort_of b"
- by (rule obtain_atom) (auto)
- then show "(a \<rightleftharpoons> b) \<bullet> x = x" using asm by (rule_tac swap_rel_trans) (simp_all)
-next
- assume "sort_of a \<noteq> sort_of b"
- then show "(a \<rightleftharpoons> b) \<bullet> x = x" by simp
-qed
-
-
-subsection {* supp and fresh are equivariant *}
-
-lemma finite_Collect_bij:
- assumes a: "bij f"
- shows "finite {x. P (f x)} = finite {x. P x}"
-by (metis a finite_vimage_iff vimage_Collect_eq)
-
-lemma fresh_permute_iff:
- shows "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> a \<sharp> x"
-proof -
- have "(p \<bullet> a) \<sharp> (p \<bullet> x) \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
- unfolding fresh_def supp_def by simp
- also have "\<dots> \<longleftrightarrow> finite {b. (p \<bullet> a \<rightleftharpoons> p \<bullet> b) \<bullet> p \<bullet> x \<noteq> p \<bullet> x}"
- using bij_permute by (rule finite_Collect_bij [symmetric])
- also have "\<dots> \<longleftrightarrow> finite {b. p \<bullet> (a \<rightleftharpoons> b) \<bullet> x \<noteq> p \<bullet> x}"
- by (simp only: permute_eqvt [of p] swap_eqvt)
- also have "\<dots> \<longleftrightarrow> finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
- by (simp only: permute_eq_iff)
- also have "\<dots> \<longleftrightarrow> a \<sharp> x"
- unfolding fresh_def supp_def by simp
- finally show ?thesis .
-qed
-
-lemma fresh_eqvt:
- shows "p \<bullet> (a \<sharp> x) = (p \<bullet> a) \<sharp> (p \<bullet> x)"
- by (simp add: permute_bool_def fresh_permute_iff)
-
-lemma supp_eqvt:
- fixes p :: "perm"
- and x :: "'a::pt"
- shows "p \<bullet> (supp x) = supp (p \<bullet> x)"
- unfolding supp_conv_fresh
- unfolding permute_fun_def Collect_def
- by (simp add: Not_eqvt fresh_eqvt)
-
-subsection {* supports *}
-
-definition
- supports :: "atom set \<Rightarrow> 'a::pt \<Rightarrow> bool" (infixl "supports" 80)
-where
- "S supports x \<equiv> \<forall>a b. (a \<notin> S \<and> b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x)"
-
-lemma supp_is_subset:
- fixes S :: "atom set"
- and x :: "'a::pt"
- assumes a1: "S supports x"
- and a2: "finite S"
- shows "(supp x) \<subseteq> S"
-proof (rule ccontr)
- assume "\<not>(supp x \<subseteq> S)"
- then obtain a where b1: "a \<in> supp x" and b2: "a \<notin> S" by auto
- from a1 b2 have "\<forall>b. b \<notin> S \<longrightarrow> (a \<rightleftharpoons> b) \<bullet> x = x" by (unfold supports_def) (auto)
- hence "{b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x} \<subseteq> S" by auto
- with a2 have "finite {b. (a \<rightleftharpoons> b)\<bullet>x \<noteq> x}" by (simp add: finite_subset)
- then have "a \<notin> (supp x)" unfolding supp_def by simp
- with b1 show False by simp
-qed
-
-lemma supports_finite:
- fixes S :: "atom set"
- and x :: "'a::pt"
- assumes a1: "S supports x"
- and a2: "finite S"
- shows "finite (supp x)"
-proof -
- have "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
- then show "finite (supp x)" using a2 by (simp add: finite_subset)
-qed
-
-lemma supp_supports:
- fixes x :: "'a::pt"
- shows "(supp x) supports x"
-proof (unfold supports_def, intro strip)
- fix a b
- assume "a \<notin> (supp x) \<and> b \<notin> (supp x)"
- then have "a \<sharp> x" and "b \<sharp> x" by (simp_all add: fresh_def)
- then show "(a \<rightleftharpoons> b) \<bullet> x = x" by (rule swap_fresh_fresh)
-qed
-
-lemma supp_is_least_supports:
- fixes S :: "atom set"
- and x :: "'a::pt"
- assumes a1: "S supports x"
- and a2: "finite S"
- and a3: "\<And>S'. finite S' \<Longrightarrow> (S' supports x) \<Longrightarrow> S \<subseteq> S'"
- shows "(supp x) = S"
-proof (rule equalityI)
- show "(supp x) \<subseteq> S" using a1 a2 by (rule supp_is_subset)
- with a2 have fin: "finite (supp x)" by (rule rev_finite_subset)
- have "(supp x) supports x" by (rule supp_supports)
- with fin a3 show "S \<subseteq> supp x" by blast
-qed
-
-lemma subsetCI:
- shows "(\<And>x. x \<in> A \<Longrightarrow> x \<notin> B \<Longrightarrow> False) \<Longrightarrow> A \<subseteq> B"
- by auto
-
-lemma finite_supp_unique:
- assumes a1: "S supports x"
- assumes a2: "finite S"
- 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"
- shows "(supp x) = S"
- using a1 a2
-proof (rule supp_is_least_supports)
- fix S'
- assume "finite S'" and "S' supports x"
- show "S \<subseteq> S'"
- proof (rule subsetCI)
- fix a
- assume "a \<in> S" and "a \<notin> S'"
- have "finite (S \<union> S')"
- using `finite S` `finite S'` by simp
- then obtain b where "b \<notin> S \<union> S'" and "sort_of b = sort_of a"
- by (rule obtain_atom)
- then have "b \<notin> S" and "b \<notin> S'" and "sort_of a = sort_of b"
- by simp_all
- then have "(a \<rightleftharpoons> b) \<bullet> x = x"
- using `a \<notin> S'` `S' supports x` by (simp add: supports_def)
- moreover have "(a \<rightleftharpoons> b) \<bullet> x \<noteq> x"
- using `a \<in> S` `b \<notin> S` `sort_of a = sort_of b`
- by (rule a3)
- ultimately show "False" by simp
- qed
-qed
-
-section {* Finitely-supported types *}
-
-class fs = pt +
- assumes finite_supp: "finite (supp x)"
-
-lemma pure_supp:
- shows "supp (x::'a::pure) = {}"
- unfolding supp_def by (simp add: permute_pure)
-
-lemma pure_fresh:
- fixes x::"'a::pure"
- shows "a \<sharp> x"
- unfolding fresh_def by (simp add: pure_supp)
-
-instance pure < fs
-by default (simp add: pure_supp)
-
-
-subsection {* Type @{typ atom} is finitely-supported. *}
-
-lemma supp_atom:
- shows "supp a = {a}"
-apply (rule finite_supp_unique)
-apply (clarsimp simp add: supports_def)
-apply simp
-apply simp
-done
-
-lemma fresh_atom:
- shows "a \<sharp> b \<longleftrightarrow> a \<noteq> b"
- unfolding fresh_def supp_atom by simp
-
-instance atom :: fs
-by default (simp add: supp_atom)
-
-
-section {* Type @{typ perm} is finitely-supported. *}
-
-lemma perm_swap_eq:
- shows "(a \<rightleftharpoons> b) \<bullet> p = p \<longleftrightarrow> (p \<bullet> (a \<rightleftharpoons> b)) = (a \<rightleftharpoons> b)"
-unfolding permute_perm_def
-by (metis add_diff_cancel minus_perm_def)
-
-lemma supports_perm:
- shows "{a. p \<bullet> a \<noteq> a} supports p"
- unfolding supports_def
- by (simp add: perm_swap_eq swap_eqvt)
-
-lemma finite_perm_lemma:
- shows "finite {a::atom. p \<bullet> a \<noteq> a}"
- using finite_Rep_perm [of p]
- unfolding permute_atom_def .
-
-lemma supp_perm:
- shows "supp p = {a. p \<bullet> a \<noteq> a}"
-apply (rule finite_supp_unique)
-apply (rule supports_perm)
-apply (rule finite_perm_lemma)
-apply (simp add: perm_swap_eq swap_eqvt)
-apply (auto simp add: expand_perm_eq swap_atom)
-done
-
-lemma fresh_perm:
- shows "a \<sharp> p \<longleftrightarrow> p \<bullet> a = a"
-unfolding fresh_def by (simp add: supp_perm)
-
-lemma supp_swap:
- shows "supp (a \<rightleftharpoons> b) = (if a = b \<or> sort_of a \<noteq> sort_of b then {} else {a, b})"
- by (auto simp add: supp_perm swap_atom)
-
-lemma fresh_zero_perm:
- shows "a \<sharp> (0::perm)"
- unfolding fresh_perm by simp
-
-lemma supp_zero_perm:
- shows "supp (0::perm) = {}"
- unfolding supp_perm by simp
-
-lemma supp_plus_perm:
- fixes p q::perm
- shows "supp (p + q) \<subseteq> supp p \<union> supp q"
- by (auto simp add: supp_perm)
-
-lemma supp_minus_perm:
- fixes p::perm
- shows "supp (- p) = supp p"
- apply(auto simp add: supp_perm)
- apply(metis permute_minus_cancel)+
- done
-
-instance perm :: fs
-by default (simp add: supp_perm finite_perm_lemma)
-
-
-section {* Finite Support instances for other types *}
-
-subsection {* Type @{typ "'a \<times> 'b"} is finitely-supported. *}
-
-lemma supp_Pair:
- shows "supp (x, y) = supp x \<union> supp y"
- by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
-
-lemma fresh_Pair:
- shows "a \<sharp> (x, y) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> y"
- by (simp add: fresh_def supp_Pair)
-
-instance "*" :: (fs, fs) fs
-apply default
-apply (induct_tac x)
-apply (simp add: supp_Pair finite_supp)
-done
-
-subsection {* Type @{typ "'a + 'b"} is finitely supported *}
-
-lemma supp_Inl:
- shows "supp (Inl x) = supp x"
- by (simp add: supp_def)
-
-lemma supp_Inr:
- shows "supp (Inr x) = supp x"
- by (simp add: supp_def)
-
-lemma fresh_Inl:
- shows "a \<sharp> Inl x \<longleftrightarrow> a \<sharp> x"
- by (simp add: fresh_def supp_Inl)
-
-lemma fresh_Inr:
- shows "a \<sharp> Inr y \<longleftrightarrow> a \<sharp> y"
- by (simp add: fresh_def supp_Inr)
-
-instance "+" :: (fs, fs) fs
-apply default
-apply (induct_tac x)
-apply (simp_all add: supp_Inl supp_Inr finite_supp)
-done
-
-subsection {* Type @{typ "'a option"} is finitely supported *}
-
-lemma supp_None:
- shows "supp None = {}"
-by (simp add: supp_def)
-
-lemma supp_Some:
- shows "supp (Some x) = supp x"
- by (simp add: supp_def)
-
-lemma fresh_None:
- shows "a \<sharp> None"
- by (simp add: fresh_def supp_None)
-
-lemma fresh_Some:
- shows "a \<sharp> Some x \<longleftrightarrow> a \<sharp> x"
- by (simp add: fresh_def supp_Some)
-
-instance option :: (fs) fs
-apply default
-apply (induct_tac x)
-apply (simp_all add: supp_None supp_Some finite_supp)
-done
-
-subsubsection {* Type @{typ "'a list"} is finitely supported *}
-
-lemma supp_Nil:
- shows "supp [] = {}"
- by (simp add: supp_def)
-
-lemma supp_Cons:
- shows "supp (x # xs) = supp x \<union> supp xs"
-by (simp add: supp_def Collect_imp_eq Collect_neg_eq)
-
-lemma fresh_Nil:
- shows "a \<sharp> []"
- by (simp add: fresh_def supp_Nil)
-
-lemma fresh_Cons:
- shows "a \<sharp> (x # xs) \<longleftrightarrow> a \<sharp> x \<and> a \<sharp> xs"
- by (simp add: fresh_def supp_Cons)
-
-instance list :: (fs) fs
-apply default
-apply (induct_tac x)
-apply (simp_all add: supp_Nil supp_Cons finite_supp)
-done
-
-section {* Support and freshness for applications *}
-
-lemma supp_fun_app:
- shows "supp (f x) \<subseteq> (supp f) \<union> (supp x)"
-proof (rule subsetCI)
- fix a::"atom"
- assume a: "a \<in> supp (f x)"
- assume b: "a \<notin> supp f \<union> supp x"
- then have "finite {b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f}" "finite {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x}"
- unfolding supp_def by auto
- then have "finite ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})" by simp
- moreover
- have "{b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x} \<subseteq> ({b. (a \<rightleftharpoons> b) \<bullet> f \<noteq> f} \<union> {b. (a \<rightleftharpoons> b) \<bullet> x \<noteq> x})"
- by auto
- ultimately have "finite {b. ((a \<rightleftharpoons> b) \<bullet> f) ((a \<rightleftharpoons> b) \<bullet> x) \<noteq> f x}"
- using finite_subset by auto
- then have "a \<notin> supp (f x)" unfolding supp_def
- by (simp add: permute_fun_app_eq)
- with a show "False" by simp
-qed
-
-lemma fresh_fun_app:
- shows "a \<sharp> (f, x) \<Longrightarrow> a \<sharp> f x"
-unfolding fresh_def
-using supp_fun_app
-by (auto simp add: supp_Pair)
-
-lemma fresh_fun_eqvt_app:
- assumes a: "\<forall>p. p \<bullet> f = f"
- shows "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
-proof -
- from a have b: "supp f = {}"
- unfolding supp_def by simp
- show "a \<sharp> x \<Longrightarrow> a \<sharp> f x"
- unfolding fresh_def
- using supp_fun_app b
- by auto
-qed
-
-end
+/home/cu200/Isabelle/nominal-huffman/Nominal2_Base.thy
\ No newline at end of file
--- a/Quot/Nominal/Nominal2_Eqvt.thy Thu Feb 04 14:55:52 2010 +0100
+++ b/Quot/Nominal/Nominal2_Eqvt.thy Thu Feb 04 15:16:34 2010 +0100
@@ -1,298 +1,1 @@
-(* Title: Nominal2_Eqvt
- Authors: Brian Huffman, Christian Urban
-
- Equivariance, Supp and Fresh Lemmas for Operators.
-*)
-theory Nominal2_Eqvt
-imports Nominal2_Base
-uses ("nominal_thmdecls.ML")
- ("nominal_permeq.ML")
-begin
-
-section {* Logical Operators *}
-
-
-lemma eq_eqvt:
- shows "p \<bullet> (x = y) \<longleftrightarrow> (p \<bullet> x) = (p \<bullet> y)"
- unfolding permute_eq_iff permute_bool_def ..
-
-lemma if_eqvt:
- shows "p \<bullet> (if b then x else y) = (if p \<bullet> b then p \<bullet> x else p \<bullet> y)"
- by (simp add: permute_fun_def permute_bool_def)
-
-lemma True_eqvt:
- shows "p \<bullet> True = True"
- unfolding permute_bool_def ..
-
-lemma False_eqvt:
- shows "p \<bullet> False = False"
- unfolding permute_bool_def ..
-
-lemma imp_eqvt:
- shows "p \<bullet> (A \<longrightarrow> B) = ((p \<bullet> A) \<longrightarrow> (p \<bullet> B))"
- by (simp add: permute_bool_def)
-
-lemma conj_eqvt:
- shows "p \<bullet> (A \<and> B) = ((p \<bullet> A) \<and> (p \<bullet> B))"
- by (simp add: permute_bool_def)
-
-lemma disj_eqvt:
- shows "p \<bullet> (A \<or> B) = ((p \<bullet> A) \<or> (p \<bullet> B))"
- by (simp add: permute_bool_def)
-
-lemma Not_eqvt:
- shows "p \<bullet> (\<not> A) = (\<not> (p \<bullet> A))"
- by (simp add: permute_bool_def)
-
-lemma all_eqvt:
- shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. (p \<bullet> P) x)"
- unfolding permute_fun_def permute_bool_def
- by (auto, drule_tac x="p \<bullet> x" in spec, simp)
-
-lemma all_eqvt2:
- shows "p \<bullet> (\<forall>x. P x) = (\<forall>x. p \<bullet> P (- p \<bullet> x))"
- unfolding permute_fun_def permute_bool_def
- by (auto, drule_tac x="p \<bullet> x" in spec, simp)
-
-lemma ex_eqvt:
- shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. (p \<bullet> P) x)"
- unfolding permute_fun_def permute_bool_def
- by (auto, rule_tac x="p \<bullet> x" in exI, simp)
-
-lemma ex_eqvt2:
- shows "p \<bullet> (\<exists>x. P x) = (\<exists>x. p \<bullet> P (- p \<bullet> x))"
- unfolding permute_fun_def permute_bool_def
- by (auto, rule_tac x="p \<bullet> x" in exI, simp)
-
-lemma ex1_eqvt:
- shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. (p \<bullet> P) x)"
- unfolding Ex1_def
- by (simp add: ex_eqvt permute_fun_def conj_eqvt all_eqvt imp_eqvt eq_eqvt)
-
-lemma ex1_eqvt2:
- shows "p \<bullet> (\<exists>!x. P x) = (\<exists>!x. p \<bullet> P (- p \<bullet> x))"
- unfolding Ex1_def ex_eqvt2 conj_eqvt all_eqvt2 imp_eqvt eq_eqvt
- by simp
-
-lemma the_eqvt:
- assumes unique: "\<exists>!x. P x"
- shows "p \<bullet> (THE x. P x) = (THE x. p \<bullet> P (- p \<bullet> x))"
- apply(rule the1_equality [symmetric])
- apply(simp add: ex1_eqvt2[symmetric])
- apply(simp add: permute_bool_def unique)
- apply(simp add: permute_bool_def)
- apply(rule theI'[OF unique])
- done
-
-section {* Set Operations *}
-
-lemma mem_eqvt:
- shows "p \<bullet> (x \<in> A) \<longleftrightarrow> (p \<bullet> x) \<in> (p \<bullet> A)"
- unfolding mem_def permute_fun_def by simp
-
-lemma not_mem_eqvt:
- shows "p \<bullet> (x \<notin> A) \<longleftrightarrow> (p \<bullet> x) \<notin> (p \<bullet> A)"
- unfolding mem_def permute_fun_def by (simp add: Not_eqvt)
-
-lemma Collect_eqvt:
- shows "p \<bullet> {x. P x} = {x. (p \<bullet> P) x}"
- unfolding Collect_def permute_fun_def ..
-
-lemma Collect_eqvt2:
- shows "p \<bullet> {x. P x} = {x. p \<bullet> (P (-p \<bullet> x))}"
- unfolding Collect_def permute_fun_def ..
-
-lemma empty_eqvt:
- shows "p \<bullet> {} = {}"
- unfolding empty_def Collect_eqvt2 False_eqvt ..
-
-lemma supp_set_empty:
- shows "supp {} = {}"
- by (simp add: supp_def empty_eqvt)
-
-lemma fresh_set_empty:
- shows "a \<sharp> {}"
- by (simp add: fresh_def supp_set_empty)
-
-lemma UNIV_eqvt:
- shows "p \<bullet> UNIV = UNIV"
- unfolding UNIV_def Collect_eqvt2 True_eqvt ..
-
-lemma union_eqvt:
- shows "p \<bullet> (A \<union> B) = (p \<bullet> A) \<union> (p \<bullet> B)"
- unfolding Un_def Collect_eqvt2 disj_eqvt mem_eqvt by simp
-
-lemma inter_eqvt:
- shows "p \<bullet> (A \<inter> B) = (p \<bullet> A) \<inter> (p \<bullet> B)"
- unfolding Int_def Collect_eqvt2 conj_eqvt mem_eqvt by simp
-
-lemma Diff_eqvt:
- fixes A B :: "'a::pt set"
- shows "p \<bullet> (A - B) = p \<bullet> A - p \<bullet> B"
- unfolding set_diff_eq Collect_eqvt2 conj_eqvt Not_eqvt mem_eqvt by simp
-
-lemma Compl_eqvt:
- fixes A :: "'a::pt set"
- shows "p \<bullet> (- A) = - (p \<bullet> A)"
- unfolding Compl_eq_Diff_UNIV Diff_eqvt UNIV_eqvt ..
-
-lemma insert_eqvt:
- shows "p \<bullet> (insert x A) = insert (p \<bullet> x) (p \<bullet> A)"
- unfolding permute_set_eq_image image_insert ..
-
-lemma vimage_eqvt:
- shows "p \<bullet> (f -` A) = (p \<bullet> f) -` (p \<bullet> A)"
- unfolding vimage_def permute_fun_def [where f=f]
- unfolding Collect_eqvt2 mem_eqvt ..
-
-lemma image_eqvt:
- shows "p \<bullet> (f ` A) = (p \<bullet> f) ` (p \<bullet> A)"
- unfolding permute_set_eq_image
- unfolding permute_fun_def [where f=f]
- by (simp add: image_image)
-
-lemma finite_permute_iff:
- shows "finite (p \<bullet> A) \<longleftrightarrow> finite A"
- unfolding permute_set_eq_vimage
- using bij_permute by (rule finite_vimage_iff)
-
-lemma finite_eqvt:
- shows "p \<bullet> finite A = finite (p \<bullet> A)"
- unfolding finite_permute_iff permute_bool_def ..
-
-
-section {* List Operations *}
-
-lemma append_eqvt:
- shows "p \<bullet> (xs @ ys) = (p \<bullet> xs) @ (p \<bullet> ys)"
- by (induct xs) auto
-
-lemma supp_append:
- shows "supp (xs @ ys) = supp xs \<union> supp ys"
- by (induct xs) (auto simp add: supp_Nil supp_Cons)
-
-lemma fresh_append:
- shows "a \<sharp> (xs @ ys) \<longleftrightarrow> a \<sharp> xs \<and> a \<sharp> ys"
- by (induct xs) (simp_all add: fresh_Nil fresh_Cons)
-
-lemma rev_eqvt:
- shows "p \<bullet> (rev xs) = rev (p \<bullet> xs)"
- by (induct xs) (simp_all add: append_eqvt)
-
-lemma supp_rev:
- shows "supp (rev xs) = supp xs"
- by (induct xs) (auto simp add: supp_append supp_Cons supp_Nil)
-
-lemma fresh_rev:
- shows "a \<sharp> rev xs \<longleftrightarrow> a \<sharp> xs"
- by (induct xs) (auto simp add: fresh_append fresh_Cons fresh_Nil)
-
-lemma set_eqvt:
- shows "p \<bullet> (set xs) = set (p \<bullet> xs)"
- by (induct xs) (simp_all add: empty_eqvt insert_eqvt)
-
-(* needs finite support premise
-lemma supp_set:
- fixes x :: "'a::pt"
- shows "supp (set xs) = supp xs"
-*)
-
-
-section {* Product Operations *}
-
-lemma fst_eqvt:
- "p \<bullet> (fst x) = fst (p \<bullet> x)"
- by (cases x) simp
-
-lemma snd_eqvt:
- "p \<bullet> (snd x) = snd (p \<bullet> x)"
- by (cases x) simp
-
-
-section {* Units *}
-
-lemma supp_unit:
- shows "supp () = {}"
- by (simp add: supp_def)
-
-lemma fresh_unit:
- shows "a \<sharp> ()"
- by (simp add: fresh_def supp_unit)
-
-section {* Equivariance automation *}
-
-text {* Setup of the theorem attributes @{text eqvt} and @{text eqvt_force} *}
-
-use "nominal_thmdecls.ML"
-setup "Nominal_ThmDecls.setup"
-
-lemmas [eqvt] =
- (* connectives *)
- eq_eqvt if_eqvt imp_eqvt disj_eqvt conj_eqvt Not_eqvt
- True_eqvt False_eqvt ex_eqvt all_eqvt
- imp_eqvt [folded induct_implies_def]
-
- (* nominal *)
- permute_eqvt supp_eqvt fresh_eqvt
- permute_pure
-
- (* datatypes *)
- permute_prod.simps
- fst_eqvt snd_eqvt
-
- (* sets *)
- empty_eqvt UNIV_eqvt union_eqvt inter_eqvt
- Diff_eqvt Compl_eqvt insert_eqvt
-
-thm eqvts
-thm eqvts_raw
-
-text {* helper lemmas for the eqvt_tac *}
-
-definition
- "unpermute p = permute (- p)"
-
-lemma eqvt_apply:
- fixes f :: "'a::pt \<Rightarrow> 'b::pt"
- and x :: "'a::pt"
- shows "p \<bullet> (f x) \<equiv> (p \<bullet> f) (p \<bullet> x)"
- unfolding permute_fun_def by simp
-
-lemma eqvt_lambda:
- fixes f :: "'a::pt \<Rightarrow> 'b::pt"
- shows "p \<bullet> (\<lambda>x. f x) \<equiv> (\<lambda>x. p \<bullet> (f (unpermute p x)))"
- unfolding permute_fun_def unpermute_def by simp
-
-lemma eqvt_bound:
- shows "p \<bullet> unpermute p x \<equiv> x"
- unfolding unpermute_def by simp
-
-use "nominal_permeq.ML"
-
-
-lemma "p \<bullet> (A \<longrightarrow> B = C)"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>(x::'a::pt). A \<longrightarrow> (B::'a \<Rightarrow> bool) x = C) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>x y. \<exists>z. x = z \<and> x = y \<longrightarrow> z \<noteq> x) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>f x. f (g (f x))) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (\<lambda>q. q \<bullet> (r \<bullet> x)) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-lemma "p \<bullet> (q \<bullet> r \<bullet> x) = foo"
-apply (tactic {* Nominal_Permeq.eqvt_tac @{context} 1 *})
-oops
-
-
-end
\ No newline at end of file
+/home/cu200/Isabelle/nominal-huffman/Nominal2_Eqvt.thy
\ No newline at end of file
--- 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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+/home/cu200/Isabelle/nominal-huffman/Nominal2_Supp.thy
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