# Theory Nominal

theory Nominal
imports Infinite_Set Old_Datatype
```theory Nominal
imports "HOL-Library.Infinite_Set" "HOL-Library.Old_Datatype"
keywords
"atom_decl" "nominal_datatype" "equivariance" :: thy_decl and
"nominal_primrec" "nominal_inductive" "nominal_inductive2" :: thy_goal and
"avoids"
begin

section ‹Permutations›
(*======================*)

type_synonym
'x prm = "('x × 'x) list"

(* polymorphic constants for permutation and swapping *)
consts
perm :: "'x prm ⇒ 'a ⇒ 'a"     (infixr "∙" 80)
swap :: "('x × 'x) ⇒ 'x ⇒ 'x"

(* a "private" copy of the option type used in the abstraction function *)
datatype 'a noption = nSome 'a | nNone

datatype_compat noption

(* a "private" copy of the product type used in the nominal induct method *)
datatype ('a, 'b) nprod = nPair 'a 'b

datatype_compat nprod

(* an auxiliary constant for the decision procedure involving *)
(* permutations (to avoid loops when using perm-compositions)  *)
definition
"perm_aux pi x = pi∙x"

perm_fun    ≡ "perm :: 'x prm ⇒ ('a⇒'b) ⇒ ('a⇒'b)"   (unchecked)
perm_bool   ≡ "perm :: 'x prm ⇒ bool ⇒ bool"           (unchecked)
perm_set    ≡ "perm :: 'x prm ⇒ 'a set ⇒ 'a set"           (unchecked)
perm_unit   ≡ "perm :: 'x prm ⇒ unit ⇒ unit"           (unchecked)
perm_prod   ≡ "perm :: 'x prm ⇒ ('a×'b) ⇒ ('a×'b)"    (unchecked)
perm_list   ≡ "perm :: 'x prm ⇒ 'a list ⇒ 'a list"     (unchecked)
perm_option ≡ "perm :: 'x prm ⇒ 'a option ⇒ 'a option" (unchecked)
perm_char   ≡ "perm :: 'x prm ⇒ char ⇒ char"           (unchecked)
perm_nat    ≡ "perm :: 'x prm ⇒ nat ⇒ nat"             (unchecked)
perm_int    ≡ "perm :: 'x prm ⇒ int ⇒ int"             (unchecked)

perm_noption ≡ "perm :: 'x prm ⇒ 'a noption ⇒ 'a noption"   (unchecked)
perm_nprod   ≡ "perm :: 'x prm ⇒ ('a, 'b) nprod ⇒ ('a, 'b) nprod" (unchecked)
begin

definition perm_fun :: "'x prm ⇒ ('a ⇒ 'b) ⇒ 'a ⇒ 'b" where
"perm_fun pi f = (λx. pi ∙ f (rev pi ∙ x))"

definition perm_bool :: "'x prm ⇒ bool ⇒ bool" where
"perm_bool pi b = b"

definition perm_set :: "'x prm ⇒ 'a set ⇒ 'a set" where
"perm_set pi X = {pi ∙ x | x. x ∈ X}"

primrec perm_unit :: "'x prm ⇒ unit ⇒ unit"  where
"perm_unit pi () = ()"

primrec perm_prod :: "'x prm ⇒ ('a×'b) ⇒ ('a×'b)" where
"perm_prod pi (x, y) = (pi∙x, pi∙y)"

primrec perm_list :: "'x prm ⇒ 'a list ⇒ 'a list" where
nil_eqvt:  "perm_list pi []     = []"
| cons_eqvt: "perm_list pi (x#xs) = (pi∙x)#(pi∙xs)"

primrec perm_option :: "'x prm ⇒ 'a option ⇒ 'a option" where
some_eqvt:  "perm_option pi (Some x) = Some (pi∙x)"
| none_eqvt:  "perm_option pi None     = None"

definition perm_char :: "'x prm ⇒ char ⇒ char" where
"perm_char pi c = c"

definition perm_nat :: "'x prm ⇒ nat ⇒ nat" where
"perm_nat pi i = i"

definition perm_int :: "'x prm ⇒ int ⇒ int" where
"perm_int pi i = i"

primrec perm_noption :: "'x prm ⇒ 'a noption ⇒ 'a noption" where
nsome_eqvt:  "perm_noption pi (nSome x) = nSome (pi∙x)"
| nnone_eqvt:  "perm_noption pi nNone     = nNone"

primrec perm_nprod :: "'x prm ⇒ ('a, 'b) nprod ⇒ ('a, 'b) nprod" where
"perm_nprod pi (nPair x y) = nPair (pi∙x) (pi∙y)"

end

(* permutations on booleans *)
lemmas perm_bool = perm_bool_def

lemma true_eqvt [simp]:
"pi ∙ True ⟷ True"

lemma false_eqvt [simp]:
"pi ∙ False ⟷ False"

lemma perm_boolI:
assumes a: "P"
shows "pi∙P"
using a by (simp add: perm_bool)

lemma perm_boolE:
assumes a: "pi∙P"
shows "P"
using a by (simp add: perm_bool)

lemma if_eqvt:
fixes pi::"'a prm"
shows "pi∙(if b then c1 else c2) = (if (pi∙b) then (pi∙c1) else (pi∙c2))"

lemma imp_eqvt:
shows "pi∙(A⟶B) = ((pi∙A)⟶(pi∙B))"

lemma conj_eqvt:
shows "pi∙(A∧B) = ((pi∙A)∧(pi∙B))"

lemma disj_eqvt:
shows "pi∙(A∨B) = ((pi∙A)∨(pi∙B))"

lemma neg_eqvt:
shows "pi∙(¬ A) = (¬ (pi∙A))"

(* permutation on sets *)
lemma empty_eqvt:
shows "pi∙{} = {}"

lemma union_eqvt:
shows "(pi∙(X∪Y)) = (pi∙X) ∪ (pi∙Y)"

lemma insert_eqvt:
shows "pi∙(insert x X) = insert (pi∙x) (pi∙X)"

(* permutations on products *)
lemma fst_eqvt:
"pi∙(fst x) = fst (pi∙x)"
by (cases x) simp

lemma snd_eqvt:
"pi∙(snd x) = snd (pi∙x)"
by (cases x) simp

(* permutation on lists *)
lemma append_eqvt:
fixes pi :: "'x prm"
and   l1 :: "'a list"
and   l2 :: "'a list"
shows "pi∙(l1@l2) = (pi∙l1)@(pi∙l2)"
by (induct l1) auto

lemma rev_eqvt:
fixes pi :: "'x prm"
and   l  :: "'a list"
shows "pi∙(rev l) = rev (pi∙l)"
by (induct l) (simp_all add: append_eqvt)

lemma set_eqvt:
fixes pi :: "'x prm"
and   xs :: "'a list"
shows "pi∙(set xs) = set (pi∙xs)"
by (induct xs) (auto simp add: empty_eqvt insert_eqvt)

(* permutation on characters and strings *)
lemma perm_string:
fixes s::"string"
shows "pi∙s = s"
by (induct s)(auto simp add: perm_char_def)

section ‹permutation equality›
(*==============================*)

definition prm_eq :: "'x prm ⇒ 'x prm ⇒ bool" (" _ ≜ _ " [80,80] 80) where
"pi1 ≜ pi2 ⟷ (∀a::'x. pi1∙a = pi2∙a)"

section ‹Support, Freshness and Supports›
(*========================================*)
definition supp :: "'a ⇒ ('x set)" where
"supp x = {a . (infinite {b . [(a,b)]∙x ≠ x})}"

definition fresh :: "'x ⇒ 'a ⇒ bool" ("_ ♯ _" [80,80] 80) where
"a ♯ x ⟷ a ∉ supp x"

definition supports :: "'x set ⇒ 'a ⇒ bool" (infixl "supports" 80) where
"S supports x ⟷ (∀a b. (a∉S ∧ b∉S ⟶ [(a,b)]∙x=x))"

lemma supp_fresh_iff:
fixes x :: "'a"
shows "(supp x) = {a::'x. ¬a♯x}"

lemma supp_unit:
shows "supp () = {}"

lemma supp_set_empty:
shows "supp {} = {}"
by (force simp add: supp_def empty_eqvt)

lemma supp_prod:
fixes x :: "'a"
and   y :: "'b"
shows "(supp (x,y)) = (supp x)∪(supp y)"
by  (force simp add: supp_def Collect_imp_eq Collect_neg_eq)

lemma supp_nprod:
fixes x :: "'a"
and   y :: "'b"
shows "(supp (nPair x y)) = (supp x)∪(supp y)"
by  (force simp add: supp_def Collect_imp_eq Collect_neg_eq)

lemma supp_list_nil:
shows "supp [] = {}"

lemma supp_list_cons:
fixes x  :: "'a"
and   xs :: "'a list"
shows "supp (x#xs) = (supp x)∪(supp xs)"
by (auto simp add: supp_def Collect_imp_eq Collect_neg_eq)

lemma supp_list_append:
fixes xs :: "'a list"
and   ys :: "'a list"
shows "supp (xs@ys) = (supp xs)∪(supp ys)"
by (induct xs) (auto simp add: supp_list_nil supp_list_cons)

lemma supp_list_rev:
fixes xs :: "'a list"
shows "supp (rev xs) = (supp xs)"
by (induct xs, auto simp add: supp_list_append supp_list_cons supp_list_nil)

lemma supp_bool:
fixes x  :: "bool"
shows "supp x = {}"
by (cases "x") (simp_all add: supp_def)

lemma supp_some:
fixes x :: "'a"
shows "supp (Some x) = (supp x)"

lemma supp_none:
fixes x :: "'a"
shows "supp (None) = {}"

lemma supp_int:
fixes i::"int"
shows "supp (i) = {}"

lemma supp_nat:
fixes n::"nat"
shows "(supp n) = {}"

lemma supp_char:
fixes c::"char"
shows "(supp c) = {}"

lemma supp_string:
fixes s::"string"
shows "(supp s) = {}"

lemma fresh_set_empty:
shows "a♯{}"

lemma fresh_unit:
shows "a♯()"

lemma fresh_prod:
fixes a :: "'x"
and   x :: "'a"
and   y :: "'b"
shows "a♯(x,y) = (a♯x ∧ a♯y)"

lemma fresh_list_nil:
fixes a :: "'x"
shows "a♯[]"

lemma fresh_list_cons:
fixes a :: "'x"
and   x :: "'a"
and   xs :: "'a list"
shows "a♯(x#xs) = (a♯x ∧ a♯xs)"

lemma fresh_list_append:
fixes a :: "'x"
and   xs :: "'a list"
and   ys :: "'a list"
shows "a♯(xs@ys) = (a♯xs ∧ a♯ys)"

lemma fresh_list_rev:
fixes a :: "'x"
and   xs :: "'a list"
shows "a♯(rev xs) = a♯xs"

lemma fresh_none:
fixes a :: "'x"
shows "a♯None"

lemma fresh_some:
fixes a :: "'x"
and   x :: "'a"
shows "a♯(Some x) = a♯x"

lemma fresh_int:
fixes a :: "'x"
and   i :: "int"
shows "a♯i"

lemma fresh_nat:
fixes a :: "'x"
and   n :: "nat"
shows "a♯n"

lemma fresh_char:
fixes a :: "'x"
and   c :: "char"
shows "a♯c"

lemma fresh_string:
fixes a :: "'x"
and   s :: "string"
shows "a♯s"

lemma fresh_bool:
fixes a :: "'x"
and   b :: "bool"
shows "a♯b"

text ‹Normalization of freshness results; cf.\ ‹nominal_induct››
lemma fresh_unit_elim:
shows "(a♯() ⟹ PROP C) ≡ PROP C"

lemma fresh_prod_elim:
shows "(a♯(x,y) ⟹ PROP C) ≡ (a♯x ⟹ a♯y ⟹ PROP C)"

(* this rule needs to be added before the fresh_prodD is *)
(* added to the simplifier with mksimps                  *)
lemma [simp]:
shows "a♯x1 ⟹ a♯x2 ⟹ a♯(x1,x2)"

lemma fresh_prodD:
shows "a♯(x,y) ⟹ a♯x"
and   "a♯(x,y) ⟹ a♯y"

ML ‹
val mksimps_pairs = (@{const_name Nominal.fresh}, @{thms fresh_prodD}) :: mksimps_pairs;
›
declaration ‹fn _ =>
Simplifier.map_ss (Simplifier.set_mksimps (mksimps mksimps_pairs))
›

section ‹Abstract Properties for Permutations and  Atoms›
(*=========================================================*)

(* properties for being a permutation type *)
definition
"pt TYPE('a) TYPE('x) ≡
(∀(x::'a). ([]::'x prm)∙x = x) ∧
(∀(pi1::'x prm) (pi2::'x prm) (x::'a). (pi1@pi2)∙x = pi1∙(pi2∙x)) ∧
(∀(pi1::'x prm) (pi2::'x prm) (x::'a). pi1 ≜ pi2 ⟶ pi1∙x = pi2∙x)"

(* properties for being an atom type *)
definition
"at TYPE('x) ≡
(∀(x::'x). ([]::'x prm)∙x = x) ∧
(∀(a::'x) (b::'x) (pi::'x prm) (x::'x). ((a,b)#(pi::'x prm))∙x = swap (a,b) (pi∙x)) ∧
(∀(a::'x) (b::'x) (c::'x). swap (a,b) c = (if a=c then b else (if b=c then a else c))) ∧
(infinite (UNIV::'x set))"

(* property of two atom-types being disjoint *)
definition
"disjoint TYPE('x) TYPE('y) ≡
(∀(pi::'x prm)(x::'y). pi∙x = x) ∧
(∀(pi::'y prm)(x::'x). pi∙x = x)"

(* composition property of two permutation on a type 'a *)
definition
"cp TYPE ('a) TYPE('x) TYPE('y) ≡
(∀(pi2::'y prm) (pi1::'x prm) (x::'a) . pi1∙(pi2∙x) = (pi1∙pi2)∙(pi1∙x))"

(* property of having finite support *)
definition
"fs TYPE('a) TYPE('x) ≡ ∀(x::'a). finite ((supp x)::'x set)"

section ‹Lemmas about the atom-type properties›
(*==============================================*)

lemma at1:
fixes x::"'x"
assumes a: "at TYPE('x)"
shows "([]::'x prm)∙x = x"
using a by (simp add: at_def)

lemma at2:
fixes a ::"'x"
and   b ::"'x"
and   x ::"'x"
and   pi::"'x prm"
assumes a: "at TYPE('x)"
shows "((a,b)#pi)∙x = swap (a,b) (pi∙x)"
using a by (simp only: at_def)

lemma at3:
fixes a ::"'x"
and   b ::"'x"
and   c ::"'x"
assumes a: "at TYPE('x)"
shows "swap (a,b) c = (if a=c then b else (if b=c then a else c))"
using a by (simp only: at_def)

(* rules to calculate simple permutations *)
lemmas at_calc = at2 at1 at3

lemma at_swap_simps:
fixes a ::"'x"
and   b ::"'x"
assumes a: "at TYPE('x)"
shows "[(a,b)]∙a = b"
and   "[(a,b)]∙b = a"
and   "⟦a≠c; b≠c⟧ ⟹ [(a,b)]∙c = c"
using a by (simp_all add: at_calc)

lemma at4:
assumes a: "at TYPE('x)"
shows "infinite (UNIV::'x set)"
using a by (simp add: at_def)

lemma at_append:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   c   :: "'x"
assumes at: "at TYPE('x)"
shows "(pi1@pi2)∙c = pi1∙(pi2∙c)"
proof (induct pi1)
case Nil show ?case by (simp add: at1[OF at])
next
case (Cons x xs)
have "(xs@pi2)∙c  =  xs∙(pi2∙c)" by fact
also have "(x#xs)@pi2 = x#(xs@pi2)" by simp
ultimately show ?case by (cases "x", simp add:  at2[OF at])
qed

lemma at_swap:
fixes a :: "'x"
and   b :: "'x"
and   c :: "'x"
assumes at: "at TYPE('x)"
shows "swap (a,b) (swap (a,b) c) = c"
by (auto simp add: at3[OF at])

lemma at_rev_pi:
fixes pi :: "'x prm"
and   c  :: "'x"
assumes at: "at TYPE('x)"
shows "(rev pi)∙(pi∙c) = c"
proof(induct pi)
case Nil show ?case by (simp add: at1[OF at])
next
case (Cons x xs) thus ?case
by (cases "x", simp add: at2[OF at] at_append[OF at] at1[OF at] at_swap[OF at])
qed

lemma at_pi_rev:
fixes pi :: "'x prm"
and   x  :: "'x"
assumes at: "at TYPE('x)"
shows "pi∙((rev pi)∙x) = x"
by (rule at_rev_pi[OF at, of "rev pi" _,simplified])

lemma at_bij1:
fixes pi :: "'x prm"
and   x  :: "'x"
and   y  :: "'x"
assumes at: "at TYPE('x)"
and     a:  "(pi∙x) = y"
shows   "x=(rev pi)∙y"
proof -
from a have "y=(pi∙x)" by (rule sym)
thus ?thesis by (simp only: at_rev_pi[OF at])
qed

lemma at_bij2:
fixes pi :: "'x prm"
and   x  :: "'x"
and   y  :: "'x"
assumes at: "at TYPE('x)"
and     a:  "((rev pi)∙x) = y"
shows   "x=pi∙y"
proof -
from a have "y=((rev pi)∙x)" by (rule sym)
thus ?thesis by (simp only: at_pi_rev[OF at])
qed

lemma at_bij:
fixes pi :: "'x prm"
and   x  :: "'x"
and   y  :: "'x"
assumes at: "at TYPE('x)"
shows "(pi∙x = pi∙y) = (x=y)"
proof
assume "pi∙x = pi∙y"
hence  "x=(rev pi)∙(pi∙y)" by (rule at_bij1[OF at])
thus "x=y" by (simp only: at_rev_pi[OF at])
next
assume "x=y"
thus "pi∙x = pi∙y" by simp
qed

lemma at_supp:
fixes x :: "'x"
assumes at: "at TYPE('x)"
shows "supp x = {x}"
by(auto simp: supp_def Collect_conj_eq Collect_imp_eq at_calc[OF at] at4[OF at])

lemma at_fresh:
fixes a :: "'x"
and   b :: "'x"
assumes at: "at TYPE('x)"
shows "(a♯b) = (a≠b)"
by (simp add: at_supp[OF at] fresh_def)

lemma at_prm_fresh1:
fixes c :: "'x"
and   pi:: "'x prm"
assumes at: "at TYPE('x)"
and     a: "c♯pi"
shows "∀(a,b)∈set pi. c≠a ∧ c≠b"
using a by (induct pi) (auto simp add: fresh_list_cons fresh_prod at_fresh[OF at])

lemma at_prm_fresh2:
fixes c :: "'x"
and   pi:: "'x prm"
assumes at: "at TYPE('x)"
and     a: "∀(a,b)∈set pi. c≠a ∧ c≠b"
shows "pi∙c = c"
using a  by(induct pi) (auto simp add: at1[OF at] at2[OF at] at3[OF at])

lemma at_prm_fresh:
fixes c :: "'x"
and   pi:: "'x prm"
assumes at: "at TYPE('x)"
and     a: "c♯pi"
shows "pi∙c = c"
by (rule at_prm_fresh2[OF at], rule at_prm_fresh1[OF at, OF a])

lemma at_prm_rev_eq:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
assumes at: "at TYPE('x)"
shows "((rev pi1) ≜ (rev pi2)) = (pi1 ≜ pi2)"
fix x
assume "∀x::'x. (rev pi1)∙x = (rev pi2)∙x"
hence "(rev (pi1::'x prm))∙(pi2∙(x::'x)) = (rev (pi2::'x prm))∙(pi2∙x)" by simp
hence "(rev (pi1::'x prm))∙((pi2::'x prm)∙x) = (x::'x)" by (simp add: at_rev_pi[OF at])
hence "(pi2::'x prm)∙x = (pi1::'x prm)∙x" by (simp add: at_bij2[OF at])
thus "pi1∙x  =  pi2∙x" by simp
next
fix x
assume "∀x::'x. pi1∙x = pi2∙x"
hence "(pi1::'x prm)∙((rev pi2)∙x) = (pi2::'x prm)∙((rev pi2)∙(x::'x))" by simp
hence "(pi1::'x prm)∙((rev pi2)∙(x::'x)) = x" by (simp add: at_pi_rev[OF at])
hence "(rev pi2)∙x = (rev pi1)∙(x::'x)" by (simp add: at_bij1[OF at])
thus "(rev pi1)∙x = (rev pi2)∙(x::'x)" by simp
qed

lemma at_prm_eq_append:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   pi3 :: "'x prm"
assumes at: "at TYPE('x)"
and     a: "pi1 ≜ pi2"
shows "(pi3@pi1) ≜ (pi3@pi2)"
using a by (simp add: prm_eq_def at_append[OF at] at_bij[OF at])

lemma at_prm_eq_append':
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   pi3 :: "'x prm"
assumes at: "at TYPE('x)"
and     a: "pi1 ≜ pi2"
shows "(pi1@pi3) ≜ (pi2@pi3)"
using a by (simp add: prm_eq_def at_append[OF at])

lemma at_prm_eq_trans:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   pi3 :: "'x prm"
assumes a1: "pi1 ≜ pi2"
and     a2: "pi2 ≜ pi3"
shows "pi1 ≜ pi3"
using a1 a2 by (auto simp add: prm_eq_def)

lemma at_prm_eq_refl:
fixes pi :: "'x prm"
shows "pi ≜ pi"

lemma at_prm_rev_eq1:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
assumes at: "at TYPE('x)"
shows "pi1 ≜ pi2 ⟹ (rev pi1) ≜ (rev pi2)"

lemma at_ds1:
fixes a  :: "'x"
assumes at: "at TYPE('x)"
shows "[(a,a)] ≜ []"
by (force simp add: prm_eq_def at_calc[OF at])

lemma at_ds2:
fixes pi :: "'x prm"
and   a  :: "'x"
and   b  :: "'x"
assumes at: "at TYPE('x)"
shows "([(a,b)]@pi) ≜ (pi@[((rev pi)∙a,(rev pi)∙b)])"
by (force simp add: prm_eq_def at_append[OF at] at_bij[OF at] at_pi_rev[OF at]
at_rev_pi[OF at] at_calc[OF at])

lemma at_ds3:
fixes a  :: "'x"
and   b  :: "'x"
and   c  :: "'x"
assumes at: "at TYPE('x)"
and     a:  "distinct [a,b,c]"
shows "[(a,c),(b,c),(a,c)] ≜ [(a,b)]"
using a by (force simp add: prm_eq_def at_calc[OF at])

lemma at_ds4:
fixes a  :: "'x"
and   b  :: "'x"
and   pi  :: "'x prm"
assumes at: "at TYPE('x)"
shows "(pi@[(a,(rev pi)∙b)]) ≜ ([(pi∙a,b)]@pi)"
by (force simp add: prm_eq_def at_append[OF at] at_calc[OF at] at_bij[OF at]
at_pi_rev[OF at] at_rev_pi[OF at])

lemma at_ds5:
fixes a  :: "'x"
and   b  :: "'x"
assumes at: "at TYPE('x)"
shows "[(a,b)] ≜ [(b,a)]"
by (force simp add: prm_eq_def at_calc[OF at])

lemma at_ds5':
fixes a  :: "'x"
and   b  :: "'x"
assumes at: "at TYPE('x)"
shows "[(a,b),(b,a)] ≜ []"
by (force simp add: prm_eq_def at_calc[OF at])

lemma at_ds6:
fixes a  :: "'x"
and   b  :: "'x"
and   c  :: "'x"
assumes at: "at TYPE('x)"
and     a: "distinct [a,b,c]"
shows "[(a,c),(a,b)] ≜ [(b,c),(a,c)]"
using a by (force simp add: prm_eq_def at_calc[OF at])

lemma at_ds7:
fixes pi :: "'x prm"
assumes at: "at TYPE('x)"
shows "((rev pi)@pi) ≜ []"
by (simp add: prm_eq_def at1[OF at] at_append[OF at] at_rev_pi[OF at])

lemma at_ds8_aux:
fixes pi :: "'x prm"
and   a  :: "'x"
and   b  :: "'x"
and   c  :: "'x"
assumes at: "at TYPE('x)"
shows "pi∙(swap (a,b) c) = swap (pi∙a,pi∙b) (pi∙c)"
by (force simp add: at_calc[OF at] at_bij[OF at])

lemma at_ds8:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   a  :: "'x"
and   b  :: "'x"
assumes at: "at TYPE('x)"
shows "(pi1@pi2) ≜ ((pi1∙pi2)@pi1)"
apply(induct_tac pi2)
apply(drule_tac x="aa" in spec)
apply(drule sym)
apply(simp)
done

lemma at_ds9:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   a  :: "'x"
and   b  :: "'x"
assumes at: "at TYPE('x)"
shows " ((rev pi2)@(rev pi1)) ≜ ((rev pi1)@(rev (pi1∙pi2)))"
apply(induct_tac pi2)
apply(simp add: at2[OF at] at1[OF at])
apply(drule_tac x="swap(pi1∙a,pi1∙b) aa" in spec)
apply(drule sym)
apply(simp)
done

lemma at_ds10:
fixes pi :: "'x prm"
and   a  :: "'x"
and   b  :: "'x"
assumes at: "at TYPE('x)"
and     a:  "b♯(rev pi)"
shows "([(pi∙a,b)]@pi) ≜ (pi@[(a,b)])"
using a
apply -
apply(rule at_prm_eq_trans)
apply(rule at_ds2[OF at])
apply(simp add: at_prm_fresh[OF at] at_rev_pi[OF at])
apply(rule at_prm_eq_refl)
done

―"there always exists an atom that is not being in a finite set"
lemma ex_in_inf:
fixes   A::"'x set"
assumes at: "at TYPE('x)"
and     fs: "finite A"
obtains c::"'x" where "c∉A"
proof -
from  fs at4[OF at] have "infinite ((UNIV::'x set) - A)"
hence "((UNIV::'x set) - A) ≠ ({}::'x set)" by (force simp only:)
then obtain c::"'x" where "c∈((UNIV::'x set) - A)" by force
then have "c∉A" by simp
then show ?thesis ..
qed

text ‹there always exists a fresh name for an object with finite support›
lemma at_exists_fresh':
fixes  x :: "'a"
assumes at: "at TYPE('x)"
and     fs: "finite ((supp x)::'x set)"
shows "∃c::'x. c♯x"
by (auto simp add: fresh_def intro: ex_in_inf[OF at, OF fs])

lemma at_exists_fresh:
fixes  x :: "'a"
assumes at: "at TYPE('x)"
and     fs: "finite ((supp x)::'x set)"
obtains c::"'x" where  "c♯x"
by (auto intro: ex_in_inf[OF at, OF fs] simp add: fresh_def)

lemma at_finite_select:
fixes S::"'a set"
assumes a: "at TYPE('a)"
and     b: "finite S"
shows "∃x. x ∉ S"
using a b
apply(drule_tac S="UNIV::'a set" in Diff_infinite_finite)
apply(subgoal_tac "UNIV - S ≠ {}")
apply(simp only: ex_in_conv [symmetric])
apply(blast)
apply(rule notI)
apply(simp)
done

lemma at_different:
assumes at: "at TYPE('x)"
shows "∃(b::'x). a≠b"
proof -
have "infinite (UNIV::'x set)" by (rule at4[OF at])
hence inf2: "infinite (UNIV-{a})" by (rule infinite_remove)
have "(UNIV-{a}) ≠ ({}::'x set)"
proof (rule_tac ccontr, drule_tac notnotD)
assume "UNIV-{a} = ({}::'x set)"
with inf2 have "infinite ({}::'x set)" by simp
then show "False" by auto
qed
hence "∃(b::'x). b∈(UNIV-{a})" by blast
then obtain b::"'x" where mem2: "b∈(UNIV-{a})" by blast
from mem2 have "a≠b" by blast
then show "∃(b::'x). a≠b" by blast
qed

―"the at-props imply the pt-props"
lemma at_pt_inst:
assumes at: "at TYPE('x)"
shows "pt TYPE('x) TYPE('x)"
apply(auto simp only: pt_def)
apply(simp only: at1[OF at])
apply(simp only: at_append[OF at])
apply(simp only: prm_eq_def)
done

section ‹finite support properties›
(*===================================*)

lemma fs1:
fixes x :: "'a"
assumes a: "fs TYPE('a) TYPE('x)"
shows "finite ((supp x)::'x set)"
using a by (simp add: fs_def)

lemma fs_at_inst:
fixes a :: "'x"
assumes at: "at TYPE('x)"
shows "fs TYPE('x) TYPE('x)"
done

lemma fs_unit_inst:
shows "fs TYPE(unit) TYPE('x)"
done

lemma fs_prod_inst:
assumes fsa: "fs TYPE('a) TYPE('x)"
and     fsb: "fs TYPE('b) TYPE('x)"
shows "fs TYPE('a×'b) TYPE('x)"
apply(unfold fs_def)
apply(rule fs1[OF fsa])
apply(rule fs1[OF fsb])
done

lemma fs_nprod_inst:
assumes fsa: "fs TYPE('a) TYPE('x)"
and     fsb: "fs TYPE('b) TYPE('x)"
shows "fs TYPE(('a,'b) nprod) TYPE('x)"
apply(unfold fs_def, rule allI)
apply(case_tac x)
apply(rule fs1[OF fsa])
apply(rule fs1[OF fsb])
done

lemma fs_list_inst:
assumes fs: "fs TYPE('a) TYPE('x)"
shows "fs TYPE('a list) TYPE('x)"
apply(induct_tac x)
apply(rule fs1[OF fs])
done

lemma fs_option_inst:
assumes fs: "fs TYPE('a) TYPE('x)"
shows "fs TYPE('a option) TYPE('x)"
apply(case_tac x)
apply(rule fs1[OF fs])
done

section ‹Lemmas about the permutation properties›
(*=================================================*)

lemma pt1:
fixes x::"'a"
assumes a: "pt TYPE('a) TYPE('x)"
shows "([]::'x prm)∙x = x"
using a by (simp add: pt_def)

lemma pt2:
fixes pi1::"'x prm"
and   pi2::"'x prm"
and   x  ::"'a"
assumes a: "pt TYPE('a) TYPE('x)"
shows "(pi1@pi2)∙x = pi1∙(pi2∙x)"
using a by (simp add: pt_def)

lemma pt3:
fixes pi1::"'x prm"
and   pi2::"'x prm"
and   x  ::"'a"
assumes a: "pt TYPE('a) TYPE('x)"
shows "pi1 ≜ pi2 ⟹ pi1∙x = pi2∙x"
using a by (simp add: pt_def)

lemma pt3_rev:
fixes pi1::"'x prm"
and   pi2::"'x prm"
and   x  ::"'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi1 ≜ pi2 ⟹ (rev pi1)∙x = (rev pi2)∙x"
by (rule pt3[OF pt], simp add: at_prm_rev_eq[OF at])

section ‹composition properties›
(* ============================== *)
lemma cp1:
fixes pi1::"'x prm"
and   pi2::"'y prm"
and   x  ::"'a"
assumes cp: "cp TYPE ('a) TYPE('x) TYPE('y)"
shows "pi1∙(pi2∙x) = (pi1∙pi2)∙(pi1∙x)"
using cp by (simp add: cp_def)

lemma cp_pt_inst:
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "cp TYPE('a) TYPE('x) TYPE('x)"
apply(auto simp add: cp_def pt2[OF pt,symmetric])
apply(rule pt3[OF pt])
apply(rule at_ds8[OF at])
done

section ‹disjointness properties›
(*=================================*)
lemma dj_perm_forget:
fixes pi::"'y prm"
and   x ::"'x"
assumes dj: "disjoint TYPE('x) TYPE('y)"
shows "pi∙x=x"
using dj by (simp_all add: disjoint_def)

lemma dj_perm_set_forget:
fixes pi::"'y prm"
and   x ::"'x set"
assumes dj: "disjoint TYPE('x) TYPE('y)"
shows "pi∙x=x"
using dj by (simp_all add: perm_set_def disjoint_def)

lemma dj_perm_perm_forget:
fixes pi1::"'x prm"
and   pi2::"'y prm"
assumes dj: "disjoint TYPE('x) TYPE('y)"
shows "pi2∙pi1=pi1"
using dj by (induct pi1, auto simp add: disjoint_def)

lemma dj_cp:
fixes pi1::"'x prm"
and   pi2::"'y prm"
and   x  ::"'a"
assumes cp: "cp TYPE ('a) TYPE('x) TYPE('y)"
and     dj: "disjoint TYPE('y) TYPE('x)"
shows "pi1∙(pi2∙x) = (pi2)∙(pi1∙x)"
by (simp add: cp1[OF cp] dj_perm_perm_forget[OF dj])

lemma dj_supp:
fixes a::"'x"
assumes dj: "disjoint TYPE('x) TYPE('y)"
shows "(supp a) = ({}::'y set)"
done

lemma at_fresh_ineq:
fixes a :: "'x"
and   b :: "'y"
assumes dj: "disjoint TYPE('y) TYPE('x)"
shows "a♯b"
by (simp add: fresh_def dj_supp[OF dj])

section ‹permutation type instances›
(* ===================================*)

lemma pt_fun_inst:
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at:  "at TYPE('x)"
shows  "pt TYPE('a⇒'b) TYPE('x)"
apply(auto simp only: pt_def)
apply(simp add: pt1[OF pta] pt1[OF ptb])
apply(simp add: pt2[OF pta] pt2[OF ptb])
apply(subgoal_tac "(rev pi1) ≜ (rev pi2)")(*A*)
apply(simp add: pt3[OF pta] pt3[OF ptb])
(*A*)
done

lemma pt_bool_inst:
shows  "pt TYPE(bool) TYPE('x)"

lemma pt_set_inst:
assumes pt: "pt TYPE('a) TYPE('x)"
shows  "pt TYPE('a set) TYPE('x)"
apply(force simp add: pt2[OF pt] pt3[OF pt])
done

lemma pt_unit_inst:
shows "pt TYPE(unit) TYPE('x)"

lemma pt_prod_inst:
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
shows  "pt TYPE('a × 'b) TYPE('x)"
apply(rule pt1[OF pta])
apply(rule pt1[OF ptb])
apply(rule pt2[OF pta])
apply(rule pt2[OF ptb])
apply(rule pt3[OF pta],assumption)
apply(rule pt3[OF ptb],assumption)
done

lemma pt_list_nil:
fixes xs :: "'a list"
assumes pt: "pt TYPE('a) TYPE ('x)"
shows "([]::'x prm)∙xs = xs"
apply(induct_tac xs)
done

lemma pt_list_append:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   xs  :: "'a list"
assumes pt: "pt TYPE('a) TYPE ('x)"
shows "(pi1@pi2)∙xs = pi1∙(pi2∙xs)"
apply(induct_tac xs)
done

lemma pt_list_prm_eq:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   xs  :: "'a list"
assumes pt: "pt TYPE('a) TYPE ('x)"
shows "pi1 ≜ pi2  ⟹ pi1∙xs = pi2∙xs"
apply(induct_tac xs)
done

lemma pt_list_inst:
assumes pt: "pt TYPE('a) TYPE('x)"
shows  "pt TYPE('a list) TYPE('x)"
apply(auto simp only: pt_def)
apply(rule pt_list_nil[OF pt])
apply(rule pt_list_append[OF pt])
apply(rule pt_list_prm_eq[OF pt],assumption)
done

lemma pt_option_inst:
assumes pta: "pt TYPE('a) TYPE('x)"
shows  "pt TYPE('a option) TYPE('x)"
apply(auto simp only: pt_def)
apply(case_tac "x")
apply(case_tac "x")
apply(case_tac "x")
done

lemma pt_noption_inst:
assumes pta: "pt TYPE('a) TYPE('x)"
shows  "pt TYPE('a noption) TYPE('x)"
apply(auto simp only: pt_def)
apply(case_tac "x")
apply(case_tac "x")
apply(case_tac "x")
done

lemma pt_nprod_inst:
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
shows  "pt TYPE(('a,'b) nprod) TYPE('x)"
apply(case_tac x)
apply(simp add: pt1[OF pta] pt1[OF ptb])
apply(case_tac x)
apply(simp add: pt2[OF pta] pt2[OF ptb])
apply(case_tac x)
apply(simp add: pt3[OF pta] pt3[OF ptb])
done

section ‹further lemmas for permutation types›
(*==============================================*)

lemma pt_rev_pi:
fixes pi :: "'x prm"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(rev pi)∙(pi∙x) = x"
proof -
have "((rev pi)@pi) ≜ ([]::'x prm)" by (simp add: at_ds7[OF at])
hence "((rev pi)@pi)∙(x::'a) = ([]::'x prm)∙x" by (simp add: pt3[OF pt])
thus ?thesis by (simp add: pt1[OF pt] pt2[OF pt])
qed

lemma pt_pi_rev:
fixes pi :: "'x prm"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙((rev pi)∙x) = x"
by (simp add: pt_rev_pi[OF pt, OF at,of "rev pi" "x",simplified])

lemma pt_bij1:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "(pi∙x) = y"
shows   "x=(rev pi)∙y"
proof -
from a have "y=(pi∙x)" by (rule sym)
thus ?thesis by (simp only: pt_rev_pi[OF pt, OF at])
qed

lemma pt_bij2:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "x = (rev pi)∙y"
shows   "(pi∙x)=y"
using a by (simp add: pt_pi_rev[OF pt, OF at])

lemma pt_bij:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙x = pi∙y) = (x=y)"
proof
assume "pi∙x = pi∙y"
hence  "x=(rev pi)∙(pi∙y)" by (rule pt_bij1[OF pt, OF at])
thus "x=y" by (simp only: pt_rev_pi[OF pt, OF at])
next
assume "x=y"
thus "pi∙x = pi∙y" by simp
qed

lemma pt_eq_eqvt:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(x=y) = (pi∙x = pi∙y)"
using pt at
by (auto simp add: pt_bij perm_bool)

lemma pt_bij3:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
assumes a:  "x=y"
shows "(pi∙x = pi∙y)"
using a by simp

lemma pt_bij4:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "pi∙x = pi∙y"
shows "x = y"
using a by (simp add: pt_bij[OF pt, OF at])

lemma pt_swap_bij:
fixes a  :: "'x"
and   b  :: "'x"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "[(a,b)]∙([(a,b)]∙x) = x"
by (rule pt_bij2[OF pt, OF at], simp)

lemma pt_swap_bij':
fixes a  :: "'x"
and   b  :: "'x"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "[(a,b)]∙([(b,a)]∙x) = x"
apply(rule trans)
apply(rule pt3[OF pt])
apply(rule at_ds5'[OF at])
apply(rule pt1[OF pt])
done

lemma pt_swap_bij'':
fixes a  :: "'x"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "[(a,a)]∙x = x"
apply(rule trans)
apply(rule pt3[OF pt])
apply(rule at_ds1[OF at])
apply(rule pt1[OF pt])
done

lemma supp_singleton:
shows "supp {x} = supp x"
by (force simp add: supp_def perm_set_def)

lemma fresh_singleton:
shows "a♯{x} = a♯x"

lemma pt_set_bij1:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "((pi∙x)∈X) = (x∈((rev pi)∙X))"
by (force simp add: perm_set_def pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])

lemma pt_set_bij1a:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(x∈(pi∙X)) = (((rev pi)∙x)∈X)"
by (force simp add: perm_set_def pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])

lemma pt_set_bij:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "((pi∙x)∈(pi∙X)) = (x∈X)"
by (simp add: perm_set_def pt_bij[OF pt, OF at])

lemma pt_in_eqvt:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(x∈X)=((pi∙x)∈(pi∙X))"
using assms
by (auto simp add:  pt_set_bij perm_bool)

lemma pt_set_bij2:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "x∈X"
shows "(pi∙x)∈(pi∙X)"
using a by (simp add: pt_set_bij[OF pt, OF at])

lemma pt_set_bij2a:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "x∈((rev pi)∙X)"
shows "(pi∙x)∈X"
using a by (simp add: pt_set_bij1[OF pt, OF at])

(* FIXME: is this lemma needed anywhere? *)
lemma pt_set_bij3:
fixes pi :: "'x prm"
and   x  :: "'a"
and   X  :: "'a set"
shows "pi∙(x∈X) = (x∈X)"

lemma pt_subseteq_eqvt:
fixes pi :: "'x prm"
and   Y  :: "'a set"
and   X  :: "'a set"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙(X⊆Y)) = ((pi∙X)⊆(pi∙Y))"
by (auto simp add: perm_set_def perm_bool pt_bij[OF pt, OF at])

lemma pt_set_diff_eqvt:
fixes X::"'a set"
and   Y::"'a set"
and   pi::"'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(X - Y) = (pi∙X) - (pi∙Y)"
by (auto simp add: perm_set_def pt_bij[OF pt, OF at])

lemma pt_Collect_eqvt:
fixes pi::"'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙{x::'a. P x} = {x. P ((rev pi)∙x)}"
apply(auto simp add: perm_set_def pt_rev_pi[OF pt, OF at])
apply(rule_tac x="(rev pi)∙x" in exI)
apply(simp add: pt_pi_rev[OF pt, OF at])
done

― "some helper lemmas for the pt_perm_supp_ineq lemma"
lemma Collect_permI:
fixes pi :: "'x prm"
and   x  :: "'a"
assumes a: "∀x. (P1 x = P2 x)"
shows "{pi∙x| x. P1 x} = {pi∙x| x. P2 x}"
using a by force

lemma Infinite_cong:
assumes a: "X = Y"
shows "infinite X = infinite Y"
using a by (simp)

lemma pt_set_eq_ineq:
fixes pi :: "'y prm"
assumes pt: "pt TYPE('x) TYPE('y)"
and     at: "at TYPE('y)"
shows "{pi∙x| x::'x. P x} = {x::'x. P ((rev pi)∙x)}"
by (force simp only: pt_rev_pi[OF pt, OF at] pt_pi_rev[OF pt, OF at])

lemma pt_inject_on_ineq:
fixes X  :: "'y set"
and   pi :: "'x prm"
assumes pt: "pt TYPE('y) TYPE('x)"
and     at: "at TYPE('x)"
shows "inj_on (perm pi) X"
proof (unfold inj_on_def, intro strip)
fix x::"'y" and y::"'y"
assume "pi∙x = pi∙y"
thus "x=y" by (simp add: pt_bij[OF pt, OF at])
qed

lemma pt_set_finite_ineq:
fixes X  :: "'x set"
and   pi :: "'y prm"
assumes pt: "pt TYPE('x) TYPE('y)"
and     at: "at TYPE('y)"
shows "finite (pi∙X) = finite X"
proof -
have image: "(pi∙X) = (perm pi ` X)" by (force simp only: perm_set_def)
show ?thesis
proof (rule iffI)
assume "finite (pi∙X)"
hence "finite (perm pi ` X)" using image by (simp)
thus "finite X" using pt_inject_on_ineq[OF pt, OF at] by (rule finite_imageD)
next
assume "finite X"
hence "finite (perm pi ` X)" by (rule finite_imageI)
thus "finite (pi∙X)" using image by (simp)
qed
qed

lemma pt_set_infinite_ineq:
fixes X  :: "'x set"
and   pi :: "'y prm"
assumes pt: "pt TYPE('x) TYPE('y)"
and     at: "at TYPE('y)"
shows "infinite (pi∙X) = infinite X"
using pt at by (simp add: pt_set_finite_ineq)

lemma pt_perm_supp_ineq:
fixes  pi  :: "'x prm"
and    x   :: "'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
shows "(pi∙((supp x)::'y set)) = supp (pi∙x)" (is "?LHS = ?RHS")
proof -
have "?LHS = {pi∙a | a. infinite {b. [(a,b)]∙x ≠ x}}" by (simp add: supp_def perm_set_def)
also have "… = {pi∙a | a. infinite {pi∙b | b. [(a,b)]∙x ≠ x}}"
proof (rule Collect_permI, rule allI, rule iffI)
fix a
assume "infinite {b::'y. [(a,b)]∙x  ≠ x}"
hence "infinite (pi∙{b::'y. [(a,b)]∙x ≠ x})" by (simp add: pt_set_infinite_ineq[OF ptb, OF at])
thus "infinite {pi∙b |b::'y. [(a,b)]∙x  ≠ x}" by (simp add: perm_set_def)
next
fix a
assume "infinite {pi∙b |b::'y. [(a,b)]∙x ≠ x}"
hence "infinite (pi∙{b::'y. [(a,b)]∙x ≠ x})" by (simp add: perm_set_def)
thus "infinite {b::'y. [(a,b)]∙x  ≠ x}"
by (simp add: pt_set_infinite_ineq[OF ptb, OF at])
qed
also have "… = {a. infinite {b::'y. [((rev pi)∙a,(rev pi)∙b)]∙x ≠ x}}"
by (simp add: pt_set_eq_ineq[OF ptb, OF at])
also have "… = {a. infinite {b. pi∙([((rev pi)∙a,(rev pi)∙b)]∙x) ≠ (pi∙x)}}"
by (simp add: pt_bij[OF pta, OF at])
also have "… = {a. infinite {b. [(a,b)]∙(pi∙x) ≠ (pi∙x)}}"
proof (rule Collect_cong, rule Infinite_cong, rule Collect_cong)
fix a::"'y" and b::"'y"
have "pi∙(([((rev pi)∙a,(rev pi)∙b)])∙x) = [(a,b)]∙(pi∙x)"
by (simp add: cp1[OF cp] pt_pi_rev[OF ptb, OF at])
thus "(pi∙([((rev pi)∙a,(rev pi)∙b)]∙x) ≠  pi∙x) = ([(a,b)]∙(pi∙x) ≠ pi∙x)" by simp
qed
finally show "?LHS = ?RHS" by (simp add: supp_def)
qed

lemma pt_perm_supp:
fixes  pi  :: "'x prm"
and    x   :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙((supp x)::'x set)) = supp (pi∙x)"
apply(rule pt_perm_supp_ineq)
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
done

lemma pt_supp_finite_pi:
fixes  pi  :: "'x prm"
and    x   :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f: "finite ((supp x)::'x set)"
shows "finite ((supp (pi∙x))::'x set)"
apply(simp add: pt_perm_supp[OF pt, OF at, symmetric])
apply(simp add: pt_set_finite_ineq[OF at_pt_inst[OF at], OF at])
apply(rule f)
done

lemma pt_fresh_left_ineq:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'y"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
shows "a♯(pi∙x) = ((rev pi)∙a)♯x"
apply(simp add: pt_set_bij1[OF ptb, OF at])
apply(simp add: pt_perm_supp_ineq[OF pta, OF ptb, OF at, OF cp])
done

lemma pt_fresh_right_ineq:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'y"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
shows "(pi∙a)♯x = a♯((rev pi)∙x)"
apply(simp add: pt_set_bij1[OF ptb, OF at])
apply(simp add: pt_perm_supp_ineq[OF pta, OF ptb, OF at, OF cp])
done

lemma pt_fresh_bij_ineq:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'y"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
shows "(pi∙a)♯(pi∙x) = a♯x"
apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
apply(simp add: pt_rev_pi[OF ptb, OF at])
done

lemma pt_fresh_left:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "a♯(pi∙x) = ((rev pi)∙a)♯x"
apply(rule pt_fresh_left_ineq)
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
done

lemma pt_fresh_right:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙a)♯x = a♯((rev pi)∙x)"
apply(rule pt_fresh_right_ineq)
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
done

lemma pt_fresh_bij:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙a)♯(pi∙x) = a♯x"
apply(rule pt_fresh_bij_ineq)
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
done

lemma pt_fresh_bij1:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "a♯x"
shows "(pi∙a)♯(pi∙x)"
using a by (simp add: pt_fresh_bij[OF pt, OF at])

lemma pt_fresh_bij2:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a:  "(pi∙a)♯(pi∙x)"
shows  "a♯x"
using a by (simp add: pt_fresh_bij[OF pt, OF at])

lemma pt_fresh_eqvt:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(a♯x) = (pi∙a)♯(pi∙x)"
by (simp add: perm_bool pt_fresh_bij[OF pt, OF at])

lemma pt_perm_fresh1:
fixes a :: "'x"
and   b :: "'x"
and   x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE ('x)"
and     a1: "¬(a♯x)"
and     a2: "b♯x"
shows "[(a,b)]∙x ≠ x"
proof
assume neg: "[(a,b)]∙x = x"
from a1 have a1':"a∈(supp x)" by (simp add: fresh_def)
from a2 have a2':"b∉(supp x)" by (simp add: fresh_def)
from a1' a2' have a3: "a≠b" by force
from a1' have "([(a,b)]∙a)∈([(a,b)]∙(supp x))"
by (simp only: pt_set_bij[OF at_pt_inst[OF at], OF at])
hence "b∈([(a,b)]∙(supp x))" by (simp add: at_calc[OF at])
hence "b∈(supp ([(a,b)]∙x))" by (simp add: pt_perm_supp[OF pt,OF at])
with a2' neg show False by simp
qed

(* the next two lemmas are needed in the proof *)
(* of the structural induction principle       *)
lemma pt_fresh_aux:
fixes a::"'x"
and   b::"'x"
and   c::"'x"
and   x::"'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE ('x)"
assumes a1: "c≠a" and  a2: "a♯x" and a3: "c♯x"
shows "c♯([(a,b)]∙x)"
using a1 a2 a3 by (simp_all add: pt_fresh_left[OF pt, OF at] at_calc[OF at])

lemma pt_fresh_perm_app:
fixes pi :: "'x prm"
and   a  :: "'x"
and   x  :: "'y"
assumes pt: "pt TYPE('y) TYPE('x)"
and     at: "at TYPE('x)"
and     h1: "a♯pi"
and     h2: "a♯x"
shows "a♯(pi∙x)"
using assms
proof -
have "a♯(rev pi)"using h1 by (simp add: fresh_list_rev)
then have "(rev pi)∙a = a" by (simp add: at_prm_fresh[OF at])
then have "((rev pi)∙a)♯x" using h2 by simp
thus "a♯(pi∙x)"  by (simp add: pt_fresh_right[OF pt, OF at])
qed

lemma pt_fresh_perm_app_ineq:
fixes pi::"'x prm"
and   c::"'y"
and   x::"'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
and     dj:  "disjoint TYPE('y) TYPE('x)"
assumes a: "c♯x"
shows "c♯(pi∙x)"
using a by (simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp] dj_perm_forget[OF dj])

lemma pt_fresh_eqvt_ineq:
fixes pi::"'x prm"
and   c::"'y"
and   x::"'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
and     dj:  "disjoint TYPE('y) TYPE('x)"
shows "pi∙(c♯x) = (pi∙c)♯(pi∙x)"
by (simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp] dj_perm_forget[OF dj] perm_bool)

―"the co-set of a finite set is infinte"
lemma finite_infinite:
assumes a: "finite {b::'x. P b}"
and     b: "infinite (UNIV::'x set)"
shows "infinite {b. ¬P b}"
proof -
from a b have "infinite (UNIV - {b::'x. P b})" by (simp add: Diff_infinite_finite)
moreover
have "{b::'x. ¬P b} = UNIV - {b::'x. P b}" by auto
ultimately show "infinite {b::'x. ¬P b}" by simp
qed

lemma pt_fresh_fresh:
fixes   x :: "'a"
and     a :: "'x"
and     b :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE ('x)"
and     a1: "a♯x" and a2: "b♯x"
shows "[(a,b)]∙x=x"
proof (cases "a=b")
assume "a=b"
hence "[(a,b)] ≜ []" by (simp add: at_ds1[OF at])
hence "[(a,b)]∙x=([]::'x prm)∙x" by (rule pt3[OF pt])
thus ?thesis by (simp only: pt1[OF pt])
next
assume c2: "a≠b"
from a1 have f1: "finite {c. [(a,c)]∙x ≠ x}" by (simp add: fresh_def supp_def)
from a2 have f2: "finite {c. [(b,c)]∙x ≠ x}" by (simp add: fresh_def supp_def)
from f1 and f2 have f3: "finite {c. perm [(a,c)] x ≠ x ∨ perm [(b,c)] x ≠ x}"
by (force simp only: Collect_disj_eq)
have "infinite {c. [(a,c)]∙x = x ∧ [(b,c)]∙x = x}"
by (simp add: finite_infinite[OF f3,OF at4[OF at], simplified])
hence "infinite ({c. [(a,c)]∙x = x ∧ [(b,c)]∙x = x}-{a,b})"
by (force dest: Diff_infinite_finite)
hence "({c. [(a,c)]∙x = x ∧ [(b,c)]∙x = x}-{a,b}) ≠ {}"
by (metis finite_set set_empty2)
hence "∃c. c∈({c. [(a,c)]∙x = x ∧ [(b,c)]∙x = x}-{a,b})" by (force)
then obtain c
where eq1: "[(a,c)]∙x = x"
and eq2: "[(b,c)]∙x = x"
and ineq: "a≠c ∧ b≠c"
by (force)
hence "[(a,c)]∙([(b,c)]∙([(a,c)]∙x)) = x" by simp
hence eq3: "[(a,c),(b,c),(a,c)]∙x = x" by (simp add: pt2[OF pt,symmetric])
from c2 ineq have "[(a,c),(b,c),(a,c)] ≜ [(a,b)]" by (simp add: at_ds3[OF at])
hence "[(a,c),(b,c),(a,c)]∙x = [(a,b)]∙x" by (rule pt3[OF pt])
thus ?thesis using eq3 by simp
qed

lemma pt_pi_fresh_fresh:
fixes   x :: "'a"
and     pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE ('x)"
and     a:  "∀(a,b)∈set pi. a♯x ∧ b♯x"
shows "pi∙x=x"
using a
proof (induct pi)
case Nil
show "([]::'x prm)∙x = x" by (rule pt1[OF pt])
next
case (Cons ab pi)
have a: "∀(a,b)∈set (ab#pi). a♯x ∧ b♯x" by fact
have ih: "(∀(a,b)∈set pi. a♯x ∧ b♯x) ⟹ pi∙x=x" by fact
obtain a b where e: "ab=(a,b)" by (cases ab) (auto)
from a have a': "a♯x" "b♯x" using e by auto
have "(ab#pi)∙x = ([(a,b)]@pi)∙x" using e by simp
also have "… = [(a,b)]∙(pi∙x)" by (simp only: pt2[OF pt])
also have "… = [(a,b)]∙x" using ih a by simp
also have "… = x" using a' by (simp add: pt_fresh_fresh[OF pt, OF at])
finally show "(ab#pi)∙x = x" by simp
qed

lemma pt_perm_compose:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi2∙(pi1∙x) = (pi2∙pi1)∙(pi2∙x)"
proof -
have "(pi2@pi1) ≜ ((pi2∙pi1)@pi2)" by (rule at_ds8 [OF at])
hence "(pi2@pi1)∙x = ((pi2∙pi1)@pi2)∙x" by (rule pt3[OF pt])
thus ?thesis by (simp add: pt2[OF pt])
qed

lemma pt_perm_compose':
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi2∙pi1)∙x = pi2∙(pi1∙((rev pi2)∙x))"
proof -
have "pi2∙(pi1∙((rev pi2)∙x)) = (pi2∙pi1)∙(pi2∙((rev pi2)∙x))"
by (rule pt_perm_compose[OF pt, OF at])
also have "… = (pi2∙pi1)∙x" by (simp add: pt_pi_rev[OF pt, OF at])
finally have "pi2∙(pi1∙((rev pi2)∙x)) = (pi2∙pi1)∙x" by simp
thus ?thesis by simp
qed

lemma pt_perm_compose_rev:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(rev pi2)∙((rev pi1)∙x) = (rev pi1)∙(rev (pi1∙pi2)∙x)"
proof -
have "((rev pi2)@(rev pi1)) ≜ ((rev pi1)@(rev (pi1∙pi2)))" by (rule at_ds9[OF at])
hence "((rev pi2)@(rev pi1))∙x = ((rev pi1)@(rev (pi1∙pi2)))∙x" by (rule pt3[OF pt])
thus ?thesis by (simp add: pt2[OF pt])
qed

section ‹equivariance for some connectives›
lemma pt_all_eqvt:
fixes  pi :: "'x prm"
and     x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(∀(x::'a). P x) = (∀(x::'a). pi∙(P ((rev pi)∙x)))"
apply(drule_tac x="pi∙x" in spec)
apply(simp add: pt_rev_pi[OF pt, OF at])
done

lemma pt_ex_eqvt:
fixes  pi :: "'x prm"
and     x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(∃(x::'a). P x) = (∃(x::'a). pi∙(P ((rev pi)∙x)))"
apply(rule_tac x="pi∙x" in exI)
apply(simp add: pt_rev_pi[OF pt, OF at])
done

lemma pt_ex1_eqvt:
fixes  pi :: "'x prm"
and     x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows  "(pi∙(∃!x. P (x::'a))) = (∃!x. pi∙(P (rev pi∙x)))"
unfolding Ex1_def
by (simp add: pt_ex_eqvt[OF pt at] conj_eqvt pt_all_eqvt[OF pt at]
imp_eqvt pt_eq_eqvt[OF pt at] pt_pi_rev[OF pt at])

lemma pt_the_eqvt:
fixes  pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     unique: "∃!x. P x"
shows "pi∙(THE(x::'a). P x) = (THE(x::'a). pi∙(P ((rev pi)∙x)))"
apply(rule the1_equality [symmetric])
apply(simp add: perm_bool pt_rev_pi [OF pt at])
apply(rule theI'[OF unique])
done

(*==============================*)

lemma supports_subset:
fixes x  :: "'a"
and   S1 :: "'x set"
and   S2 :: "'x set"
assumes  a: "S1 supports x"
and      b: "S1 ⊆ S2"
shows "S2 supports x"
using a b

lemma supp_is_subset:
fixes S :: "'x set"
and   x :: "'a"
assumes a1: "S supports x"
and     a2: "finite S"
shows "(supp x)⊆S"
proof (rule ccontr)
assume "¬(supp x ⊆ S)"
hence "∃a. a∈(supp x) ∧ a∉S" by force
then obtain a where b1: "a∈supp x" and b2: "a∉S" by force
from a1 b2 have "∀b. (b∉S ⟶ ([(a,b)]∙x = x))" by (unfold supports_def, force)
hence "{b. [(a,b)]∙x ≠ x}⊆S" by force
with a2 have "finite {b. [(a,b)]∙x ≠ x}" by (simp add: finite_subset)
hence "a∉(supp x)" by (unfold supp_def, auto)
with b1 show False by simp
qed

lemma supp_supports:
fixes x :: "'a"
assumes  pt: "pt TYPE('a) TYPE('x)"
and      at: "at TYPE ('x)"
shows "((supp x)::'x set) supports x"
proof (unfold supports_def, intro strip)
fix a b
assume "(a::'x)∉(supp x) ∧ (b::'x)∉(supp x)"
hence "a♯x" and "b♯x" by (auto simp add: fresh_def)
thus "[(a,b)]∙x = x" by (rule pt_fresh_fresh[OF pt, OF at])
qed

lemma supports_finite:
fixes S :: "'x set"
and   x :: "'a"
assumes a1: "S supports x"
and     a2: "finite S"
shows "finite ((supp x)::'x set)"
proof -
have "(supp x)⊆S" using a1 a2 by (rule supp_is_subset)
thus ?thesis using a2 by (simp add: finite_subset)
qed

lemma supp_is_inter:
fixes  x :: "'a"
assumes  pt: "pt TYPE('a) TYPE('x)"
and      at: "at TYPE ('x)"
and      fs: "fs TYPE('a) TYPE('x)"
shows "((supp x)::'x set) = (⋂{S. finite S ∧ S supports x})"
proof (rule equalityI)
show "((supp x)::'x set) ⊆ (⋂{S. finite S ∧ S supports x})"
proof (clarify)
fix S c
assume b: "c∈((supp x)::'x set)" and "finite (S::'x set)" and "S supports x"
hence  "((supp x)::'x set)⊆S" by (simp add: supp_is_subset)
with b show "c∈S" by force
qed
next
show "(⋂{S. finite S ∧ S supports x}) ⊆ ((supp x)::'x set)"
proof (clarify, simp)
fix c
assume d: "∀(S::'x set). finite S ∧ S supports x ⟶ c∈S"
have "((supp x)::'x set) supports x" by (rule supp_supports[OF pt, OF at])
with d fs1[OF fs] show "c∈supp x" by force
qed
qed

lemma supp_is_least_supports:
fixes S :: "'x set"
and   x :: "'a"
assumes  pt: "pt TYPE('a) TYPE('x)"
and      at: "at TYPE ('x)"
and      a1: "S supports x"
and      a2: "finite S"
and      a3: "∀S'. (S' supports x) ⟶ S⊆S'"
shows "S = (supp x)"
proof (rule equalityI)
show "((supp x)::'x set)⊆S" using a1 a2 by (rule supp_is_subset)
next
have "((supp x)::'x set) supports x" by (rule supp_supports[OF pt, OF at])
with a3 show "S⊆supp x" by force
qed

lemma supports_set:
fixes S :: "'x set"
and   X :: "'a set"
assumes  pt: "pt TYPE('a) TYPE('x)"
and      at: "at TYPE ('x)"
and      a: "∀x∈X. (∀(a::'x) (b::'x). a∉S∧b∉S ⟶ ([(a,b)]∙x)∈X)"
shows  "S supports X"
using a
apply(simp add: pt_set_bij1a[OF pt, OF at])
apply(force simp add: pt_swap_bij[OF pt, OF at])
apply(simp add: pt_set_bij1a[OF pt, OF at])
done

lemma supports_fresh:
fixes S :: "'x set"
and   a :: "'x"
and   x :: "'a"
assumes a1: "S supports x"
and     a2: "finite S"
and     a3: "a∉S"
shows "a♯x"
have "(supp x)⊆S" using a1 a2 by (rule supp_is_subset)
thus "a∉(supp x)" using a3 by force
qed

lemma at_fin_set_supports:
fixes X::"'x set"
assumes at: "at TYPE('x)"
shows "X supports X"
proof -
have "∀a b. a∉X ∧ b∉X ⟶ [(a,b)]∙X = X"
by (auto simp add: perm_set_def at_calc[OF at])
then show ?thesis by (simp add: supports_def)
qed

lemma infinite_Collection:
assumes a1:"infinite X"
and     a2:"∀b∈X. P(b)"
shows "infinite {b∈X. P(b)}"
using a1 a2
apply auto
apply (subgoal_tac "infinite (X - {b∈X. P b})")
done

lemma at_fin_set_supp:
fixes X::"'x set"
assumes at: "at TYPE('x)"
and     fs: "finite X"
shows "(supp X) = X"
proof (rule subset_antisym)
show "(supp X) ⊆ X" using at_fin_set_supports[OF at] using fs by (simp add: supp_is_subset)
next
have inf: "infinite (UNIV-X)" using at4[OF at] fs by (auto simp add: Diff_infinite_finite)
{ fix a::"'x"
assume asm: "a∈X"
hence "∀b∈(UNIV-X). [(a,b)]∙X≠X"
by (auto simp add: perm_set_def at_calc[OF at])
with inf have "infinite {b∈(UNIV-X). [(a,b)]∙X≠X}" by (rule infinite_Collection)
hence "infinite {b. [(a,b)]∙X≠X}" by (rule_tac infinite_super, auto)
hence "a∈(supp X)" by (simp add: supp_def)
}
then show "X⊆(supp X)" by blast
qed

lemma at_fin_set_fresh:
fixes X::"'x set"
assumes at: "at TYPE('x)"
and     fs: "finite X"
shows "(x ♯ X) = (x ∉ X)"
by (simp add: at_fin_set_supp fresh_def at fs)

section ‹Permutations acting on Functions›
(*==========================================*)

lemma pt_fun_app_eq:
fixes f  :: "'a⇒'b"
and   x  :: "'a"
and   pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(f x) = (pi∙f)(pi∙x)"
by (simp add: perm_fun_def pt_rev_pi[OF pt, OF at])

―"sometimes pt_fun_app_eq does too much; this lemma 'corrects it'"
lemma pt_perm:
fixes x  :: "'a"
and   pi1 :: "'x prm"
and   pi2 :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE ('x)"
shows "(pi1∙perm pi2)(pi1∙x) = pi1∙(pi2∙x)"
by (simp add: pt_fun_app_eq[OF pt, OF at])

lemma pt_fun_eq:
fixes f  :: "'a⇒'b"
and   pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙f = f) = (∀ x. pi∙(f x) = f (pi∙x))" (is "?LHS = ?RHS")
proof
assume a: "?LHS"
show "?RHS"
proof
fix x
have "pi∙(f x) = (pi∙f)(pi∙x)" by (simp add: pt_fun_app_eq[OF pt, OF at])
also have "… = f (pi∙x)" using a by simp
finally show "pi∙(f x) = f (pi∙x)" by simp
qed
next
assume b: "?RHS"
show "?LHS"
proof (rule ccontr)
assume "(pi∙f) ≠ f"
hence "∃x. (pi∙f) x ≠ f x" by (simp add: fun_eq_iff)
then obtain x where b1: "(pi∙f) x ≠ f x" by force
from b have "pi∙(f ((rev pi)∙x)) = f (pi∙((rev pi)∙x))" by force
hence "(pi∙f)(pi∙((rev pi)∙x)) = f (pi∙((rev pi)∙x))"
by (simp add: pt_fun_app_eq[OF pt, OF at])
hence "(pi∙f) x = f x" by (simp add: pt_pi_rev[OF pt, OF at])
with b1 show "False" by simp
qed
qed

― "two helper lemmas for the equivariance of functions"
lemma pt_swap_eq_aux:
fixes   y :: "'a"
and    pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     a: "∀(a::'x) (b::'x). [(a,b)]∙y = y"
shows "pi∙y = y"
proof(induct pi)
case Nil show ?case by (simp add: pt1[OF pt])
next
case (Cons x xs)
have ih: "xs∙y = y" by fact
obtain a b where p: "x=(a,b)" by force
have "((a,b)#xs)∙y = ([(a,b)]@xs)∙y" by simp
also have "… = [(a,b)]∙(xs∙y)" by (simp only: pt2[OF pt])
finally show ?case using a ih p by simp
qed

lemma pt_swap_eq:
fixes   y :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
shows "(∀(a::'x) (b::'x). [(a,b)]∙y = y) = (∀pi::'x prm. pi∙y = y)"
by (force intro: pt_swap_eq_aux[OF pt])

lemma pt_eqvt_fun1a:
fixes f     :: "'a⇒'b"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at:  "at TYPE('x)"
and     a:   "((supp f)::'x set)={}"
shows "∀(pi::'x prm). pi∙f = f"
proof (intro strip)
fix pi
have "∀a b. a∉((supp f)::'x set) ∧ b∉((supp f)::'x set) ⟶ (([(a,b)]∙f) = f)"
by (intro strip, fold fresh_def,
simp add: pt_fresh_fresh[OF pt_fun_inst[OF pta, OF ptb, OF at],OF at])
with a have "∀(a::'x) (b::'x). ([(a,b)]∙f) = f" by force
hence "∀(pi::'x prm). pi∙f = f"
by (simp add: pt_swap_eq[OF pt_fun_inst[OF pta, OF ptb, OF at]])
thus "(pi::'x prm)∙f = f" by simp
qed

lemma pt_eqvt_fun1b:
fixes f     :: "'a⇒'b"
assumes a: "∀(pi::'x prm). pi∙f = f"
shows "((supp f)::'x set)={}"
using a by (simp add: supp_def)

lemma pt_eqvt_fun1:
fixes f     :: "'a⇒'b"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at: "at TYPE('x)"
shows "(((supp f)::'x set)={}) = (∀(pi::'x prm). pi∙f = f)" (is "?LHS = ?RHS")
by (rule iffI, simp add: pt_eqvt_fun1a[OF pta, OF ptb, OF at], simp add: pt_eqvt_fun1b)

lemma pt_eqvt_fun2a:
fixes f     :: "'a⇒'b"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at: "at TYPE('x)"
assumes a: "((supp f)::'x set)={}"
shows "∀(pi::'x prm) (x::'a). pi∙(f x) = f(pi∙x)"
proof (intro strip)
fix pi x
from a have b: "∀(pi::'x prm). pi∙f = f" by (simp add: pt_eqvt_fun1[OF pta, OF ptb, OF at])
have "(pi::'x prm)∙(f x) = (pi∙f)(pi∙x)" by (simp add: pt_fun_app_eq[OF pta, OF at])
with b show "(pi::'x prm)∙(f x) = f (pi∙x)" by force
qed

lemma pt_eqvt_fun2b:
fixes f     :: "'a⇒'b"
assumes pt1: "pt TYPE('a) TYPE('x)"
and     pt2: "pt TYPE('b) TYPE('x)"
and     at: "at TYPE('x)"
assumes a: "∀(pi::'x prm) (x::'a). pi∙(f x) = f(pi∙x)"
shows "((supp f)::'x set)={}"
proof -
from a have "∀(pi::'x prm). pi∙f = f" by (simp add: pt_fun_eq[OF pt1, OF at, symmetric])
thus ?thesis by (simp add: supp_def)
qed

lemma pt_eqvt_fun2:
fixes f     :: "'a⇒'b"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at: "at TYPE('x)"
shows "(((supp f)::'x set)={}) = (∀(pi::'x prm) (x::'a). pi∙(f x) = f(pi∙x))"
by (rule iffI,
simp add: pt_eqvt_fun2a[OF pta, OF ptb, OF at],
simp add: pt_eqvt_fun2b[OF pta, OF ptb, OF at])

lemma pt_supp_fun_subset:
fixes f :: "'a⇒'b"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at: "at TYPE('x)"
and     f1: "finite ((supp f)::'x set)"
and     f2: "finite ((supp x)::'x set)"
shows "supp (f x) ⊆ (((supp f)∪(supp x))::'x set)"
proof -
have s1: "((supp f)∪((supp x)::'x set)) supports (f x)"
proof (simp add: supports_def, fold fresh_def, auto)
fix a::"'x" and b::"'x"
assume "a♯f" and "b♯f"
hence a1: "[(a,b)]∙f = f"
by (rule pt_fresh_fresh[OF pt_fun_inst[OF pta, OF ptb, OF at], OF at])
assume "a♯x" and "b♯x"
hence a2: "[(a,b)]∙x = x" by (rule pt_fresh_fresh[OF pta, OF at])
from a1 a2 show "[(a,b)]∙(f x) = (f x)" by (simp add: pt_fun_app_eq[OF pta, OF at])
qed
from f1 f2 have "finite ((supp f)∪((supp x)::'x set))" by force
with s1 show ?thesis by (rule supp_is_subset)
qed

lemma pt_empty_supp_fun_subset:
fixes f :: "'a⇒'b"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('b) TYPE('x)"
and     at:  "at TYPE('x)"
and     e:   "(supp f)=({}::'x set)"
shows "supp (f x) ⊆ ((supp x)::'x set)"
proof (unfold supp_def, auto)
fix a::"'x"
assume a1: "finite {b. [(a, b)]∙x ≠ x}"
assume "infinite {b. [(a, b)]∙(f x) ≠ f x}"
hence a2: "infinite {b. f ([(a, b)]∙x) ≠ f x}" using e
by (simp add: pt_eqvt_fun2[OF pta, OF ptb, OF at])
have a3: "{b. f ([(a,b)]∙x) ≠ f x}⊆{b. [(a,b)]∙x ≠ x}" by force
from a1 a2 a3 show False by (force dest: finite_subset)
qed

section ‹Facts about the support of finite sets of finitely supported things›
(*=============================================================================*)

definition X_to_Un_supp :: "('a set) ⇒ 'x set" where
"X_to_Un_supp X ≡ ⋃x∈X. ((supp x)::'x set)"

lemma UNION_f_eqvt:
fixes X::"('a set)"
and   f::"'a ⇒ 'x set"
and   pi::"'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(⋃x∈X. f x) = (⋃x∈(pi∙X). (pi∙f) x)"
proof -
have pt_x: "pt TYPE('x) TYPE('x)" by (force intro: at_pt_inst at)
show ?thesis
proof (rule equalityI)
show "pi∙(⋃x∈X. f x) ⊆ (⋃x∈(pi∙X). (pi∙f) x)"
apply(rule_tac x="pi∙xb" in exI)
apply(rule conjI)
apply(rule_tac x="xb" in exI)
apply(simp)
apply(subgoal_tac "(pi∙f) (pi∙xb) = pi∙(f xb)")(*A*)
apply(simp)
apply(rule pt_set_bij2[OF pt_x, OF at])
apply(assumption)
(*A*)
apply(rule sym)
apply(rule pt_fun_app_eq[OF pt, OF at])
done
next
show "(⋃x∈(pi∙X). (pi∙f) x) ⊆ pi∙(⋃x∈X. f x)"
apply(rule_tac x="(rev pi)∙x" in exI)
apply(rule conjI)
apply(simp add: pt_pi_rev[OF pt_x, OF at])
apply(rule_tac x="xb" in bexI)
apply(simp add: pt_set_bij1[OF pt_x, OF at])
apply(simp add: pt_fun_app_eq[OF pt, OF at])
apply(assumption)
done
qed
qed

lemma X_to_Un_supp_eqvt:
fixes X::"('a set)"
and   pi::"'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(X_to_Un_supp X) = ((X_to_Un_supp (pi∙X))::'x set)"
apply(simp add: UNION_f_eqvt[OF pt, OF at] perm_fun_def)
apply(simp add: pt_perm_supp[OF pt, OF at])
apply(simp add: pt_pi_rev[OF pt, OF at])
done

lemma Union_supports_set:
fixes X::"('a set)"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(⋃x∈X. ((supp x)::'x set)) supports X"
apply(rule allI)+
apply(rule impI)
apply(erule conjE)
apply(auto)
apply(subgoal_tac "[(a,b)]∙xa = xa")(*A*)
apply(simp)
apply(rule pt_fresh_fresh[OF pt, OF at])
apply(force)
apply(force)
apply(rule_tac x="x" in exI)
apply(simp)
apply(rule sym)
apply(rule pt_fresh_fresh[OF pt, OF at])
apply(force)+
done

lemma Union_of_fin_supp_sets:
fixes X::"('a set)"
assumes fs: "fs TYPE('a) TYPE('x)"
and     fi: "finite X"
shows "finite (⋃x∈X. ((supp x)::'x set))"
using fi by (induct, auto simp add: fs1[OF fs])

lemma Union_included_in_supp:
fixes X::"('a set)"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     fi: "finite X"
shows "(⋃x∈X. ((supp x)::'x set)) ⊆ supp X"
proof -
have "supp ((X_to_Un_supp X)::'x set) ⊆ ((supp X)::'x set)"
apply(rule pt_empty_supp_fun_subset)
apply(force intro: pt_set_inst at_pt_inst pt at)+
apply(rule pt_eqvt_fun2b)
apply(force intro: pt_set_inst at_pt_inst pt at)+
apply(rule allI)+
apply(rule X_to_Un_supp_eqvt[OF pt, OF at])
done
hence "supp (⋃x∈X. ((supp x)::'x set)) ⊆ ((supp X)::'x set)" by (simp add: X_to_Un_supp_def)
moreover
have "supp (⋃x∈X. ((supp x)::'x set)) = (⋃x∈X. ((supp x)::'x set))"
apply(rule at_fin_set_supp[OF at])
apply(rule Union_of_fin_supp_sets[OF fs, OF fi])
done
ultimately show ?thesis by force
qed

lemma supp_of_fin_sets:
fixes X::"('a set)"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     fi: "finite X"
shows "(supp X) = (⋃x∈X. ((supp x)::'x set))"
apply(rule equalityI)
apply(rule supp_is_subset)
apply(rule Union_supports_set[OF pt, OF at])
apply(rule Union_of_fin_supp_sets[OF fs, OF fi])
apply(rule Union_included_in_supp[OF pt, OF at, OF fs, OF fi])
done

lemma supp_fin_union:
fixes X::"('a set)"
and   Y::"('a set)"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     f1: "finite X"
and     f2: "finite Y"
shows "(supp (X∪Y)) = (supp X)∪((supp Y)::'x set)"
using f1 f2 by (force simp add: supp_of_fin_sets[OF pt, OF at, OF fs])

lemma supp_fin_insert:
fixes X::"('a set)"
and   x::"'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     f:  "finite X"
shows "(supp (insert x X)) = (supp x)∪((supp X)::'x set)"
proof -
have "(supp (insert x X)) = ((supp ({x}∪(X::'a set)))::'x set)" by simp
also have "… = (supp {x})∪(supp X)"
by (rule supp_fin_union[OF pt, OF at, OF fs], simp_all add: f)
finally show "(supp (insert x X)) = (supp x)∪((supp X)::'x set)"
qed

lemma fresh_fin_union:
fixes X::"('a set)"
and   Y::"('a set)"
and   a::"'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     f1: "finite X"
and     f2: "finite Y"
shows "a♯(X∪Y) = (a♯X ∧ a♯Y)"
apply(simp add: supp_fin_union[OF pt, OF at, OF fs, OF f1, OF f2])
done

lemma fresh_fin_insert:
fixes X::"('a set)"
and   x::"'a"
and   a::"'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     f:  "finite X"
shows "a♯(insert x X) = (a♯x ∧ a♯X)"
apply(simp add: supp_fin_insert[OF pt, OF at, OF fs, OF f])
done

lemma fresh_fin_insert1:
fixes X::"('a set)"
and   x::"'a"
and   a::"'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
and     f:  "finite X"
and     a1:  "a♯x"
and     a2:  "a♯X"
shows "a♯(insert x X)"
using a1 a2
by (simp add: fresh_fin_insert[OF pt, OF at, OF fs, OF f])

lemma pt_list_set_supp:
fixes xs :: "'a list"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
shows "supp (set xs) = ((supp xs)::'x set)"
proof -
have "supp (set xs) = (⋃x∈(set xs). ((supp x)::'x set))"
by (rule supp_of_fin_sets[OF pt, OF at, OF fs], rule finite_set)
also have "(⋃x∈(set xs). ((supp x)::'x set)) = (supp xs)"
proof(induct xs)
case Nil show ?case by (simp add: supp_list_nil)
next
case (Cons h t) thus ?case by (simp add: supp_list_cons)
qed
finally show ?thesis by simp
qed

lemma pt_list_set_fresh:
fixes a :: "'x"
and   xs :: "'a list"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     fs: "fs TYPE('a) TYPE('x)"
shows "a♯(set xs) = a♯xs"
by (simp add: fresh_def pt_list_set_supp[OF pt, OF at, OF fs])

section ‹generalisation of freshness to lists and sets of atoms›
(*================================================================*)

consts
fresh_star :: "'b ⇒ 'a ⇒ bool" ("_ ♯* _" [100,100] 100)

begin
definition fresh_star_set: "xs♯*c ≡ ∀x::'b∈xs. x♯(c::'a)"
end

begin
definition fresh_star_list: "xs♯*c ≡ ∀x::'b∈set xs. x♯(c::'a)"
end

lemmas fresh_star_def = fresh_star_list fresh_star_set

lemma fresh_star_prod_set:
fixes xs::"'a set"
shows "xs♯*(a,b) = (xs♯*a ∧ xs♯*b)"
by (auto simp add: fresh_star_def fresh_prod)

lemma fresh_star_prod_list:
fixes xs::"'a list"
shows "xs♯*(a,b) = (xs♯*a ∧ xs♯*b)"
by (auto simp add: fresh_star_def fresh_prod)

lemmas fresh_star_prod = fresh_star_prod_list fresh_star_prod_set

lemma fresh_star_set_eq: "set xs ♯* c = xs ♯* c"

lemma fresh_star_Un_elim:
"((S ∪ T) ♯* c ⟹ PROP C) ≡ (S ♯* c ⟹ T ♯* c ⟹ PROP C)"
apply rule
apply (erule meta_mp)
apply blast
done

lemma fresh_star_insert_elim:
"(insert x S ♯* c ⟹ PROP C) ≡ (x ♯ c ⟹ S ♯* c ⟹ PROP C)"

lemma fresh_star_empty_elim:
"({} ♯* c ⟹ PROP C) ≡ PROP C"

text ‹Normalization of freshness results; see \ ‹nominal_induct››

lemma fresh_star_unit_elim:
shows "((a::'a set)♯*() ⟹ PROP C) ≡ PROP C"
and "((b::'a list)♯*() ⟹ PROP C) ≡ PROP C"
by (simp_all add: fresh_star_def fresh_def supp_unit)

lemma fresh_star_prod_elim:
shows "((a::'a set)♯*(x,y) ⟹ PROP C) ≡ (a♯*x ⟹ a♯*y ⟹ PROP C)"
and "((b::'a list)♯*(x,y) ⟹ PROP C) ≡ (b♯*x ⟹ b♯*y ⟹ PROP C)"

lemma pt_fresh_star_bij_ineq:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'y set"
and     b :: "'y list"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
shows "(pi∙a)♯*(pi∙x) = a♯*x"
and   "(pi∙b)♯*(pi∙x) = b♯*x"
apply(unfold fresh_star_def)
apply(auto)
apply(drule_tac x="pi∙xa" in bspec)
apply(erule pt_set_bij2[OF ptb, OF at])
apply(simp add: fresh_star_def pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
apply(drule_tac x="(rev pi)∙xa" in bspec)
apply(simp add: pt_set_bij1[OF ptb, OF at])
apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
apply(drule_tac x="pi∙xa" in bspec)
apply(simp add: pt_set_bij1[OF ptb, OF at])
apply(simp add: set_eqvt pt_rev_pi[OF pt_list_inst[OF ptb], OF at])
apply(simp add: pt_fresh_bij_ineq[OF pta, OF ptb, OF at, OF cp])
apply(drule_tac x="(rev pi)∙xa" in bspec)
apply(simp add: pt_set_bij1[OF ptb, OF at] set_eqvt)
apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
done

lemma pt_fresh_star_bij:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x set"
and     b :: "'x list"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙a)♯*(pi∙x) = a♯*x"
and   "(pi∙b)♯*(pi∙x) = b♯*x"
apply(rule pt_fresh_star_bij_ineq(1))
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
apply(rule pt_fresh_star_bij_ineq(2))
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
done

lemma pt_fresh_star_eqvt:
fixes  pi :: "'x prm"
and     x :: "'a"
and     a :: "'x set"
and     b :: "'x list"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙(a♯*x) = (pi∙a)♯*(pi∙x)"
and   "pi∙(b♯*x) = (pi∙b)♯*(pi∙x)"
by (simp_all add: perm_bool pt_fresh_star_bij[OF pt, OF at])

lemma pt_fresh_star_eqvt_ineq:
fixes pi::"'x prm"
and   a::"'y set"
and   b::"'y list"
and   x::"'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
and     dj:  "disjoint TYPE('y) TYPE('x)"
shows "pi∙(a♯*x) = (pi∙a)♯*(pi∙x)"
and   "pi∙(b♯*x) = (pi∙b)♯*(pi∙x)"
by (simp_all add: pt_fresh_star_bij_ineq[OF pta, OF ptb, OF at, OF cp] dj_perm_forget[OF dj] perm_bool)

lemma pt_freshs_freshs:
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE ('x)"
and pi: "set (pi::'x prm) ⊆ Xs × Ys"
and Xs: "Xs ♯* (x::'a)"
and Ys: "Ys ♯* x"
shows "pi∙x = x"
using pi
proof (induct pi)
case Nil
show ?case by (simp add: pt1 [OF pt])
next
case (Cons p pi)
obtain a b where p: "p = (a, b)" by (cases p)
with Cons Xs Ys have "a ♯ x" "b ♯ x"
with Cons p show ?case
by (simp add: pt_fresh_fresh [OF pt at]
pt2 [OF pt, of "[(a, b)]" pi, simplified])
qed

lemma pt_fresh_star_pi:
fixes x::"'a"
and   pi::"'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     a: "((supp x)::'x set)♯* pi"
shows "pi∙x = x"
using a
apply(induct pi)
apply(auto simp add: fresh_star_def fresh_list_cons fresh_prod pt1[OF pt])
apply(subgoal_tac "((a,b)#pi)∙x = ([(a,b)]@pi)∙x")
apply(simp only: pt2[OF pt])
apply(rule pt_fresh_fresh[OF pt at])
apply(blast)
apply(blast)
done

section ‹Infrastructure lemmas for strong rule inductions›
(*==========================================================*)

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::"'a set"
and   As::"'a set"
assumes at: "at TYPE('a)"
and     b: "Xs ⊆ As"
and     c: "finite As"
and     d: "finite ((supp c)::'a set)"
shows "∃(pi::'a prm). (pi∙Xs)♯*c ∧ (pi∙Xs) ∩ As = {} ∧ set pi ⊆ Xs × (pi∙Xs)"
proof -
from b c have "finite Xs" by (simp add: finite_subset)
then show ?thesis using b
proof (induct)
case empty
have "({}::'a set)♯*c" by (simp add: fresh_star_def)
moreover
have "({}::'a set) ∩ As = {}" by simp
moreover
have "set ([]::'a prm) ⊆ {} × {}" by simp
ultimately show ?case by (simp add: empty_eqvt)
next
case (insert x Xs)
then have ih: "∃pi. (pi∙Xs)♯*c ∧ (pi∙Xs) ∩ As = {} ∧ set pi ⊆ Xs × (pi∙Xs)" by simp
then obtain pi where a1: "(pi∙Xs)♯*c" and a2: "(pi∙Xs) ∩ As = {}" and
a4: "set pi ⊆ Xs × (pi∙Xs)" by blast
have b: "x∉Xs" by fact
have d1: "finite As" by fact
have d2: "finite Xs" by fact
have d3: "({x} ∪ Xs) ⊆ As" using insert(4) by simp
from d d1 d2
obtain y::"'a" where fr: "y♯(c,pi∙Xs,As)"
apply(rule_tac at_exists_fresh[OF at, where x="(c,pi∙Xs,As)"])
apply(auto simp add: supp_prod at_supp[OF at] at_fin_set_supp[OF at]
pt_supp_finite_pi[OF pt_set_inst[OF at_pt_inst[OF at]] at])
done
have "({y}∪(pi∙Xs))♯*c" using a1 fr by (simp add: fresh_star_def)
moreover
have "({y}∪(pi∙Xs))∩As = {}" using a2 d1 fr
by (simp add: fresh_prod at_fin_set_fresh[OF at])
moreover
have "pi∙x=x" using a4 b a2 d3
by (rule_tac at_prm_fresh2[OF at]) (auto)
then have "set ((pi∙x,y)#pi) ⊆ ({x} ∪ Xs) × ({y}∪(pi∙Xs))" using a4 by auto
moreover
have "(((pi∙x,y)#pi)∙({x} ∪ Xs)) = {y}∪(pi∙Xs)"
proof -
have eq: "[(pi∙x,y)]∙(pi∙Xs) = (pi∙Xs)"
proof -
have "(pi∙x)♯(pi∙Xs)" using b d2
by (simp add: pt_fresh_bij [OF pt_set_inst [OF at_pt_inst [OF at]], OF at]
at_fin_set_fresh [OF at])
moreover
have "y♯(pi∙Xs)" using fr by simp
ultimately show "[(pi∙x,y)]∙(pi∙Xs) = (pi∙Xs)"
[OF at_pt_inst[OF at]], OF at])
qed
have "(((pi∙x,y)#pi)∙({x}∪Xs)) = ([(pi∙x,y)]∙(pi∙({x}∪Xs)))"
by (simp add: pt2[symmetric, OF pt_set_inst [OF at_pt_inst[OF at]]])
also have "… = {y}∪([(pi∙x,y)]∙(pi∙Xs))"
by (simp only: union_eqvt perm_set_def at_calc[OF at])(auto)
finally show "(((pi∙x,y)#pi)∙({x} ∪ Xs)) = {y}∪(pi∙Xs)" using eq by simp
qed
ultimately
show ?case by (rule_tac x="(pi∙x,y)#pi" in exI) (auto)
qed
qed

lemma at_set_avoiding:
fixes Xs::"'a set"
assumes at: "at TYPE('a)"
and     a: "finite Xs"
and     b: "finite ((supp c)::'a set)"
obtains pi::"'a prm" where "(pi∙Xs)♯*c" and "set pi ⊆ Xs × (pi∙Xs)"
using a b at_set_avoiding_aux[OF at, where Xs="Xs" and As="Xs" and c="c"]
by (blast)

section ‹composition instances›
(* ============================= *)

lemma cp_list_inst:
assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
shows "cp TYPE ('a list) TYPE('x) TYPE('y)"
using c1
apply(auto)
apply(induct_tac x)
apply(auto)
done

lemma cp_set_inst:
assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
shows "cp TYPE ('a set) TYPE('x) TYPE('y)"
using c1
apply(auto)
apply(rule_tac x="pi2∙xc" in exI)
apply(auto)
done

lemma cp_option_inst:
assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
shows "cp TYPE ('a option) TYPE('x) TYPE('y)"
using c1
apply(auto)
apply(case_tac x)
apply(auto)
done

lemma cp_noption_inst:
assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
shows "cp TYPE ('a noption) TYPE('x) TYPE('y)"
using c1
apply(auto)
apply(case_tac x)
apply(auto)
done

lemma cp_unit_inst:
shows "cp TYPE (unit) TYPE('x) TYPE('y)"
done

lemma cp_bool_inst:
shows "cp TYPE (bool) TYPE('x) TYPE('y)"
apply(rule allI)+
apply(induct_tac x)
apply(simp_all)
done

lemma cp_prod_inst:
assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
and     c2: "cp TYPE ('b) TYPE('x) TYPE('y)"
shows "cp TYPE ('a×'b) TYPE('x) TYPE('y)"
using c1 c2
done

lemma cp_fun_inst:
assumes c1: "cp TYPE ('a) TYPE('x) TYPE('y)"
and     c2: "cp TYPE ('b) TYPE('x) TYPE('y)"
and     pt: "pt TYPE ('y) TYPE('x)"
and     at: "at TYPE ('x)"
shows "cp TYPE ('a⇒'b) TYPE('x) TYPE('y)"
using c1 c2
apply(auto simp add: cp_def perm_fun_def fun_eq_iff)
apply(simp add: pt_rev_pi[OF pt_list_inst[OF pt_prod_inst[OF pt, OF pt]], OF at])
done

section ‹Andy's freshness lemma›
(*================================*)

lemma freshness_lemma:
fixes h :: "'x⇒'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     at:  "at TYPE('x)"
and     f1:  "finite ((supp h)::'x set)"
and     a: "∃a::'x. a♯(h,h a)"
shows  "∃fr::'a. ∀a::'x. a♯h ⟶ (h a) = fr"
proof -
have ptb: "pt TYPE('x) TYPE('x)" by (simp add: at_pt_inst[OF at])
have ptc: "pt TYPE('x⇒'a) TYPE('x)" by (simp add: pt_fun_inst[OF ptb, OF pta, OF at])
from a obtain a0 where a1: "a0♯h" and a2: "a0♯(h a0)" by (force simp add: fresh_prod)
show ?thesis
proof
let ?fr = "h (a0::'x)"
show "∀(a::'x). (a♯h ⟶ ((h a) = ?fr))"
proof (intro strip)
fix a
assume a3: "(a::'x)♯h"
show "h (a::'x) = h a0"
proof (cases "a=a0")
case True thus "h (a::'x) = h a0" by simp
next
case False
assume "a≠a0"
hence c1: "a∉((supp a0)::'x set)" by  (simp add: fresh_def[symmetric] at_fresh[OF at])
have c2: "a∉((supp h)::'x set)" using a3 by (simp add: fresh_def)
from c1 c2 have c3: "a∉((supp h)∪((supp a0)::'x set))" by force
have f2: "finite ((supp a0)::'x set)" by (simp add: at_supp[OF at])
from f1 f2 have "((supp (h a0))::'x set)⊆((supp h)∪(supp a0))"
by (simp add: pt_supp_fun_subset[OF ptb, OF pta, OF at])
hence "a∉((supp (h a0))::'x set)" using c3 by force
hence "a♯(h a0)" by (simp add: fresh_def)
with a2 have d1: "[(a0,a)]∙(h a0) = (h a0)" by (rule pt_fresh_fresh[OF pta, OF at])
from a1 a3 have d2: "[(a0,a)]∙h = h" by (rule pt_fresh_fresh[OF ptc, OF at])
from d1 have "h a0 = [(a0,a)]∙(h a0)" by simp
also have "…= ([(a0,a)]∙h)([(a0,a)]∙a0)" by (simp add: pt_fun_app_eq[OF ptb, OF at])
also have "… = h ([(a0,a)]∙a0)" using d2 by simp
also have "… = h a" by (simp add: at_calc[OF at])
finally show "h a = h a0" by simp
qed
qed
qed
qed

lemma freshness_lemma_unique:
fixes h :: "'x⇒'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f1: "finite ((supp h)::'x set)"
and     a: "∃(a::'x). a♯(h,h a)"
shows  "∃!(fr::'a). ∀(a::'x). a♯h ⟶ (h a) = fr"
proof (rule ex_ex1I)
from pt at f1 a show "∃fr::'a. ∀a::'x. a♯h ⟶ h a = fr" by (simp add: freshness_lemma)
next
fix fr1 fr2
assume b1: "∀a::'x. a♯h ⟶ h a = fr1"
assume b2: "∀a::'x. a♯h ⟶ h a = fr2"
from a obtain a where "(a::'x)♯h" by (force simp add: fresh_prod)
with b1 b2 have "h a = fr1 ∧ h a = fr2" by force
thus "fr1 = fr2" by force
qed

― "packaging the freshness lemma into a function"
definition fresh_fun :: "('x⇒'a)⇒'a" where
"fresh_fun (h) ≡ THE fr. (∀(a::'x). a♯h ⟶ (h a) = fr)"

lemma fresh_fun_app:
fixes h :: "'x⇒'a"
and   a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f1: "finite ((supp h)::'x set)"
and     a: "∃(a::'x). a♯(h,h a)"
and     b: "a♯h"
shows "(fresh_fun h) = (h a)"
proof (unfold fresh_fun_def, rule the_equality)
show "∀(a'::'x). a'♯h ⟶ h a' = h a"
proof (intro strip)
fix a'::"'x"
assume c: "a'♯h"
from pt at f1 a have "∃(fr::'a). ∀(a::'x). a♯h ⟶ (h a) = fr" by (rule freshness_lemma)
with b c show "h a' = h a" by force
qed
next
fix fr::"'a"
assume "∀a. a♯h ⟶ h a = fr"
with b show "fr = h a" by force
qed

lemma fresh_fun_app':
fixes h :: "'x⇒'a"
and   a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f1: "finite ((supp h)::'x set)"
and     a: "a♯h" "a♯h a"
shows "(fresh_fun h) = (h a)"
apply(rule fresh_fun_app[OF pt, OF at, OF f1])
apply(auto simp add: fresh_prod intro: a)
done

lemma fresh_fun_equiv_ineq:
fixes h :: "'y⇒'a"
and   pi:: "'x prm"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     ptb':"pt TYPE('a) TYPE('y)"
and     at:  "at TYPE('x)"
and     at': "at TYPE('y)"
and     cpa: "cp TYPE('a) TYPE('x) TYPE('y)"
and     cpb: "cp TYPE('y) TYPE('x) TYPE('y)"
and     f1: "finite ((supp h)::'y set)"
and     a1: "∃(a::'y). a♯(h,h a)"
shows "pi∙(fresh_fun h) = fresh_fun(pi∙h)" (is "?LHS = ?RHS")
proof -
have ptd: "pt TYPE('y) TYPE('y)" by (simp add: at_pt_inst[OF at'])
have ptc: "pt TYPE('y⇒'a) TYPE('x)" by (simp add: pt_fun_inst[OF ptb, OF pta, OF at])
have cpc: "cp TYPE('y⇒'a) TYPE ('x) TYPE ('y)" by (rule cp_fun_inst[OF cpb cpa ptb at])
have f2: "finite ((supp (pi∙h))::'y set)"
proof -
from f1 have "finite (pi∙((supp h)::'y set))"
by (simp add: pt_set_finite_ineq[OF ptb, OF at])
thus ?thesis
by (simp add: pt_perm_supp_ineq[OF ptc, OF ptb, OF at, OF cpc])
qed
from a1 obtain a' where c0: "a'♯(h,h a')" by force
hence c1: "a'♯h" and c2: "a'♯(h a')" by (simp_all add: fresh_prod)
have c3: "(pi∙a')♯(pi∙h)" using c1
by (simp add: pt_fresh_bij_ineq[OF ptc, OF ptb, OF at, OF cpc])
have c4: "(pi∙a')♯(pi∙h) (pi∙a')"
proof -
from c2 have "(pi∙a')♯(pi∙(h a'))"
by (simp add: pt_fresh_bij_ineq[OF pta, OF ptb, OF at,OF cpa])
thus ?thesis by (simp add: pt_fun_app_eq[OF ptb, OF at])
qed
have a2: "∃(a::'y). a♯(pi∙h,(pi∙h) a)" using c3 c4 by (force simp add: fresh_prod)
have d1: "?LHS = pi∙(h a')" using c1 a1 by (simp add: fresh_fun_app[OF ptb', OF at', OF f1])
have d2: "?RHS = (pi∙h) (pi∙a')" using c3 a2
by (simp add: fresh_fun_app[OF ptb', OF at', OF f2])
show ?thesis using d1 d2 by (simp add: pt_fun_app_eq[OF ptb, OF at])
qed

lemma fresh_fun_equiv:
fixes h :: "'x⇒'a"
and   pi:: "'x prm"
assumes pta: "pt TYPE('a) TYPE('x)"
and     at:  "at TYPE('x)"
and     f1:  "finite ((supp h)::'x set)"
and     a1: "∃(a::'x). a♯(h,h a)"
shows "pi∙(fresh_fun h) = fresh_fun(pi∙h)" (is "?LHS = ?RHS")
proof -
have ptb: "pt TYPE('x) TYPE('x)" by (simp add: at_pt_inst[OF at])
have ptc: "pt TYPE('x⇒'a) TYPE('x)" by (simp add: pt_fun_inst[OF ptb, OF pta, OF at])
have f2: "finite ((supp (pi∙h))::'x set)"
proof -
from f1 have "finite (pi∙((supp h)::'x set))" by (simp add: pt_set_finite_ineq[OF ptb, OF at])
thus ?thesis by (simp add: pt_perm_supp[OF ptc, OF at])
qed
from a1 obtain a' where c0: "a'♯(h,h a')" by force
hence c1: "a'♯h" and c2: "a'♯(h a')" by (simp_all add: fresh_prod)
have c3: "(pi∙a')♯(pi∙h)" using c1 by (simp add: pt_fresh_bij[OF ptc, OF at])
have c4: "(pi∙a')♯(pi∙h) (pi∙a')"
proof -
from c2 have "(pi∙a')♯(pi∙(h a'))" by (simp add: pt_fresh_bij[OF pta, OF at])
thus ?thesis by (simp add: pt_fun_app_eq[OF ptb, OF at])
qed
have a2: "∃(a::'x). a♯(pi∙h,(pi∙h) a)" using c3 c4 by (force simp add: fresh_prod)
have d1: "?LHS = pi∙(h a')" using c1 a1 by (simp add: fresh_fun_app[OF pta, OF at, OF f1])
have d2: "?RHS = (pi∙h) (pi∙a')" using c3 a2 by (simp add: fresh_fun_app[OF pta, OF at, OF f2])
show ?thesis using d1 d2 by (simp add: pt_fun_app_eq[OF ptb, OF at])
qed

lemma fresh_fun_supports:
fixes h :: "'x⇒'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f1: "finite ((supp h)::'x set)"
and     a: "∃(a::'x). a♯(h,h a)"
shows "((supp h)::'x set) supports (fresh_fun h)"
apply(auto)
apply(simp add: fresh_fun_equiv[OF pt, OF at, OF f1, OF a])
apply(simp add: pt_fresh_fresh[OF pt_fun_inst[OF at_pt_inst[OF at], OF pt], OF at, OF at])
done

section ‹Abstraction function›
(*==============================*)

lemma pt_abs_fun_inst:
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pt TYPE('x⇒('a noption)) TYPE('x)"
by (rule pt_fun_inst[OF at_pt_inst[OF at],OF pt_noption_inst[OF pt],OF at])

definition abs_fun :: "'x⇒'a⇒('x⇒('a noption))" ("[_]._" [100,100] 100) where
"[a].x ≡ (λb. (if b=a then nSome(x) else (if b♯x then nSome([(a,b)]∙x) else nNone)))"

(* FIXME: should be called perm_if and placed close to the definition of permutations on bools *)
lemma abs_fun_if:
fixes pi :: "'x prm"
and   x  :: "'a"
and   y  :: "'a"
and   c  :: "bool"
shows "pi∙(if c then x else y) = (if c then (pi∙x) else (pi∙y))"
by force

lemma abs_fun_pi_ineq:
fixes a  :: "'y"
and   x  :: "'a"
and   pi :: "'x prm"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
shows "pi∙([a].x) = [(pi∙a)].(pi∙x)"
apply(simp only: fun_eq_iff)
apply(rule allI)
apply(subgoal_tac "(((rev pi)∙(xa::'y)) = (a::'y)) = (xa = pi∙a)")(*A*)
apply(subgoal_tac "(((rev pi)∙xa)♯x) = (xa♯(pi∙x))")(*B*)
apply(subgoal_tac "pi∙([(a,(rev pi)∙xa)]∙x) = [(pi∙a,xa)]∙(pi∙x)")(*C*)
apply(simp)
(*C*)
apply(simp add: pt_pi_rev[OF ptb, OF at])
(*B*)
apply(simp add: pt_fresh_left_ineq[OF pta, OF ptb, OF at, OF cp])
(*A*)
apply(rule iffI)
apply(rule pt_bij2[OF ptb, OF at, THEN sym])
apply(simp)
apply(rule pt_bij2[OF ptb, OF at])
apply(simp)
done

lemma abs_fun_pi:
fixes a  :: "'x"
and   x  :: "'a"
and   pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi∙([a].x) = [(pi∙a)].(pi∙x)"
apply(rule abs_fun_pi_ineq)
apply(rule pt)
apply(rule at_pt_inst)
apply(rule at)+
apply(rule cp_pt_inst)
apply(rule pt)
apply(rule at)
done

lemma abs_fun_eq1:
fixes x  :: "'a"
and   y  :: "'a"
and   a  :: "'x"
shows "([a].x = [a].y) = (x = y)"
apply(drule_tac x="a" in spec)
apply(simp)
done

lemma abs_fun_eq2:
fixes x  :: "'a"
and   y  :: "'a"
and   a  :: "'x"
and   b  :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
and a1: "a≠b"
and a2: "[a].x = [b].y"
shows "x=[(a,b)]∙y ∧ a♯y"
proof -
from a2 have "∀c::'x. ([a].x) c = ([b].y) c" by (force simp add: fun_eq_iff)
hence "([a].x) a = ([b].y) a" by simp
hence a3: "nSome(x) = ([b].y) a" by (simp add: abs_fun_def)
show "x=[(a,b)]∙y ∧ a♯y"
proof (cases "a♯y")
assume a4: "a♯y"
hence "x=[(b,a)]∙y" using a3 a1 by (simp add: abs_fun_def)
moreover
have "[(a,b)]∙y = [(b,a)]∙y" by (rule pt3[OF pt], rule at_ds5[OF at])
ultimately show ?thesis using a4 by simp
next
assume "¬a♯y"
hence "nSome(x) = nNone" using a1 a3 by (simp add: abs_fun_def)
hence False by simp
thus ?thesis by simp
qed
qed

lemma abs_fun_eq3:
fixes x  :: "'a"
and   y  :: "'a"
and   a   :: "'x"
and   b   :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
and a1: "a≠b"
and a2: "x=[(a,b)]∙y"
and a3: "a♯y"
shows "[a].x =[b].y"
proof -
show ?thesis
proof (simp only: abs_fun_def fun_eq_iff, intro strip)
fix c::"'x"
let ?LHS = "if c=a then nSome(x) else if c♯x then nSome([(a,c)]∙x) else nNone"
and ?RHS = "if c=b then nSome(y) else if c♯y then nSome([(b,c)]∙y) else nNone"
show "?LHS=?RHS"
proof -
have "(c=a) ∨ (c=b) ∨ (c≠a ∧ c≠b)" by blast
moreover  ―"case c=a"
{ have "nSome(x) = nSome([(a,b)]∙y)" using a2 by simp
also have "… = nSome([(b,a)]∙y)" by (simp, rule pt3[OF pt], rule at_ds5[OF at])
finally have "nSome(x) = nSome([(b,a)]∙y)" by simp
moreover
assume "c=a"
ultimately have "?LHS=?RHS" using a1 a3 by simp
}
moreover  ― "case c=b"
{ have a4: "y=[(a,b)]∙x" using a2 by (simp only: pt_swap_bij[OF pt, OF at])
hence "a♯([(a,b)]∙x)" using a3 by simp
hence "b♯x" by (simp add: at_calc[OF at] pt_fresh_left[OF pt, OF at])
moreover
assume "c=b"
ultimately have "?LHS=?RHS" using a1 a4 by simp
}
moreover  ― "case c≠a ∧ c≠b"
{ assume a5: "c≠a ∧ c≠b"
moreover
have "c♯x = c♯y" using a2 a5 by (force simp add: at_calc[OF at] pt_fresh_left[OF pt, OF at])
moreover
have "c♯y ⟶ [(a,c)]∙x = [(b,c)]∙y"
proof (intro strip)
assume a6: "c♯y"
have "[(a,c),(b,c),(a,c)] ≜ [(a,b)]" using a1 a5 by (force intro: at_ds3[OF at])
hence "[(a,c)]∙([(b,c)]∙([(a,c)]∙y)) = [(a,b)]∙y"
by (simp add: pt2[OF pt, symmetric] pt3[OF pt])
hence "[(a,c)]∙([(b,c)]∙y) = [(a,b)]∙y" using a3 a6
by (simp add: pt_fresh_fresh[OF pt, OF at])
hence "[(a,c)]∙([(b,c)]∙y) = x" using a2 by simp
hence "[(b,c)]∙y = [(a,c)]∙x" by (drule_tac pt_bij1[OF pt, OF at], simp)
thus "[(a,c)]∙x = [(b,c)]∙y" by simp
qed
ultimately have "?LHS=?RHS" by simp
}
ultimately show "?LHS = ?RHS" by blast
qed
qed
qed

(* alpha equivalence *)
lemma abs_fun_eq:
fixes x  :: "'a"
and   y  :: "'a"
and   a  :: "'x"
and   b  :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
shows "([a].x = [b].y) = ((a=b ∧ x=y)∨(a≠b ∧ x=[(a,b)]∙y ∧ a♯y))"
proof (rule iffI)
assume b: "[a].x = [b].y"
show "(a=b ∧ x=y)∨(a≠b ∧ x=[(a,b)]∙y ∧ a♯y)"
proof (cases "a=b")
case True with b show ?thesis by (simp add: abs_fun_eq1)
next
case False with b show ?thesis by (simp add: abs_fun_eq2[OF pt, OF at])
qed
next
assume "(a=b ∧ x=y)∨(a≠b ∧ x=[(a,b)]∙y ∧ a♯y)"
thus "[a].x = [b].y"
proof
assume "a=b ∧ x=y" thus ?thesis by simp
next
assume "a≠b ∧ x=[(a,b)]∙y ∧ a♯y"
thus ?thesis by (simp add: abs_fun_eq3[OF pt, OF at])
qed
qed

(* symmetric version of alpha-equivalence *)
lemma abs_fun_eq':
fixes x  :: "'a"
and   y  :: "'a"
and   a  :: "'x"
and   b  :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
shows "([a].x = [b].y) = ((a=b ∧ x=y)∨(a≠b ∧ [(b,a)]∙x=y ∧ b♯x))"
by (auto simp add: abs_fun_eq[OF pt, OF at] pt_swap_bij'[OF pt, OF at]
pt_fresh_left[OF pt, OF at]
at_calc[OF at])

(* alpha_equivalence with a fresh name *)
lemma abs_fun_fresh:
fixes x :: "'a"
and   y :: "'a"
and   c :: "'x"
and   a :: "'x"
and   b :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
and fr: "c≠a" "c≠b" "c♯x" "c♯y"
shows "([a].x = [b].y) = ([(a,c)]∙x = [(b,c)]∙y)"
proof (rule iffI)
assume eq0: "[a].x = [b].y"
show "[(a,c)]∙x = [(b,c)]∙y"
proof (cases "a=b")
case True then show ?thesis using eq0 by (simp add: pt_bij[OF pt, OF at] abs_fun_eq[OF pt, OF at])
next
case False
have ineq: "a≠b" by fact
with eq0 have eq: "x=[(a,b)]∙y" and fr': "a♯y" by (simp_all add: abs_fun_eq[OF pt, OF at])
from eq have "[(a,c)]∙x = [(a,c)]∙[(a,b)]∙y" by (simp add: pt_bij[OF pt, OF at])
also have "… = ([(a,c)]∙[(a,b)])∙([(a,c)]∙y)" by (rule pt_perm_compose[OF pt, OF at])
also have "… = [(c,b)]∙y" using ineq fr fr'
by (simp add: pt_fresh_fresh[OF pt, OF at] at_calc[OF at])
also have "… = [(b,c)]∙y" by (rule pt3[OF pt], rule at_ds5[OF at])
finally show ?thesis by simp
qed
next
assume eq: "[(a,c)]∙x = [(b,c)]∙y"
thus "[a].x = [b].y"
proof (cases "a=b")
case True then show ?thesis using eq by (simp add: pt_bij[OF pt, OF at] abs_fun_eq[OF pt, OF at])
next
case False
have ineq: "a≠b" by fact
from fr have "([(a,c)]∙c)♯([(a,c)]∙x)" by (simp add: pt_fresh_bij[OF pt, OF at])
hence "a♯([(b,c)]∙y)" using eq fr by (simp add: at_calc[OF at])
hence fr0: "a♯y" using ineq fr by (simp add: pt_fresh_left[OF pt, OF at] at_calc[OF at])
from eq have "x = (rev [(a,c)])∙([(b,c)]∙y)" by (rule pt_bij1[OF pt, OF at])
also have "… = [(a,c)]∙([(b,c)]∙y)" by simp
also have "… = ([(a,c)]∙[(b,c)])∙([(a,c)]∙y)" by (rule pt_perm_compose[OF pt, OF at])
also have "… = [(b,a)]∙y" using ineq fr fr0
by (simp add: pt_fresh_fresh[OF pt, OF at] at_calc[OF at])
also have "… = [(a,b)]∙y" by (rule pt3[OF pt], rule at_ds5[OF at])
finally show ?thesis using ineq fr0 by (simp add: abs_fun_eq[OF pt, OF at])
qed
qed

lemma abs_fun_fresh':
fixes x :: "'a"
and   y :: "'a"
and   c :: "'x"
and   a :: "'x"
and   b :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and at: "at TYPE('x)"
and as: "[a].x = [b].y"
and fr: "c≠a" "c≠b" "c♯x" "c♯y"
shows "x = [(a,c)]∙[(b,c)]∙y"
using as fr
apply(drule_tac sym)
apply(simp add: abs_fun_fresh[OF pt, OF at] pt_swap_bij[OF pt, OF at])
done

lemma abs_fun_supp_approx:
fixes x :: "'a"
and   a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "((supp ([a].x))::'x set) ⊆ (supp (x,a))"
proof
fix c
assume "c∈((supp ([a].x))::'x set)"
hence "infinite {b. [(c,b)]∙([a].x) ≠ [a].x}" by (simp add: supp_def)
hence "infinite {b. [([(c,b)]∙a)].([(c,b)]∙x) ≠ [a].x}" by (simp add: abs_fun_pi[OF pt, OF at])
moreover
have "{b. [([(c,b)]∙a)].([(c,b)]∙x) ≠ [a].x} ⊆ {b. ([(c,b)]∙x,[(c,b)]∙a) ≠ (x, a)}" by force
ultimately have "infinite {b. ([(c,b)]∙x,[(c,b)]∙a) ≠ (x, a)}" by (simp add: infinite_super)
thus "c∈(supp (x,a))" by (simp add: supp_def)
qed

lemma abs_fun_finite_supp:
fixes x :: "'a"
and   a :: "'x"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f:  "finite ((supp x)::'x set)"
shows "finite ((supp ([a].x))::'x set)"
proof -
from f have "finite ((supp (x,a))::'x set)" by (simp add: supp_prod at_supp[OF at])
moreover
have "((supp ([a].x))::'x set) ⊆ (supp (x,a))" by (rule abs_fun_supp_approx[OF pt, OF at])
ultimately show ?thesis by (simp add: finite_subset)
qed

lemma fresh_abs_funI1:
fixes  x :: "'a"
and    a :: "'x"
and    b :: "'x"
assumes pt:  "pt TYPE('a) TYPE('x)"
and     at:   "at TYPE('x)"
and f:  "finite ((supp x)::'x set)"
and a1: "b♯x"
and a2: "a≠b"
shows "b♯([a].x)"
proof -
have "∃c::'x. c♯(b,a,x,[a].x)"
proof (rule at_exists_fresh'[OF at], auto simp add: supp_prod at_supp[OF at] f)
show "finite ((supp ([a].x))::'x set)" using f
by (simp add: abs_fun_finite_supp[OF pt, OF at])
qed
then obtain c where fr1: "c≠b"
and   fr2: "c≠a"
and   fr3: "c♯x"
and   fr4: "c♯([a].x)"
by (force simp add: fresh_prod at_fresh[OF at])
have e: "[(c,b)]∙([a].x) = [a].([(c,b)]∙x)" using a2 fr1 fr2
by (force simp add: abs_fun_pi[OF pt, OF at] at_calc[OF at])
from fr4 have "([(c,b)]∙c)♯ ([(c,b)]∙([a].x))"
by (simp add: pt_fresh_bij[OF pt_abs_fun_inst[OF pt, OF at], OF at])
hence "b♯([a].([(c,b)]∙x))" using fr1 fr2 e
thus ?thesis using a1 fr3
by (simp add: pt_fresh_fresh[OF pt, OF at])
qed

lemma fresh_abs_funE:
fixes a :: "'x"
and   b :: "'x"
and   x :: "'a"
assumes pt:  "pt TYPE('a) TYPE('x)"
and     at:  "at TYPE('x)"
and     f:  "finite ((supp x)::'x set)"
and     a1: "b♯([a].x)"
and     a2: "b≠a"
shows "b♯x"
proof -
have "∃c::'x. c♯(b,a,x,[a].x)"
proof (rule at_exists_fresh'[OF at], auto simp add: supp_prod at_supp[OF at] f)
show "finite ((supp ([a].x))::'x set)" using f
by (simp add: abs_fun_finite_supp[OF pt, OF at])
qed
then obtain c where fr1: "b≠c"
and   fr2: "c≠a"
and   fr3: "c♯x"
and   fr4: "c♯([a].x)" by (force simp add: fresh_prod at_fresh[OF at])
have "[a].x = [(b,c)]∙([a].x)" using a1 fr4
by (simp add: pt_fresh_fresh[OF pt_abs_fun_inst[OF pt, OF at], OF at])
hence "[a].x = [a].([(b,c)]∙x)" using fr2 a2
by (force simp add: abs_fun_pi[OF pt, OF at] at_calc[OF at])
hence b: "([(b,c)]∙x) = x" by (simp add: abs_fun_eq1)
from fr3 have "([(b,c)]∙c)♯([(b,c)]∙x)"
by (simp add: pt_fresh_bij[OF pt, OF at])
thus ?thesis using b fr1 by (simp add: at_calc[OF at])
qed

lemma fresh_abs_funI2:
fixes a :: "'x"
and   x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f: "finite ((supp x)::'x set)"
shows "a♯([a].x)"
proof -
have "∃c::'x. c♯(a,x)"
by  (rule at_exists_fresh'[OF at], auto simp add: supp_prod at_supp[OF at] f)
then obtain c where fr1: "a≠c" and fr1_sym: "c≠a"
and   fr2: "c♯x" by (force simp add: fresh_prod at_fresh[OF at])
have "c♯([a].x)" using f fr1 fr2 by (simp add: fresh_abs_funI1[OF pt, OF at])
hence "([(c,a)]∙c)♯([(c,a)]∙([a].x))" using fr1
by (simp only: pt_fresh_bij[OF pt_abs_fun_inst[OF pt, OF at], OF at])
hence a: "a♯([c].([(c,a)]∙x))" using fr1_sym
by (simp add: abs_fun_pi[OF pt, OF at] at_calc[OF at])
have "[c].([(c,a)]∙x) = ([a].x)" using fr1_sym fr2
by (simp add: abs_fun_eq[OF pt, OF at])
thus ?thesis using a by simp
qed

lemma fresh_abs_fun_iff:
fixes a :: "'x"
and   b :: "'x"
and   x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f: "finite ((supp x)::'x set)"
shows "(b♯([a].x)) = (b=a ∨ b♯x)"
by (auto  dest: fresh_abs_funE[OF pt, OF at,OF f]
intro: fresh_abs_funI1[OF pt, OF at,OF f]
fresh_abs_funI2[OF pt, OF at,OF f])

lemma abs_fun_supp:
fixes a :: "'x"
and   x :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
and     f: "finite ((supp x)::'x set)"
shows "supp ([a].x) = (supp x)-{a}"
by (force simp add: supp_fresh_iff fresh_abs_fun_iff[OF pt, OF at, OF f])

(* maybe needs to be better stated as supp intersection supp *)
lemma abs_fun_supp_ineq:
fixes a :: "'y"
and   x :: "'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
and     dj:  "disjoint TYPE('y) TYPE('x)"
shows "((supp ([a].x))::'x set) = (supp x)"
apply(auto simp add: abs_fun_pi_ineq[OF pta, OF ptb, OF at, OF cp])
done

lemma fresh_abs_fun_iff_ineq:
fixes a :: "'y"
and   b :: "'x"
and   x :: "'a"
assumes pta: "pt TYPE('a) TYPE('x)"
and     ptb: "pt TYPE('y) TYPE('x)"
and     at:  "at TYPE('x)"
and     cp:  "cp TYPE('a) TYPE('x) TYPE('y)"
and     dj:  "disjoint TYPE('y) TYPE('x)"
shows "b♯([a].x) = b♯x"
by (simp add: fresh_def abs_fun_supp_ineq[OF pta, OF ptb, OF at, OF cp, OF dj])

section ‹abstraction type for the parsing in nominal datatype›
(*==============================================================*)

inductive_set ABS_set :: "('x⇒('a noption)) set"
where
ABS_in: "(abs_fun a x)∈ABS_set"

definition "ABS = ABS_set"

typedef ('x, 'a) ABS ("«_»_" [1000,1000] 1000) =
"ABS::('x⇒('a noption)) set"
morphisms Rep_ABS Abs_ABS
unfolding ABS_def
proof
fix x::"'a" and a::"'x"
show "(abs_fun a x)∈ ABS_set" by (rule ABS_in)
qed

section ‹lemmas for deciding permutation equations›
(*===================================================*)

lemma perm_aux_fold:
shows "perm_aux pi x = pi∙x" by (simp only: perm_aux_def)

lemma pt_perm_compose_aux:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   x  :: "'a"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "pi2∙(pi1∙x) = perm_aux (pi2∙pi1) (pi2∙x)"
proof -
have "(pi2@pi1) ≜ ((pi2∙pi1)@pi2)" by (rule at_ds8[OF at])
hence "(pi2@pi1)∙x = ((pi2∙pi1)@pi2)∙x" by (rule pt3[OF pt])
thus ?thesis by (simp add: pt2[OF pt] perm_aux_def)
qed

lemma cp1_aux:
fixes pi1::"'x prm"
and   pi2::"'y prm"
and   x  ::"'a"
assumes cp: "cp TYPE ('a) TYPE('x) TYPE('y)"
shows "pi1∙(pi2∙x) = perm_aux (pi1∙pi2) (pi1∙x)"
using cp by (simp add: cp_def perm_aux_def)

lemma perm_eq_app:
fixes f  :: "'a⇒'b"
and   x  :: "'a"
and   pi :: "'x prm"
assumes pt: "pt TYPE('a) TYPE('x)"
and     at: "at TYPE('x)"
shows "(pi∙(f x)=y) = ((pi∙f)(pi∙x)=y)"
by (simp add: pt_fun_app_eq[OF pt, OF at])

lemma perm_eq_lam:
fixes f  :: "'a⇒'b"
and   x  :: "'a"
and   pi :: "'x prm"
shows "((pi∙(λx. f x))=y) = ((λx. (pi∙(f ((rev pi)∙x))))=y)"

section ‹test›
lemma at_prm_eq_compose:
fixes pi1 :: "'x prm"
and   pi2 :: "'x prm"
and   pi3 :: "'x prm"
assumes at: "at TYPE('x)"
and     a: "pi1 ≜ pi2"
shows "(pi3∙pi1) ≜ (pi3∙pi2)"
proof -
have pt: "pt TYPE('x) TYPE('x)" by (rule at_pt_inst[OF at])
have pt_prm: "pt TYPE('x prm) TYPE('x)"
by (rule pt_list_inst[OF pt_prod_inst[OF pt, OF pt]])
from a show ?thesis
apply -
apply(rule_tac pi="rev pi3" in pt_bij4[OF pt, OF at])
apply(rule trans)
apply(rule pt_perm_compose[OF pt, OF at])
apply(simp add: pt_rev_pi[OF pt_prm, OF at])
apply(rule sym)
apply(rule trans)
apply(rule pt_perm_compose[OF pt, OF at])
apply(simp add: pt_rev_pi[OF pt_prm, OF at])
done
qed

(************************)
(* Various eqvt-lemmas  *)

lemma Zero_nat_eqvt:
shows "pi∙(0::nat) = 0"

lemma One_nat_eqvt:
shows "pi∙(1::nat) = 1"

lemma Suc_eqvt:
shows "pi∙(Suc x) = Suc (pi∙x)"

lemma numeral_nat_eqvt:
shows "pi∙((numeral n)::nat) = numeral n"

lemma max_nat_eqvt:
fixes x::"nat"
shows "pi∙(max x y) = max (pi∙x) (pi∙y)"

lemma min_nat_eqvt:
fixes x::"nat"
shows "pi∙(min x y) = min (pi∙x) (pi∙y)"

lemma plus_nat_eqvt:
fixes x::"nat"
shows "pi∙(x + y) = (pi∙x) + (pi∙y)"

lemma minus_nat_eqvt:
fixes x::"nat"
shows "pi∙(x - y) = (pi∙x) - (pi∙y)"

lemma mult_nat_eqvt:
fixes x::"nat"
shows "pi∙(x * y) = (pi∙x) * (pi∙y)"

lemma div_nat_eqvt:
fixes x::"nat"
shows "pi∙(x div y) = (pi∙x) div (pi∙y)"

lemma Zero_int_eqvt:
shows "pi∙(0::int) = 0"

lemma One_int_eqvt:
shows "pi∙(1::int) = 1"

lemma numeral_int_eqvt:
shows "pi∙((numeral n)::int) = numeral n"

lemma neg_numeral_int_eqvt:
shows "pi∙((- numeral n)::int) = - numeral n"

lemma max_int_eqvt:
fixes x::"int"
shows "pi∙(max (x::int) y) = max (pi∙x) (pi∙y)"

lemma min_int_eqvt:
fixes x::"int"
shows "pi∙(min x y) = min (pi∙x) (pi∙y)"

lemma plus_int_eqvt:
fixes x::"int"
shows "pi∙(x + y) = (pi∙x) + (pi∙y)"

lemma minus_int_eqvt:
fixes x::"int"
shows "pi∙(x - y) = (pi∙x) - (pi∙y)"

lemma mult_int_eqvt:
fixes x::"int"
shows "pi∙(x * y) = (pi∙x) * (pi∙y)"

lemma div_int_eqvt:
fixes x::"int"
shows "pi∙(x div y) = (pi∙x) div (pi∙y)"

(*******************************************************)
(* Setup of the theorem attributes eqvt and eqvt_force *)
ML_file "nominal_thmdecls.ML"
setup "NominalThmDecls.setup"

lemmas [eqvt] =
(* connectives *)
if_eqvt imp_eqvt disj_eqvt conj_eqvt neg_eqvt
true_eqvt false_eqvt
imp_eqvt [folded HOL.induct_implies_def]

(* datatypes *)
perm_unit.simps
perm_list.simps append_eqvt
perm_prod.simps
fst_eqvt snd_eqvt
perm_option.simps

(* nats *)
Suc_eqvt Zero_nat_eqvt One_nat_eqvt min_nat_eqvt max_nat_eqvt
plus_nat_eqvt minus_nat_eqvt mult_nat_eqvt div_nat_eqvt

(* ints *)
Zero_int_eqvt One_int_eqvt min_int_eqvt max_int_eqvt
plus_int_eqvt minus_int_eqvt mult_int_eqvt div_int_eqvt

(* sets *)
union_eqvt empty_eqvt insert_eqvt set_eqvt

(* the lemmas numeral_nat_eqvt numeral_int_eqvt do not conform with the *)
(* usual form of an eqvt-lemma, but they are needed for analysing       *)
(* permutations on nats and ints *)
lemmas [eqvt_force] = numeral_nat_eqvt numeral_int_eqvt neg_numeral_int_eqvt

(***************************************)
(* setup for the individial atom-kinds *)
(* and nominal datatypes               *)
ML_file "nominal_atoms.ML"

(************************************************************)
(* various tactics for analysing permutations, supports etc *)
ML_file "nominal_permeq.ML"

method_setup perm_simp =
‹NominalPermeq.perm_simp_meth›
‹simp rules and simprocs for analysing permutations›

method_setup perm_simp_debug =
‹NominalPermeq.perm_simp_meth_debug›
‹simp rules and simprocs for analysing permutations including debugging facilities›

method_setup perm_extend_simp =
‹NominalPermeq.perm_extend_simp_meth›
‹tactic for deciding equalities involving permutations›

method_setup perm_extend_simp_debug =
‹NominalPermeq.perm_extend_simp_meth_debug›
‹tactic for deciding equalities involving permutations including debugging facilities›

method_setup supports_simp =
‹NominalPermeq.supports_meth›
‹tactic for deciding whether something supports something else›

method_setup supports_simp_debug =
‹NominalPermeq.supports_meth_debug›
‹tactic for deciding whether something supports something else including debugging facilities›

method_setup finite_guess =
‹NominalPermeq.finite_guess_meth›
‹tactic for deciding whether something has finite support›

method_setup finite_guess_debug =
‹NominalPermeq.finite_guess_meth_debug›
‹tactic for deciding whether something has finite support including debugging facilities›

method_setup fresh_guess =
‹NominalPermeq.fresh_guess_meth›
‹tactic for deciding whether an atom is fresh for something›

method_setup fresh_guess_debug =
‹NominalPermeq.fresh_guess_meth_debug›
‹tactic for deciding whether an atom is fresh for something including debugging facilities›

(*****************************************************************)
(* tactics for generating fresh names and simplifying fresh_funs *)
ML_file "nominal_fresh_fun.ML"

method_setup generate_fresh = ‹
Args.type_name {proper = true, strict = true} >>
(fn s => fn ctxt => SIMPLE_METHOD (generate_fresh_tac ctxt s))
› "generate a name fresh for all the variables in the goal"

method_setup fresh_fun_simp = ‹
Scan.lift (Args.parens (Args.\$\$\$ "no_asm") >> K true || Scan.succeed false) >>
(fn b => fn ctxt => SIMPLE_METHOD' (fresh_fun_tac ctxt b))
› "delete one inner occurrence of fresh_fun"

(************************************************)
(* main file for constructing nominal datatypes *)
lemma allE_Nil: assumes "∀x. P x" obtains "P []"
using assms ..

ML_file "nominal_datatype.ML"

(******************************************************)
(* primitive recursive functions on nominal datatypes *)
ML_file "nominal_primrec.ML"

(****************************************************)
(* inductive definition involving nominal datatypes *)
ML_file "nominal_inductive.ML"
ML_file "nominal_inductive2.ML"

(*****************************************)
(* setup for induction principles method *)
ML_file "nominal_induct.ML"
method_setup nominal_induct =
‹NominalInduct.nominal_induct_method›
‹nominal induction›

end
```