theory Abs

imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "../Quotient" "../Quotient_Product"

begin

(* the next three lemmas that should be in Nominal \<dots>\<dots>must be cleaned *)

lemma ball_image: 

  shows "(\<forall>x \<in> p \<bullet> S. P x) = (\<forall>x \<in> S. P (p \<bullet> x))"

apply(auto)

apply(drule_tac x="p \<bullet> x" in bspec)

apply(simp add: mem_permute_iff)

apply(simp)

apply(drule_tac x="(- p) \<bullet> x" in bspec)

apply(rule_tac p1="p" in mem_permute_iff[THEN iffD1])

apply(simp)

apply(simp)

done

lemma fresh_star_plus:

  fixes p q::perm

  shows "\<lbrakk>a \<sharp>* p;  a \<sharp>* q\<rbrakk> \<Longrightarrow> a \<sharp>* (p + q)"

  unfolding fresh_star_def

  by (simp add: fresh_plus_perm)

lemma fresh_star_permute_iff:

  shows "(p \<bullet> a) \<sharp>* (p \<bullet> x) \<longleftrightarrow> a \<sharp>* x"

apply(simp add: fresh_star_def)

apply(simp add: ball_image)

apply(simp add: fresh_permute_iff)

done

fun

  alpha_gen 

where

  alpha_gen[simp del]:

  "alpha_gen (bs, x) R f pi (cs, y) \<longleftrightarrow> f x - bs = f y - cs \<and> (f x - bs) \<sharp>* pi \<and> R (pi \<bullet> x) y"

notation

  alpha_gen ("_ \<approx>gen _ _ _ _" [100, 100, 100, 100, 100] 100)

lemma [mono]: "R1 \<le> R2 \<Longrightarrow> alpha_gen x R1 \<le> alpha_gen x R2"

  by (cases x) (auto simp add: le_fun_def le_bool_def alpha_gen.simps)

lemma alpha_gen_refl:

  assumes a: "R x x"

  shows "(bs, x) \<approx>gen R f 0 (bs, x)"

  using a by (simp add: alpha_gen fresh_star_def fresh_zero_perm)

lemma alpha_gen_sym:

  assumes a: "(bs, x) \<approx>gen R f p (cs, y)"

  and     b: "R (p \<bullet> x) y \<Longrightarrow> R (- p \<bullet> y) x"

  shows "(cs, y) \<approx>gen R f (- p) (bs, x)"

  using a b by (simp add: alpha_gen fresh_star_def fresh_def supp_minus_perm)

lemma alpha_gen_trans:

  assumes a: "(bs, x) \<approx>gen R f p1 (cs, y)"

  and     b: "(cs, y) \<approx>gen R f p2 (ds, z)"

  and     c: "\<lbrakk>R (p1 \<bullet> x) y; R (p2 \<bullet> y) z\<rbrakk> \<Longrightarrow> R ((p2 + p1) \<bullet> x) z"

  shows "(bs, x) \<approx>gen R f (p2 + p1) (ds, z)"

  using a b c using supp_plus_perm

  apply(simp add: alpha_gen fresh_star_def fresh_def)

  apply(blast)

  done

lemma alpha_gen_eqvt:

  assumes a: "(bs, x) \<approx>gen R f q (cs, y)"

  and     b: "R (q \<bullet> x) y \<Longrightarrow> R (p \<bullet> (q \<bullet> x)) (p \<bullet> y)"

  and     c: "p \<bullet> (f x) = f (p \<bullet> x)"

  and     d: "p \<bullet> (f y) = f (p \<bullet> y)"

  shows "(p \<bullet> bs, p \<bullet> x) \<approx>gen R f (p \<bullet> q) (p \<bullet> cs, p \<bullet> y)"

  using a b

  apply(simp add: alpha_gen c[symmetric] d[symmetric] Diff_eqvt[symmetric])

  apply(simp add: permute_eqvt[symmetric])

  apply(simp add: fresh_star_permute_iff)

  apply(clarsimp)

  done

(* introduced for examples *)

lemma alpha_gen_atom_sym:

  assumes a: "\<And>pi t s. (R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s))"

  shows "\<exists>pi. ({atom a}, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> R x2 x1) f pi ({atom b}, s) \<Longrightarrow>

        \<exists>pi. ({atom b}, s) \<approx>gen R f pi ({atom a}, t)"

  apply(erule exE)

  apply(rule_tac x="- pi" in exI)

  apply(simp add: alpha_gen.simps)

  apply(erule conjE)+

  apply(rule conjI)

  apply(simp add: fresh_star_def fresh_minus_perm)

  apply(subgoal_tac "R (- pi \<bullet> s) ((- pi) \<bullet> (pi \<bullet> t))")

  apply simp

  apply(rule a)

  apply assumption

  done

lemma alpha_gen_atom_trans:

  assumes a: "\<And>pi t s. (R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s))"

  shows "\<lbrakk>\<exists>pi\<Colon>perm. ({atom a}, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> (\<forall>x. R x2 x \<longrightarrow> R x1 x)) f pi ({atom aa}, ta);

        \<exists>pi\<Colon>perm. ({atom aa}, ta) \<approx>gen R f pi ({atom ba}, sa)\<rbrakk>

    \<Longrightarrow> \<exists>pi\<Colon>perm. ({atom a}, t) \<approx>gen R f pi ({atom ba}, sa)"

  apply(simp add: alpha_gen.simps)

  apply(erule conjE)+

  apply(erule exE)+

  apply(erule conjE)+

  apply(rule_tac x="pia + pi" in exI)

  apply(simp add: fresh_star_plus)

  apply(drule_tac x="- pia \<bullet> sa" in spec)

  apply(drule mp)

  apply(rotate_tac 4)

  apply(drule_tac pi="- pia" in a)

  apply(simp)

  apply(rotate_tac 6)

  apply(drule_tac pi="pia" in a)

  apply(simp)

  done

lemma alpha_gen_atom_eqvt:

  assumes a: "\<And>x. pi \<bullet> (f x) = f (pi \<bullet> x)"

  and     b: "\<exists>pia. ({atom a}, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> R (pi \<bullet> x1) (pi \<bullet> x2)) f pia ({atom b}, s)"

  shows  "\<exists>pia. ({atom (pi \<bullet> a)}, pi \<bullet> t) \<approx>gen R f pia ({atom (pi \<bullet> b)}, pi \<bullet> s)"

  using b 

  apply -

  apply(erule exE)

  apply(rule_tac x="pi \<bullet> pia" in exI)

  apply(simp add: alpha_gen.simps)

  apply(erule conjE)+

  apply(rule conjI)

  apply(rule_tac ?p1="- pi" in permute_eq_iff[THEN iffD1])

  apply(simp add: a[symmetric] atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt)

  apply(rule conjI)

  apply(rule_tac ?p1="- pi" in fresh_star_permute_iff[THEN iffD1])

  apply(simp add: a[symmetric] atom_eqvt Diff_eqvt insert_eqvt set_eqvt empty_eqvt)

  apply(subst permute_eqvt[symmetric])

  apply(simp)

  done

fun

  alpha_abs 

where

  "alpha_abs (bs, x) (cs, y) = (\<exists>p. (bs, x) \<approx>gen (op=) supp p (cs, y))"

notation

  alpha_abs ("_ \<approx>abs _")

lemma alpha_abs_swap:

  assumes a1: "a \<notin> (supp x) - bs"

  and     a2: "b \<notin> (supp x) - bs"

  shows "(bs, x) \<approx>abs ((a \<rightleftharpoons> b) \<bullet> bs, (a \<rightleftharpoons> b) \<bullet> x)"

  apply(simp)

  apply(rule_tac x="(a \<rightleftharpoons> b)" in exI)

  apply(simp add: alpha_gen)

  apply(simp add: supp_eqvt[symmetric] Diff_eqvt[symmetric])

  apply(simp add: swap_set_not_in[OF a1 a2])

  apply(subgoal_tac "supp (a \<rightleftharpoons> b) \<subseteq> {a, b}")

  using a1 a2

  apply(simp add: fresh_star_def fresh_def)

  apply(blast)

  apply(simp add: supp_swap)

  done

fun

  supp_abs_fun

where

  "supp_abs_fun (bs, x) = (supp x) - bs"

lemma supp_abs_fun_lemma:

  assumes a: "x \<approx>abs y" 

  shows "supp_abs_fun x = supp_abs_fun y"

  using a

  apply(induct rule: alpha_abs.induct)

  apply(simp add: alpha_gen)

  done

quotient_type 'a abs = "(atom set \<times> 'a::pt)" / "alpha_abs"

  apply(rule equivpI)

  unfolding reflp_def symp_def transp_def

  apply(simp_all)

  (* refl *)

  apply(clarify)

  apply(rule exI)

  apply(rule alpha_gen_refl)

  apply(simp)

  (* symm *)

  apply(clarify)

  apply(rule exI)

  apply(rule alpha_gen_sym)

  apply(assumption)

  apply(clarsimp)

  (* trans *)

  apply(clarify)

  apply(rule exI)

  apply(rule alpha_gen_trans)

  apply(assumption)

  apply(assumption)

  apply(simp)

  done

quotient_definition

  "Abs::atom set \<Rightarrow> ('a::pt) \<Rightarrow> 'a abs"

is

  "Pair::atom set \<Rightarrow> ('a::pt) \<Rightarrow> (atom set \<times> 'a)"

lemma [quot_respect]:

  shows "((op =) ===> (op =) ===> alpha_abs) Pair Pair"

  apply(clarsimp)

  apply(rule exI)

  apply(rule alpha_gen_refl)

  apply(simp)

  done

lemma [quot_respect]:

  shows "((op =) ===> alpha_abs ===> alpha_abs) permute permute"

  apply(clarsimp)

  apply(rule exI)

  apply(rule alpha_gen_eqvt)

  apply(assumption)

  apply(simp_all add: supp_eqvt)

  done

lemma [quot_respect]:

  shows "(alpha_abs ===> (op =)) supp_abs_fun supp_abs_fun"

  apply(simp add: supp_abs_fun_lemma)

  done

lemma abs_induct:

  "\<lbrakk>\<And>as (x::'a::pt). P (Abs as x)\<rbrakk> \<Longrightarrow> P t"

  apply(lifting prod.induct[where 'a="atom set" and 'b="'a"])

  done

(* TEST case *)

lemmas abs_induct2 = prod.induct[where 'a="atom set" and 'b="'a::pt", quot_lifted]

thm abs_induct abs_induct2

instantiation abs :: (pt) pt

begin

quotient_definition

  "permute_abs::perm \<Rightarrow> ('a::pt abs) \<Rightarrow> 'a abs"

is

  "permute:: perm \<Rightarrow> (atom set \<times> 'a::pt) \<Rightarrow> (atom set \<times> 'a::pt)"

lemma permute_ABS [simp]:

  fixes x::"'a::pt"  (* ??? has to be 'a \<dots> 'b does not work *)

  shows "(p \<bullet> (Abs as x)) = Abs (p \<bullet> as) (p \<bullet> x)"

  by (lifting permute_prod.simps(1)[where 'a="atom set" and 'b="'a"])

instance

  apply(default)

  apply(induct_tac [!] x rule: abs_induct)

  apply(simp_all)

  done

end

quotient_definition

  "supp_Abs_fun :: ('a::pt) abs \<Rightarrow> atom \<Rightarrow> bool"

is

  "supp_abs_fun"

lemma supp_Abs_fun_simp:

  shows "supp_Abs_fun (Abs bs x) = (supp x) - bs"

  by (lifting supp_abs_fun.simps(1))

lemma supp_Abs_fun_eqvt [eqvt]:

  shows "(p \<bullet> supp_Abs_fun x) = supp_Abs_fun (p \<bullet> x)"

  apply(induct_tac x rule: abs_induct)

  apply(simp add: supp_Abs_fun_simp supp_eqvt Diff_eqvt)

  done

lemma supp_Abs_fun_fresh:

  shows "a \<sharp> Abs bs x \<Longrightarrow> a \<sharp> supp_Abs_fun (Abs bs x)"

  apply(rule fresh_fun_eqvt_app)

  apply(simp add: eqvts_raw)

  apply(simp)

  done

lemma Abs_swap:

  assumes a1: "a \<notin> (supp x) - bs"

  and     a2: "b \<notin> (supp x) - bs"

  shows "(Abs bs x) = (Abs ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x))"

  using a1 a2 by (lifting alpha_abs_swap)

lemma Abs_supports:

  shows "((supp x) - as) supports (Abs as x)"

  unfolding supports_def

  apply(clarify)

  apply(simp (no_asm))

  apply(subst Abs_swap[symmetric])

  apply(simp_all)

  done

lemma supp_Abs_subset1:

  fixes x::"'a::fs"

  shows "(supp x) - as \<subseteq> supp (Abs as x)"

  apply(simp add: supp_conv_fresh)

  apply(auto)

  apply(drule_tac supp_Abs_fun_fresh)

  apply(simp only: supp_Abs_fun_simp)

  apply(simp add: fresh_def)

  apply(simp add: supp_finite_atom_set finite_supp)

  done

lemma supp_Abs_subset2:

  fixes x::"'a::fs"

  shows "supp (Abs as x) \<subseteq> (supp x) - as"

  apply(rule supp_is_subset)

  apply(rule Abs_supports)

  apply(simp add: finite_supp)

  done

lemma supp_Abs:

  fixes x::"'a::fs"

  shows "supp (Abs as x) = (supp x) - as"

  apply(rule_tac subset_antisym)

  apply(rule supp_Abs_subset2)

  apply(rule supp_Abs_subset1)

  done

instance abs :: (fs) fs

  apply(default)

  apply(induct_tac x rule: abs_induct)

  apply(simp add: supp_Abs)

  apply(simp add: finite_supp)

  done

lemma Abs_fresh_iff:

  fixes x::"'a::fs"

  shows "a \<sharp> Abs bs x \<longleftrightarrow> a \<in> bs \<or> (a \<notin> bs \<and> a \<sharp> x)"

  apply(simp add: fresh_def)

  apply(simp add: supp_Abs)

  apply(auto)

  done

lemma Abs_eq_iff:

  shows "Abs bs x = Abs cs y \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>gen (op =) supp p (cs, y))"

  by (lifting alpha_abs.simps(1))

(* 

  below is a construction site for showing that in the

  single-binder case, the old and new alpha equivalence 

  coincide

*)

fun

  alpha1

where

  "alpha1 (a, x) (b, y) \<longleftrightarrow> (a = b \<and> x = y) \<or> (a \<noteq> b \<and> x = (a \<rightleftharpoons> b) \<bullet> y \<and> a \<sharp> y)"

notation 

  alpha1 ("_ \<approx>abs1 _")

thm swap_set_not_in

lemma qq:

  fixes S::"atom set"

  assumes a: "supp p \<inter> S = {}"

  shows "p \<bullet> S = S"

using a

apply(simp add: supp_perm permute_set_eq)

apply(auto)

apply(simp only: disjoint_iff_not_equal)

apply(simp)

apply (metis permute_atom_def_raw)

apply(rule_tac x="(- p) \<bullet> x" in exI)

apply(simp)

apply(simp only: disjoint_iff_not_equal)

apply(simp)

apply(metis permute_minus_cancel)

done

lemma alpha_abs_swap:

  assumes a1: "(supp x - bs) \<sharp>* p"

  and     a2: "(supp x - bs) \<sharp>* p"

  shows "(bs, x) \<approx>abs (p \<bullet> bs, p \<bullet> x)"

  apply(simp)

  apply(rule_tac x="p" in exI)

  apply(simp add: alpha_gen)

  apply(simp add: supp_eqvt[symmetric] Diff_eqvt[symmetric])

  apply(rule conjI)

  apply(rule sym)

  apply(rule qq)

  using a1 a2

  apply(auto)[1]

  oops

lemma

  assumes a: "(a, x) \<approx>abs1 (b, y)" "sort_of a = sort_of b"

  shows "({a}, x) \<approx>abs ({b}, y)"

using a