theory Abs

imports "Nominal2_Atoms" 

        "Nominal2_Eqvt" 

        "Nominal2_Supp" 

        "Nominal2_FSet"

        "../Quotient" 

        "../Quotient_Product" 

begin

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 \<and>

     pi \<bullet> bs = cs"

fun

  alpha_res

where

  alpha_res[simp del]:

  "alpha_res (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"

fun

  alpha_lst

where

  alpha_lst[simp del]:

  "alpha_lst (bs, x) R f pi (cs, y) \<longleftrightarrow> 

     f x - set bs = f y - set cs \<and> 

     (f x - set bs) \<sharp>* pi \<and> 

     R (pi \<bullet> x) y \<and>

     pi \<bullet> bs = cs"

lemmas alphas = alpha_gen.simps alpha_res.simps alpha_lst.simps

notation

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

  alpha_res ("_ \<approx>res _ _ _ _" [100, 100, 100, 100, 100] 100) and

  alpha_lst ("_ \<approx>lst _ _ _ _" [100, 100, 100, 100, 100] 100) 

(* monos *)

lemma [mono]: 

  shows "R1 \<le> R2 \<Longrightarrow> alpha_gen bs R1 \<le> alpha_gen bs R2"

  and   "R1 \<le> R2 \<Longrightarrow> alpha_res bs R1 \<le> alpha_res bs R2"

  and   "R1 \<le> R2 \<Longrightarrow> alpha_lst cs R1 \<le> alpha_lst cs R2"

  by (case_tac [!] bs, case_tac [!] cs) 

     (auto simp add: le_fun_def le_bool_def alphas)

fun

  alpha_abs 

where

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

fun

  alpha_abs_lst

where

  "alpha_abs_lst (bs, x) (cs, y) \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>lst (op=) supp p (cs, y))"

fun

  alpha_abs_res

where

  "alpha_abs_res (bs, x) (cs, y) \<longleftrightarrow> (\<exists>p. (bs, x) \<approx>res (op=) supp p (cs, y))"

notation

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

  alpha_abs_lst ("_ \<approx>abs'_lst _") and

  alpha_abs_res ("_ \<approx>abs'_res _")

lemmas alphas_abs = alpha_abs.simps alpha_abs_res.simps alpha_abs_lst.simps

lemma alphas_abs_refl:

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

  and   "(bs, x) \<approx>abs_res (bs, x)"

  and   "(cs, x) \<approx>abs_lst (cs, x)" 

  unfolding alphas_abs

  unfolding alphas

  unfolding fresh_star_def

  by (rule_tac [!] x="0" in exI)

     (simp_all add: fresh_zero_perm)

lemma alphas_abs_sym:

  shows "(bs, x) \<approx>abs (cs, y) \<Longrightarrow> (cs, y) \<approx>abs (bs, x)"

  and   "(bs, x) \<approx>abs_res (cs, y) \<Longrightarrow> (cs, y) \<approx>abs_res (bs, x)"

  and   "(ds, x) \<approx>abs_lst (es, y) \<Longrightarrow> (es, y) \<approx>abs_lst (ds, x)"

  unfolding alphas_abs

  unfolding alphas

  unfolding fresh_star_def

  by (erule_tac [!] exE, rule_tac [!] x="-p" in exI)

     (auto simp add: fresh_minus_perm)

lemma alphas_abs_trans:

  shows "\<lbrakk>(bs, x) \<approx>abs (cs, y); (cs, y) \<approx>abs (ds, z)\<rbrakk> \<Longrightarrow> (bs, x) \<approx>abs (ds, z)"

  and   "\<lbrakk>(bs, x) \<approx>abs_res (cs, y); (cs, y) \<approx>abs_res (ds, z)\<rbrakk> \<Longrightarrow> (bs, x) \<approx>abs_res (ds, z)"

  and   "\<lbrakk>(es, x) \<approx>abs_lst (gs, y); (gs, y) \<approx>abs_lst (hs, z)\<rbrakk> \<Longrightarrow> (es, x) \<approx>abs_lst (hs, z)"

  unfolding alphas_abs

  unfolding alphas

  unfolding fresh_star_def

  apply(erule_tac [!] exE, erule_tac [!] exE)

  apply(rule_tac [!] x="pa + p" in exI)

  by (simp_all add: fresh_plus_perm)

lemma alphas_abs_eqvt:

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

  and   "(bs, x) \<approx>abs_res (cs, y) \<Longrightarrow> (p \<bullet> bs, p \<bullet> x) \<approx>abs_res (p \<bullet> cs, p \<bullet> y)"

  and   "(ds, x) \<approx>abs_lst (es, y) \<Longrightarrow> (p \<bullet> ds, p \<bullet> x) \<approx>abs_lst (p \<bullet> es, p \<bullet> y)"

  unfolding alphas_abs

  unfolding alphas

  unfolding set_eqvt[symmetric]

  unfolding supp_eqvt[symmetric]

  unfolding Diff_eqvt[symmetric]

  apply(erule_tac [!] exE)

  apply(rule_tac [!] x="p \<bullet> pa" in exI)

  by (auto simp add: fresh_star_permute_iff permute_eqvt[symmetric])

lemma alphas_abs_swap1:

  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)"

  and   "(bs, x) \<approx>abs_res ((a \<rightleftharpoons> b) \<bullet> bs, (a \<rightleftharpoons> b) \<bullet> x)"

  using a1 a2

  unfolding alphas_abs

  unfolding alphas

  unfolding supp_eqvt[symmetric] Diff_eqvt[symmetric] 

  unfolding fresh_star_def fresh_def

  unfolding swap_set_not_in[OF a1 a2] 

  by (rule_tac [!] x="(a \<rightleftharpoons> b)" in exI)

     (auto simp add: supp_perm swap_atom)

lemma alphas_abs_swap2:

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

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

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

  using a1 a2

  unfolding alphas_abs

  unfolding alphas

  unfolding supp_eqvt[symmetric] Diff_eqvt[symmetric] set_eqvt[symmetric]

  unfolding fresh_star_def fresh_def

  unfolding swap_set_not_in[OF a1 a2] 

  by (rule_tac [!] x="(a \<rightleftharpoons> b)" in exI)

     (auto simp add: supp_perm swap_atom)

fun

  aux_set 

where

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

fun

  aux_list

where

  "aux_list (cs, x) = (supp x) - (set cs)"

lemma aux_abs_lemma:

  assumes a: "(bs, x) \<approx>abs (cs, y)" 

  shows "aux_set (bs, x) = aux_set (cs, y)"

  using a

  by (induct rule: alpha_abs.induct)

     (simp add: alphas_abs alphas)

lemma aux_abs_res_lemma:

  assumes a: "(bs, x) \<approx>abs_res (cs, y)" 

  shows "aux_set (bs, x) = aux_set (cs, y)"

  using a

  by (induct rule: alpha_abs_res.induct)

     (simp add: alphas_abs alphas)

lemma aux_abs_list_lemma:

  assumes a: "(bs, x) \<approx>abs_lst (cs, y)" 

  shows "aux_list (bs, x) = aux_list (cs, y)"

  using a

  by (induct rule: alpha_abs_lst.induct)

     (simp add: alphas_abs alphas)

quotient_type 

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

and 'b abs_res = "(atom set \<times> 'b::pt)" / "alpha_abs_res"

and 'c abs_lst = "(atom list \<times> 'c::pt)" / "alpha_abs_lst"

  apply(rule_tac [!] equivpI)

  unfolding reflp_def symp_def transp_def

  by (auto intro: alphas_abs_sym alphas_abs_refl alphas_abs_trans simp only:)

quotient_definition

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

is

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

quotient_definition

  "Abs_res::atom set \<Rightarrow> ('a::pt) \<Rightarrow> 'a abs_res"

is

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

quotient_definition

  "Abs_lst::atom list \<Rightarrow> ('a::pt) \<Rightarrow> 'a abs_lst"

is

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

lemma [quot_respect]:

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

  and   "(op= ===> op= ===> alpha_abs_res) Pair Pair"

  and   "(op= ===> op= ===> alpha_abs_lst) Pair Pair"

  unfolding fun_rel_def

  by (auto intro: alphas_abs_refl simp only:)

lemma [quot_respect]:

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

  and   "(op= ===> alpha_abs_res ===> alpha_abs_res) permute permute"

  and   "(op= ===> alpha_abs_lst ===> alpha_abs_lst) permute permute"

  unfolding fun_rel_def

  by (auto intro: alphas_abs_eqvt simp only: Pair_eqvt)

lemma [quot_respect]:

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

  and   "(alpha_abs_res ===> op=) aux_set aux_set"

  and   "(alpha_abs_lst ===> op=) aux_list aux_list"

  unfolding fun_rel_def

  apply(rule_tac [!] allI)

  apply(rule_tac [!] allI)

  apply(case_tac [!] x, case_tac [!] y)

  apply(rule_tac [!] impI)

  by (simp_all only: aux_abs_lemma aux_abs_res_lemma aux_abs_list_lemma)

lemma abs_inducts:

  shows "(\<And>as (x::'a::pt). P1 (Abs as x)) \<Longrightarrow> P1 x1"

  and   "(\<And>as (x::'a::pt). P2 (Abs_res as x)) \<Longrightarrow> P2 x2"

  and   "(\<And>as (x::'a::pt). P3 (Abs_lst as x)) \<Longrightarrow> P3 x3"

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

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

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

  done

instantiation abs_gen :: (pt) pt

begin

quotient_definition

  "permute_abs_gen::perm \<Rightarrow> ('a::pt abs_gen) \<Rightarrow> 'a abs_gen"

is

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

lemma permute_Abs[simp]:

  fixes x::"'a::pt"  

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

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

instance

  apply(default)

  apply(induct_tac [!] x rule: abs_inducts(1))

  apply(simp_all)

  done

end

instantiation abs_res :: (pt) pt

begin

quotient_definition

  "permute_abs_res::perm \<Rightarrow> ('a::pt abs_res) \<Rightarrow> 'a abs_res"

is

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

lemma permute_Abs_res[simp]:

  fixes x::"'a::pt"  

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

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

instance

  apply(default)

  apply(induct_tac [!] x rule: abs_inducts(2))

  apply(simp_all)

  done

end

instantiation abs_lst :: (pt) pt

begin

quotient_definition

  "permute_abs_lst::perm \<Rightarrow> ('a::pt abs_lst) \<Rightarrow> 'a abs_lst"

is

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

lemma permute_Abs_lst[simp]:

  fixes x::"'a::pt"  

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

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

instance

  apply(default)

  apply(induct_tac [!] x rule: abs_inducts(3))

  apply(simp_all)

  done

end

lemmas permute_abs = permute_Abs permute_Abs_res permute_Abs_lst

quotient_definition

  "supp_gen :: ('a::pt) abs_gen \<Rightarrow> atom set"

is

  "aux_set"

quotient_definition

  "supp_res :: ('a::pt) abs_res \<Rightarrow> atom set"

is

  "aux_set"

quotient_definition

  "supp_lst :: ('a::pt) abs_lst \<Rightarrow> atom set"

is

  "aux_list"

lemma aux_supps:

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

  and   "supp_res (Abs_res bs x) = (supp x) - bs"

  and   "supp_lst (Abs_lst cs x) = (supp x) - (set cs)"

  apply(lifting aux_set.simps)

  apply(lifting aux_set.simps)

  apply(lifting aux_list.simps)

  done

lemma aux_supp_eqvt[eqvt]:

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

  and   "(p \<bullet> supp_res y) = supp_res (p \<bullet> y)"

  and   "(p \<bullet> supp_lst z) = supp_lst (p \<bullet> z)"

  apply(induct_tac x rule: abs_inducts(1))

  apply(simp add: aux_supps supp_eqvt Diff_eqvt)

  apply(induct_tac y rule: abs_inducts(2))

  apply(simp add: aux_supps supp_eqvt Diff_eqvt)

  apply(induct_tac z rule: abs_inducts(3))

  apply(simp add: aux_supps supp_eqvt Diff_eqvt set_eqvt)

  done

lemma aux_fresh:

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

  and   "a \<sharp> Abs_res bs x \<Longrightarrow> a \<sharp> supp_res (Abs_res bs x)"

  and   "a \<sharp> Abs_lst cs x \<Longrightarrow> a \<sharp> supp_lst (Abs_lst cs x)"

  apply(rule_tac [!] fresh_fun_eqvt_app)

  apply(simp_all add: eqvts_raw)

  done

lemma abs_swap1:

  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)"

  and   "Abs_res bs x = Abs_res ((a \<rightleftharpoons> b) \<bullet> bs) ((a \<rightleftharpoons> b) \<bullet> x)"

  using a1 a2 

  apply(lifting alphas_abs_swap1(1))

  apply(lifting alphas_abs_swap1(2))

  done

lemma abs_swap2:

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

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

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

  using a1 a2 by (lifting alphas_abs_swap2)

lemma abs_supports:

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

  and   "((supp x) - as) supports (Abs_res as x)"

  and   "((supp x) - (set bs)) supports (Abs_lst bs x)"

  unfolding supports_def

  unfolding permute_abs

  by (simp_all add: abs_swap1[symmetric] abs_swap2[symmetric])

lemma supp_abs_subset1:

  assumes a: "finite (supp x)"

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

  and   "(supp x) - as \<subseteq> supp (Abs_res as x)"

  and   "(supp x) - (set bs) \<subseteq> supp (Abs_lst bs x)"

  unfolding supp_conv_fresh

  apply(auto dest!: aux_fresh simp add: aux_supps)

  apply(simp_all add: fresh_def supp_finite_atom_set a)

  done

lemma supp_abs_subset2:

  assumes a: "finite (supp x)"

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

  and   "supp (Abs_res as x) \<subseteq> (supp x) - as"

  and   "supp (Abs_lst bs x) \<subseteq> (supp x) - (set bs)"

  apply(rule_tac [!] supp_is_subset)

  apply(simp_all add: abs_supports a)

  done

lemma abs_finite_supp:

  assumes a: "finite (supp x)"

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

  and   "supp (Abs_res as x) = (supp x) - as"

  and   "supp (Abs_lst bs x) = (supp x) - (set bs)"

  apply(rule_tac [!] subset_antisym)

  apply(simp_all add: supp_abs_subset1[OF a] supp_abs_subset2[OF a])

  done

lemma supp_abs:

  fixes x::"'a::fs"

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

  and   "supp (Abs_res as x) = (supp x) - as"

  and   "supp (Abs_lst bs x) = (supp x) - (set bs)"

  apply(rule_tac [!] abs_finite_supp)

  apply(simp_all add: finite_supp)

  done

instance abs_gen :: (fs) fs

  apply(default)

  apply(induct_tac x rule: abs_inducts(1))

  apply(simp add: supp_abs finite_supp)

  done

instance abs_res :: (fs) fs

  apply(default)

  apply(induct_tac x rule: abs_inducts(2))

  apply(simp add: supp_abs finite_supp)

  done

instance abs_lst :: (fs) fs

  apply(default)

  apply(induct_tac x rule: abs_inducts(3))

  apply(simp add: supp_abs 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)"

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

  and   "a \<sharp> Abs_lst cs x \<longleftrightarrow> a \<in> (set cs) \<or> (a \<notin> (set cs) \<and> a \<sharp> x)"

  unfolding fresh_def

  unfolding supp_abs

  by auto

lemma abs_eq_iff:

  shows "Abs bs x = Abs cs y \<longleftrightarrow> (bs, x) \<approx>abs (cs, y)"

  and   "Abs_res bs x = Abs_res cs y \<longleftrightarrow> (bs, x) \<approx>abs_res (cs, y)"

  and   "Abs_lst ds x = Abs_lst hs y \<longleftrightarrow> (ds, x) \<approx>abs_lst (hs, y)"

  apply(simp_all)

  apply(lifting alphas_abs)

  done

section {* BELOW is stuff that may or may not be needed *}

(* support of concrete atom sets *)

lemma supp_atom_image:

  fixes as::"'a::at_base set"

  shows "supp (atom ` as) = supp as"

apply(simp add: supp_def)

apply(simp add: image_eqvt)

apply(simp add: atom_eqvt_raw)

apply(simp add: atom_image_cong)

done

lemma swap_atom_image_fresh: "\<lbrakk>a \<sharp> atom ` (fn :: ('a :: at_base set)); b \<sharp> atom ` fn\<rbrakk> \<Longrightarrow> (a \<rightleftharpoons> b) \<bullet> fn = fn"

  apply (simp add: fresh_def)

  apply (simp add: supp_atom_image)

  apply (fold fresh_def)

  apply (simp add: swap_fresh_fresh)

  done

(* TODO: The following lemmas can be moved somewhere... *)

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))

lemma split_rsp2[quot_respect]: "((R1 ===> R2 ===> prod_rel R1 R2 ===> op =) ===>

  prod_rel R1 R2 ===> prod_rel R1 R2 ===> op =) split split"

  by auto

lemma split_prs2[quot_preserve]:

  assumes q1: "Quotient R1 Abs1 Rep1"

  and q2: "Quotient R2 Abs2 Rep2"

  shows "((Abs1 ---> Abs2 ---> prod_fun Abs1 Abs2 ---> id) ---> prod_fun Rep1 Rep2 ---> prod_fun Rep1 Rep2 ---> id) split = split"

  by (simp add: expand_fun_eq Quotient_abs_rep[OF q1] Quotient_abs_rep[OF q2])

lemma alpha_gen2:

  "(bs, x1, x2) \<approx>gen (\<lambda>(x1, y1) (x2, y2). R1 x1 x2 \<and> R2 y1 y2) (\<lambda>(a, b). f1 a \<union> f2 b) pi (cs, y1, y2) =

  (f1 x1 \<union> f2 x2 - bs = f1 y1 \<union> f2 y2 - cs \<and> (f1 x1 \<union> f2 x2 - bs) \<sharp>* pi \<and> R1 (pi \<bullet> x1) y1 \<and> R2 (pi \<bullet> x2) y2

  \<and> pi \<bullet> bs = cs)"

by (simp add: alpha_gen)

lemma alpha_gen_compose_sym:

  fixes pi

  assumes b: "(aa, t) \<approx>gen (\<lambda>x1 x2. R x1 x2 \<and> R x2 x1) f pi (ab, s)"

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

  shows "(ab, s) \<approx>gen R f (- pi) (aa, t)"

  using b apply -

  apply(simp add: alpha_gen)

  apply(erule conjE)+

  apply(rule conjI)