header {* Constant definitions *}

theory Consts imports Utils begin

fun Umn :: "nat \<Rightarrow> nat \<Rightarrow> lam"
where
  [simp del]: "Umn 0 n = \<integral>(cn 0). V (cn n)"
| [simp del]: "Umn (Suc m) n = \<integral>(cn (Suc m)). Umn m n"

lemma [simp]: "2 = Suc 1"
  by auto

lemma Lam_U:
  "x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> z \<Longrightarrow> Umn 2 0 = \<integral>x. \<integral>y. \<integral>z. V z"
  "x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> z \<Longrightarrow> Umn 2 1 = \<integral>x. \<integral>y. \<integral>z. V y"
  "x \<noteq> y \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> z \<Longrightarrow> Umn 2 2 = \<integral>x. \<integral>y. \<integral>z. V x"
  apply (simp_all add: Umn.simps Abs1_eq_iff lam.fresh fresh_at_base flip_def[symmetric] Umn.simps)
  apply (smt Zero_not_Suc cnd flip_at_base_simps flip_at_simps)+
  done

lemma a: "n \<le> m \<Longrightarrow> atom (cn n) \<notin> supp (Umn m n)"
  apply (induct m)
  apply (auto simp add: lam.supp supp_at_base Umn.simps)
  by smt

lemma b: "supp (Umn m n) \<subseteq> {atom (cn n)}"
  by (induct m) (auto simp add: lam.supp supp_at_base Umn.simps)

lemma supp_U[simp]: "n \<le> m \<Longrightarrow> supp (Umn m n) = {}"
  using a b
  by blast

lemma U_eqvt:
  "n \<le> m \<Longrightarrow> p \<bullet> (Umn m n) = Umn m n"
  by (rule_tac [!] perm_supp_eq) (simp_all add: fresh_star_def fresh_def)

definition Var where "Var \<equiv> \<integral>cx. \<integral>cy. (V cy \<cdot> (Umn 2 2) \<cdot> V cx \<cdot> V cy)"
definition "App \<equiv> \<integral>cx. \<integral>cy. \<integral>cz. (V cz \<cdot> Umn 2 1 \<cdot> V cx \<cdot> V cy \<cdot> V cz)"
definition "Abs \<equiv> \<integral>cx. \<integral>cy. (V cy \<cdot> Umn 2 0 \<cdot> V cx \<cdot> V cy)"

lemma Var_App_Abs:
  "x \<noteq> e \<Longrightarrow> Var = \<integral>x. \<integral>e. (V e \<cdot> Umn 2 2 \<cdot> V x \<cdot> V e)"
  "e \<noteq> x \<Longrightarrow> e \<noteq> y \<Longrightarrow> x \<noteq> y \<Longrightarrow> App = \<integral>x. \<integral>y. \<integral>e. (V e \<cdot> Umn 2 1 \<cdot> V x \<cdot> V y \<cdot> V e)"
  "x \<noteq> e \<Longrightarrow> Abs = \<integral>x. \<integral>e. (V e \<cdot> Umn 2 0 \<cdot> V x \<cdot> V e)"
  unfolding Var_def App_def Abs_def
  by (simp_all add: Abs1_eq_iff lam.fresh flip_def[symmetric] U_eqvt fresh_def lam.supp supp_at_base)
     (smt cx_cy_cz permute_flip_at Zero_not_Suc cnd flip_at_base_simps flip_at_simps)+

lemma Var_app:
  "Var \<cdot> x \<cdot> e \<approx> e \<cdot> Umn 2 2 \<cdot> x \<cdot> e"
  by (rule lam2_fast_app) (simp_all add: Var_App_Abs)

lemma App_app:
  "App \<cdot> x \<cdot> y \<cdot> e \<approx> e \<cdot> Umn 2 1 \<cdot> x \<cdot> y \<cdot> e"
  by (rule lam3_fast_app[OF Var_App_Abs(2)]) (simp_all)

lemma Abs_app:
  "Abs \<cdot> x \<cdot> e \<approx> e \<cdot> Umn 2 0 \<cdot> x \<cdot> e"
  by (rule lam2_fast_app) (simp_all add: Var_App_Abs)

lemma supp_Var_App_Abs[simp]:
  "supp Var = {}" "supp App = {}" "supp Abs = {}"
  by (simp_all add: Var_def App_def Abs_def lam.supp supp_at_base) blast+

lemma Var_App_Abs_eqvt[eqvt]:
  "p \<bullet> Var = Var" "p \<bullet> App = App" "p \<bullet> Abs = Abs"
  by (rule_tac [!] perm_supp_eq) (simp_all add: fresh_star_def fresh_def)

nominal_primrec
  Numeral :: "lam \<Rightarrow> lam" ("\<lbrace>_\<rbrace>" 1000)
where
  "\<lbrace>V x\<rbrace> = Var \<cdot> (V x)"
| Ap: "\<lbrace>M \<cdot> N\<rbrace> = App \<cdot> \<lbrace>M\<rbrace> \<cdot> \<lbrace>N\<rbrace>"
| "\<lbrace>\<integral>x. M\<rbrace> = Abs \<cdot> (\<integral>x. \<lbrace>M\<rbrace>)"
proof auto
  fix x :: lam and P
  assume "\<And>xa. x = V xa \<Longrightarrow> P" "\<And>M N. x = M \<cdot> N \<Longrightarrow> P" "\<And>xa M. x = \<integral> xa. M \<Longrightarrow> P"
  then show "P"
    by (rule_tac y="x" and c="0 :: perm" in lam.strong_exhaust)
       (auto simp add: Abs1_eq_iff fresh_star_def)[3]
next
  fix x :: var and M and xa :: var and Ma
  assume "[[atom x]]lst. M = [[atom xa]]lst. Ma"
    "eqvt_at Numeral_sumC M"
  then show "[[atom x]]lst. Numeral_sumC M = [[atom xa]]lst. Numeral_sumC Ma"
    apply -
    apply (erule Abs_lst1_fcb)
    apply (simp_all add: Abs_fresh_iff)
    apply (erule fresh_eqvt_at)
    apply (simp_all add: finite_supp Abs1_eq_iff eqvt_at_def)
    done
next
  show "eqvt Numeral_graph" unfolding eqvt_def Numeral_graph_def
    by (rule, perm_simp, rule)
qed

termination
  by (relation "measure (\<lambda>(t). size t)")
     (simp_all add: lam.size)

lemma numeral_eqvt[eqvt]: "p \<bullet> \<lbrace>x\<rbrace> = \<lbrace>p \<bullet> x\<rbrace>"
  by (induct x rule: lam.induct)
     (simp_all add: Var_App_Abs_eqvt)

lemma supp_numeral[simp]:
  "supp \<lbrace>x\<rbrace> = supp x"
  by (induct x rule: lam.induct)
     (simp_all add: lam.supp)

lemma fresh_numeral[simp]:
  "x \<sharp> \<lbrace>y\<rbrace> = x \<sharp> y"
  unfolding fresh_def by simp

fun app_lst :: "var \<Rightarrow> lam list \<Rightarrow> lam" where
  "app_lst n [] = V n"
| "app_lst n (h # t) = Ap (app_lst n t) h"

lemma app_lst_eqvt[eqvt]: "p \<bullet> (app_lst t ts) = app_lst (p \<bullet> t) (p \<bullet> ts)"
  by (induct ts arbitrary: t p) (simp_all add: eqvts)

lemma supp_app_lst: "supp (app_lst x l) = {atom x} \<union> supp l"
  apply (induct l)
  apply (simp_all add: supp_Nil lam.supp supp_at_base supp_Cons)
  by blast

lemma app_lst_eq_iff: "app_lst n M = app_lst n N \<Longrightarrow> M = N"
  by (induct M N rule: list_induct2') simp_all

lemma app_lst_rev_eq_iff: "app_lst n (rev M) = app_lst n (rev N) \<Longrightarrow> M = N"
  by (drule app_lst_eq_iff) simp

nominal_primrec
  Ltgt :: "lam list \<Rightarrow> lam" ("\<guillemotleft>_\<guillemotright>" 1000)
where
  [simp del]: "atom x \<sharp> l \<Longrightarrow> \<guillemotleft>l\<guillemotright> = \<integral>x. (app_lst x (rev l))"
  unfolding eqvt_def Ltgt_graph_def
  apply (rule, perm_simp, rule, rule)
  apply (rule_tac x="x" and ?'a="var" in obtain_fresh)
  apply (simp_all add: Abs1_eq_iff lam.fresh swap_fresh_fresh fresh_at_base)
  apply (simp add: eqvts swap_fresh_fresh)
  apply (case_tac "x = xa")
  apply simp_all
  apply (subgoal_tac "eqvt app_lst")
  apply (erule fresh_fun_eqvt_app2)
  apply (simp_all add: fresh_at_base lam.fresh eqvt_def eqvts_raw fresh_rev)
  done

termination
  by (relation "measure (\<lambda>t. size t)")
     (simp_all add: lam.size)

lemma ltgt_eqvt[eqvt]:
  "p \<bullet> \<guillemotleft>t\<guillemotright> = \<guillemotleft>p \<bullet> t\<guillemotright>"
proof -
  obtain x :: var where "atom x \<sharp> (t, p \<bullet> t)" using obtain_fresh by auto
  then have *: "atom x \<sharp> t" "atom x \<sharp> (p \<bullet> t)" using fresh_Pair by simp_all
  then show ?thesis using *[unfolded fresh_def]
    apply (simp add: Abs1_eq_iff lam.fresh app_lst_eqvt Ltgt.simps)
    apply (case_tac "p \<bullet> x = x")
    apply (simp_all add: eqvts)
    apply rule
    apply (subst swap_fresh_fresh)
    apply (simp_all add: fresh_at_base_permute_iff fresh_def[symmetric] fresh_at_base)
    apply (subgoal_tac "eqvt app_lst")
    apply (erule fresh_fun_eqvt_app2)
    apply (simp_all add: fresh_at_base lam.fresh eqvt_def eqvts_raw fresh_rev)
    done
qed

lemma ltgt_eq_iff[simp]:
  "\<guillemotleft>M\<guillemotright> = \<guillemotleft>N\<guillemotright> \<longleftrightarrow> M = N"
proof auto
  obtain x :: var where "atom x \<sharp> (M, N)" using obtain_fresh by auto
  then have *: "atom x \<sharp> M" "atom x \<sharp> N" using fresh_Pair by simp_all
  then show "(\<guillemotleft>M\<guillemotright> = \<guillemotleft>N\<guillemotright>) \<Longrightarrow> (M = N)" by (simp add: Abs1_eq_iff app_lst_rev_eq_iff Ltgt.simps)
qed

lemma Ltgt1_app: "\<guillemotleft>[M]\<guillemotright> \<cdot> N \<approx> N \<cdot> M"
proof -
  obtain x :: var where "atom x \<sharp> (M, N)" using obtain_fresh by auto
  then have "atom x \<sharp> M" "atom x \<sharp> N" using fresh_Pair by simp_all
  then show ?thesis
  apply (subst Ltgt.simps)
  apply (simp add: fresh_Cons fresh_Nil)
  apply (rule b3, rule bI, simp add: b1)
  done
qed

lemma Ltgt3_app: "\<guillemotleft>[M,N,P]\<guillemotright> \<cdot> R \<approx> R \<cdot> M \<cdot> N \<cdot> P"
proof -
  obtain x :: var where "atom x \<sharp> (M, N, P, R)" using obtain_fresh by auto
  then have *: "atom x \<sharp> (M,N,P)" "atom x \<sharp> R" using fresh_Pair by simp_all
  then have s: "V x \<cdot> M \<cdot> N \<cdot> P [x ::= R] \<approx> R \<cdot> M \<cdot> N \<cdot> P" using b1 by simp
  show ?thesis using *
    apply (subst Ltgt.simps)
  apply (simp add: fresh_Cons fresh_Nil fresh_Pair_elim)
  apply auto[1]
  apply (rule b3, rule bI, simp add: b1)
  done
qed

lemma supp_ltgt[simp]:
  "supp \<guillemotleft>t\<guillemotright> = supp t"
proof -
  obtain x :: var where *:"atom x \<sharp> t" using obtain_fresh by auto
  show ?thesis using *
  by (simp_all add: Ltgt.simps lam.supp supp_at_base supp_Nil supp_app_lst supp_rev fresh_def)
qed

lemma fresh_ltgt[simp]:
  "x \<sharp> \<guillemotleft>[y]\<guillemotright> = x \<sharp> y"
  "x \<sharp> \<guillemotleft>[t,r,s]\<guillemotright> = x \<sharp> (t,r,s)"
  by (simp_all add: fresh_def supp_Cons supp_Nil supp_Pair)

lemma Ltgt1_subst[simp]:
  "\<guillemotleft>[M]\<guillemotright> [y ::= A] = \<guillemotleft>[M [y ::= A]]\<guillemotright>"
proof -
  obtain x :: var where a: "atom x \<sharp> (M, A, y, M [y ::= A])" using obtain_fresh by blast
  have "x \<noteq> y" using a[simplified fresh_Pair fresh_at_base] by simp
  then show ?thesis
    apply (subst Ltgt.simps)
    using a apply (simp add: fresh_Nil fresh_Cons fresh_Pair_elim)
    apply (subst Ltgt.simps)
    using a apply (simp add: fresh_Pair_elim fresh_Nil fresh_Cons)
    apply (simp add: a)
    done
qed

lemma U_app:
  "\<guillemotleft>[A,B,C]\<guillemotright> \<cdot> Umn 2 2 \<approx> A" "\<guillemotleft>[A,B,C]\<guillemotright> \<cdot> Umn 2 1 \<approx> B" "\<guillemotleft>[A,B,C]\<guillemotright> \<cdot> Umn 2 0 \<approx> C"
  by (rule b3, rule Ltgt3_app, rule lam3_fast_app, rule Lam_U, simp_all)
     (rule b3, rule Ltgt3_app, rule lam3_fast_app, rule Lam_U[simplified], simp_all)+

definition "F1 \<equiv> \<integral>cx. (App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> V cx))"
definition "F2 \<equiv> \<integral>cx. \<integral>cy. \<integral>cz. ((App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (V cz \<cdot> V cx))) \<cdot> (V cz \<cdot> V cy))"
definition "F3 \<equiv> \<integral>cx. \<integral>cy. (App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> (\<integral>cz. (V cy \<cdot> (V cx \<cdot> V cz)))))"


lemma Lam_F:
  "F1 = \<integral>x. (App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> V x))"
  "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow> c \<noteq> b \<Longrightarrow> F2 = \<integral>a. \<integral>b. \<integral>c. ((App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (V c \<cdot> V a))) \<cdot> (V c \<cdot> V b))"
  "a \<noteq> b \<Longrightarrow> a \<noteq> x \<Longrightarrow> x \<noteq> b \<Longrightarrow> F3 = \<integral>a. \<integral>b. (App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> (\<integral>x. (V b \<cdot> (V a \<cdot> V x)))))"
  apply (simp_all add: F1_def F2_def F3_def Abs1_eq_iff lam.fresh supp_at_base Var_App_Abs_eqvt numeral_eqvt flip_def[symmetric] fresh_at_base)
  apply (smt cx_cy_cz permute_flip_at)+
  done

lemma supp_F[simp]:
  "supp F1 = {}" "supp F2 = {}" "supp F3 = {}"
  by (simp_all add: F1_def F2_def F3_def lam.supp supp_at_base)
     blast+

lemma F_eqvt[eqvt]:
  "p \<bullet> F1 = F1" "p \<bullet> F2 = F2" "p \<bullet> F3 = F3"
  by (rule_tac [!] perm_supp_eq)
     (simp_all add: fresh_star_def fresh_def)

lemma F_app:
  "F1 \<cdot> A \<approx> App \<cdot> \<lbrace>Var\<rbrace> \<cdot> (Var \<cdot> A)"
  "F2 \<cdot> A \<cdot> B \<cdot> C \<approx> (App \<cdot> (App \<cdot> \<lbrace>App\<rbrace> \<cdot> (C \<cdot> A))) \<cdot> (C \<cdot> B)"
  by (rule lam1_fast_app, rule Lam_F, simp_all)
     (rule lam3_fast_app, rule Lam_F, simp_all)

lemma F3_app:
  assumes f: "atom x \<sharp> A" "atom x \<sharp> B" (* or A and B have empty support *)
  shows "F3 \<cdot> A \<cdot> B \<approx> App \<cdot> \<lbrace>Abs\<rbrace> \<cdot> (Abs \<cdot> (\<integral>x. (B \<cdot> (A \<cdot> V x))))"
proof -
  obtain y :: var where b: "atom y \<sharp> (x, A, B)" using obtain_fresh by blast
  obtain z :: var where c: "atom z \<sharp> (x, y, A, B)" using obtain_fresh by blast
  have *: "x \<noteq> z" "x \<noteq> y" "y \<noteq> z"
    using b c by (simp_all add: fresh_Pair fresh_at_base) blast+
  have **:
    "atom y \<sharp> z" "atom x \<sharp> z" "atom y \<sharp> x"
    "atom z \<sharp> y" "atom z \<sharp> x" "atom x \<sharp> y"
    "atom x \<sharp> A" "atom y \<sharp> A" "atom z \<sharp> A"
    "atom x \<sharp> B" "atom y \<sharp> B" "atom z \<sharp> B"
    using b c f by (simp_all add: fresh_Pair fresh_at_base) blast+
  show ?thesis
    apply (simp add: Lam_F(3)[of y z x] * *[symmetric])
    apply (rule b3) apply (rule b5) apply (rule bI)
    apply (simp add: ** fresh_Pair * *[symmetric])
    apply (rule b3) apply (rule bI)
    apply (simp add: ** fresh_Pair * *[symmetric])
    apply (rule b1)
    done
qed

definition Lam_A1_pre : "A1 \<equiv> \<integral>cx. \<integral>cy. (F1 \<cdot> V cx)"
definition Lam_A2_pre : "A2 \<equiv> \<integral>cx. \<integral>cy. \<integral>cz. (F2 \<cdot> V cx \<cdot> V cy \<cdot> \<guillemotleft>[V cz]\<guillemotright>)"
definition Lam_A3_pre : "A3 \<equiv> \<integral>cx. \<integral>cy. (F3 \<cdot> V cx \<cdot> \<guillemotleft>[V cy]\<guillemotright>)"
lemma Lam_A:
  "x \<noteq> y \<Longrightarrow> A1 = \<integral>x. \<integral>y. (F1 \<cdot> V x)"
  "a \<noteq> b \<Longrightarrow> a \<noteq> c \<Longrightarrow> c \<noteq> b \<Longrightarrow> A2 = \<integral>a. \<integral>b. \<integral>c. (F2 \<cdot> V a \<cdot> V b \<cdot> \<guillemotleft>[V c]\<guillemotright>)"
  "a \<noteq> b \<Longrightarrow> A3 = \<integral>a. \<integral>b. (F3 \<cdot> V a \<cdot> \<guillemotleft>[V b]\<guillemotright>)"
  apply (simp_all add: Lam_A1_pre Lam_A2_pre Lam_A3_pre Abs1_eq_iff lam.fresh supp_at_base Var_App_Abs_eqvt numeral_eqvt flip_def[symmetric] fresh_at_base F_eqvt ltgt_eqvt)
  apply (smt cx_cy_cz permute_flip_at)+
  done

lemma supp_A[simp]:
  "supp A1 = {}" "supp A2 = {}" "supp A3 = {}"
  by (auto simp add: Lam_A1_pre Lam_A2_pre Lam_A3_pre lam.supp supp_at_base supp_Cons supp_Nil)

lemma A_app:
  "A1 \<cdot> A \<cdot> B \<approx> F1 \<cdot> A"
  "A2 \<cdot> A \<cdot> B \<cdot> C \<approx> F2 \<cdot> A \<cdot> B \<cdot> \<guillemotleft>[C]\<guillemotright>"
  "A3 \<cdot> A \<cdot> B \<approx> F3 \<cdot> A \<cdot> \<guillemotleft>[B]\<guillemotright>"
  apply (rule lam2_fast_app, rule Lam_A, simp_all)
  apply (rule lam3_fast_app, rule Lam_A, simp_all)
  apply (rule lam2_fast_app, rule Lam_A, simp_all)
  done

definition "Num \<equiv> \<guillemotleft>[\<guillemotleft>[A1,A2,A3]\<guillemotright>]\<guillemotright>"

lemma supp_Num[simp]:
  "supp Num = {}"
  by (auto simp only: Num_def supp_ltgt supp_Pair supp_A supp_Cons supp_Nil)

end