theory LamEx

imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs"

begin

atom_decl name

datatype rlam =

  rVar "name"

| rApp "rlam" "rlam"

| rLam "name" "rlam"

fun

  rfv :: "rlam \<Rightarrow> atom set"

where

  rfv_var: "rfv (rVar a) = {atom a}"

| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"

| rfv_lam: "rfv (rLam a t) = (rfv t) - {atom a}"

instantiation rlam :: pt

begin

primrec

  permute_rlam

where

  "permute_rlam pi (rVar a) = rVar (pi \<bullet> a)"

| "permute_rlam pi (rApp t1 t2) = rApp (permute_rlam pi t1) (permute_rlam pi t2)"

| "permute_rlam pi (rLam a t) = rLam (pi \<bullet> a) (permute_rlam pi t)"

instance

apply default

apply(induct_tac [!] x)

apply(simp_all)

done

end

instantiation rlam :: fs

begin

lemma neg_conj:

  "\<not>(P \<and> Q) \<longleftrightarrow> (\<not>P) \<or> (\<not>Q)"

  by simp

lemma infinite_Un:

  "infinite (S \<union> T) \<longleftrightarrow> infinite S \<or> infinite T"

  by simp

instance

apply default

apply(induct_tac x)

(* var case *)

apply(simp add: supp_def)

apply(fold supp_def)[1]

apply(simp add: supp_at_base)

(* app case *)

apply(simp only: supp_def)

apply(simp only: permute_rlam.simps) 

apply(simp only: rlam.inject) 

apply(simp only: neg_conj)

apply(simp only: Collect_disj_eq)

apply(simp only: infinite_Un)

apply(simp only: Collect_disj_eq)

apply(simp)

(* lam case *)

apply(simp only: supp_def)

apply(simp only: permute_rlam.simps) 

apply(simp only: rlam.inject) 

apply(simp only: neg_conj)

apply(simp only: Collect_disj_eq)

apply(simp only: infinite_Un)

apply(simp only: Collect_disj_eq)

apply(simp)

apply(fold supp_def)[1]

apply(simp add: supp_at_base)

done

end

(* for the eqvt proof of the alpha-equivalence *)

declare permute_rlam.simps[eqvt]

lemma rfv_eqvt[eqvt]:

  shows "(pi\<bullet>rfv t) = rfv (pi\<bullet>t)"

apply(induct t)

apply(simp_all)

apply(simp add: permute_set_eq atom_eqvt)

apply(simp add: union_eqvt)

apply(simp add: Diff_eqvt)

apply(simp add: permute_set_eq atom_eqvt)

done

inductive

    alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)

where

  a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"

| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"

| a3: "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s)) \<Longrightarrow> rLam a t \<approx> rLam b s"

print_theorems

thm alpha.induct

lemma a3_inverse:

  assumes "rLam a t \<approx> rLam b s"

  shows "\<exists>pi. (({atom a}, t) \<approx>gen alpha rfv pi ({atom b}, s))"

using assms

apply(erule_tac alpha.cases)

apply(auto)

done

text {* should be automatic with new version of eqvt-machinery *}

lemma alpha_eqvt:

  shows "t \<approx> s \<Longrightarrow> (pi \<bullet> t) \<approx> (pi \<bullet> s)"

apply(induct rule: alpha.induct)

apply(simp add: a1)

apply(simp add: a2)

apply(simp)

apply(rule a3)

apply(rule alpha_gen_atom_eqvt)

apply(rule rfv_eqvt)

apply assumption

done

lemma alpha_refl:

  shows "t \<approx> t"

apply(induct t rule: rlam.induct)

apply(simp add: a1)

apply(simp add: a2)

apply(rule a3)

apply(rule_tac x="0" in exI)

apply(rule alpha_gen_refl)

apply(assumption)

done

lemma alpha_sym:

  shows "t \<approx> s \<Longrightarrow> s \<approx> t"

  apply(induct rule: alpha.induct)

  apply(simp add: a1)

  apply(simp add: a2)

  apply(rule a3)

  apply(rule alpha_gen_atom_sym)

  apply(rule alpha_eqvt)

  apply(assumption)+

  done

lemma alpha_trans:

  shows "t1 \<approx> t2 \<Longrightarrow> t2 \<approx> t3 \<Longrightarrow> t1 \<approx> t3"

apply(induct arbitrary: t3 rule: alpha.induct)

apply(simp add: a1)

apply(rotate_tac 4)

apply(erule alpha.cases)

apply(simp_all add: a2)

apply(erule alpha.cases)

apply(simp_all)

apply(rule a3)

apply(rule alpha_gen_atom_trans)

apply(rule alpha_eqvt)

apply(assumption)+

done

lemma alpha_equivp:

  shows "equivp alpha"

  apply(rule equivpI)

  unfolding reflp_def symp_def transp_def

  apply(auto intro: alpha_refl alpha_sym alpha_trans)

  done

lemma alpha_rfv:

  shows "t \<approx> s \<Longrightarrow> rfv t = rfv s"

  apply(induct rule: alpha.induct)

  apply(simp_all add: alpha_gen.simps)

  done

quotient_type lam = rlam / alpha

  by (rule alpha_equivp)

quotient_definition

  "Var :: name \<Rightarrow> lam"

  "rVar"

quotient_definition

   "App :: lam \<Rightarrow> lam \<Rightarrow> lam"

  "rApp"

quotient_definition

  "Lam :: name \<Rightarrow> lam \<Rightarrow> lam"

  "rLam"

quotient_definition

  "fv :: lam \<Rightarrow> atom set"

  "rfv"

lemma perm_rsp[quot_respect]:

  "(op = ===> alpha ===> alpha) permute permute"

  apply(auto)

  apply(rule alpha_eqvt)

  apply(simp)

  done

lemma rVar_rsp[quot_respect]:

  "(op = ===> alpha) rVar rVar"

  by (auto intro: a1)

lemma rApp_rsp[quot_respect]: 

  "(alpha ===> alpha ===> alpha) rApp rApp"

  by (auto intro: a2)

lemma rLam_rsp[quot_respect]: 

  "(op = ===> alpha ===> alpha) rLam rLam"

  apply(auto)

  apply(rule a3)

  apply(rule_tac x="0" in exI)

  unfolding fresh_star_def 

  apply(simp add: fresh_star_def fresh_zero_perm alpha_gen.simps)

  apply(simp add: alpha_rfv)

  done

lemma rfv_rsp[quot_respect]: 

  "(alpha ===> op =) rfv rfv"

apply(simp add: alpha_rfv)

done

section {* lifted theorems *}

lemma lam_induct:

  "\<lbrakk>\<And>name. P (Var name);

    \<And>lam1 lam2. \<lbrakk>P lam1; P lam2\<rbrakk> \<Longrightarrow> P (App lam1 lam2);

    \<And>name lam. P lam \<Longrightarrow> P (Lam name lam)\<rbrakk> 

    \<Longrightarrow> P lam"

  apply (lifting rlam.induct)

  done

instantiation lam :: pt

begin

quotient_definition

  "permute_lam :: perm \<Rightarrow> lam \<Rightarrow> lam"

  "permute :: perm \<Rightarrow> rlam \<Rightarrow> rlam"

lemma permute_lam [simp]:

  shows "pi \<bullet> Var a = Var (pi \<bullet> a)"

  and   "pi \<bullet> App t1 t2 = App (pi \<bullet> t1) (pi \<bullet> t2)"

  and   "pi \<bullet> Lam a t = Lam (pi \<bullet> a) (pi \<bullet> t)"

apply(lifting permute_rlam.simps)

done

instance

apply default

apply(induct_tac [!] x rule: lam_induct)

apply(simp_all)

done

end

lemma fv_lam [simp]: 

  shows "fv (Var a) = {atom a}"

  and   "fv (App t1 t2) = fv t1 \<union> fv t2"

  and   "fv (Lam a t) = fv t - {atom a}"

apply(lifting rfv_var rfv_app rfv_lam)

done

lemma fv_eqvt:

  shows "(p \<bullet> fv t) = fv (p \<bullet> t)"

apply(lifting rfv_eqvt)

done

lemma a1: 

  "a = b \<Longrightarrow> Var a = Var b"

  by  (lifting a1)

lemma a2: 

  "\<lbrakk>x = xa; xb = xc\<rbrakk> \<Longrightarrow> App x xb = App xa xc"

  by  (lifting a2)

lemma alpha_gen_rsp_pre:

  assumes a5: "\<And>t s. R t s \<Longrightarrow> R (pi \<bullet> t) (pi \<bullet> s)"

  and     a1: "R s1 t1"

  and     a2: "R s2 t2"

  and     a3: "\<And>a b c d. R a b \<Longrightarrow> R c d \<Longrightarrow> R1 a c = R2 b d"

  and     a4: "\<And>x y. R x y \<Longrightarrow> fv1 x = fv2 y"

  shows   "(a, s1) \<approx>gen R1 fv1 pi (b, s2) = (a, t1) \<approx>gen R2 fv2 pi (b, t2)"

apply (simp add: alpha_gen.simps)

apply (simp only: a4[symmetric, OF a1] a4[symmetric, OF a2])

apply auto

apply (subst a3[symmetric])

apply (rule a5)

apply (rule a1)

apply (rule a2)

apply (assumption)

apply (subst a3)

apply (rule a5)

apply (rule a1)

apply (rule a2)

apply (assumption)

done

lemma [quot_respect]: "(prod_rel op = alpha ===>

           (alpha ===> alpha ===> op =) ===> (alpha ===> op =) ===> op = ===> prod_rel op = alpha ===> op =)

           alpha_gen alpha_gen"

apply simp

apply clarify

apply (rule alpha_gen_rsp_pre[of "alpha",OF alpha_eqvt])

apply auto

done

(* pi_abs would be also sufficient to prove the next lemma *)

lemma replam_eqvt: "pi \<bullet> (rep_lam x) = rep_lam (pi \<bullet> x)"

apply (unfold rep_lam_def)

sorry

lemma [quot_preserve]: "(prod_fun id rep_lam --->

           (abs_lam ---> abs_lam ---> id) ---> (abs_lam ---> id) ---> id ---> (prod_fun id rep_lam) ---> id)

           alpha_gen = alpha_gen"

apply (simp add: expand_fun_eq alpha_gen.simps Quotient_abs_rep[OF Quotient_lam])

apply (simp add: replam_eqvt)

apply (simp only: Quotient_abs_rep[OF Quotient_lam])

apply auto

done

lemma alpha_prs [quot_preserve]: "(rep_lam ---> rep_lam ---> id) alpha = (op =)"

apply (simp add: expand_fun_eq)

apply (simp add: Quotient_rel_rep[OF Quotient_lam])

done

lemma a3:

  "\<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s) \<Longrightarrow> Lam a t = Lam b s"

  apply (unfold alpha_gen)

  apply (lifting a3[unfolded alpha_gen])

  done

lemma a3_inv:

  "Lam a t = Lam b s \<Longrightarrow> \<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s)"

  apply (unfold alpha_gen)

  apply (lifting a3_inverse[unfolded alpha_gen])

  done

lemma alpha_cases: 

  "\<lbrakk>a1 = a2; \<And>a b. \<lbrakk>a1 = Var a; a2 = Var b; a = b\<rbrakk> \<Longrightarrow> P;

    \<And>t1 t2 s1 s2. \<lbrakk>a1 = App t1 s1; a2 = App t2 s2; t1 = t2; s1 = s2\<rbrakk> \<Longrightarrow> P;

    \<And>a t b s. \<lbrakk>a1 = Lam a t; a2 = Lam b s; \<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s)\<rbrakk>

   \<Longrightarrow> P\<rbrakk>

    \<Longrightarrow> P"

unfolding alpha_gen

apply (lifting alpha.cases[unfolded alpha_gen])

done

(* not sure whether needed *)

lemma alpha_induct: 

  "\<lbrakk>qx = qxa; \<And>a b. a = b \<Longrightarrow> qxb (Var a) (Var b);

    \<And>x xa xb xc. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);

 \<And>a t b s. \<exists>pi. ({atom a}, t) \<approx>gen (\<lambda>x1 x2. x1 = x2 \<and> qxb x1 x2) fv pi ({atom b}, s) \<Longrightarrow> qxb (Lam a t) (Lam b s)\<rbrakk>

    \<Longrightarrow> qxb qx qxa"

unfolding alpha_gen by (lifting alpha.induct[unfolded alpha_gen])

(* should they lift automatically *)

lemma lam_inject [simp]: 

  shows "(Var a = Var b) = (a = b)"

  and   "(App t1 t2 = App s1 s2) = (t1 = s1 \<and> t2 = s2)"

apply(lifting rlam.inject(1) rlam.inject(2))

apply(regularize)

prefer 2

apply(regularize)

prefer 2

apply(auto)

apply(drule alpha.cases)

apply(simp_all)

apply(simp add: alpha.a1)

apply(drule alpha.cases)

apply(simp_all)

apply(drule alpha.cases)

apply(simp_all)

apply(rule alpha.a2)

apply(simp_all)

done

thm a3_inv

lemma Lam_pseudo_inject:

  shows "(Lam a t = Lam b s) = (\<exists>pi. ({atom a}, t) \<approx>gen (op =) fv pi ({atom b}, s))"

apply(rule iffI)

apply(rule a3_inv)

apply(assumption)

apply(rule a3)

apply(assumption)

done

lemma rlam_distinct:

  shows "\<not>(rVar nam \<approx> rApp rlam1' rlam2')"

  and   "\<not>(rApp rlam1' rlam2' \<approx> rVar nam)"

  and   "\<not>(rVar nam \<approx> rLam nam' rlam')"

  and   "\<not>(rLam nam' rlam' \<approx> rVar nam)"

  and   "\<not>(rApp rlam1 rlam2 \<approx> rLam nam' rlam')"

  and   "\<not>(rLam nam' rlam' \<approx> rApp rlam1 rlam2)"

apply auto

apply (erule alpha.cases)

apply (simp_all only: rlam.distinct)

apply (erule alpha.cases)

apply (simp_all only: rlam.distinct)

apply (erule alpha.cases)

apply (simp_all only: rlam.distinct)

apply (erule alpha.cases)

apply (simp_all only: rlam.distinct)

apply (erule alpha.cases)

apply (simp_all only: rlam.distinct)

apply (erule alpha.cases)

apply (simp_all only: rlam.distinct)

done

lemma lam_distinct[simp]:

  shows "Var nam \<noteq> App lam1' lam2'"

  and   "App lam1' lam2' \<noteq> Var nam"

  and   "Var nam \<noteq> Lam nam' lam'"

  and   "Lam nam' lam' \<noteq> Var nam"

  and   "App lam1 lam2 \<noteq> Lam nam' lam'"

  and   "Lam nam' lam' \<noteq> App lam1 lam2"

apply(lifting rlam_distinct(1) rlam_distinct(2) rlam_distinct(3) rlam_distinct(4) rlam_distinct(5) rlam_distinct(6))

done

lemma var_supp1:

  shows "(supp (Var a)) = (supp a)"

  apply (simp add: supp_def)

  done

lemma var_supp:

  shows "(supp (Var a)) = {a:::name}"

  using var_supp1 by (simp add: supp_at_base)

lemma app_supp:

  shows "supp (App t1 t2) = (supp t1) \<union> (supp t2)"

apply(simp only: supp_def lam_inject)

apply(simp add: Collect_imp_eq Collect_neg_eq)

done

(* supp for lam *)

lemma lam_supp1:

  shows "(supp (atom x, t)) supports (Lam x t) "

apply(simp add: supports_def)

apply(fold fresh_def)

apply(simp add: fresh_Pair swap_fresh_fresh)

apply(clarify)

apply(subst swap_at_base_simps(3))

apply(simp_all add: fresh_atom)

done

lemma lam_fsupp1:

  assumes a: "finite (supp t)"

  shows "finite (supp (Lam x t))"

apply(rule supports_finite)

apply(rule lam_supp1)

apply(simp add: a supp_Pair supp_atom)

done

instance lam :: fs

apply(default)

apply(induct_tac x rule: lam_induct)

apply(simp add: var_supp)

apply(simp add: app_supp)

apply(simp add: lam_fsupp1)

done

lemma supp_fv:

  shows "supp t = fv t"

apply(induct t rule: lam_induct)

apply(simp add: var_supp)

apply(simp add: app_supp)

apply(subgoal_tac "supp (Lam name lam) = supp (Abs {atom name} lam)")

apply(simp add: supp_Abs)

apply(simp (no_asm) add: supp_def permute_set_eq atom_eqvt)

apply(simp add: Lam_pseudo_inject)

apply(simp add: Abs_eq_iff)

apply(simp add: alpha_gen.simps)

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

done

lemma lam_supp2:

  shows "supp (Lam x t) = supp (Abs {atom x} t)"

apply(simp add: supp_def permute_set_eq atom_eqvt)

apply(simp add: Lam_pseudo_inject)

apply(simp add: Abs_eq_iff)

apply(simp add: alpha_gen  supp_fv)

done

lemma lam_supp:

  shows "supp (Lam x t) = ((supp t) - {atom x})"

apply(simp add: lam_supp2)

apply(simp add: supp_Abs)

done

lemma fresh_lam:

  "(atom a \<sharp> Lam b t) \<longleftrightarrow> (a = b) \<or> (a \<noteq> b \<and> atom a \<sharp> t)"

apply(simp add: fresh_def)

apply(simp add: lam_supp)