theory Lambda

imports "../Nominal2" 

begin

lemma Abs_lst_fcb2:

  fixes as bs :: "atom list"

    and x y :: "'b :: fs"

    and c::"'c::fs"

  assumes eq: "[as]lst. x = [bs]lst. y"

  and fcb1: "(set as) \<sharp>* f as x c"

  and fresh1: "set as \<sharp>* c"

  and fresh2: "set bs \<sharp>* c"

  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f as x c) = f (p \<bullet> as) (p \<bullet> x) c"

  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f bs y c) = f (p \<bullet> bs) (p \<bullet> y) c"

  shows "f as x c = f bs y c"

proof -

  have "supp (as, x, c) supports (f as x c)"

    unfolding  supports_def fresh_def[symmetric]

    by (simp add: fresh_Pair perm1 fresh_star_def supp_swap swap_fresh_fresh)

  then have fin1: "finite (supp (f as x c))"

    by (auto intro: supports_finite simp add: finite_supp)

  have "supp (bs, y, c) supports (f bs y c)"

    unfolding  supports_def fresh_def[symmetric]

    by (simp add: fresh_Pair perm2 fresh_star_def supp_swap swap_fresh_fresh)

  then have fin2: "finite (supp (f bs y c))"

    by (auto intro: supports_finite simp add: finite_supp)

  obtain q::"perm" where 

    fr1: "(q \<bullet> (set as)) \<sharp>* (x, c, f as x c, f bs y c)" and 

    fr2: "supp q \<sharp>* Abs_lst as x" and 

    inc: "supp q \<subseteq> (set as) \<union> q \<bullet> (set as)"

    using at_set_avoiding3[where xs="set as" and c="(x, c, f as x c, f bs y c)" and x="[as]lst. x"]  

      fin1 fin2

    by (auto simp add: supp_Pair finite_supp Abs_fresh_star dest: fresh_star_supp_conv)

  have "Abs_lst (q \<bullet> as) (q \<bullet> x) = q \<bullet> Abs_lst as x" by simp

  also have "\<dots> = Abs_lst as x"

    by (simp only: fr2 perm_supp_eq)

  finally have "Abs_lst (q \<bullet> as) (q \<bullet> x) = Abs_lst bs y" using eq by simp

  then obtain r::perm where 

    qq1: "q \<bullet> x = r \<bullet> y" and 

    qq2: "q \<bullet> as = r \<bullet> bs" and 

    qq3: "supp r \<subseteq> (q \<bullet> (set as)) \<union> set bs"

    apply(drule_tac sym)

    apply(simp only: Abs_eq_iff2 alphas)

    apply(erule exE)

    apply(erule conjE)+

    apply(drule_tac x="p" in meta_spec)

    apply(simp add: set_eqvt)

    apply(blast)

    done

  have "(set as) \<sharp>* f as x c" by (rule fcb1)

  then have "q \<bullet> ((set as) \<sharp>* f as x c)"

    by (simp add: permute_bool_def)

  then have "set (q \<bullet> as) \<sharp>* f (q \<bullet> as) (q \<bullet> x) c"

    apply(simp add: fresh_star_eqvt set_eqvt)

    apply(subst (asm) perm1)

    using inc fresh1 fr1

    apply(auto simp add: fresh_star_def fresh_Pair)

    done

  then have "set (r \<bullet> bs) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp

  then have "r \<bullet> ((set bs) \<sharp>* f bs y c)"

    apply(simp add: fresh_star_eqvt set_eqvt)

    apply(subst (asm) perm2[symmetric])

    using qq3 fresh2 fr1

    apply(auto simp add: set_eqvt fresh_star_def fresh_Pair)

    done

  then have fcb2: "(set bs) \<sharp>* f bs y c" by (simp add: permute_bool_def)

  have "f as x c = q \<bullet> (f as x c)"

    apply(rule perm_supp_eq[symmetric])

    using inc fcb1 fr1 by (auto simp add: fresh_star_def)

  also have "\<dots> = f (q \<bullet> as) (q \<bullet> x) c" 

    apply(rule perm1)

    using inc fresh1 fr1 by (auto simp add: fresh_star_def)

  also have "\<dots> = f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 by simp

  also have "\<dots> = r \<bullet> (f bs y c)"

    apply(rule perm2[symmetric])

    using qq3 fresh2 fr1 by (auto simp add: fresh_star_def)

  also have "... = f bs y c"

    apply(rule perm_supp_eq)

    using qq3 fr1 fcb2 by (auto simp add: fresh_star_def)

  finally show ?thesis by simp

qed

lemma Abs_lst1_fcb2:

  fixes a b :: "atom"

    and x y :: "'b :: fs"

    and c::"'c :: fs"

  assumes e: "(Abs_lst [a] x) = (Abs_lst [b] y)"

  and fcb1: "a \<sharp> f a x c"

  and fresh: "{a, b} \<sharp>* c"

  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"

  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"

  shows "f a x c = f b y c"

using e

apply(drule_tac Abs_lst_fcb2[where c="c" and f="\<lambda>(as::atom list) . f (hd as)"])

apply(simp_all)

using fcb1 fresh perm1 perm2

apply(simp_all add: fresh_star_def)

done

lemma Abs_lst1_fcb2':

  fixes a b :: "'a::at"

    and x y :: "'b :: fs"

    and c::"'c :: fs"

  assumes e: "(Abs_lst [atom a] x) = (Abs_lst [atom b] y)"

  and fcb1: "atom a \<sharp> f a x c"

  and fresh: "{atom a, atom b} \<sharp>* c"

  and perm1: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f a x c) = f (p \<bullet> a) (p \<bullet> x) c"

  and perm2: "\<And>p. supp p \<sharp>* c \<Longrightarrow> p \<bullet> (f b y c) = f (p \<bullet> b) (p \<bullet> y) c"

  shows "f a x c = f b y c"

using e

apply(drule_tac Abs_lst1_fcb2[where c="c" and f="\<lambda>a . f ((inv atom) a)"])

using  fcb1 fresh perm1 perm2

apply(simp_all add: fresh_star_def inv_f_f inj_on_def atom_eqvt)

done

atom_decl name

nominal_datatype lam =

  Var "name"

| App "lam" "lam"

| Lam x::"name" l::"lam"  bind x in l ("Lam [_]. _" [100, 100] 100)

ML {* Method.SIMPLE_METHOD' *}

ML {* Sign.intern_const *}

ML {*

val test:((Proof.context -> Method.method) context_parser) =

Scan.succeed (fn ctxt =>

 let

   val _ = Inductive.the_inductive ctxt "local.frees_lst_graph"

 in 

   Method.SIMPLE_METHOD' (K all_tac)

 end)

*}

method_setup test = {* test *} {* test *}

section {* free name function *}

text {* first returns an atom list *}

ML Thm.implies_intr

nominal_primrec 

  frees_lst :: "lam \<Rightarrow> atom list"

where

  "frees_lst (Var x) = [atom x]"

| "frees_lst (App t1 t2) = frees_lst t1 @ frees_lst t2"

| "frees_lst (Lam [x]. t) = removeAll (atom x) (frees_lst t)"

  unfolding eqvt_def

  unfolding frees_lst_graph_def

  apply (rule, perm_simp, rule)

apply(rule TrueI)

apply(rule_tac y="x" in lam.exhaust)

apply(auto)

apply (erule_tac c="()" in Abs_lst1_fcb2)

apply(simp add: supp_removeAll fresh_def)

apply(simp add: fresh_star_def fresh_Unit)

apply(simp add: eqvt_at_def removeAll_eqvt atom_eqvt)

apply(simp add: eqvt_at_def removeAll_eqvt atom_eqvt)

done

termination

  by lexicographic_order

text {* a small test lemma *}

lemma shows "supp t = set (frees_lst t)"

  by (induct t rule: frees_lst.induct) (simp_all add: lam.supp supp_at_base)

text {* second returns an atom set - therefore needs an invariant *}

nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")

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

where

  "frees_set (Var x) = {atom x}"

| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"

| "frees_set (Lam [x]. t) = (frees_set t) - {atom x}"

  apply(simp add: eqvt_def frees_set_graph_def)

  apply(rule, perm_simp, rule)

  apply(erule frees_set_graph.induct)

  apply(auto)[9]

  apply(rule_tac y="x" in lam.exhaust)

  apply(auto)[3]

  apply(simp)

  apply(erule_tac c="()" in Abs_lst1_fcb2)

  apply(simp add: fresh_minus_atom_set)

  apply(simp add: fresh_star_def fresh_Unit)

  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)

  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)

  done

termination 

  by lexicographic_order

lemma "frees_set t = supp t"

  by (induct rule: frees_set.induct) (simp_all add: lam.supp supp_at_base)

lemma fresh_fun_eqvt_app3:

  assumes a: "eqvt f"

  and b: "a \<sharp> x" "a \<sharp> y" "a \<sharp> z"

  shows "a \<sharp> f x y z"

  using fresh_fun_eqvt_app[OF a b(1)] a b

  by (metis fresh_fun_app)

section {* A test with a locale and an interpretation. *}

text {* conclusion: it is no necessary *}

locale test =

   fixes f1::"name \<Rightarrow> ('a::pt)"

     and f2::"lam \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> 'a \<Rightarrow> ('a::pt)"

     and f3::"name \<Rightarrow> lam \<Rightarrow> 'a \<Rightarrow> ('a::pt)"

   assumes fs: "finite (supp (f1, f2, f3))"

       and eq: "eqvt f1" "eqvt f2" "eqvt f3"

       and fcb: "\<forall>x t r. atom x \<sharp> f3 x t r"

begin

nominal_primrec

  f :: "lam \<Rightarrow> ('a::pt)"

where

  "f (Var x) = f1 x"

| "f (App t1 t2) = f2 t1 t2 (f t1) (f t2)"

| "f (Lam [x].t) = f3 x t (f t)"

  apply (simp add: eqvt_def f_graph_def)

  apply (perm_simp)

  apply(simp add: eq[simplified eqvt_def])

  apply(rule TrueI)

  apply(rule_tac y="x" in lam.exhaust)

  apply(auto simp add: fresh_star_def)

  apply(erule Abs_lst1_fcb)

  apply simp_all

  apply(simp add: fcb)

  apply (rule fresh_fun_eqvt_app3[OF eq(3)])

  apply (simp add: fresh_at_base)

  apply assumption

  apply (erule fresh_eqvt_at)

  apply (simp add: finite_supp)

  apply assumption

  apply (subgoal_tac "\<And>p y r. p \<bullet> (f3 x y r) = f3 (p \<bullet> x) (p \<bullet> y) (p \<bullet> r)")

  apply (simp add: eqvt_at_def)

  apply (simp add: permute_fun_app_eq eq[unfolded eqvt_def])

  done

termination

  by lexicographic_order

end

interpretation hei: test

  "%n. (1 :: nat)"

  "%t1 t2 r1 r2. (r1 + r2)"

  "%n t r. r + 1"

  apply default

  apply (auto simp add: pure_fresh supp_Pair)

  apply (simp_all add: fresh_def supp_def permute_fun_def permute_pure)[3]

  apply (simp_all add: eqvt_def permute_fun_def permute_pure)

  done

section {* height function *}

nominal_primrec

  height :: "lam \<Rightarrow> int"

where

  "height (Var x) = 1"

| "height (App t1 t2) = max (height t1) (height t2) + 1"

| "height (Lam [x].t) = height t + 1"

  apply(simp add: eqvt_def height_graph_def)

  apply (rule, perm_simp, rule)

  apply(rule TrueI)

  apply(rule_tac y="x" in lam.exhaust)

  apply(auto)

  apply (erule_tac c="()" in Abs_lst1_fcb2)

  apply(simp_all add: fresh_def pure_supp eqvt_at_def fresh_star_def)

  done

termination

  by lexicographic_order

thm height.simps

section {* capture-avoiding substitution *}

nominal_primrec

  subst :: "lam \<Rightarrow> name \<Rightarrow> lam \<Rightarrow> lam"  ("_ [_ ::= _]" [90, 90, 90] 90)

where

  "(Var x)[y ::= s] = (if x = y then s else (Var x))"

| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"

| "atom x \<sharp> (y, s) \<Longrightarrow> (Lam [x]. t)[y ::= s] = Lam [x].(t[y ::= s])"

  unfolding eqvt_def subst_graph_def

  apply (rule, perm_simp, rule)

  apply(rule TrueI)

  apply(auto simp add: lam.distinct lam.eq_iff)

  apply(rule_tac y="a" and c="(aa, b)" in lam.strong_exhaust)

  apply(blast)+

  apply(simp_all add: fresh_star_def fresh_Pair_elim)

  apply (erule_tac c="(ya,sa)" in Abs_lst1_fcb2)

  apply(simp_all add: Abs_fresh_iff)

  apply(simp add: fresh_star_def fresh_Pair)

  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)

  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)

done

termination

  by lexicographic_order

lemma subst_eqvt[eqvt]:

  shows "(p \<bullet> t[x ::= s]) = (p \<bullet> t)[(p \<bullet> x) ::= (p \<bullet> s)]"

by (induct t x s rule: subst.induct) (simp_all)

lemma forget:

  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"

  by (nominal_induct t avoiding: x s rule: lam.strong_induct)

     (auto simp add: lam.fresh fresh_at_base)

text {* same lemma but with subst.induction *}

lemma forget2:

  shows "atom x \<sharp> t \<Longrightarrow> t[x ::= s] = t"

  by (induct t x s rule: subst.induct)

     (auto simp add: lam.fresh fresh_at_base fresh_Pair)

lemma fresh_fact:

  fixes z::"name"

  assumes a: "atom z \<sharp> s"

      and b: "z = y \<or> atom z \<sharp> t"

  shows "atom z \<sharp> t[y ::= s]"

  using a b

  by (nominal_induct t avoiding: z y s rule: lam.strong_induct)

      (auto simp add: lam.fresh fresh_at_base)

lemma substitution_lemma:  

  assumes a: "x \<noteq> y" "atom x \<sharp> u"

  shows "t[x ::= s][y ::= u] = t[y ::= u][x ::= s[y ::= u]]"

using a 

by (nominal_induct t avoiding: x y s u rule: lam.strong_induct)

   (auto simp add: fresh_fact forget)

lemma subst_rename: 

  assumes a: "atom y \<sharp> t"

  shows "t[x ::= s] = ((y \<leftrightarrow> x) \<bullet>t)[y ::= s]"

using a 

apply (nominal_induct t avoiding: x y s rule: lam.strong_induct)

apply (auto simp add: lam.fresh fresh_at_base)

done

lemma height_ge_one:

  shows "1 \<le> (height e)"

by (induct e rule: lam.induct) (simp_all)

theorem height_subst:

  shows "height (e[x::=e']) \<le> ((height e) - 1) + (height e')"

proof (nominal_induct e avoiding: x e' rule: lam.strong_induct)

  case (Var y)

  have "1 \<le> height e'" by (rule height_ge_one)

  then show "height (Var y[x::=e']) \<le> height (Var y) - 1 + height e'" by simp

next

  case (Lam y e1)

  hence ih: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')" by simp

  moreover

  have vc: "atom y\<sharp>x" "atom y\<sharp>e'" by fact+ (* usual variable convention *)

  ultimately show "height ((Lam [y]. e1)[x::=e']) \<le> height (Lam [y]. e1) - 1 + height e'" by simp

next

  case (App e1 e2)

  hence ih1: "height (e1[x::=e']) \<le> ((height e1) - 1) + (height e')"

    and ih2: "height (e2[x::=e']) \<le> ((height e2) - 1) + (height e')" by simp_all

  then show "height ((App e1 e2)[x::=e']) \<le> height (App e1 e2) - 1 + height e'"  by simp

qed

subsection {* single-step beta-reduction *}

inductive 

  beta :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>b _" [80,80] 80)

where

  b1[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> App t1 s \<longrightarrow>b App t2 s"

| b2[intro]: "s1 \<longrightarrow>b s2 \<Longrightarrow> App t s1 \<longrightarrow>b App t s2"

| b3[intro]: "t1 \<longrightarrow>b t2 \<Longrightarrow> Lam [x]. t1 \<longrightarrow>b Lam [x]. t2"

| b4[intro]: "atom x \<sharp> s \<Longrightarrow> App (Lam [x]. t) s \<longrightarrow>b t[x ::= s]"

equivariance beta

nominal_inductive beta

  avoids b4: "x"

  by (simp_all add: fresh_star_def fresh_Pair lam.fresh fresh_fact)

text {* One-Reduction *}

inductive 

  One :: "lam \<Rightarrow> lam \<Rightarrow> bool" (" _ \<longrightarrow>1 _" [80,80] 80)

where