theory ExLet
imports "../Parser" "../../Attic/Prove"
begin

text {* example 3 or example 5 from Terms.thy *}

atom_decl name

ML {* val _ = recursive := false *}
ML {* val _ = alpha_type := AlphaLst *}
nominal_datatype trm =
  Vr "name"
| Ap "trm" "trm"
| Lm x::"name" t::"trm"  bind x in t
| Lt a::"lts" t::"trm"   bind "bn a" in t
(*| L a::"lts" t1::"trm" t2::"trm"  bind "bn a" in t1, bind "bn a" in t2*)
and lts =
  Lnil
| Lcons "name" "trm" "lts"
binder
  bn
where
  "bn Lnil = []"
| "bn (Lcons x t l) = (atom x) # (bn l)"


thm alpha_trm_raw_alpha_lts_raw_alpha_bn_raw.intros

thm trm_lts.fv
thm trm_lts.eq_iff
thm trm_lts.bn
thm trm_lts.perm
thm trm_lts.induct[no_vars]
thm trm_lts.inducts[no_vars]
thm trm_lts.distinct
(*thm trm_lts.supp*)
thm trm_lts.fv[simplified trm_lts.supp(1-2)]


primrec
  permute_bn_raw
where
  "permute_bn_raw pi (Lnil_raw) = Lnil_raw"
| "permute_bn_raw pi (Lcons_raw a t l) = Lcons_raw (pi \<bullet> a) t (permute_bn_raw pi l)"

quotient_definition
  "permute_bn :: perm \<Rightarrow> lts \<Rightarrow> lts"
is
  "permute_bn_raw"

lemma [quot_respect]: "((op =) ===> alpha_lts_raw ===> alpha_lts_raw) permute_bn_raw permute_bn_raw"
  apply simp
  apply clarify
  apply (erule alpha_trm_raw_alpha_lts_raw_alpha_bn_raw.inducts)
  apply simp_all
  apply (rule alpha_trm_raw_alpha_lts_raw_alpha_bn_raw.intros)
  apply simp
  apply (rule alpha_trm_raw_alpha_lts_raw_alpha_bn_raw.intros)
  apply simp
  done

lemmas permute_bn = permute_bn_raw.simps[quot_lifted]

lemma permute_bn_zero:
  "permute_bn 0 a = a"
  apply(induct a rule: trm_lts.inducts(2))
  apply(rule TrueI)
  apply(simp_all add:permute_bn eqvts)
  done

lemma permute_bn_add:
  "permute_bn (p + q) a = permute_bn p (permute_bn q a)"
  oops

lemma permute_bn_alpha_bn: "alpha_bn lts (permute_bn q lts)"
  apply(induct lts rule: trm_lts.inducts(2))
  apply(rule TrueI)
  apply(simp_all add:permute_bn eqvts trm_lts.eq_iff)
  done

lemma perm_bn:
  "p \<bullet> bn l = bn(permute_bn p l)"
  apply(induct l rule: trm_lts.inducts(2))
  apply(rule TrueI)
  apply(simp_all add:permute_bn eqvts)
  done

lemma fv_perm_bn:
  "fv_bn l = fv_bn (permute_bn p l)"
  apply(induct l rule: trm_lts.inducts(2))
  apply(rule TrueI)
  apply(simp_all add:permute_bn eqvts)
  done

lemma fv_fv_bn:
  "fv_bn l - set (bn l) = fv_lts l - set (bn l)"
  apply(induct l rule: trm_lts.inducts(2))
  apply(rule TrueI)
  apply(simp_all add:permute_bn eqvts)
  by blast

lemma Lt_subst:
  "supp (Abs_lst (bn lts) trm) \<sharp>* q \<Longrightarrow> (Lt lts trm) = Lt (permute_bn q lts) (q \<bullet> trm)"
  apply (simp only: trm_lts.eq_iff)
  apply (rule_tac x="q" in exI)
  apply (simp add: alphas)
  apply (simp add: permute_bn_alpha_bn)
  apply (simp add: perm_bn[symmetric])
  apply (simp add: eqvts[symmetric])
  apply (simp add: supp_abs)
  apply (simp add: trm_lts.supp)
  apply (rule supp_perm_eq[symmetric])
  apply (subst supp_finite_atom_set)
  apply (rule finite_Diff)
  apply (simp add: finite_supp)
  apply (assumption)
  done


lemma fin_bn:
  "finite (set (bn l))"
  apply(induct l rule: trm_lts.inducts(2))
  apply(simp_all add:permute_bn eqvts)
  done

thm trm_lts.inducts[no_vars]

lemma 
  fixes t::trm
  and   l::lts
  and   c::"'a::fs"
  assumes a1: "\<And>name c. P1 c (Vr name)"
  and     a2: "\<And>trm1 trm2 c. \<lbrakk>\<And>d. P1 d trm1; \<And>d. P1 d trm2\<rbrakk> \<Longrightarrow> P1 c (Ap trm1 trm2)"
  and     a3: "\<And>name trm c. \<lbrakk>atom name \<sharp> c; \<And>d. P1 d trm\<rbrakk> \<Longrightarrow> P1 c (Lm name trm)"
  and     a4: "\<And>lts trm c. \<lbrakk>set (bn lts) \<sharp>* c; \<And>d. P2 d lts; \<And>d. P1 d trm\<rbrakk> \<Longrightarrow> P1 c (Lt lts trm)"
  and     a5: "\<And>c. P2 c Lnil"
  and     a6: "\<And>name trm lts c. \<lbrakk>\<And>d. P1 d trm; \<And>d. P2 d lts\<rbrakk> \<Longrightarrow> P2 c (Lcons name trm lts)"
  shows "P1 c t" and "P2 c l"
proof -
  have "(\<And>(p::perm) (c::'a::fs). P1 c (p \<bullet> t))" and
       b': "(\<And>(p::perm) (q::perm) (c::'a::fs). P2 c (permute_bn p (q \<bullet> l)))"
    apply(induct rule: trm_lts.inducts)
    apply(simp)
    apply(rule a1)
    apply(simp)
    apply(rule a2)
    apply(simp)
    apply(simp)
    apply(simp)
    apply(subgoal_tac "\<exists>q. (q \<bullet> (atom (p \<bullet> name))) \<sharp> c \<and> supp (Lm (p \<bullet> name) (p \<bullet> trm)) \<sharp>* q")
    apply(erule exE)
    apply(rule_tac t="Lm (p \<bullet> name) (p \<bullet> trm)" 
               and s="q\<bullet> Lm (p \<bullet> name) (p \<bullet> trm)" in subst)
    apply(rule supp_perm_eq)
    apply(simp)
    apply(simp)
    apply(rule a3)
    apply(simp add: atom_eqvt)
    apply(subst permute_plus[symmetric])
    apply(blast)
    apply(rule at_set_avoiding2_atom)
    apply(simp add: finite_supp)
    apply(simp add: finite_supp)
    apply(simp add: fresh_def)
    apply(simp add: trm_lts.fv[simplified trm_lts.supp])
    apply(simp)
    apply(subgoal_tac "\<exists>q. (q \<bullet> set (bn (p \<bullet> lts))) \<sharp>* c \<and> supp (Abs_lst (bn (p \<bullet> lts)) (p \<bullet> trm)) \<sharp>* q")
    apply(erule exE)
    apply(erule conjE)
    thm Lt_subst
    apply(subst Lt_subst)
    apply assumption
    apply(rule a4)
    apply(simp add:perm_bn[symmetric])
    apply(simp add: eqvts)
    apply (simp add: fresh_star_def fresh_def)
    apply(rotate_tac 1)
    apply(drule_tac x="q + p" in meta_spec)
    apply(simp)
    apply(rule at_set_avoiding2)
    apply(rule fin_bn)
    apply(simp add: finite_supp)
    apply(simp add: finite_supp)
    apply(simp add: fresh_star_def fresh_def supp_abs)
    apply(simp add: eqvts permute_bn)
    apply(rule a5)
    apply(simp add: permute_bn)
    apply(rule a6)
    apply simp
    apply simp
    done
  then have a: "P1 c (0 \<bullet> t)" by blast
  have "P2 c (permute_bn 0 (0 \<bullet> l))" using b' by blast
  then show "P1 c t" and "P2 c l" using a permute_bn_zero by simp_all
qed



lemma lets_bla:
  "x \<noteq> z \<Longrightarrow> y \<noteq> z \<Longrightarrow> x \<noteq> y \<Longrightarrow>(Lt (Lcons x (Vr y) Lnil) (Vr x)) \<noteq> (Lt (Lcons x (Vr z) Lnil) (Vr x))"
  by (simp add: trm_lts.eq_iff)

lemma lets_ok:
  "(Lt (Lcons x (Vr y) Lnil) (Vr x)) = (Lt (Lcons y (Vr y) Lnil) (Vr y))"
  apply (simp add: trm_lts.eq_iff)
  apply (rule_tac x="(x \<leftrightarrow> y)" in exI)
  apply (simp_all add: alphas)
  apply (simp add: fresh_star_def eqvts)
  done

lemma lets_ok3:
  "x \<noteq> y \<Longrightarrow>
   (Lt (Lcons x (Ap (Vr y) (Vr x)) (Lcons y (Vr y) Lnil)) (Ap (Vr x) (Vr y))) \<noteq>
   (Lt (Lcons y (Ap (Vr x) (Vr y)) (Lcons x (Vr x) Lnil)) (Ap (Vr x) (Vr y)))"
  apply (simp add: alphas trm_lts.eq_iff)
  done


lemma lets_not_ok1:
  "x \<noteq> y \<Longrightarrow>
   (Lt (Lcons x (Vr x) (Lcons y (Vr y) Lnil)) (Ap (Vr x) (Vr y))) \<noteq>
   (Lt (Lcons y (Vr x) (Lcons x (Vr y) Lnil)) (Ap (Vr x) (Vr y)))"
  apply (simp add: alphas trm_lts.eq_iff fresh_star_def eqvts)
  done

lemma lets_nok:
  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>
   (Lt (Lcons x (Ap (Vr z) (Vr z)) (Lcons y (Vr z) Lnil)) (Ap (Vr x) (Vr y))) \<noteq>
   (Lt (Lcons y (Vr z) (Lcons x (Ap (Vr z) (Vr z)) Lnil)) (Ap (Vr x) (Vr y)))"
  apply (simp add: alphas trm_lts.eq_iff fresh_star_def)
  done


end