theory ExLet

imports "Parser"

begin

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

atom_decl name

ML {* val _ = recursive := false  *}

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

and lts =

  Lnil

| Lcons "name" "trm" "lts"

binder

  bn

where

  "bn Lnil = {}"

| "bn (Lcons x t l) = {atom x} \<union> (bn l)"

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.fv[simplified trm_lts.supp]

consts

  permute_bn :: "perm \<Rightarrow> lts \<Rightarrow> lts"

lemma test:

  "permute_bn pi (Lnil) = Lnil"

  "permute_bn pi (Lcons a t l) = Lcons (pi \<bullet> a) t (permute_bn pi l)"

  sorry

lemma 

  fixes t::trm

  and   l::lts

  and   c::"'a::fs"

  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     a4: "\<And>lts trm c. \<lbrakk>bn lts \<sharp>* (c, Lt lts trm); \<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

    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> bn (p \<bullet> lts)) \<sharp>* c \<and> supp (Abs (bn (p \<bullet> lts)) (p \<bullet> trm)) \<sharp>* q")

    apply(erule exE)

    (* HERE *)

    apply(rule_tac t="Lt (p \<bullet> lts) (p \<bullet> trm)" 

               and s="Lt (permute_bn q (p \<bullet> lts)) (q \<bullet> p \<bullet> trm)" in subst)

    defer

    apply(rule a4)

    defer

    apply(simp add: eqvts)

    apply(rule at_set_avoiding2)

    apply(simp add: finite_supp)

    defer

    apply(simp add: finite_supp)

    apply(simp add: finite_supp)

    apply(simp add: fresh_star_def)

    apply(simp add: fresh_def)

    thm trm_lts.fv[simplified trm_lts.supp]

    apply(simp add: trm_lts.fv[simplified trm_lts.supp])

    apply(simp add: alpha_bn_eq_iff)

    defer

    apply(simp)

    apply(rule a5)

    apply(simp)

    apply(rule a6)

    apply(simp)

    apply(simp)

    oops

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:

  "(Lt (Lcons x (Vr x) (Lcons y (Vr y) Lnil)) (Ap (Vr x) (Vr y))) =

   (Lt (Lcons y (Vr x) (Lcons x (Vr y) Lnil)) (Ap (Vr x) (Vr y)))"

  apply (simp add: alphas trm_lts.eq_iff)

  apply (rule_tac x="0::perm" in exI)

  apply (simp add: fresh_star_def eqvts)

  apply blast

  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