theory ExLet

imports "Parser" "../Attic/Prove"

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]

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 Lt_subst:

  "supp (Abs (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: 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 (bn l)"

  apply(induct l rule: trm_lts.inducts(2))

  apply(simp_all add:permute_bn eqvts)

  done

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

    apply(erule exE)

    apply(erule conjE)

    apply(subst Lt_subst)

    apply assumption

    apply(rule a4)

    apply(simp add:perm_bn)

    apply assumption

    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(rule finite_Diff)

    apply(simp add: finite_supp)

    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:

  "(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