theory Let

imports "../Nominal2" 

begin

atom_decl name

nominal_datatype trm =

  Var "name"

| App "trm" "trm"

| Lam x::"name" t::"trm"    binds  x in t

| Let as::"assn" t::"trm"   binds "bn as" in t

and assn =

  ANil

| ACons "name" "trm" "assn"

binder

  bn

where

  "bn ANil = []"

| "bn (ACons x t as) = (atom x) # (bn as)"

print_theorems

thm alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.intros

thm bn_raw.simps

thm permute_bn_raw.simps

thm trm_assn.perm_bn_alpha

thm trm_assn.permute_bn

thm trm_assn.fv_defs

thm trm_assn.eq_iff 

thm trm_assn.bn_defs

thm trm_assn.bn_inducts

thm trm_assn.perm_simps

thm trm_assn.induct

thm trm_assn.inducts

thm trm_assn.distinct

thm trm_assn.supp

thm trm_assn.fresh

thm trm_assn.exhaust

thm trm_assn.strong_exhaust

thm trm_assn.perm_bn_simps

lemma alpha_bn_inducts_raw[consumes 1]:

  "\<lbrakk>alpha_bn_raw a b; P3 ANil_raw ANil_raw;

 \<And>trm_raw trm_rawa assn_raw assn_rawa name namea.

    \<lbrakk>alpha_trm_raw trm_raw trm_rawa; alpha_bn_raw assn_raw assn_rawa;

     P3 assn_raw assn_rawa\<rbrakk>

    \<Longrightarrow> P3 (ACons_raw name trm_raw assn_raw)

        (ACons_raw namea trm_rawa assn_rawa)\<rbrakk> \<Longrightarrow> P3 a b"

  by (erule alpha_trm_raw_alpha_assn_raw_alpha_bn_raw.inducts(3)[of _ _ "\<lambda>x y. True" _ "\<lambda>x y. True", simplified]) auto

lemmas alpha_bn_inducts[consumes 1] = alpha_bn_inducts_raw[quot_lifted]

lemma alpha_bn_refl: "alpha_bn x x"

  by (induct x rule: trm_assn.inducts(2))

     (rule TrueI, auto simp add: trm_assn.eq_iff)

lemma alpha_bn_sym: "alpha_bn x y \<Longrightarrow> alpha_bn y x"

  sorry

lemma alpha_bn_trans: "alpha_bn x y \<Longrightarrow> alpha_bn y z \<Longrightarrow> alpha_bn x z"

  sorry

lemma bn_inj[rule_format]:

  assumes a: "alpha_bn x y"

  shows "bn x = bn y \<longrightarrow> x = y"

  by (rule alpha_bn_inducts[OF a]) (simp_all add: trm_assn.bn_defs)

lemma bn_inj2:

  assumes a: "alpha_bn x y"

  shows "\<And>q r. (q \<bullet> bn x) = (r \<bullet> bn y) \<Longrightarrow> permute_bn q x = permute_bn r y"

using a

apply(induct rule: alpha_bn_inducts)

apply(simp add: trm_assn.perm_bn_simps)

apply(simp add: trm_assn.perm_bn_simps)

apply(simp add: trm_assn.bn_defs)

apply(simp add: atom_eqvt)

done

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>* c \<Longrightarrow> (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" 

    apply(rule fcb1)

    apply(rule fresh1)

    done

  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[OF fresh1] 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> c \<Longrightarrow> 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

function

  apply_assn2 :: "(trm \<Rightarrow> trm) \<Rightarrow> assn \<Rightarrow> assn"

where

  "apply_assn2 f ANil = ANil"

| "apply_assn2 f (ACons x t as) = ACons x (f t) (apply_assn2 f as)"

  apply(case_tac x)

  apply(case_tac b rule: trm_assn.exhaust(2))

  apply(simp_all)

  apply(blast)

  done

termination by lexicographic_order

lemma apply_assn_eqvt[eqvt]:

  "p \<bullet> (apply_assn2 f a) = apply_assn2 (p \<bullet> f) (p \<bullet> a)"

  apply(induct f a rule: apply_assn2.induct)

  apply simp_all

  apply(perm_simp)

  apply rule

  done

lemma

  fixes x y :: "'a :: fs"

  shows "[a # as]lst. x = [b # bs]lst. y \<Longrightarrow> [[a]]lst. [as]lst. x = [[b]]lst. [bs]lst. y"

  apply (simp add: Abs_eq_iff)

  apply (elim exE)

  apply (rule_tac x="p" in exI)

  apply (simp add: alphas)

  apply clarify

  apply rule

  apply (simp add: supp_Abs)

  apply blast

  apply (simp add: supp_Abs fresh_star_def)

  apply blast

  done

lemma

  assumes neq: "a \<noteq> b" "sort_of a = sort_of b"

  shows "[[a]]lst. [[a]]lst. a = [[a]]lst. [[b]]lst. b \<and> [[a, a]]lst. a \<noteq> [[a, b]]lst. b"

  apply (simp add: Abs1_eq_iff)

  apply rule

  apply (simp add: Abs_eq_iff alphas supp_atom fresh_star_def)

  apply (rule_tac x="(a \<rightleftharpoons> b)" in exI)

  apply (simp add: neq)

  apply (simp add: Abs_eq_iff alphas supp_atom fresh_star_def neq)

  done

nominal_primrec

    subst  :: "name \<Rightarrow> trm \<Rightarrow> trm \<Rightarrow> trm"

where

  "subst s t (Var x) = (if (s = x) then t else (Var x))"

| "subst s t (App l r) = App (subst s t l) (subst s t r)"

| "atom v \<sharp> (s, t) \<Longrightarrow> subst s t (Lam v b) = Lam v (subst s t b)"

| "set (bn as) \<sharp>* (s, t) \<Longrightarrow> subst s t (Let as b) = Let (apply_assn2 (subst s t) as) (subst s t b)"

  apply (simp only: eqvt_def subst_graph_def)

  apply (rule, perm_simp, rule)

  apply (rule TrueI)

  apply (case_tac x)

  apply (rule_tac y="c" and c="(a,b)" in trm_assn.strong_exhaust(1))

  apply (auto simp add: fresh_star_def)[3]

  apply (drule_tac x="assn" in meta_spec)

  apply (simp add: Abs1_eq_iff alpha_bn_refl)

  apply auto[7]

  prefer 2

  apply(simp)

  thm subst_sumC_def

  thm THE_default_def

  thm theI

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

  apply (simp add: Abs_fresh_iff)

  apply (simp add: fresh_star_def)

  apply (simp_all add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def)[2]

  apply (subgoal_tac "apply_assn2 (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) asa

    = apply_assn2 (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) as")

  prefer 2

  apply (erule alpha_bn_inducts)

  apply simp

  apply (simp only: apply_assn2.simps)

  apply simp

--"We know nothing about names; not true; but we can apply fcb2"

  defer

  apply (simp only: )

  apply (erule_tac c="(sa, ta)" in Abs_lst_fcb2)

--"We again need induction for fcb assumption; this time true"

  apply (induct_tac as rule: trm_assn.inducts(2))

  apply (rule TrueI)+

  apply (simp_all add: trm_assn.bn_defs fresh_star_def)[2]

  apply (auto simp add: Abs_fresh_iff)[1]

  apply assumption+

--"But eqvt is not going to be true as well"

  apply (simp add: fresh_star_Pair_elim perm_supp_eq eqvt_at_def trm_assn.fv_bn_eqvt)

  apply (simp add: apply_assn_eqvt)

  apply (drule sym)

  apply (subgoal_tac "p \<bullet> (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2)) = (\<lambda>x2\<Colon>trm. subst_sumC (sa, ta, x2))")

  apply (simp)

  apply (erule alpha_bn_inducts)

  apply simp

  apply simp

  apply (simp add: trm_assn.bn_defs)

--"Again we cannot relate 'namea' with 'p \<bullet> name'"

  prefer 4

  apply (erule alpha_bn_inducts)

  apply simp_all[2]

  oops

end