theory LetRecB

imports "../Nominal2"

begin

atom_decl name

nominal_datatype let_rec:

 trm =

  Var "name"

| App "trm" "trm"

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

| Let_Rec bp::"bp" t::"trm"  bind "bn bp" in bp t

and bp =

  Bp "name" "trm"

binder

  bn::"bp \<Rightarrow> atom list"

where

  "bn (Bp x t) = [atom x]"

thm let_rec.distinct

thm let_rec.induct

thm let_rec.exhaust

thm let_rec.fv_defs

thm let_rec.bn_defs

thm let_rec.perm_simps

thm let_rec.eq_iff

thm let_rec.fv_bn_eqvt

thm let_rec.size_eqvt

lemma Abs_lst_fcb2:

  fixes as bs :: "'a :: fs"

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

    and c::"'c::fs"

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

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

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

  and fresh2: "set (ba 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"

  and props: "eqvt ba" "inj ba"

  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 (ba as))) \<sharp>* (x, c, f as x c, f bs y c)" and 

    fr2: "supp q \<sharp>* ([ba as]lst. x)" and 

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

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

      fin1 fin2

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

  have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = q \<bullet> ([ba as]lst. x)" by simp

  also have "\<dots> = [ba as]lst. x"

    by (simp only: fr2 perm_supp_eq)

  finally have "[q \<bullet> (ba as)]lst. (q \<bullet> x) = [ba bs]lst. y" using eq by simp

  then obtain r::perm where 

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

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

    qq3: "supp r \<subseteq> (q \<bullet> (set (ba as))) \<union> set (ba 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 qq4: "q \<bullet> as = r \<bullet> bs" using qq2 props unfolding eqvt_def inj_on_def

    apply(perm_simp)

    apply(simp)

    done

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

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

    by (simp add: permute_bool_def)

  then have "set (q \<bullet> (ba 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> (ba bs)) \<sharp>* f (r \<bullet> bs) (r \<bullet> y) c" using qq1 qq2 qq4

    by simp

  then have "r \<bullet> ((set (ba 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 (ba 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 qq4 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 max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"

  by (simp add: permute_pure)

nominal_primrec

    height_trm :: "trm \<Rightarrow> nat"

and height_bp :: "bp \<Rightarrow> nat"

where

  "height_trm (Var x) = 1"

| "height_trm (App l r) = max (height_trm l) (height_trm r)"

| "height_trm (Lam v b) = 1 + (height_trm b)"

| "height_trm (Let_Rec bp b) = max (height_bp bp) (height_trm b)"

| "height_bp (Bp v t) = height_trm t"

  --"eqvt"

  apply (simp only: eqvt_def height_trm_height_bp_graph_def)

  apply (rule, perm_simp, rule, rule TrueI)

  --"completeness"

  apply (case_tac x)

  apply (case_tac a rule: let_rec.exhaust(1))

  apply (auto)[4]

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

  apply blast

  apply(simp_all)

  apply (erule_tac c="()" in Abs_lst_fcb2)

  apply (simp_all add: fresh_star_def pure_fresh)[3]

  apply (simp add: eqvt_at_def)

  apply (simp add: eqvt_at_def)

  apply(simp add: eqvt_def)

  apply(perm_simp)

  apply(simp)

  apply(simp add: inj_on_def)

  --"The following could be done by nominal"

  apply (simp add: meta_eq_to_obj_eq[OF height_trm_def, symmetric, unfolded fun_eq_iff])

  apply (simp add: meta_eq_to_obj_eq[OF height_bp_def, symmetric, unfolded fun_eq_iff])

  apply (subgoal_tac "eqvt_at height_bp bp")

  apply (subgoal_tac "eqvt_at height_bp bpa")

  apply (subgoal_tac "eqvt_at height_trm b")

  apply (subgoal_tac "eqvt_at height_trm ba")

  apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bp)")

  apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inr bpa)")

  apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl b)")

  apply (thin_tac "eqvt_at height_trm_height_bp_sumC (Inl ba)")

  defer

  apply (simp add: eqvt_at_def height_trm_def)

  apply (simp add: eqvt_at_def height_trm_def)

  apply (simp add: eqvt_at_def height_bp_def)

  apply (simp add: eqvt_at_def height_bp_def)

  apply (subgoal_tac "height_bp bp = height_bp bpa")

  apply (subgoal_tac "height_trm b = height_trm ba")

  apply simp

  apply (subgoal_tac "(\<lambda>as x c. height_trm (snd (bp, b))) as x c = (\<lambda>as x c. height_trm (snd (bpa, ba))) as x c")

  apply simp

  apply (erule_tac c="()" in Abs_lst_fcb2)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: eqvt_at_def)

  apply (simp add: eqvt_at_def)

  defer defer

  apply (subgoal_tac "(\<lambda>as x c. height_bp (fst (bp, b))) as x c = (\<lambda>as x c. height_bp (fst (bpa, ba))) as x c")

  apply simp

  apply (erule_tac c="()" in Abs_lst_fcb2)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: fresh_star_def pure_fresh)

  apply (simp add: eqvt_at_def)

  apply (simp add: eqvt_at_def)

--""

  apply(simp_all add: eqvt_def inj_on_def)

  apply(perm_simp)

  apply(simp)

  apply(perm_simp)

  apply(simp)

  done

termination by lexicographic_order

end