theory Let

imports "../Nominal2" 

begin

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

atom_decl name

nominal_datatype trm =

  Var "name"

| App "trm" "trm"

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

and assn =

  Assn "name" "trm"

binder

  bn

where

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

print_theorems

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

thm alpha_bn_raw.cases

lemmas alpha_bn_cases[consumes 1] = alpha_bn_raw.cases[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"

  apply(erule alpha_bn_cases)

  apply(auto)

  done

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

  sorry

lemma k: "A \<Longrightarrow> A \<and> A" by blast

lemma max_eqvt[eqvt]: "p \<bullet> (max (a :: _ :: pure) b) = max (p \<bullet> a) (p \<bullet> b)"

  by (simp add: permute_pure)

section {* definition with helper functions *}

function 

  apply_assn

where

  "apply_assn f (Assn x t) = (f t)"

apply(case_tac x)

apply(simp)

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

apply(blast)

apply(simp)

done

termination

  by lexicographic_order

function 

  apply_assn2

where

  "apply_assn2 f (Assn x t) = Assn x (f t)"

apply(case_tac x)

apply(simp)

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

apply(blast)

apply(simp)

done

termination

  by lexicographic_order

lemma [eqvt]:

  shows "p \<bullet> (apply_assn f as) = apply_assn (p \<bullet> f) (p \<bullet> as)"

apply(induct f as rule: apply_assn.induct)

apply(simp)

apply(perm_simp)

apply(rule)

done

lemma [eqvt]:

  shows "p \<bullet> (apply_assn2 f as) = apply_assn2 (p \<bullet> f) (p \<bullet> as)"

apply(induct f as rule: apply_assn.induct)

apply(simp)

apply(perm_simp)

apply(rule)

done

nominal_primrec

    height_trm :: "trm \<Rightarrow> nat"

where

  "height_trm (Var x) = 1"

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

| "height_trm (Let as b) = max (apply_assn height_trm as) (height_trm b)"

  apply (simp only: eqvt_def height_trm_graph_def)

  apply (rule, perm_simp)

  apply(rule)

  apply(rule TrueI)

  apply (case_tac x rule: trm_assn.exhaust(1))

  apply (auto simp add: alpha_bn_refl)[3]

  apply (drule_tac x="assn" in meta_spec)

  apply (drule_tac x="trm" in meta_spec)

  apply(simp add: alpha_bn_refl)

  apply(simp_all)[5]

  apply(simp)

  apply(erule conjE)+

  thm alpha_bn_cases

  apply(erule alpha_bn_cases)

  apply(simp)

  apply (subgoal_tac "height_trm_sumC b = height_trm_sumC ba")

  apply simp

  apply(simp add: trm_assn.bn_defs)

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

  apply(simp_all add: pure_fresh fresh_star_def)[3]

  apply(simp_all add: eqvt_at_def)

  done

nominal_primrec 

  subst_trm :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm"  ("_ [_ ::= _]" [90, 90, 90] 90) 

where

  "(Var x)[y ::= s] = (if x = y then s else (Var x))"

| "(App t1 t2)[y ::= s] = App (t1[y ::= s]) (t2[y ::= s])"

| "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow> 

      (Let as t)[y ::= s] = Let (apply_assn2 (\<lambda>t. t[y ::=s]) as) (t[y ::= s])"

  apply (simp only: eqvt_def subst_trm_graph_def)

  apply (rule, perm_simp)

  apply(rule)

  apply(rule TrueI)

  apply(case_tac x)

  apply(simp)

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

  apply (auto simp add: alpha_bn_refl)[3]

  apply(simp_all)[5]

  apply(simp)

  apply(erule conjE)+

  thm alpha_bn_cases

  apply(erule alpha_bn_cases)

  apply(simp)

  apply(simp add: trm_assn.bn_defs)

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

  apply(simp add: Abs_fresh_iff fresh_star_def)

  apply(simp add: fresh_star_def)

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

  done

section {* direct definitions --- problems *}

nominal_primrec

    height_trm :: "trm \<Rightarrow> nat"

and height_assn :: "assn \<Rightarrow> nat"

where

  "height_trm (Var x) = 1"

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

| "height_trm (Let as b) = max (height_assn as) (height_trm b)"

| "height_assn (Assn x t) = (height_trm t)"

  apply (simp only: eqvt_def height_trm_height_assn_graph_def)

  apply (rule, perm_simp, rule, rule TrueI)

  apply (case_tac x)

  apply(simp)

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

  apply (auto simp add: alpha_bn_refl)[3]

  apply (drule_tac x="assn" in meta_spec)

  apply (drule_tac x="trm" in meta_spec)

  apply(simp add: alpha_bn_refl)

  apply(simp)

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

  apply (auto)[1]

  apply(simp_all)[7]

  prefer 2

  apply(simp)

  --"let case"

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

  apply (simp only: meta_eq_to_obj_eq[OF height_assn_def, symmetric, unfolded fun_eq_iff])

  apply (subgoal_tac "eqvt_at height_assn as")

  apply (subgoal_tac "eqvt_at height_assn asa")

  apply (subgoal_tac "eqvt_at height_trm b")

  apply (subgoal_tac "eqvt_at height_trm ba")

  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr as)")

  apply (thin_tac "eqvt_at height_trm_height_assn_sumC (Inr asa)")

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

  apply (thin_tac "eqvt_at height_trm_height_assn_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_assn_def)

  apply (simp add: eqvt_at_def height_assn_def)

  prefer 2

  apply (subgoal_tac "height_assn as = height_assn asa")

  apply (subgoal_tac "height_trm b = height_trm ba")

  apply simp

  apply(simp)

  apply(erule conjE)+

  apply(erule alpha_bn_cases)

  apply(simp)

  apply(simp add: trm_assn.bn_defs)

  thm Abs_lst_fcb2

  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)

  defer

  apply(simp)

  apply(frule Inl_inject)

  apply(subst (asm) trm_assn.eq_iff)

  apply(drule Inl_inject)

  apply(clarify)

  apply(erule alpha_bn_cases)

  apply(simp del: trm_assn.eq_iff)

  apply(rename_tac as s as' s' t' t x x')

  apply(simp only: trm_assn.bn_defs)

  (* HERE *)

  oops

lemma ww1:

  shows "finite (fv_trm t)"

  and "finite (fv_bn as)"

apply(induct t and as rule: trm_assn.inducts)

apply(simp_all add: trm_assn.fv_defs supp_at_base)

done

text {* works, but only because no recursion in as *}

nominal_primrec (invariant "\<lambda>x (y::atom set). finite y")

  frees_set :: "trm \<Rightarrow> atom set"

where

  "frees_set (Var x) = {atom x}"

| "frees_set (App t1 t2) = frees_set t1 \<union> frees_set t2"

| "frees_set (Let as t) = (frees_set t) - (set (bn as)) \<union> (fv_bn as)"

  apply(simp add: eqvt_def frees_set_graph_def)

  apply(rule, perm_simp, rule)

  apply(erule frees_set_graph.induct)

  apply(auto simp add: ww1)[3]

  apply(rule_tac y="x" in trm_assn.exhaust(1))

  apply(auto simp add: alpha_bn_refl)[3]

  apply(drule_tac x="assn" in meta_spec)

  apply(drule_tac x="trm" in meta_spec)

  apply(simp add: alpha_bn_refl)

  apply(simp_all)[5]

  apply(simp)

  apply(erule conjE)

  apply(erule alpha_bn_cases)

  apply(simp add: trm_assn.bn_defs)

  apply(simp add: trm_assn.fv_defs)

  (* apply(erule_tac c="(trm_rawa)" in Abs_lst1_fcb2) *)

  apply(subgoal_tac " frees_set_sumC t - {atom name} = frees_set_sumC ta - {atom namea}")

  apply(simp)

  apply(erule_tac c="()" in Abs_lst1_fcb2)

  apply(simp add: fresh_minus_atom_set)

  apply(simp add: fresh_star_def fresh_Unit)

  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)

  apply(simp add: Diff_eqvt eqvt_at_def, perm_simp, rule refl)

  done

termination

  by lexicographic_order

lemma test:

  assumes a: "\<exists>y. f x = Inl y"

  shows "(p \<bullet> (Sum_Type.Projl (f x))) = Sum_Type.Projl ((p \<bullet> f) (p \<bullet> x))"

using a

apply clarify

apply(frule_tac p="p" in permute_boolI)

apply(simp (no_asm_use) only: eqvts)

apply(subst (asm) permute_fun_app_eq)

back

apply(simp)

done

nominal_primrec (default "sum_case (\<lambda>x. Inl undefined) (\<lambda>x. Inr undefined)")

  subst_trm :: "trm \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> trm"  ("_ [_ ::trm= _]" [90, 90, 90] 90) and

  subst_assn :: "assn \<Rightarrow> name \<Rightarrow> trm \<Rightarrow> assn"  ("_ [_ ::assn= _]" [90, 90, 90] 90)

where

  "(Var x)[y ::trm= s] = (if x = y then s else (Var x))"

| "(App t1 t2)[y ::trm= s] = App (t1[y ::trm= s]) (t2[y ::trm= s])"

| "(set (bn as)) \<sharp>* (y, s) \<Longrightarrow> (Let as t)[y ::trm= s] = Let (ast[y ::assn= s]) (t[y ::trm= s])"

| "(Assn x t)[y ::assn= s] = Assn x (t[y ::trm= s])"

apply(subgoal_tac "\<And>p x r. subst_trm_subst_assn_graph x r \<Longrightarrow> subst_trm_subst_assn_graph (p \<bullet> x) (p \<bullet> r)")

apply(simp add: eqvt_def)

apply(rule allI)

apply(simp add: permute_fun_def permute_bool_def)

apply(rule ext)

apply(rule ext)

apply(rule iffI)

apply(drule_tac x="p" in meta_spec)

apply(drule_tac x="- p \<bullet> x" in meta_spec)

apply(drule_tac x="- p \<bullet> xa" in meta_spec)

apply(simp)

apply(drule_tac x="-p" in meta_spec)

apply(drule_tac x="x" in meta_spec)

apply(drule_tac x="xa" in meta_spec)

apply(simp)

--"Eqvt One way"

defer

  apply(rule TrueI)

  apply(case_tac x)

  apply(simp)

  apply(case_tac a)

  apply(simp)

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

  apply(blast)+

  apply(simp)

  apply(case_tac b)

  apply(simp)

  apply(rule_tac y="a" in trm_assn.exhaust(2))

  apply(simp)

  apply(blast)

  apply(simp_all)[7]

  prefer 2

  apply(simp)

  prefer 2

  apply(simp)

  apply(simp)

  apply (simp only: meta_eq_to_obj_eq[OF subst_trm_def, symmetric, unfolded fun_eq_iff])

  apply (simp only: meta_eq_to_obj_eq[OF subst_assn_def, symmetric, unfolded fun_eq_iff])

  apply (subgoal_tac "eqvt_at (\<lambda>ast. subst_assn ast ya sa) ast")

  apply (subgoal_tac "eqvt_at (\<lambda>asta. subst_assn asta ya sa) asta")

  apply (subgoal_tac "eqvt_at (\<lambda>t. subst_trm t ya sa) t")

  apply (subgoal_tac "eqvt_at (\<lambda>ta. subst_trm ta ya sa) ta")

  apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inr (ast, ya, sa))")

  apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inr (asta, ya, sa))")

  apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inl (t, ya, sa))")

  apply (thin_tac "eqvt_at subst_trm_subst_assn_sumC (Inl (ta, ya, sa))")

  defer

  defer

  defer

  defer

  defer

  defer

  apply(rule conjI)

  apply (subgoal_tac "subst_assn ast ya sa= subst_assn asta ya sa")

  apply (subgoal_tac "subst_trm t ya sa = subst_trm ta ya sa")

  apply(simp)

  apply(erule_tac conjE)+

  apply(erule alpha_bn_cases)

  apply(simp add: trm_assn.bn_defs)

  apply(rotate_tac 7)

  apply(drule k)

  apply(erule conjE)

  apply(subst (asm) Abs1_eq_iff)

  apply(rule sort_of_atom_eq)

  apply(rule sort_of_atom_eq)

  apply(erule disjE)

  apply(simp)

  apply(rotate_tac 12)

  apply(drule sym)

  apply(rule sym)

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

  apply(erule fresh_eqvt_at)

  thm fresh_eqvt_at

  apply(simp add: Abs_fresh_iff)

  apply(simp add: fresh_star_def fresh_Pair)

  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)

  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)

  apply(simp_all add: fresh_star_def fresh_Pair_elim)[1]

  apply(blast)

  apply(simp_all)[5]

  apply(simp (no_asm_use))

  apply(simp)

  apply(erule conjE)+

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

  apply(simp add: Abs_fresh_iff)

  apply(simp add: fresh_star_def fresh_Pair)

  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)

  apply(simp add: eqvt_at_def atom_eqvt fresh_star_Pair perm_supp_eq)

done

end