theory Term5

imports "Nominal2_Atoms" "Nominal2_Eqvt" "Nominal2_Supp" "Abs" "Perm" "Fv" "Rsp" "../Attic/Prove"

begin

atom_decl name

datatype rtrm5 =

  rVr5 "name"

| rAp5 "rtrm5" "rtrm5"

| rLt5 "rlts" "rtrm5" --"bind (bv5 lts) in (rtrm5)"

and rlts =

  rLnil

| rLcons "name" "rtrm5" "rlts"

primrec

  rbv5

where

  "rbv5 rLnil = {}"

| "rbv5 (rLcons n t ltl) = {atom n} \<union> (rbv5 ltl)"

print_theorems

  [[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[], [], []]]  ] *}

print_theorems

(* Alternate version with additional binding of name in rlts in rLcons *)

(*local_setup {* snd o define_fv_alpha "Term5.rtrm5" [

  [[[]], [[], []], [[(SOME @{term rbv5}, 0)], [(SOME @{term rbv5}, 0)]]], [[], [[(NONE,0)], [], [(NONE,0)]]]  ] *}

print_theorems*)

notation

  alpha_rtrm5 ("_ \<approx>5 _" [100, 100] 100) and

  alpha_rlts ("_ \<approx>l _" [100, 100] 100)

thm alpha_rtrm5_alpha_rlts.intros

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha5_inj}, []), (build_alpha_inj @{thms alpha_rtrm5_alpha_rlts.intros} @{thms rtrm5.distinct rtrm5.inject rlts.distinct rlts.inject} @{thms alpha_rtrm5.cases alpha_rlts.cases} ctxt)) ctxt)) *}

thm alpha5_inj

lemma rbv5_eqvt[eqvt]:

  "pi \<bullet> (rbv5 x) = rbv5 (pi \<bullet> x)"

  apply (induct x)

  apply (simp_all add: eqvts atom_eqvt)

  done

lemma fv_rtrm5_rlts_eqvt[eqvt]:

  "pi \<bullet> (fv_rtrm5 x) = fv_rtrm5 (pi \<bullet> x)"

  "pi \<bullet> (fv_rlts l) = fv_rlts (pi \<bullet> l)"

  apply (induct x and l)

  apply (simp_all add: eqvts atom_eqvt)

  done

lemma alpha5_eqvt:

  "xa \<approx>5 y \<Longrightarrow> (x \<bullet> xa) \<approx>5 (x \<bullet> y)"

  "xb \<approx>l ya \<Longrightarrow> (x \<bullet> xb) \<approx>l (x \<bullet> ya)"

  apply (induct rule: alpha_rtrm5_alpha_rlts.inducts)

  apply (simp_all add: alpha5_inj)

  apply (tactic {* 

    ALLGOALS (

      TRY o REPEAT_ALL_NEW (CHANGED o rtac conjI) THEN_ALL_NEW

      (etac @{thm alpha_gen_compose_eqvt})

    ) *})

  apply (simp_all only: eqvts atom_eqvt)

  done

local_setup {* (fn ctxt => snd (Local_Theory.note ((@{binding alpha5_equivp}, []),

  (build_equivps [@{term alpha_rtrm5}, @{term alpha_rlts}] @{thm rtrm5_rlts.induct} @{thm alpha_rtrm5_alpha_rlts.induct} @{thms rtrm5.inject rlts.inject} @{thms alpha5_inj} @{thms rtrm5.distinct rlts.distinct} @{thms alpha_rtrm5.cases alpha_rlts.cases} @{thms alpha5_eqvt} ctxt)) ctxt)) *}

thm alpha5_equivp

quotient_type

  trm5 = rtrm5 / alpha_rtrm5

and

  lts = rlts / alpha_rlts

  by (auto intro: alpha5_equivp)

local_setup {*

(fn ctxt => ctxt

 |> snd o (Quotient_Def.quotient_lift_const ("Vr5", @{term rVr5}))

 |> snd o (Quotient_Def.quotient_lift_const ("Ap5", @{term rAp5}))

 |> snd o (Quotient_Def.quotient_lift_const ("Lt5", @{term rLt5}))

 |> snd o (Quotient_Def.quotient_lift_const ("Lnil", @{term rLnil}))

 |> snd o (Quotient_Def.quotient_lift_const ("Lcons", @{term rLcons}))

 |> snd o (Quotient_Def.quotient_lift_const ("fv_trm5", @{term fv_rtrm5}))

 |> snd o (Quotient_Def.quotient_lift_const ("fv_lts", @{term fv_rlts}))

 |> snd o (Quotient_Def.quotient_lift_const ("bv5", @{term rbv5})))

*}

print_theorems

lemma alpha5_rfv:

  "(t \<approx>5 s \<Longrightarrow> fv_rtrm5 t = fv_rtrm5 s)"

  "(l \<approx>l m \<Longrightarrow> fv_rlts l = fv_rlts m)"

  apply(induct rule: alpha_rtrm5_alpha_rlts.inducts)

  apply(simp_all add: alpha_gen)

  done

lemma bv_list_rsp:

  shows "x \<approx>l y \<Longrightarrow> rbv5 x = rbv5 y"

  apply(induct rule: alpha_rtrm5_alpha_rlts.inducts(2))

  apply(simp_all)

  done

lemma [quot_respect]:

  "(alpha_rlts ===> op =) fv_rlts fv_rlts"

  "(alpha_rtrm5 ===> op =) fv_rtrm5 fv_rtrm5"

  "(alpha_rlts ===> op =) rbv5 rbv5"

  "(op = ===> alpha_rtrm5) rVr5 rVr5"

  "(alpha_rtrm5 ===> alpha_rtrm5 ===> alpha_rtrm5) rAp5 rAp5"

  "(alpha_rlts ===> alpha_rtrm5 ===> alpha_rtrm5) rLt5 rLt5"

  "(op = ===> alpha_rtrm5 ===> alpha_rlts ===> alpha_rlts) rLcons rLcons"

  "(op = ===> alpha_rtrm5 ===> alpha_rtrm5) permute permute"

  "(op = ===> alpha_rlts ===> alpha_rlts) permute permute"

  apply (simp_all add: alpha5_inj alpha5_rfv alpha5_eqvt bv_list_rsp)

  apply (clarify) apply (rule conjI)

  apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)

  apply (rule_tac x="0" in exI) apply (simp add: fresh_star_def fresh_zero_perm alpha_gen alpha5_rfv)

  done

lemma

  shows "(alpha_rlts ===> op =) rbv5 rbv5"

  by (simp add: bv_list_rsp)

lemmas trm5_lts_inducts = rtrm5_rlts.inducts[quot_lifted]

instantiation trm5 and lts :: pt

begin

quotient_definition

  "permute_trm5 :: perm \<Rightarrow> trm5 \<Rightarrow> trm5"

is

  "permute :: perm \<Rightarrow> rtrm5 \<Rightarrow> rtrm5"

quotient_definition

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

is

  "permute :: perm \<Rightarrow> rlts \<Rightarrow> rlts"

instance by default

  (simp_all add: permute_rtrm5_permute_rlts_zero[quot_lifted] permute_rtrm5_permute_rlts_append[quot_lifted])

end

lemmas

    permute_trm5_lts = permute_rtrm5_permute_rlts.simps[quot_lifted]

and alpha5_INJ = alpha5_inj[unfolded alpha_gen, quot_lifted, folded alpha_gen]

and bv5[simp] = rbv5.simps[quot_lifted]

and fv_trm5_lts[simp] = fv_rtrm5_fv_rlts.simps[quot_lifted]

lemma lets_ok:

  "(Lt5 (Lcons x (Vr5 x) Lnil) (Vr5 x)) = (Lt5 (Lcons y (Vr5 y) Lnil) (Vr5 y))"

apply (subst alpha5_INJ)

apply (rule conjI)

apply (rule_tac x="(x \<leftrightarrow> y)" in exI)

apply (simp only: alpha_gen)

apply (simp add: permute_trm5_lts fresh_star_def)

apply (rule_tac x="(x \<leftrightarrow> y)" in exI)

apply (simp only: alpha_gen)

apply (simp add: permute_trm5_lts fresh_star_def)

done

lemma lets_ok2:

  "(Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) =

   (Lt5 (Lcons y (Vr5 y) (Lcons x (Vr5 x) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"

apply (subst alpha5_INJ)

apply (rule conjI)

apply (rule_tac x="(x \<leftrightarrow> y)" in exI)

apply (simp only: alpha_gen)

apply (simp add: permute_trm5_lts fresh_star_def)

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

apply (simp only: alpha_gen)

apply (simp add: permute_trm5_lts fresh_star_def)

done

lemma lets_not_ok1:

  "x \<noteq> y \<Longrightarrow> (Lt5 (Lcons x (Vr5 x) (Lcons y (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>

             (Lt5 (Lcons y (Vr5 x) (Lcons x (Vr5 y) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"

apply (simp add: alpha5_INJ(3) alpha_gen)

apply (simp add: permute_trm5_lts fresh_star_def alpha5_INJ(5) alpha5_INJ(2) alpha5_INJ(1))

done

lemma distinct_helper:

  shows "\<not>(rVr5 x \<approx>5 rAp5 y z)"

  apply auto

  apply (erule alpha_rtrm5.cases)

  apply (simp_all only: rtrm5.distinct)

  done

lemma distinct_helper2:

  shows "(Vr5 x) \<noteq> (Ap5 y z)"

  by (lifting distinct_helper)

lemma lets_nok:

  "x \<noteq> y \<Longrightarrow> x \<noteq> z \<Longrightarrow> z \<noteq> y \<Longrightarrow>

   (Lt5 (Lcons x (Ap5 (Vr5 z) (Vr5 z)) (Lcons y (Vr5 z) Lnil)) (Ap5 (Vr5 x) (Vr5 y))) \<noteq>

   (Lt5 (Lcons y (Vr5 z) (Lcons x (Ap5 (Vr5 z) (Vr5 z)) Lnil)) (Ap5 (Vr5 x) (Vr5 y)))"

apply (simp only: alpha5_INJ(3) alpha5_INJ(5) alpha_gen permute_trm5_lts fresh_star_def)

apply (simp add: distinct_helper2)

done

end