theory LamEx

imports Nominal QuotMain

begin

atom_decl name

thm abs_fresh(1)

nominal_datatype rlam =

  rVar "name"

| rApp "rlam" "rlam"

| rLam "name" "rlam"

print_theorems

function

  rfv :: "rlam \<Rightarrow> name set"

where

  rfv_var: "rfv (rVar a) = {a}"

| rfv_app: "rfv (rApp t1 t2) = (rfv t1) \<union> (rfv t2)"

| rfv_lam: "rfv (rLam a t) = (rfv t) - {a}"

sorry

termination rfv sorry

inductive 

  alpha :: "rlam \<Rightarrow> rlam \<Rightarrow> bool" ("_ \<approx> _" [100, 100] 100)

where

  a1: "a = b \<Longrightarrow> (rVar a) \<approx> (rVar b)"

| a2: "\<lbrakk>t1 \<approx> t2; s1 \<approx> s2\<rbrakk> \<Longrightarrow> rApp t1 s1 \<approx> rApp t2 s2"

| a3: "\<lbrakk>t \<approx> ([(a,b)]\<bullet>s); a \<notin> rfv (rLam b t)\<rbrakk> \<Longrightarrow> rLam a t \<approx> rLam b s"

print_theorems

quotient lam = rlam / alpha

sorry

print_quotients

quotient_def (for lam)

  Var :: "name \<Rightarrow> lam"

where

  "Var \<equiv> rVar"

quotient_def (for lam)

  App :: "lam \<Rightarrow> lam \<Rightarrow> lam"

where

  "App \<equiv> rApp"

quotient_def (for lam)

  Lam :: "name \<Rightarrow> lam \<Rightarrow> lam"

where

  "Lam \<equiv> rLam"

thm Var_def

thm App_def

thm Lam_def

quotient_def (for lam)

  fv :: "lam \<Rightarrow> name set"

where

  "fv \<equiv> rfv"

thm fv_def

(* definition of overloaded permutation function *)

(* for the lifted type lam                       *)

overloading

  perm_lam    \<equiv> "perm :: 'x prm \<Rightarrow> lam \<Rightarrow> lam"   (unchecked)

begin

quotient_def (for lam)

  perm_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"

where

  "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"

end

(*quotient_def (for lam)

  abs_fun_lam :: "'x prm \<Rightarrow> lam \<Rightarrow> lam"

where

  "perm_lam \<equiv> (perm::'x prm \<Rightarrow> rlam \<Rightarrow> rlam)"*)

thm perm_lam_def

(* lemmas that need to lift *)

lemma pi_var_com:

  fixes pi::"'x prm"

  shows "(pi\<bullet>rVar a) \<approx> rVar (pi\<bullet>a)"

  sorry

lemma pi_app_com:

  fixes pi::"'x prm"

  shows "(pi\<bullet>rApp t1 t2) \<approx> rApp (pi\<bullet>t1) (pi\<bullet>t2)"

  sorry

lemma pi_lam_com:

  fixes pi::"'x prm"

  shows "(pi\<bullet>rLam a t) \<approx> rLam (pi\<bullet>a) (pi\<bullet>t)"

  sorry

lemma fv_var:

  shows "fv (Var a) = {a}"

sorry

lemma fv_app:

  shows "fv (App t1 t2) = (fv t1) \<union> (fv t2)"

sorry

lemma fv_lam:

  shows "fv (Lam a t) = (fv t) - {a}"

sorry

lemma real_alpha:

  assumes "t = [(a,b)]\<bullet>s" "a\<sharp>[b].s"

  shows "Lam a t = Lam b s"

sorry

(* Construction Site code *)

lemma perm_rsp: "(op = ===> alpha ===> alpha) op \<bullet> op \<bullet>"

  apply(auto)

  (* this is propably true if some type conditions are imposed ;o) *)

  sorry

lemma fresh_rsp: "(op = ===> alpha ===> op =) fresh fresh" 

  apply(auto)

  (* this is probably only true if some type conditions are imposed *)

  sorry

lemma rVar_rsp: "(op = ===> alpha) rVar rVar"

  apply(auto)

  apply(rule a1)

  apply(simp)

  done

lemma rApp_rsp: "(alpha ===> alpha ===> alpha) rApp rApp"

  apply(auto)

  apply(rule a2)

  apply (assumption)

  apply (assumption)

  done

lemma rLam_rsp: "(op = ===> alpha ===> alpha) rLam rLam"

  apply(auto)

  apply(rule a3)

  apply(rule_tac t="[(x,x)]\<bullet>y" and s="y" in subst)

  apply(rule sym)

  apply(rule trans)

  apply(rule pt_name3)

  apply(rule at_ds1[OF at_name_inst])

  apply(simp add: pt_name1)

  apply(assumption)

  apply(simp add: abs_fresh)

  done

lemma rfv_rsp: "(alpha ===> op =) rfv rfv"

  sorry

ML {* val qty = @{typ "lam"} *}

ML {* val (rty, rel, rel_refl, rel_eqv) = lookup_quot_data @{context} qty *}

ML {* val defs = @{thms Var_def App_def Lam_def perm_lam_def fv_def} *}

ML {* val consts = lookup_quot_consts defs *}

ML {* val (trans2, reps_same, quot) = lookup_quot_thms @{context} "lam" *}

ML {* val rsp_thms = @{thms perm_rsp fresh_rsp rVar_rsp rApp_rsp rLam_rsp rfv_rsp} @ @{thms ho_all_prs ho_ex_prs} *}

ML {* fun lift_thm_lam lthy t = lift_thm lthy qty "lam" rsp_thms defs t *}

ML {* val pi_var = lift_thm_lam @{context} @{thm pi_var_com} *}

ML {* val pi_app = lift_thm_lam @{context} @{thm pi_app_com} *}

ML {* val pi_lam = lift_thm_lam @{context} @{thm pi_lam_com} *}

ML {* val fv_var = lift_thm_lam @{context} @{thm rfv_var} *}

ML {* val fv_app = lift_thm_lam @{context} @{thm rfv_app} *}

ML {* val fv_lam = lift_thm_lam @{context} @{thm rfv_lam} *}

ML {* val a1 = lift_thm_lam @{context} @{thm a1} *}

ML {* val a2 = lift_thm_lam @{context} @{thm a2} *}

ML {* val a3 = lift_thm_lam @{context} @{thm a3} *}

ML {* val alpha_cases = lift_thm_lam @{context} @{thm alpha.cases} *}

local_setup {*

  Quotient.note (@{binding "pi_var"}, pi_var) #> snd #>

  Quotient.note (@{binding "pi_app"}, pi_app) #> snd #>

  Quotient.note (@{binding "pi_lam"}, pi_lam) #> snd #>

  Quotient.note (@{binding "a1"}, a1) #> snd #>

  Quotient.note (@{binding "a2"}, a2) #> snd #>

  Quotient.note (@{binding "a3"}, a3) #> snd #>

  Quotient.note (@{binding "alpha_cases"}, alpha_cases) #> snd

*}

thm alpha.cases

thm alpha_cases

thm rlam.inject

thm pi_var

lemma var_inject:

  shows "(Var a = Var b) = (a = b)"

sorry

lemma var_supp:

  shows "supp (Var a) = ((supp a)::name set)"

  apply(simp add: supp_def)

  apply(simp add: pi_var)

  apply(simp add: var_inject)

  done

lemma var_fresh:

  fixes a::"name"

  shows "(a\<sharp>(Var b)) = (a\<sharp>b)"

  apply(simp add: fresh_def)

  apply(simp add: var_supp)

  done

ML {* val t_a = atomize_thm @{thm alpha.induct} *}

(* prove {* build_regularize_goal t_a rty rel @{context}  *}

ML_prf {*  fun tac ctxt =

     (FIRST' [

      rtac rel_refl,

      atac,

      rtac @{thm universal_twice},

      (rtac @{thm impI} THEN' atac),

      rtac @{thm implication_twice},

      EqSubst.eqsubst_tac ctxt [0]

        [(@{thm equiv_res_forall} OF [rel_eqv]),

         (@{thm equiv_res_exists} OF [rel_eqv])],

      (rtac @{thm impI} THEN' (asm_full_simp_tac (Simplifier.context ctxt HOL_ss)) THEN' rtac rel_refl),

      (rtac @{thm RIGHT_RES_FORALL_REGULAR})

     ]);

 *}

  apply (tactic {* tac @{context} 1 *}) *)

ML {* val t_r = regularize t_a rty rel rel_eqv rel_refl @{context} *}

(*ML {*

  val rt = build_repabs_term @{context} t_r consts rty qty

  val rg = Logic.mk_equals ((Thm.prop_of t_r), rt);

*}

prove rg

apply(atomize(full))

ML_prf {*

fun r_mk_comb_tac ctxt rty quot_thm reflex_thm trans_thm rsp_thms =

  (FIRST' [

    rtac trans_thm,

    LAMBDA_RES_TAC ctxt,

    res_forall_rsp_tac ctxt,

    res_exists_rsp_tac ctxt,

    (

     (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps rsp_thms))

     THEN_ALL_NEW (fn _ => no_tac)

    ),

    (instantiate_tac @{thm REP_ABS_RSP(1)} ctxt THEN' (RANGE [quotient_tac quot_thm])),

    rtac refl,

    (APPLY_RSP_TAC rty ctxt THEN' (RANGE [quotient_tac quot_thm, quotient_tac quot_thm])),

    Cong_Tac.cong_tac @{thm cong},

    rtac @{thm ext},

    rtac reflex_thm,

    atac,

    (

     (simp_tac ((Simplifier.context ctxt HOL_ss) addsimps @{thms FUN_REL.simps}))

     THEN_ALL_NEW (fn _ => no_tac)

    ),

    WEAK_LAMBDA_RES_TAC ctxt

    ]);

  fun r_mk_comb_tac_lam lthy = r_mk_comb_tac lthy rty quot rel_refl trans2 rsp_thms

*}

apply (tactic {* REPEAT_ALL_NEW (r_mk_comb_tac_lam @{context}) 1 *})

*)

ML {* val t_t = repabs @{context} t_r consts rty qty quot rel_refl trans2 rsp_thms *}

ML {* val abs = findabs rty (prop_of (atomize_thm @{thm alpha.induct})) *}

ML {* val aps = findaps rty (prop_of (atomize_thm @{thm alpha.induct})) *}

ML {* val aps = @{typ "LamEx.rlam \<Rightarrow> bool"} :: aps; *}

  @{thm FORALL_PRS[OF FUN_QUOTIENT[OF QUOTIENT_lam FUN_QUOTIENT[OF QUOTIENT_lam IDENTITY_QUOTIENT]]]} *}

ML {* val t_a = simp_allex_prs quot [thm] t_t *}

ML {* val defs_sym = add_lower_defs @{context} defs; *}

ML {* val defs_sym_eq = map (fn x => eq_reflection OF [x]) defs_sym *}

ML t_l

ML {* val t_d0 = MetaSimplifier.rewrite_rule defs_sym_eq t_l *}

ML {* val t_d = repeat_eqsubst_thm @{context} defs_sym t_d0 *}

ML {* val t_r = MetaSimplifier.rewrite_rule [reps_same] t_d *}

ML {* val t_r1 = repeat_eqsubst_thm @{context} @{thms fun_map.simps} t_r *}

ML {* val t_r2 = MetaSimplifier.rewrite_rule @{thms QUOT_TYPE_I_lam.thm10} t_r1 *}

ML {* val t_r3 = MetaSimplifier.rewrite_rule @{thms eq_reflection[OF id_apply]} t_r2 *}

ML {* val alpha_induct = ObjectLogic.rulify t_r3 *}

local_setup {*

  Quotient.note (@{binding "alpha_induct"}, alpha_induct) #> snd

*}

thm alpha_induct

thm alpha.induct

lemma rvar_inject: "rVar a \<approx> rVar b \<Longrightarrow> (a = b)"

apply (rule alpha.cases)

apply (assumption)

apply (assumption)

apply (simp_all)

apply (cases "a = b")

apply (simp_all)

apply (cases "ba = b")

apply (simp_all)

lemma var_inject_test:

  fixes a b

  assumes a: "Var a = Var b"

  shows "(a = b)"

  using a   apply (cases "a = b")

  apply (simp_all)

  apply (rule_tac x="Var a" and xa = "Var b" in alpha_cases)

  apply (rule a)

lemma

  assumes a: "(x::lam) = y"

  shows "P x y"

  apply (induct rule: alpha_induct)

  apply (rule a)

thm alpha.induct

thm alpha_induct

fun

  option_map::"('a \<Rightarrow> 'b) \<Rightarrow> ('a noption) \<Rightarrow> ('b noption)"

where

  "option_map f (nSome x) = nSome (f x)"

| "option_map f nNone = nNone"

fun

  option_rel

where

  "option_rel r (nSome x) (nSome y) = r x y"

| "option_rel r _ _ = False"

declare [[map noption = (option_map, option_rel)]]

lemma OPT_QUOTIENT:

  assumes q: "QUOTIENT R Abs Rep"

  shows "QUOTIENT (option_rel R) (option_map Abs) (option_map Rep)"

  apply (unfold QUOTIENT_def)

  apply (auto)

  using q

  apply (unfold QUOTIENT_def)

  apply (case_tac "a :: 'b noption")

  apply (simp)

  apply (simp)

  apply (case_tac "a :: 'b noption")

  apply (simp only: option_map.simps)

  apply (subst option_rel.simps)

  (* Simp starts hanging so don't know how to continue *)

  sorry