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

lemma alpha_refl:

  fixes t::"rlam"

  shows "t \<approx> t"

  apply(induct t rule: rlam.induct)

  apply(simp add: a1)

  apply(simp add: a2)

  apply(rule a3)

  apply(subst pt_swap_bij'')

  apply(rule pt_name_inst)

  apply(rule at_name_inst)

  apply(simp)

  apply(simp)

  done

lemma alpha_EQUIV:

  shows "EQUIV alpha"

sorry

quotient lam = rlam / alpha

  apply(rule alpha_EQUIV)

  done

print_quotients

quotient_def 

  Var :: "name \<Rightarrow> lam"

where

  "Var \<equiv> rVar"

quotient_def 

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

where

  "App \<equiv> rApp"

quotient_def 

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

where

  "Lam \<equiv> rLam"

thm Var_def

thm App_def

thm Lam_def

quotient_def 

  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 

  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 real_alpha:

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

  shows "Lam a t = Lam b s"

sorry

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

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

apply (auto)

apply (erule alpha.cases)

apply (simp_all add: rlam.inject alpha_refl)

done

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

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

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

ML {* val consts = lookup_quot_consts defs *}

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

ML {* fun lift_tac_lam lthy t = lift_tac lthy t rel_eqv rel_refl rty quot trans2 rsp_thms reps_same defs *}

apply (tactic {* lift_tac_lam @{context} @{thm pi_var_com} 1 *})

done

apply (tactic {* lift_tac_lam @{context} @{thm pi_app_com} 1 *})

done

apply (tactic {* lift_tac_lam @{context} @{thm pi_lam_com} 1 *})

done

apply (tactic {* lift_tac_lam @{context} @{thm rfv_var} 1 *})

done

apply (tactic {* lift_tac_lam @{context} @{thm a1} 1 *})

done

apply (tactic {* lift_tac_lam @{context} @{thm a2} 1 *})

done

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

lemma "\<lbrakk>(x\<Colon>lam) = (xa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. \<lbrakk>x = Var a; xa = Var b; a = b\<rbrakk> \<Longrightarrow> P\<Colon>bool;

     \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = App x xb; xa = App xa xc; x = xa; xb = xc\<rbrakk> \<Longrightarrow> P;

     \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam. \<lbrakk>x = Lam a x; xa = Lam b xa; x = [(a, b)] \<bullet> xa; a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> P\<rbrakk>

    \<Longrightarrow> P"

sorry

ML {* val alpha_induct = lift_thm_lam @{context} @{thm alpha.induct} *}

lemma "\<lbrakk>(qx\<Colon>lam) = (qxa\<Colon>lam); \<And>(a\<Colon>name) b\<Colon>name. a = b \<Longrightarrow> (qxb\<Colon>lam \<Rightarrow> lam \<Rightarrow> bool) (Var a) (Var b);

     \<And>(x\<Colon>lam) (xa\<Colon>lam) (xb\<Colon>lam) xc\<Colon>lam. \<lbrakk>x = xa; qxb x xa; xb = xc; qxb xb xc\<rbrakk> \<Longrightarrow> qxb (App x xb) (App xa xc);

     \<And>(x\<Colon>lam) (a\<Colon>name) (b\<Colon>name) xa\<Colon>lam.

        \<lbrakk>x = [(a, b)] \<bullet> xa; qxb x ([(a, b)] \<bullet> xa); a \<notin> fv (Lam b x)\<rbrakk> \<Longrightarrow> qxb (Lam a x) (Lam b xa)\<rbrakk>

    \<Longrightarrow> qxb qx qxa"

(* apply (tactic {* lift_tac_lam @{context} @{thm alpha.induct} 1 *}) *)

sorry

apply (tactic {* lift_tac_lam @{context} @{thm rvar_inject} 1 *})

done

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

(* Construction Site code *)

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

ML {* val consts = lookup_quot_consts defs *}

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

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

(* 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

    ]);

*}*)

ML_prf {*

  fun r_mk_comb_tac_lam lthy = r_mk_comb_tac lthy rty quot rel_refl trans2 rsp_thms

*}

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

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 (alls, exs) = findallex rty qty (prop_of (atomize_thm @{thm alpha.induct})) *}

ML {* val allthms = map (make_allex_prs_thm @{context} quot @{thm FORALL_PRS} ) alls *}

ML {* val exthms = map (make_allex_prs_thm @{context} quot @{thm EXISTS_PRS} ) exs *}

ML {* val t_a = MetaSimplifier.rewrite_rule allthms t_t *}

ML {* val simp_app_prs_thms = map (make_simp_prs_thm @{context} quot @{thm APP_PRS}) aps *}

ML {* val simp_lam_prs_thms = map (make_simp_prs_thm @{context} quot @{thm LAMBDA_PRS}) abs *}

ML {* val t_l = repeat_eqsubst_thm @{context} (simp_lam_prs_thms) t_a *}

ML {* val t_l1 = repeat_eqsubst_thm @{context} simp_app_prs_thms t_l *}

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

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

ML {* val t_d0 = MetaSimplifier.rewrite_rule defs_sym_eq t_l1 *}

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

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