src/HOL/NSA/transfer.ML

more algebraic properties for gcd/lcm

(*  Title:      HOL/NSA/transfer.ML
    Author:     Brian Huffman

Transfer principle tactic for nonstandard analysis.
*)

signature TRANSFER_PRINCIPLE =
sig
  val transfer_tac: Proof.context -> thm list -> int -> tactic
  val add_const: string -> theory -> theory
end;

structure Transfer_Principle: TRANSFER_PRINCIPLE =
struct

structure TransferData = Generic_Data
(
  type T = {
    intros: thm list,
    unfolds: thm list,
    refolds: thm list,
    consts: string list
  };
  val empty = {intros = [], unfolds = [], refolds = [], consts = []};
  val extend = I;
  fun merge
    ({intros = intros1, unfolds = unfolds1,
      refolds = refolds1, consts = consts1},
     {intros = intros2, unfolds = unfolds2,
      refolds = refolds2, consts = consts2}) : T =
   {intros = Thm.merge_thms (intros1, intros2),
    unfolds = Thm.merge_thms (unfolds1, unfolds2),
    refolds = Thm.merge_thms (refolds1, refolds2),
    consts = Library.merge (op =) (consts1, consts2)};
);

fun unstar_typ (Type (@{type_name star}, [t])) = unstar_typ t
  | unstar_typ (Type (a, Ts)) = Type (a, map unstar_typ Ts)
  | unstar_typ T = T

fun unstar_term consts term =
  let
    fun unstar (Const(a,T) $ t) = if member (op =) consts a then (unstar t)
          else (Const(a,unstar_typ T) $ unstar t)
      | unstar (f $ t) = unstar f $ unstar t
      | unstar (Const(a,T)) = Const(a,unstar_typ T)
      | unstar (Abs(a,T,t)) = Abs(a,unstar_typ T,unstar t)
      | unstar t = t
  in
    unstar term
  end

fun transfer_thm_of ctxt ths t =
  let
    val {intros,unfolds,refolds,consts} = TransferData.get (Context.Proof ctxt);
    val meta = Local_Defs.meta_rewrite_rule ctxt;
    val ths' = map meta ths;
    val unfolds' = map meta unfolds and refolds' = map meta refolds;
    val (_$_$t') = Thm.concl_of (Raw_Simplifier.rewrite ctxt true unfolds' (Thm.cterm_of ctxt t))
    val u = unstar_term consts t'
    val tac =
      rewrite_goals_tac ctxt (ths' @ refolds' @ unfolds') THEN
      ALLGOALS (Object_Logic.full_atomize_tac ctxt) THEN
      match_tac ctxt [transitive_thm] 1 THEN
      resolve_tac ctxt [@{thm transfer_start}] 1 THEN
      REPEAT_ALL_NEW (resolve_tac ctxt intros) 1 THEN
      match_tac ctxt [reflexive_thm] 1
  in Goal.prove ctxt [] [] (Logic.mk_equals (t,u)) (K tac) end

fun transfer_tac ctxt ths =
    SUBGOAL (fn (t, _) =>
        (fn th =>
            let
              val tr = transfer_thm_of ctxt ths t
              val (_$l$r) = Thm.concl_of tr;
              val trs = if l aconv r then [] else [tr];
            in rewrite_goals_tac ctxt trs th end))

local

fun map_intros f = TransferData.map
  (fn {intros,unfolds,refolds,consts} =>
    {intros=f intros, unfolds=unfolds, refolds=refolds, consts=consts})

fun map_unfolds f = TransferData.map
  (fn {intros,unfolds,refolds,consts} =>
    {intros=intros, unfolds=f unfolds, refolds=refolds, consts=consts})

fun map_refolds f = TransferData.map
  (fn {intros,unfolds,refolds,consts} =>
    {intros=intros, unfolds=unfolds, refolds=f refolds, consts=consts})
in
val intro_add = Thm.declaration_attribute (map_intros o Thm.add_thm);
val intro_del = Thm.declaration_attribute (map_intros o Thm.del_thm);

val unfold_add = Thm.declaration_attribute (map_unfolds o Thm.add_thm);
val unfold_del = Thm.declaration_attribute (map_unfolds o Thm.del_thm);

val refold_add = Thm.declaration_attribute (map_refolds o Thm.add_thm);
val refold_del = Thm.declaration_attribute (map_refolds o Thm.del_thm);
end

fun add_const c = Context.theory_map (TransferData.map
  (fn {intros,unfolds,refolds,consts} =>
    {intros=intros, unfolds=unfolds, refolds=refolds, consts=c::consts}))

val _ =
  Theory.setup
   (Attrib.setup @{binding transfer_intro} (Attrib.add_del intro_add intro_del)
      "declaration of transfer introduction rule" #>
    Attrib.setup @{binding transfer_unfold} (Attrib.add_del unfold_add unfold_del)
      "declaration of transfer unfolding rule" #>
    Attrib.setup @{binding transfer_refold} (Attrib.add_del refold_add refold_del)
      "declaration of transfer refolding rule")

end;