(* Title: HOL/Tools/transfer.ML
Author: Brian Huffman, TU Muenchen
Generic theorem transfer method.
*)
signature TRANSFER =
sig
val prep_conv: conv
val get_relator_eq: Proof.context -> thm list
val transfer_add: attribute
val transfer_del: attribute
val transfer_tac: bool -> Proof.context -> int -> tactic
val transfer_prover_tac: Proof.context -> int -> tactic
val setup: theory -> theory
end
structure Transfer : TRANSFER =
struct
structure Data = Named_Thms
(
val name = @{binding transfer_raw}
val description = "raw transfer rule for transfer method"
)
structure Relator_Eq = Named_Thms
(
val name = @{binding relator_eq}
val description = "relator equality rule (used by transfer method)"
)
fun get_relator_eq ctxt =
map (Thm.symmetric o mk_meta_eq) (Relator_Eq.get ctxt)
(** Conversions **)
val Rel_rule = Thm.symmetric @{thm Rel_def}
fun dest_funcT cT =
(case Thm.dest_ctyp cT of [T, U] => (T, U)
| _ => raise TYPE ("dest_funcT", [Thm.typ_of cT], []))
fun Rel_conv ct =
let val (cT, cT') = dest_funcT (Thm.ctyp_of_term ct)
val (cU, _) = dest_funcT cT'
in Drule.instantiate' [SOME cT, SOME cU] [SOME ct] Rel_rule end
fun Trueprop_conv cv ct =
(case Thm.term_of ct of
Const (@{const_name Trueprop}, _) $ _ => Conv.arg_conv cv ct
| _ => raise CTERM ("Trueprop_conv", [ct]))
(* Conversion to preprocess a transfer rule *)
fun prep_conv ct = (
Conv.implies_conv Conv.all_conv prep_conv
else_conv
Trueprop_conv (Conv.fun_conv (Conv.fun_conv Rel_conv))
else_conv
Conv.all_conv) ct
(** Transfer proof method **)
fun RelT (Const (@{const_name Rel}, _) $ _ $ _ $ y) = fastype_of y
| RelT t = raise TERM ("RelT", [t])
(* Tactic to correspond a value to itself *)
fun eq_tac ctxt = SUBGOAL (fn (t, i) =>
let
val T = RelT (HOLogic.dest_Trueprop (Logic.strip_assums_concl t))
val cT = ctyp_of (Proof_Context.theory_of ctxt) T
val rews = get_relator_eq ctxt
val thm1 = Drule.instantiate' [SOME cT] [] @{thm Rel_eq_refl}
val thm2 = Raw_Simplifier.rewrite_rule rews thm1
in
rtac thm2 i
end handle Match => no_tac | TERM _ => no_tac)
val post_simps =
@{thms transfer_forall_eq [symmetric]
transfer_implies_eq [symmetric] transfer_bforall_unfold}
fun gen_frees_tac keepers ctxt = SUBGOAL (fn (t, i) =>
let
val vs = rev (Term.add_frees t [])
val vs' = filter_out (member (op =) keepers) vs
in
Induct.arbitrary_tac ctxt 0 vs' i
end)
(* create a lambda term of the same shape as the given term *)
fun skeleton (Bound i) ctxt = (Bound i, ctxt)
| skeleton (Abs (x, _, t)) ctxt =
let
val (t', ctxt) = skeleton t ctxt
in
(Abs (x, dummyT, t'), ctxt)
end
| skeleton (t $ u) ctxt =
let
val (t', ctxt) = skeleton t ctxt
val (u', ctxt) = skeleton u ctxt
in
(t' $ u', ctxt)
end
| skeleton _ ctxt =
let
val (c, ctxt) = yield_singleton Variable.variant_fixes "a" ctxt
in
(Free (c, dummyT), ctxt)
end
fun mk_relT (T, U) = T --> U --> HOLogic.boolT
fun mk_Rel t =
let val T = fastype_of t
in Const (@{const_name Transfer.Rel}, T --> T) $ t end
fun transfer_rule_of_terms ctxt tab t u =
let
val thy = Proof_Context.theory_of ctxt
(* precondition: T must consist of only TFrees and function space *)
fun rel (T as TFree (a, _)) U =
Free (the (AList.lookup (op =) tab a), mk_relT (T, U))
| rel (T as Type ("fun", [T1, T2])) (U as Type ("fun", [U1, U2])) =
let
val r1 = rel T1 U1
val r2 = rel T2 U2
val rT = fastype_of r1 --> fastype_of r2 --> mk_relT (T, U)
in
Const (@{const_name fun_rel}, rT) $ r1 $ r2
end
| rel T U = raise TYPE ("rel", [T, U], [])
fun zip _ thms (Bound i) (Bound _) = (nth thms i, [])
| zip ctxt thms (Abs (x, T, t)) (Abs (y, U, u)) =
let
val ([x', y'], ctxt') = Variable.variant_fixes [x, y] ctxt
val prop = mk_Rel (rel T U) $ Free (x', T) $ Free (y', U)
val cprop = Thm.cterm_of thy (HOLogic.mk_Trueprop prop)
val thm0 = Thm.assume cprop
val (thm1, hyps) = zip ctxt' (thm0 :: thms) t u
val ((r1, x), y) = apfst Thm.dest_comb (Thm.dest_comb (Thm.dest_arg cprop))
val r2 = Thm.dest_fun2 (Thm.dest_arg (cprop_of thm1))
val (a1, (b1, _)) = apsnd dest_funcT (dest_funcT (ctyp_of_term r1))
val (a2, (b2, _)) = apsnd dest_funcT (dest_funcT (ctyp_of_term r2))
val tinsts = [SOME a1, SOME b1, SOME a2, SOME b2]
val insts = [SOME (Thm.dest_arg r1), SOME (Thm.dest_arg r2)]
val rule = Drule.instantiate' tinsts insts @{thm Rel_abs}
val thm2 = Thm.forall_intr x (Thm.forall_intr y (Thm.implies_intr cprop thm1))
in
(thm2 COMP rule, hyps)
end
| zip ctxt thms (f $ t) (g $ u) =
let
val (thm1, hyps1) = zip ctxt thms f g
val (thm2, hyps2) = zip ctxt thms t u
in
(thm2 RS (thm1 RS @{thm Rel_app}), hyps1 @ hyps2)
end
| zip _ _ (t as Free (_, T)) u =
let
val U = fastype_of u
val prop = mk_Rel (rel T U) $ t $ u
val cprop = Thm.cterm_of thy (HOLogic.mk_Trueprop prop)
in
(Thm.assume cprop, [cprop])
end
| zip _ _ t u = raise TERM ("zip_relterm", [t, u])
val r = mk_Rel (rel (fastype_of t) (fastype_of u))
val goal = HOLogic.mk_Trueprop (r $ t $ u)
val rename = Thm.trivial (cterm_of thy goal)
val (thm, hyps) = zip ctxt [] t u
in
Drule.implies_intr_list hyps (thm RS rename)
end
fun transfer_rule_of_term ctxt t =
let
val s = skeleton t ctxt |> fst |> Syntax.check_term ctxt |>
map_types (map_type_tfree (fn (a, _) => TFree (a, HOLogic.typeS)))
val frees = map fst (Term.add_frees s [])
val tfrees = map fst (Term.add_tfrees s [])
fun prep a = "R" ^ Library.unprefix "'" a
val (rnames, ctxt') = Variable.variant_fixes (map prep tfrees) ctxt
val thm = transfer_rule_of_terms ctxt' (tfrees ~~ rnames) s t
in
Thm.generalize (tfrees, rnames @ frees) (Thm.maxidx_of thm + 1) thm
end
fun transfer_tac equiv ctxt i =
let
val pre_simps = @{thms transfer_forall_eq transfer_implies_eq}
val start_rule =
if equiv then @{thm transfer_start} else @{thm transfer_start'}
val rules = Data.get ctxt
in
EVERY
[rewrite_goal_tac pre_simps i THEN
SUBGOAL (fn (t, i) =>
rtac start_rule i THEN
(rtac (transfer_rule_of_term ctxt (HOLogic.dest_Trueprop t))
THEN_ALL_NEW
(SOLVED' (REPEAT_ALL_NEW (resolve_tac rules))
ORELSE' eq_tac ctxt)) (i + 1)) i,
(* FIXME: rewrite_goal_tac does unwanted eta-contraction *)
rewrite_goal_tac post_simps i,
rtac @{thm _} i]
end
fun transfer_prover_tac ctxt = SUBGOAL (fn (t, i) =>
let
val rhs = (snd o Term.dest_comb o HOLogic.dest_Trueprop) t
val rule1 = transfer_rule_of_term ctxt rhs
val rules = Data.get ctxt
in
EVERY
[CONVERSION prep_conv i,
rtac @{thm transfer_prover_start} i,
(rtac rule1 THEN_ALL_NEW
REPEAT_ALL_NEW
(resolve_tac rules ORELSE' eq_tac ctxt)) (i+1),
rtac @{thm refl} i]
end)
(** Methods and attributes **)
val free = Args.context -- Args.term >> (fn (_, Free v) => v | (ctxt, t) =>
error ("Bad free variable: " ^ Syntax.string_of_term ctxt t))
val fixing = Scan.optional (Scan.lift (Args.$$$ "fixing" -- Args.colon)
|-- Scan.repeat free) []
fun transfer_method equiv : (Proof.context -> Method.method) context_parser =
fixing >> (fn vs => fn ctxt =>
SIMPLE_METHOD' (gen_frees_tac vs ctxt THEN' transfer_tac equiv ctxt))
val transfer_prover_method : (Proof.context -> Method.method) context_parser =
Scan.succeed (fn ctxt => SIMPLE_METHOD' (transfer_prover_tac ctxt))
(* Attribute for transfer rules *)
val prep_rule = Conv.fconv_rule prep_conv
val transfer_add =
Thm.declaration_attribute (Data.add_thm o prep_rule)
val transfer_del =
Thm.declaration_attribute (Data.del_thm o prep_rule)
val transfer_attribute =
Attrib.add_del transfer_add transfer_del
(* Theory setup *)
val setup =
Data.setup
#> Relator_Eq.setup
#> Attrib.setup @{binding transfer_rule} transfer_attribute
"transfer rule for transfer method"
#> Method.setup @{binding transfer} (transfer_method true)
"generic theorem transfer method"
#> Method.setup @{binding transfer'} (transfer_method false)
"generic theorem transfer method"
#> Method.setup @{binding transfer_prover} transfer_prover_method
"for proving transfer rules"
end