(* Title: HOL/Tools/Lifting/lifting_def.ML
Author: Ondrej Kuncar
Definitions for constants on quotient types.
*)
signature LIFTING_DEF =
sig
val add_lift_def:
(binding * mixfix) -> typ -> term -> thm -> local_theory -> local_theory
val lift_def_cmd:
(binding * string option * mixfix) * string -> local_theory -> Proof.state
val can_generate_code_cert: thm -> bool
end;
structure Lifting_Def: LIFTING_DEF =
struct
(** Interface and Syntax Setup **)
(* Generation of the code certificate from the rsp theorem *)
infix 0 MRSL
fun ants MRSL thm = fold (fn rl => fn thm => rl RS thm) ants thm
fun get_body_types (Type ("fun", [_, U]), Type ("fun", [_, V])) = get_body_types (U, V)
| get_body_types (U, V) = (U, V)
fun get_binder_types (Type ("fun", [T, U]), Type ("fun", [V, W])) = (T, V) :: get_binder_types (U, W)
| get_binder_types _ = []
fun force_rty_type ctxt rty rhs =
let
val thy = Proof_Context.theory_of ctxt
val rhs_schematic = singleton (Variable.polymorphic ctxt) rhs
val rty_schematic = fastype_of rhs_schematic
val match = Sign.typ_match thy (rty_schematic, rty) Vartab.empty
in
Envir.subst_term_types match rhs_schematic
end
fun unabs_def ctxt def =
let
val (_, rhs) = Thm.dest_equals (cprop_of def)
fun dest_abs (Abs (var_name, T, _)) = (var_name, T)
| dest_abs tm = raise TERM("get_abs_var",[tm])
val (var_name, T) = dest_abs (term_of rhs)
val (new_var_names, ctxt') = Variable.variant_fixes [var_name] ctxt
val thy = Proof_Context.theory_of ctxt'
val refl_thm = Thm.reflexive (cterm_of thy (Free (hd new_var_names, T)))
in
Thm.combination def refl_thm |>
singleton (Proof_Context.export ctxt' ctxt)
end
fun unabs_all_def ctxt def =
let
val (_, rhs) = Thm.dest_equals (cprop_of def)
val xs = strip_abs_vars (term_of rhs)
in
fold (K (unabs_def ctxt)) xs def
end
val map_fun_unfolded =
@{thm map_fun_def[abs_def]} |>
unabs_def @{context} |>
unabs_def @{context} |>
Local_Defs.unfold @{context} [@{thm comp_def}]
fun unfold_fun_maps ctm =
let
fun unfold_conv ctm =
case (Thm.term_of ctm) of
Const (@{const_name "map_fun"}, _) $ _ $ _ =>
(Conv.arg_conv unfold_conv then_conv Conv.rewr_conv map_fun_unfolded) ctm
| _ => Conv.all_conv ctm
val try_beta_conv = Conv.try_conv (Thm.beta_conversion false)
in
(Conv.arg_conv (Conv.fun_conv unfold_conv then_conv try_beta_conv)) ctm
end
fun prove_rel ctxt rsp_thm (rty, qty) =
let
val ty_args = get_binder_types (rty, qty)
fun disch_arg args_ty thm =
let
val quot_thm = Lifting_Term.prove_quot_theorem ctxt args_ty
in
[quot_thm, thm] MRSL @{thm apply_rsp''}
end
in
fold disch_arg ty_args rsp_thm
end
exception CODE_CERT_GEN of string
fun simplify_code_eq ctxt def_thm =
Local_Defs.unfold ctxt [@{thm o_def}, @{thm map_fun_def}, @{thm id_def}] def_thm
fun can_generate_code_cert quot_thm =
case Lifting_Term.quot_thm_rel quot_thm of
Const (@{const_name HOL.eq}, _) => true
| Const (@{const_name invariant}, _) $ _ => true
| _ => false
fun generate_code_cert ctxt def_thm rsp_thm (rty, qty) =
let
val thy = Proof_Context.theory_of ctxt
val quot_thm = Lifting_Term.prove_quot_theorem ctxt (get_body_types (rty, qty))
val fun_rel = prove_rel ctxt rsp_thm (rty, qty)
val abs_rep_thm = [quot_thm, fun_rel] MRSL @{thm Quotient_rep_abs}
val abs_rep_eq =
case (HOLogic.dest_Trueprop o prop_of) fun_rel of
Const (@{const_name HOL.eq}, _) $ _ $ _ => abs_rep_thm
| Const (@{const_name invariant}, _) $ _ $ _ $ _ => abs_rep_thm RS @{thm invariant_to_eq}
| _ => raise CODE_CERT_GEN "relation is neither equality nor invariant"
val unfolded_def = Conv.fconv_rule unfold_fun_maps def_thm
val unabs_def = unabs_all_def ctxt unfolded_def
val rep = (cterm_of thy o Lifting_Term.quot_thm_rep) quot_thm
val rep_refl = Thm.reflexive rep RS @{thm meta_eq_to_obj_eq}
val repped_eq = [rep_refl, unabs_def RS @{thm meta_eq_to_obj_eq}] MRSL @{thm cong}
val code_cert = [repped_eq, abs_rep_eq] MRSL @{thm trans}
in
simplify_code_eq ctxt code_cert
end
fun define_code_cert code_eqn_thm_name def_thm rsp_thm (rty, qty) lthy =
let
val quot_thm = Lifting_Term.prove_quot_theorem lthy (get_body_types (rty, qty))
in
if can_generate_code_cert quot_thm then
let
val code_cert = generate_code_cert lthy def_thm rsp_thm (rty, qty)
val add_abs_eqn_attribute =
Thm.declaration_attribute (fn thm => Context.mapping (Code.add_abs_eqn thm) I)
val add_abs_eqn_attrib = Attrib.internal (K add_abs_eqn_attribute);
in
lthy
|> (snd oo Local_Theory.note) ((code_eqn_thm_name, [add_abs_eqn_attrib]), [code_cert])
end
else
lthy
end
fun define_code_eq code_eqn_thm_name def_thm lthy =
let
val unfolded_def = Conv.fconv_rule unfold_fun_maps def_thm
val code_eq = unabs_all_def lthy unfolded_def
val simp_code_eq = simplify_code_eq lthy code_eq
in
lthy
|> (snd oo Local_Theory.note) ((code_eqn_thm_name, [Code.add_default_eqn_attrib]), [simp_code_eq])
end
fun define_code code_eqn_thm_name def_thm rsp_thm (rty, qty) lthy =
if body_type rty = body_type qty then
define_code_eq code_eqn_thm_name def_thm lthy
else
define_code_cert code_eqn_thm_name def_thm rsp_thm (rty, qty) lthy
fun add_lift_def var qty rhs rsp_thm lthy =
let
val rty = fastype_of rhs
val quotient_thm = Lifting_Term.prove_quot_theorem lthy (rty, qty)
val absrep_trm = Lifting_Term.quot_thm_abs quotient_thm
val rty_forced = (domain_type o fastype_of) absrep_trm
val forced_rhs = force_rty_type lthy rty_forced rhs
val lhs = Free (Binding.print (#1 var), qty)
val prop = Logic.mk_equals (lhs, absrep_trm $ forced_rhs)
val (_, prop') = Local_Defs.cert_def lthy prop
val (_, newrhs) = Local_Defs.abs_def prop'
val ((_, (_ , def_thm)), lthy') =
Local_Theory.define (var, ((Thm.def_binding (#1 var), []), newrhs)) lthy
val transfer_thm = [quotient_thm, rsp_thm, def_thm] MRSL @{thm Quotient_to_transfer}
fun qualify defname suffix = Binding.name suffix
|> Binding.qualify true defname
val lhs_name = Binding.name_of (#1 var)
val rsp_thm_name = qualify lhs_name "rsp"
val code_eqn_thm_name = qualify lhs_name "rep_eq"
val transfer_thm_name = qualify lhs_name "transfer"
val transfer_attr = Attrib.internal (K Transfer.transfer_add)
in
lthy'
|> (snd oo Local_Theory.note) ((rsp_thm_name, []), [rsp_thm])
|> (snd oo Local_Theory.note) ((transfer_thm_name, [transfer_attr]), [transfer_thm])
|> define_code code_eqn_thm_name def_thm rsp_thm (rty_forced, qty)
end
fun mk_readable_rsp_thm_eq tm lthy =
let
val ctm = cterm_of (Proof_Context.theory_of lthy) tm
fun norm_fun_eq ctm =
let
fun abs_conv2 cv = Conv.abs_conv (K (Conv.abs_conv (K cv) lthy)) lthy
fun erase_quants ctm' =
case (Thm.term_of ctm') of
Const ("HOL.eq", _) $ _ $ _ => Conv.all_conv ctm'
| _ => (Conv.binder_conv (K erase_quants) lthy then_conv
Conv.rewr_conv @{thm fun_eq_iff[symmetric, THEN eq_reflection]}) ctm'
in
(abs_conv2 erase_quants then_conv Thm.eta_conversion) ctm
end
fun simp_arrows_conv ctm =
let
val unfold_conv = Conv.rewrs_conv
[@{thm fun_rel_eq_invariant[THEN eq_reflection]}, @{thm fun_rel_eq_rel[THEN eq_reflection]},
@{thm fun_rel_def[THEN eq_reflection]}]
val left_conv = simp_arrows_conv then_conv Conv.try_conv norm_fun_eq
fun binop_conv2 cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
in
case (Thm.term_of ctm) of
Const (@{const_name "fun_rel"}, _) $ _ $ _ =>
(binop_conv2 left_conv simp_arrows_conv then_conv unfold_conv) ctm
| _ => Conv.all_conv ctm
end
val unfold_ret_val_invs = Conv.bottom_conv
(K (Conv.try_conv (Conv.rewr_conv @{thm invariant_same_args}))) lthy
val simp_conv = Conv.arg_conv (Conv.fun2_conv simp_arrows_conv)
val univq_conv = Conv.rewr_conv @{thm HOL.all_simps(6)[symmetric, THEN eq_reflection]}
val univq_prenex_conv = Conv.top_conv (K (Conv.try_conv univq_conv)) lthy
val beta_conv = Thm.beta_conversion true
val eq_thm =
(simp_conv then_conv univq_prenex_conv then_conv beta_conv then_conv unfold_ret_val_invs) ctm
in
Object_Logic.rulify(eq_thm RS Drule.equal_elim_rule2)
end
fun lift_def_cmd (raw_var, rhs_raw) lthy =
let
val ((binding, SOME qty, mx), ctxt) = yield_singleton Proof_Context.read_vars raw_var lthy
val rhs = (Syntax.check_term ctxt o Syntax.parse_term ctxt) rhs_raw
fun try_to_prove_refl thm =
let
val lhs_eq =
thm
|> prop_of
|> Logic.dest_implies
|> fst
|> strip_all_body
|> try HOLogic.dest_Trueprop
in
case lhs_eq of
SOME (Const ("HOL.eq", _) $ _ $ _) => SOME (@{thm refl} RS thm)
| _ => NONE
end
val rsp_rel = Lifting_Term.equiv_relation lthy (fastype_of rhs, qty)
val rty_forced = (domain_type o fastype_of) rsp_rel;
val forced_rhs = force_rty_type lthy rty_forced rhs;
val internal_rsp_tm = HOLogic.mk_Trueprop (rsp_rel $ forced_rhs $ forced_rhs)
val readable_rsp_thm_eq = mk_readable_rsp_thm_eq internal_rsp_tm lthy
val maybe_proven_rsp_thm = try_to_prove_refl readable_rsp_thm_eq
val (readable_rsp_tm, _) = Logic.dest_implies (prop_of readable_rsp_thm_eq)
fun after_qed thm_list lthy =
let
val internal_rsp_thm =
case thm_list of
[] => the maybe_proven_rsp_thm
| [[thm]] => Goal.prove ctxt [] [] internal_rsp_tm
(fn _ => rtac readable_rsp_thm_eq 1 THEN Proof_Context.fact_tac [thm] 1)
in
add_lift_def (binding, mx) qty rhs internal_rsp_thm lthy
end
in
case maybe_proven_rsp_thm of
SOME _ => Proof.theorem NONE after_qed [] lthy
| NONE => Proof.theorem NONE after_qed [[(readable_rsp_tm,[])]] lthy
end
(* parser and command *)
val liftdef_parser =
((Parse.binding -- (@{keyword "::"} |-- (Parse.typ >> SOME) -- Parse.opt_mixfix')) >> Parse.triple2)
--| @{keyword "is"} -- Parse.term
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "lift_definition"}
"definition for constants over the quotient type"
(liftdef_parser >> lift_def_cmd)
end; (* structure *)