(* Title: HOL/Tools/Quotient/quotient_def.ML
Author: Cezary Kaliszyk and Christian Urban
Definitions for constants on quotient types.
*)
signature QUOTIENT_DEF =
sig
val add_quotient_def:
((binding * mixfix) * Attrib.binding) * (term * term) -> thm ->
local_theory -> Quotient_Info.quotconsts * local_theory
val quotient_def:
(binding * typ option * mixfix) option * (Attrib.binding * (term * term)) ->
local_theory -> Proof.state
val quotient_def_cmd:
(binding * string option * mixfix) option * (Attrib.binding * (string * string)) ->
local_theory -> Proof.state
val lift_raw_const: typ list -> (string * term * mixfix) -> local_theory ->
Quotient_Info.quotconsts * local_theory
end;
structure Quotient_Def: QUOTIENT_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 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 = Quotient_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 generate_code_cert ctxt def_thm rsp_thm (rty, qty) =
let
val quot_thm = Quotient_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 = (snd o Thm.dest_comb o snd o Thm.dest_comb o cprop_of) 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 = Quotient_Term.prove_quot_theorem lthy (get_body_types (rty, qty))
in
if Quotient_Type.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
(* The ML-interface for a quotient definition takes
as argument:
- an optional binding and mixfix annotation
- attributes
- the new constant as term
- the rhs of the definition as term
- respectfulness theorem for the rhs
It stores the qconst_info in the quotconsts data slot.
Restriction: At the moment the left- and right-hand
side of the definition must be a constant.
*)
fun error_msg bind str =
let
val name = Binding.name_of bind
val pos = Position.str_of (Binding.pos_of bind)
in
error ("Head of quotient_definition " ^
quote str ^ " differs from declaration " ^ name ^ pos)
end
fun add_quotient_def ((var, (name, atts)), (lhs, rhs)) rsp_thm lthy =
let
val rty = fastype_of rhs
val qty = fastype_of lhs
val absrep_trm =
Quotient_Term.absrep_fun lthy Quotient_Term.AbsF (rty, qty) $ rhs
val prop = Syntax.check_term lthy (Logic.mk_equals (lhs, absrep_trm))
val (_, prop') = Local_Defs.cert_def lthy prop
val (_, newrhs) = Local_Defs.abs_def prop'
val ((trm, (_ , def_thm)), lthy') =
Local_Theory.define (var, ((Thm.def_binding_optional (#1 var) name, atts), newrhs)) lthy
(* data storage *)
val qconst_data = {qconst = trm, rconst = rhs, def = def_thm}
val lhs_name = #1 var
val rsp_thm_name = Binding.suffix_name "_rsp" lhs_name
val code_eqn_thm_name = Binding.suffix_name "_code_eqn" lhs_name
val lthy'' = lthy'
|> Local_Theory.declaration {syntax = false, pervasive = true}
(fn phi =>
(case Quotient_Info.transform_quotconsts phi qconst_data of
qcinfo as {qconst = Const (c, _), ...} =>
Quotient_Info.update_quotconsts c qcinfo
| _ => I))
|> (snd oo Local_Theory.note)
((rsp_thm_name, [Attrib.internal (K Quotient_Info.rsp_rules_add)]),
[rsp_thm])
|> define_code code_eqn_thm_name def_thm rsp_thm (rty, qty)
in
(qconst_data, lthy'')
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 (@{const_name 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 gen_quotient_def prep_vars prep_term (raw_var, (attr, (lhs_raw, rhs_raw))) lthy =
let
val (vars, ctxt) = prep_vars (the_list raw_var) lthy
val T_opt = (case vars of [(_, SOME T, _)] => SOME T | _ => NONE)
val lhs = prep_term T_opt ctxt lhs_raw
val rhs = prep_term NONE ctxt rhs_raw
val (lhs_str, lhs_ty) = dest_Free lhs handle TERM _ => error "Constant already defined."
val _ = if null (strip_abs_vars rhs) then () else error "The definiens cannot be an abstraction"
val _ = if is_Const rhs then () else warning "The definiens is not a constant"
val var =
(case vars of
[] => (Binding.name lhs_str, NoSyn)
| [(binding, _, mx)] =>
if Variable.check_name binding = lhs_str then (binding, mx)
else error_msg binding lhs_str
| _ => raise Match)
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 (@{const_name HOL.eq}, _) $ _ $ _) => SOME (@{thm refl} RS thm)
| SOME _ => (case body_type (fastype_of lhs) of
Type (typ_name, _) => ( SOME
(#equiv_thm (the (Quotient_Info.lookup_quotients lthy typ_name))
RS @{thm Equiv_Relations.equivp_reflp} RS thm)
handle _ => NONE)
| _ => NONE
)
| _ => NONE
end
val rsp_rel = Quotient_Term.equiv_relation lthy (fastype_of rhs, lhs_ty)
val internal_rsp_tm = HOLogic.mk_Trueprop (Syntax.check_term lthy (rsp_rel $ rhs $ 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
snd (add_quotient_def ((var, attr), (lhs, 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
fun check_term' cnstr ctxt =
Syntax.check_term ctxt o (case cnstr of SOME T => Type.constraint T | _ => I)
fun read_term' cnstr ctxt =
check_term' cnstr ctxt o Syntax.parse_term ctxt
val quotient_def = gen_quotient_def Proof_Context.cert_vars check_term'
val quotient_def_cmd = gen_quotient_def Proof_Context.read_vars read_term'
(* a wrapper for automatically lifting a raw constant *)
fun lift_raw_const qtys (qconst_name, rconst, mx) ctxt =
let
val rty = fastype_of rconst
val qty = Quotient_Term.derive_qtyp ctxt qtys rty
val lhs = Free (qconst_name, qty)
in
(*quotient_def (SOME (Binding.name qconst_name, NONE, mx), (Attrib.empty_binding, (lhs, rconst))) ctxt*)
({qconst = lhs, rconst = lhs, def = @{thm refl}}, ctxt)
end
(* parser and command *)
val quotdef_parser =
Scan.option Parse_Spec.constdecl --
Parse.!!! (Parse_Spec.opt_thm_name ":" -- (Parse.term --| @{keyword "is"} -- Parse.term))
val _ =
Outer_Syntax.local_theory_to_proof @{command_spec "quotient_definition"}
"definition for constants over the quotient type"
(quotdef_parser >> quotient_def_cmd)
end; (* structure *)