(* 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
end;
structure Quotient_Def: QUOTIENT_DEF =
struct
(** Interface and Syntax Setup **)
(* Generation of the code certificate from the rsp theorem *)
open Lifting_Util
infix 0 MRSL
(* 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.here (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}
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 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, @{attributes [quot_respect]}), [rsp_thm])
in
(qconst_data, lthy'')
end
fun mk_readable_rsp_thm_eq tm lthy =
let
val ctm = Thm.cterm_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 rel_fun_eq_eq_onp[THEN eq_reflection]}, @{thm rel_fun_eq_rel[THEN eq_reflection]},
@{thm rel_fun_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 rel_fun}, _) $ _ $ _ =>
(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 eq_onp_same_args[THEN eq_reflection]}))) 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 lthy (eq_thm RS Drule.equal_elim_rule2)
end
fun gen_quotient_def prep_var parse_term (raw_var, (attr, (raw_lhs, raw_rhs))) lthy =
let
val (opt_var, ctxt) =
(case raw_var of
NONE => (NONE, lthy)
| SOME var => prep_var var lthy |>> SOME)
val lhs_constraint = (case opt_var of SOME (_, SOME T, _) => T | _ => dummyT)
fun prep_term T = parse_term ctxt #> Type.constraint T #> Syntax.check_term ctxt;
val lhs = prep_term lhs_constraint raw_lhs
val rhs = prep_term dummyT raw_rhs
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 opt_var of
NONE => (Binding.name lhs_str, NoSyn)
| SOME (binding, _, mx) =>
if Variable.check_name binding = lhs_str then (binding, mx)
else error_msg binding lhs_str);
fun try_to_prove_refl thm =
let
val lhs_eq =
thm
|> 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, _) =>
try (fn () =>
#equiv_thm (the (Quotient_Info.lookup_quotients lthy typ_name))
RS @{thm Equiv_Relations.equivp_reflp} RS thm) ()
| _ => 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 (Thm.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 ctxt [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
val quotient_def = gen_quotient_def Proof_Context.cert_var (K I)
val quotient_def_cmd = gen_quotient_def Proof_Context.read_var Syntax.parse_term
(* command syntax *)
val _ =
Outer_Syntax.local_theory_to_proof @{command_keyword quotient_definition}
"definition for constants over the quotient type"
(Scan.option Parse_Spec.constdecl --
Parse.!!! (Parse_Spec.opt_thm_name ":" -- (Parse.term -- (@{keyword "is"} |-- Parse.term)))
>> quotient_def_cmd);
end;