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(* Title: HOL/Library/conditional_parametricity.ML
Author: Jan Gilcher, Andreas Lochbihler, Dmitriy Traytel, ETH Zürich
A conditional parametricity prover
*)
signature CONDITIONAL_PARAMETRICITY =
sig
exception WARNING of string
type settings =
{suppress_print_theorem: bool,
suppress_warnings: bool,
warnings_as_errors: bool,
use_equality_heuristic: bool}
val default_settings: settings
val quiet_settings: settings
val parametric_constant: settings -> Attrib.binding * thm -> Proof.context ->
(thm * Proof.context)
val get_parametricity_theorems: Proof.context -> thm list
val prove_goal: settings -> Proof.context -> thm option -> term -> thm
val prove_find_goal_cond: settings -> Proof.context -> thm list -> thm option -> term -> thm
val mk_goal: Proof.context -> term -> term
val mk_cond_goal: Proof.context -> thm -> term * thm
val mk_param_goal_from_eq_def: Proof.context -> thm -> term
val step_tac: settings -> Proof.context -> thm list -> int -> tactic
end
structure Conditional_Parametricity: CONDITIONAL_PARAMETRICITY =
struct
type settings =
{suppress_print_theorem: bool,
suppress_warnings: bool,
warnings_as_errors: bool (* overrides suppress_warnings! *),
use_equality_heuristic: bool};
val quiet_settings =
{suppress_print_theorem = true,
suppress_warnings = true,
warnings_as_errors = false,
use_equality_heuristic = false};
val default_settings =
{suppress_print_theorem = false,
suppress_warnings = false,
warnings_as_errors = false,
use_equality_heuristic = false};
(* helper functions *)
fun strip_imp_prems_concl (Const("Pure.imp", _) $ A $ B) = A :: strip_imp_prems_concl B
| strip_imp_prems_concl C = [C];
fun strip_prop_safe t = Logic.unprotect t handle TERM _ => t;
fun get_class_of ctxt t =
Axclass.class_of_param (Proof_Context.theory_of ctxt) (fst (dest_Const t));
fun is_class_op ctxt t =
let
val t' = t |> Envir.eta_contract;
in
Term.is_Const t' andalso is_some (get_class_of ctxt t')
end;
fun apply_Var_to_bounds t =
let
val (t, ts) = strip_comb t;
in
(case t of
Var (xi, _) =>
let
val (bounds, tail) = chop_prefix is_Bound ts;
in
list_comb (Var (xi, fastype_of (betapplys (t, bounds))), map apply_Var_to_bounds tail)
end
| _ => list_comb (t, map apply_Var_to_bounds ts))
end;
fun theorem_format_error ctxt thm =
let
val msg = Pretty.string_of (Pretty.chunks [(Pretty.para
"Unexpected format of definition. Must be an unconditional equation."), Thm.pretty_thm ctxt thm]);
in error msg end;
(* Tacticals and Tactics *)
exception FINISH of thm;
(* Tacticals *)
fun REPEAT_TRY_ELSE_DEFER tac st =
let
fun COMB' tac count st = (
let
val n = Thm.nprems_of st;
in
(if n = 0 then all_tac st else
(case Seq.pull ((tac THEN COMB' tac 0) st) of
NONE =>
if count+1 = n
then raise FINISH st
else (defer_tac 1 THEN (COMB' tac (count+1))) st
| some => Seq.make (fn () => some)))
end)
in COMB' tac 0 st end;
(* Tactics *)
(* helper tactics for printing *)
fun error_tac ctxt msg st =
(error(msg ^ "\n" ^ Pretty.string_of (Pretty.chunks (Goal_Display.pretty_goals ctxt st)));
Seq.single st);
fun error_tac' ctxt msg = SELECT_GOAL (error_tac ctxt msg);
(* finds assumption of the form "Rel ?B Bound x Bound y", rotates it in front,
applies rel_app arity times and uses ams_rl *)
fun rel_app_tac ctxt t x y arity =
let
val rel_app = [@{thm Rel_app}];
val assume = [asm_rl];
fun find_and_rotate_tac t i =
let
fun is_correct_rule t =
(case t of
Const (\<^const_name>\<open>HOL.Trueprop\<close>, _) $ (Const (\<^const_name>\<open>Transfer.Rel\<close>, _) $
_ $ Bound x' $ Bound y') => x = x' andalso y = y'
| _ => false);
val idx = find_index is_correct_rule (t |> Logic.strip_assums_hyp);
in
if idx < 0 then no_tac else rotate_tac idx i
end;
fun rotate_and_dresolve_tac ctxt arity i = REPEAT_DETERM_N (arity - 1)
(EVERY' [rotate_tac ~1, dresolve_tac ctxt rel_app, defer_tac] i);
in
SELECT_GOAL (EVERY' [find_and_rotate_tac t, forward_tac ctxt rel_app, defer_tac,
rotate_and_dresolve_tac ctxt arity, rotate_tac ~1, eresolve_tac ctxt assume] 1)
end;
exception WARNING of string;
fun transform_rules 0 thms = thms
| transform_rules n thms = transform_rules (n - 1) (curry (Drule.RL o swap)
@{thms Rel_app Rel_match_app} thms);
fun assume_equality_tac settings ctxt t arity i st =
let
val quiet = #suppress_warnings settings;
val errors = #warnings_as_errors settings;
val T = fastype_of t;
val is_eq_lemma = @{thm is_equality_Rel} |> Thm.incr_indexes ((Term.maxidx_of_term t) + 1) |>
Drule.infer_instantiate' ctxt [NONE, SOME (Thm.cterm_of ctxt t)];
val msg = Pretty.string_of (Pretty.chunks [Pretty.paragraph ((Pretty.text
"No rule found for constant \"") @ [Syntax.pretty_term ctxt t, Pretty.str " :: " ,
Syntax.pretty_typ ctxt T] @ (Pretty.text "\". Using is_eq_lemma:")), Pretty.quote
(Thm.pretty_thm ctxt is_eq_lemma)]);
fun msg_tac st = (if errors then raise WARNING msg else if quiet then () else warning msg;
Seq.single st)
val tac = resolve_tac ctxt (transform_rules arity [is_eq_lemma]) i;
in
(if fold_atyps (K (K true)) T false then msg_tac THEN tac else tac) st
end;
fun mark_class_as_match_tac ctxt const const' arity =
let
val rules = transform_rules arity [@{thm Rel_match_Rel} |> Thm.incr_indexes ((Int.max o
apply2 Term.maxidx_of_term) (const, const') + 1) |> Drule.infer_instantiate' ctxt [NONE,
SOME (Thm.cterm_of ctxt const), SOME (Thm.cterm_of ctxt const')]];
in resolve_tac ctxt rules end;
(* transforms the parametricity theorems to fit a given arity and uses them for resolution *)
fun parametricity_thm_tac settings ctxt parametricity_thms const arity =
let
val rules = transform_rules arity parametricity_thms;
in resolve_tac ctxt rules ORELSE' assume_equality_tac settings ctxt const arity end;
(* variant of parametricity_thm_tac to use matching *)
fun parametricity_thm_match_tac ctxt parametricity_thms arity =
let
val rules = transform_rules arity parametricity_thms;
in match_tac ctxt rules end;
fun rel_abs_tac ctxt = resolve_tac ctxt [@{thm Rel_abs}];
fun step_tac' settings ctxt parametricity_thms (tm, i) =
(case tm |> Logic.strip_assums_concl of
Const (\<^const_name>\<open>HOL.Trueprop\<close>, _) $ (Const (rel, _) $ _ $ t $ u) =>
let
val (arity_of_t, arity_of_u) = apply2 (strip_comb #> snd #> length) (t, u);
in
(case rel of
\<^const_name>\<open>Transfer.Rel\<close> =>
(case (head_of t, head_of u) of
(Abs _, _) => rel_abs_tac ctxt
| (_, Abs _) => rel_abs_tac ctxt
| (const as (Const _), const' as (Const _)) =>
if #use_equality_heuristic settings andalso t aconv u
then
assume_equality_tac quiet_settings ctxt t 0
else if arity_of_t = arity_of_u
then if is_class_op ctxt const orelse is_class_op ctxt const'
then mark_class_as_match_tac ctxt const const' arity_of_t
else parametricity_thm_tac settings ctxt parametricity_thms const arity_of_t
else error_tac' ctxt "Malformed term. Arities of t and u don't match."
| (Bound x, Bound y) =>
if arity_of_t = arity_of_u then if arity_of_t > 0 then rel_app_tac ctxt tm x y arity_of_t
else assume_tac ctxt
else error_tac' ctxt "Malformed term. Arities of t and u don't match."
| _ => error_tac' ctxt
"Unexpected format. Expected (Abs _, _), (_, Abs _), (Const _, Const _) or (Bound _, Bound _).")
| \<^const_name>\<open>Conditional_Parametricity.Rel_match\<close> =>
parametricity_thm_match_tac ctxt parametricity_thms arity_of_t
| _ => error_tac' ctxt "Unexpected format. Expected Transfer.Rel or Rel_match marker." ) i
end
| Const (\<^const_name>\<open>HOL.Trueprop\<close>, _) $ (Const (\<^const_name>\<open>Transfer.is_equality\<close>, _) $ _) =>
Transfer.eq_tac ctxt i
| _ => error_tac' ctxt "Unexpected format. Not of form Const (HOL.Trueprop, _) $ _" i);
fun step_tac settings = SUBGOAL oo step_tac' settings;
fun apply_theorem_tac ctxt thm =
HEADGOAL (resolve_tac ctxt [Local_Defs.unfold ctxt @{thms Pure.prop_def} thm] THEN_ALL_NEW
assume_tac ctxt);
(* Goal Generation *)
fun strip_boundvars_from_rel_match t =
(case t of
(Tp as Const (\<^const_name>\<open>HOL.Trueprop\<close>, _)) $
((Rm as Const (\<^const_name>\<open>Conditional_Parametricity.Rel_match\<close>, _)) $ R $ t $ t') =>
Tp $ (Rm $ apply_Var_to_bounds R $ t $ t')
| _ => t);
val extract_conditions =
let
val filter_bounds = filter_out Term.is_open;
val prem_to_conditions =
map (map strip_boundvars_from_rel_match o strip_imp_prems_concl o strip_all_body);
val remove_duplicates = distinct Term.aconv;
in remove_duplicates o filter_bounds o flat o prem_to_conditions end;
fun mk_goal ctxt t =
let
val ctxt = fold (Variable.declare_typ o snd) (Term.add_frees t []) ctxt;
val t = singleton (Variable.polymorphic ctxt) t;
val i = maxidx_of_term t + 1;
fun tvar_to_tfree ((name, _), sort) = (name, sort);
val tvars = Term.add_tvars t [];
val new_frees = map TFree (Term.variant_frees t (map tvar_to_tfree tvars));
val u = subst_atomic_types ((map TVar tvars) ~~ new_frees) t;
val T = fastype_of t;
val U = fastype_of u;
val R = [T,U] ---> \<^typ>\<open>bool\<close>
val r = Var (("R", 2 * i), R);
val transfer_rel = Const (\<^const_name>\<open>Transfer.Rel\<close>, [R,T,U] ---> \<^typ>\<open>bool\<close>);
in HOLogic.mk_Trueprop (transfer_rel $ r $ t $ u) end;
fun mk_abs_helper T t =
let
val U = fastype_of t;
fun mk_abs_helper' T U =
if T = U then t else
let
val (T2, T1) = Term.dest_funT T;
in
Term.absdummy T2 (mk_abs_helper' T1 U)
end;
in mk_abs_helper' T U end;
fun compare_ixs ((name, i):indexname, (name', i'):indexname) = if name < name' then LESS
else if name > name' then GREATER
else if i < i' then LESS
else if i > i' then GREATER
else EQUAL;
fun mk_cond_goal ctxt thm =
let
val conclusion = (hd o strip_imp_prems_concl o strip_prop_safe o Thm.concl_of) thm;
val conditions = (extract_conditions o Thm.prems_of) thm;
val goal = Logic.list_implies (conditions, conclusion);
fun compare ((ix, _), (ix', _)) = compare_ixs (ix, ix');
val goal_vars = Term.add_vars goal [] |> Ord_List.make compare;
val (ixs, Ts) = split_list goal_vars;
val (_, Ts') = Term.add_vars (Thm.prop_of thm) [] |> Ord_List.make compare
|> Ord_List.inter compare goal_vars |> split_list;
val (As, _) = Ctr_Sugar_Util.mk_Frees "A" Ts ctxt;
val goal_subst = ixs ~~ As;
val thm_subst = ixs ~~ (map2 mk_abs_helper Ts' As);
val thm' = thm |> Drule.infer_instantiate ctxt (map (apsnd (Thm.cterm_of ctxt)) thm_subst);
in (goal |> Term.subst_Vars goal_subst, thm') end;
fun mk_param_goal_from_eq_def ctxt thm =
let
val t =
(case Thm.full_prop_of thm of
(Const (\<^const_name>\<open>Pure.eq\<close>, _) $ t' $ _) => t'
| _ => theorem_format_error ctxt thm);
in mk_goal ctxt t end;
(* Transformations and parametricity theorems *)
fun transform_class_rule ctxt thm =
(case Thm.concl_of thm of
Const (\<^const_name>\<open>HOL.Trueprop\<close>, _) $ (Const (\<^const_name>\<open>Transfer.Rel\<close>, _) $ _ $ t $ u ) =>
(if curry Term.aconv_untyped t u andalso is_class_op ctxt t then
thm RS @{thm Rel_Rel_match}
else thm)
| _ => thm);
fun is_parametricity_theorem thm =
(case Thm.concl_of thm of
Const (\<^const_name>\<open>HOL.Trueprop\<close>, _) $ (Const (rel, _) $ _ $ t $ u ) =>
if rel = \<^const_name>\<open>Transfer.Rel\<close> orelse
rel = \<^const_name>\<open>Conditional_Parametricity.Rel_match\<close>
then curry Term.aconv_untyped t u
else false
| _ => false);
(* Pre- and postprocessing of theorems *)
fun mk_Domainp_assm (T, R) =
HOLogic.mk_eq ((Const (\<^const_name>\<open>Domainp\<close>, Term.fastype_of T --> Term.fastype_of R) $ T), R);
val Domainp_lemma =
@{lemma "(!!R. Domainp T = R ==> PROP (P R)) == PROP (P (Domainp T))"
by (rule, drule meta_spec,
erule meta_mp, rule HOL.refl, simp)};
fun fold_Domainp f (t as Const (\<^const_name>\<open>Domainp\<close>,_) $ (Var (_,_))) = f t
| fold_Domainp f (t $ u) = fold_Domainp f t #> fold_Domainp f u
| fold_Domainp f (Abs (_, _, t)) = fold_Domainp f t
| fold_Domainp _ _ = I;
fun subst_terms tab t =
let
val t' = Termtab.lookup tab t
in
(case t' of
SOME t' => t'
| NONE =>
(case t of
u $ v => (subst_terms tab u) $ (subst_terms tab v)
| Abs (a, T, t) => Abs (a, T, subst_terms tab t)
| t => t))
end;
fun gen_abstract_domains ctxt (dest : term -> term * (term -> term)) thm =
let
val prop = Thm.prop_of thm
val (t, mk_prop') = dest prop
val Domainp_ts = rev (fold_Domainp (fn t => insert op= t) t [])
val Domainp_Ts = map (snd o dest_funT o snd o dest_Const o fst o dest_comb) Domainp_ts
val used = Term.add_free_names t []
val rels = map (snd o dest_comb) Domainp_ts
val rel_names = map (fst o fst o dest_Var) rels
val names = map (fn name => ("D" ^ name)) rel_names |> Name.variant_list used
val frees = map Free (names ~~ Domainp_Ts)
val prems = map (HOLogic.mk_Trueprop o mk_Domainp_assm) (rels ~~ frees);
val t' = subst_terms (fold Termtab.update (Domainp_ts ~~ frees) Termtab.empty) t
val prop1 = fold Logic.all frees (Logic.list_implies (prems, mk_prop' t'))
val prop2 = Logic.list_rename_params (rev names) prop1
val cprop = Thm.cterm_of ctxt prop2
val equal_thm = Raw_Simplifier.rewrite ctxt false [Domainp_lemma] cprop
fun forall_elim thm = Thm.forall_elim_vars (Thm.maxidx_of thm + 1) thm;
in
forall_elim (thm COMP (equal_thm COMP @{thm equal_elim_rule2}))
end
handle TERM _ => thm;
fun abstract_domains_transfer ctxt thm =
let
fun dest prop =
let
val prems = Logic.strip_imp_prems prop
val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop)
val ((rel, x), y) = apfst Term.dest_comb (Term.dest_comb concl)
in
(x, fn x' =>
Logic.list_implies (prems, HOLogic.mk_Trueprop (rel $ x' $ y)))
end
in
gen_abstract_domains ctxt dest thm
end;
fun transfer_rel_conv conv =
Conv.concl_conv ~1 (HOLogic.Trueprop_conv (Conv.fun2_conv (Conv.arg_conv conv)));
fun fold_relator_eqs_conv ctxt ct = (Transfer.bottom_rewr_conv (Transfer.get_relator_eq ctxt)) ct;
fun mk_is_equality t =
Const (\<^const_name>\<open>is_equality\<close>, Term.fastype_of t --> HOLogic.boolT) $ t;
val is_equality_lemma =
@{lemma "(!!R. is_equality R ==> PROP (P R)) == PROP (P (=))"
by (unfold is_equality_def, rule, drule meta_spec,
erule meta_mp, rule HOL.refl, simp)};
fun gen_abstract_equalities ctxt (dest : term -> term * (term -> term)) thm =
let
val prop = Thm.prop_of thm
val (t, mk_prop') = dest prop
(* Only consider "(=)" at non-base types *)
fun is_eq (Const (\<^const_name>\<open>HOL.eq\<close>, Type ("fun", [T, _]))) =
(case T of Type (_, []) => false | _ => true)
| is_eq _ = false
val add_eqs = Term.fold_aterms (fn t => if is_eq t then insert (op =) t else I)
val eq_consts = rev (add_eqs t [])
val eqTs = map (snd o dest_Const) eq_consts
val used = Term.add_free_names prop []
val names = map (K "") eqTs |> Name.variant_list used
val frees = map Free (names ~~ eqTs)
val prems = map (HOLogic.mk_Trueprop o mk_is_equality) frees
val prop1 = mk_prop' (Term.subst_atomic (eq_consts ~~ frees) t)
val prop2 = fold Logic.all frees (Logic.list_implies (prems, prop1))
val cprop = Thm.cterm_of ctxt prop2
val equal_thm = Raw_Simplifier.rewrite ctxt false [is_equality_lemma] cprop
fun forall_elim thm = Thm.forall_elim_vars (Thm.maxidx_of thm + 1) thm
in
forall_elim (thm COMP (equal_thm COMP @{thm equal_elim_rule2}))
end
handle TERM _ => thm;
fun abstract_equalities_transfer ctxt thm =
let
fun dest prop =
let
val prems = Logic.strip_imp_prems prop
val concl = HOLogic.dest_Trueprop (Logic.strip_imp_concl prop)
val ((rel, x), y) = apfst Term.dest_comb (Term.dest_comb concl)
in
(rel, fn rel' =>
Logic.list_implies (prems, HOLogic.mk_Trueprop (rel' $ x $ y)))
end
val contracted_eq_thm =
Conv.fconv_rule (transfer_rel_conv (fold_relator_eqs_conv ctxt)) thm
handle CTERM _ => thm
in
gen_abstract_equalities ctxt dest contracted_eq_thm
end;
fun prep_rule ctxt = abstract_equalities_transfer ctxt #> abstract_domains_transfer ctxt;
fun get_preprocess_theorems ctxt =
Named_Theorems.get ctxt \<^named_theorems>\<open>parametricity_preprocess\<close>;
fun preprocess_theorem ctxt =
Local_Defs.unfold0 ctxt (get_preprocess_theorems ctxt)
#> transform_class_rule ctxt;
fun postprocess_theorem ctxt =
Local_Defs.fold ctxt (@{thm Rel_Rel_match_eq} :: get_preprocess_theorems ctxt)
#> prep_rule ctxt
#> Local_Defs.unfold ctxt @{thms Rel_def};
fun get_parametricity_theorems ctxt =
let
val parametricity_thm_map_filter =
Option.filter (is_parametricity_theorem andf (not o curry Term.could_unify
(Thm.full_prop_of @{thm is_equality_Rel})) o Thm.full_prop_of) o preprocess_theorem ctxt;
in
map_filter (parametricity_thm_map_filter o Thm.transfer' ctxt)
(Transfer.get_transfer_raw ctxt)
end;
(* Provers *)
(* Tries to prove a parametricity theorem without conditions, returns the last goal_state as thm *)
fun prove_find_goal_cond settings ctxt rules def_thm t =
let
fun find_conditions_tac {context = ctxt, prems = _} = unfold_tac ctxt (the_list def_thm) THEN
(REPEAT_TRY_ELSE_DEFER o HEADGOAL) (step_tac settings ctxt rules);
in
Goal.prove ctxt [] [] t find_conditions_tac handle FINISH st => st
end;
(* Simplifies and proves thm *)
fun prove_cond_goal ctxt thm =
let
val (goal, thm') = mk_cond_goal ctxt thm;
val vars = Variable.add_free_names ctxt goal [];
fun prove_conditions_tac {context = ctxt, prems = _} = apply_theorem_tac ctxt thm';
val vars = Variable.add_free_names ctxt (Thm.prop_of thm') vars;
in
Goal.prove ctxt vars [] goal prove_conditions_tac
end;
(* Finds necessary conditions for t and proofs conditional parametricity of t under those conditions *)
fun prove_goal settings ctxt def_thm t =
let
val parametricity_thms = get_parametricity_theorems ctxt;
val found_thm = prove_find_goal_cond settings ctxt parametricity_thms def_thm t;
val thm = prove_cond_goal ctxt found_thm;
in
postprocess_theorem ctxt thm
end;
(* Commands *)
fun gen_parametric_constant settings prep_att prep_thm (raw_b : Attrib.binding, raw_eq) lthy =
let
val b = apsnd (map (prep_att lthy)) raw_b;
val def_thm = (prep_thm lthy raw_eq);
val eq = Ctr_Sugar_Util.mk_abs_def def_thm handle TERM _ => theorem_format_error lthy def_thm;
val goal= mk_param_goal_from_eq_def lthy eq;
val thm = prove_goal settings lthy (SOME eq) goal;
val (res, lthy') = Local_Theory.note (b, [thm]) lthy;
val _ = if #suppress_print_theorem settings then () else
Proof_Display.print_results true (Position.thread_data ()) lthy' (("theorem",""), [res]);
in
(the_single (snd res), lthy')
end;
fun parametric_constant settings = gen_parametric_constant settings (K I) (K I);
val parametric_constant_cmd = snd oo gen_parametric_constant default_settings (Attrib.check_src)
(singleton o Attrib.eval_thms);
val _ =
Outer_Syntax.local_theory \<^command_keyword>\<open>parametric_constant\<close> "proves parametricity"
((Parse_Spec.opt_thm_name ":" -- Parse.thm) >> parametric_constant_cmd);
end;
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