do not embed 'nat' into 'int's in 'smt2' method -- this is highly inefficient and decreases the Sledgehammer success rate significantly
(* Title: HOL/Tools/SMT2/smt2_normalize.ML
Author: Sascha Boehme, TU Muenchen
Normalization steps on theorems required by SMT solvers.
*)
signature SMT2_NORMALIZE =
sig
val drop_fact_warning: Proof.context -> thm -> unit
val atomize_conv: Proof.context -> conv
val special_quant_table: (string * thm) list
val case_bool_entry: string * thm
val abs_min_max_table: (string * thm) list
type extra_norm = Proof.context -> thm list * thm list -> thm list * thm list
val add_extra_norm: SMT2_Util.class * extra_norm -> Context.generic -> Context.generic
val normalize: Proof.context -> thm list -> (int * thm) list
end;
structure SMT2_Normalize: SMT2_NORMALIZE =
struct
fun drop_fact_warning ctxt =
SMT2_Config.verbose_msg ctxt (prefix "Warning: dropping assumption: " o
Display.string_of_thm ctxt)
(* general theorem normalizations *)
(** instantiate elimination rules **)
local
val (cpfalse, cfalse) = `SMT2_Util.mk_cprop (Thm.cterm_of @{theory} @{const False})
fun inst f ct thm =
let val cv = f (Drule.strip_imp_concl (Thm.cprop_of thm))
in Thm.instantiate ([], [(cv, ct)]) thm end
in
fun instantiate_elim thm =
(case Thm.concl_of thm of
@{const Trueprop} $ Var (_, @{typ bool}) => inst Thm.dest_arg cfalse thm
| Var _ => inst I cpfalse thm
| _ => thm)
end
(** normalize definitions **)
fun norm_def thm =
(case Thm.prop_of thm of
@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ _ $ Abs _) =>
norm_def (thm RS @{thm fun_cong})
| Const (@{const_name Pure.eq}, _) $ _ $ Abs _ => norm_def (thm RS @{thm meta_eq_to_obj_eq})
| _ => thm)
(** atomization **)
fun atomize_conv ctxt ct =
(case Thm.term_of ct of
@{const Pure.imp} $ _ $ _ =>
Conv.binop_conv (atomize_conv ctxt) then_conv Conv.rewr_conv @{thm atomize_imp}
| Const (@{const_name Pure.eq}, _) $ _ $ _ =>
Conv.binop_conv (atomize_conv ctxt) then_conv Conv.rewr_conv @{thm atomize_eq}
| Const (@{const_name Pure.all}, _) $ Abs _ =>
Conv.binder_conv (atomize_conv o snd) ctxt then_conv Conv.rewr_conv @{thm atomize_all}
| _ => Conv.all_conv) ct
val setup_atomize =
fold SMT2_Builtin.add_builtin_fun_ext'' [@{const_name Pure.imp}, @{const_name Pure.eq},
@{const_name Pure.all}, @{const_name Trueprop}]
(** unfold special quantifiers **)
val special_quant_table = [
(@{const_name Ex1}, @{thm Ex1_def_raw}),
(@{const_name Ball}, @{thm Ball_def_raw}),
(@{const_name Bex}, @{thm Bex_def_raw})]
local
fun special_quant (Const (n, _)) = AList.lookup (op =) special_quant_table n
| special_quant _ = NONE
fun special_quant_conv _ ct =
(case special_quant (Thm.term_of ct) of
SOME thm => Conv.rewr_conv thm
| NONE => Conv.all_conv) ct
in
fun unfold_special_quants_conv ctxt =
SMT2_Util.if_exists_conv (is_some o special_quant) (Conv.top_conv special_quant_conv ctxt)
val setup_unfolded_quants = fold (SMT2_Builtin.add_builtin_fun_ext'' o fst) special_quant_table
end
(** trigger inference **)
local
(*** check trigger syntax ***)
fun dest_trigger (Const (@{const_name pat}, _) $ _) = SOME true
| dest_trigger (Const (@{const_name nopat}, _) $ _) = SOME false
| dest_trigger _ = NONE
fun eq_list [] = false
| eq_list (b :: bs) = forall (equal b) bs
fun proper_trigger t =
t
|> these o try SMT2_Util.dest_symb_list
|> map (map_filter dest_trigger o these o try SMT2_Util.dest_symb_list)
|> (fn [] => false | bss => forall eq_list bss)
fun proper_quant inside f t =
(case t of
Const (@{const_name All}, _) $ Abs (_, _, u) => proper_quant true f u
| Const (@{const_name Ex}, _) $ Abs (_, _, u) => proper_quant true f u
| @{const trigger} $ p $ u =>
(if inside then f p else false) andalso proper_quant false f u
| Abs (_, _, u) => proper_quant false f u
| u1 $ u2 => proper_quant false f u1 andalso proper_quant false f u2
| _ => true)
fun check_trigger_error ctxt t =
error ("SMT triggers must only occur under quantifier and multipatterns " ^
"must have the same kind: " ^ Syntax.string_of_term ctxt t)
fun check_trigger_conv ctxt ct =
if proper_quant false proper_trigger (SMT2_Util.term_of ct) then Conv.all_conv ct
else check_trigger_error ctxt (Thm.term_of ct)
(*** infer simple triggers ***)
fun dest_cond_eq ct =
(case Thm.term_of ct of
Const (@{const_name HOL.eq}, _) $ _ $ _ => Thm.dest_binop ct
| @{const HOL.implies} $ _ $ _ => dest_cond_eq (Thm.dest_arg ct)
| _ => raise CTERM ("no equation", [ct]))
fun get_constrs thy (Type (n, _)) = these (Datatype.get_constrs thy n)
| get_constrs _ _ = []
fun is_constr thy (n, T) =
let fun match (m, U) = m = n andalso Sign.typ_instance thy (T, U)
in can (the o find_first match o get_constrs thy o Term.body_type) T end
fun is_constr_pat thy t =
(case Term.strip_comb t of
(Free _, []) => true
| (Const c, ts) => is_constr thy c andalso forall (is_constr_pat thy) ts
| _ => false)
fun is_simp_lhs ctxt t =
(case Term.strip_comb t of
(Const c, ts as _ :: _) =>
not (SMT2_Builtin.is_builtin_fun_ext ctxt c ts) andalso
forall (is_constr_pat (Proof_Context.theory_of ctxt)) ts
| _ => false)
fun has_all_vars vs t =
subset (op aconv) (vs, map Free (Term.add_frees t []))
fun minimal_pats vs ct =
if has_all_vars vs (Thm.term_of ct) then
(case Thm.term_of ct of
_ $ _ =>
(case pairself (minimal_pats vs) (Thm.dest_comb ct) of
([], []) => [[ct]]
| (ctss, ctss') => union (eq_set (op aconvc)) ctss ctss')
| _ => [])
else []
fun proper_mpat _ _ _ [] = false
| proper_mpat thy gen u cts =
let
val tps = (op ~~) (`gen (map Thm.term_of cts))
fun some_match u = tps |> exists (fn (t', t) =>
Pattern.matches thy (t', u) andalso not (t aconv u))
in not (Term.exists_subterm some_match u) end
val pat = SMT2_Util.mk_const_pat @{theory} @{const_name pat} SMT2_Util.destT1
fun mk_pat ct = Thm.apply (SMT2_Util.instT' ct pat) ct
fun mk_clist T =
pairself (Thm.cterm_of @{theory}) (SMT2_Util.symb_cons_const T, SMT2_Util.symb_nil_const T)
fun mk_list (ccons, cnil) f cts = fold_rev (Thm.mk_binop ccons o f) cts cnil
val mk_pat_list = mk_list (mk_clist @{typ pattern})
val mk_mpat_list = mk_list (mk_clist @{typ "pattern symb_list"})
fun mk_trigger ctss = mk_mpat_list (mk_pat_list mk_pat) ctss
val trigger_eq = mk_meta_eq @{lemma "p = trigger t p" by (simp add: trigger_def)}
fun insert_trigger_conv [] ct = Conv.all_conv ct
| insert_trigger_conv ctss ct =
let val (ctr, cp) = Thm.dest_binop (Thm.rhs_of trigger_eq) ||> rpair ct
in Thm.instantiate ([], [cp, (ctr, mk_trigger ctss)]) trigger_eq end
fun infer_trigger_eq_conv outer_ctxt (ctxt, cvs) ct =
let
val (lhs, rhs) = dest_cond_eq ct
val vs = map Thm.term_of cvs
val thy = Proof_Context.theory_of ctxt
fun get_mpats ct =
if is_simp_lhs ctxt (Thm.term_of ct) then minimal_pats vs ct
else []
val gen = Variable.export_terms ctxt outer_ctxt
val filter_mpats = filter (proper_mpat thy gen (Thm.term_of rhs))
in insert_trigger_conv (filter_mpats (get_mpats lhs)) ct end
fun has_trigger (@{const trigger} $ _ $ _) = true
| has_trigger _ = false
fun try_trigger_conv cv ct =
if SMT2_Util.under_quant has_trigger (SMT2_Util.term_of ct) then Conv.all_conv ct
else Conv.try_conv cv ct
fun infer_trigger_conv ctxt =
if Config.get ctxt SMT2_Config.infer_triggers then
try_trigger_conv (SMT2_Util.under_quant_conv (infer_trigger_eq_conv ctxt) ctxt)
else Conv.all_conv
in
fun trigger_conv ctxt =
SMT2_Util.prop_conv (check_trigger_conv ctxt then_conv infer_trigger_conv ctxt)
val setup_trigger =
fold SMT2_Builtin.add_builtin_fun_ext''
[@{const_name pat}, @{const_name nopat}, @{const_name trigger}]
end
(** combined general normalizations **)
fun gen_normalize1_conv ctxt =
atomize_conv ctxt then_conv
unfold_special_quants_conv ctxt then_conv
Thm.beta_conversion true then_conv
trigger_conv ctxt
fun gen_normalize1 ctxt =
instantiate_elim #>
norm_def #>
Conv.fconv_rule (Thm.beta_conversion true then_conv Thm.eta_conversion) #>
Drule.forall_intr_vars #>
Conv.fconv_rule (gen_normalize1_conv ctxt) #>
(* Z3 4.3.1 silently normalizes "P --> Q --> R" to "P & Q --> R" *)
Raw_Simplifier.rewrite_rule ctxt @{thms HOL.imp_conjL[symmetric, THEN eq_reflection]}
fun gen_norm1_safe ctxt (i, thm) =
(case try (gen_normalize1 ctxt) thm of
SOME thm' => SOME (i, thm')
| NONE => (drop_fact_warning ctxt thm; NONE))
fun gen_normalize ctxt iwthms = map_filter (gen_norm1_safe ctxt) iwthms
(* unfolding of definitions and theory-specific rewritings *)
fun expand_head_conv cv ct =
(case Thm.term_of ct of
_ $ _ =>
Conv.fun_conv (expand_head_conv cv) then_conv
Conv.try_conv (Thm.beta_conversion false)
| _ => cv) ct
(** rewrite bool case expressions as if expressions **)
val case_bool_entry = (@{const_name "bool.case_bool"}, @{thm case_bool_if})
local
fun is_case_bool (Const (@{const_name "bool.case_bool"}, _)) = true
| is_case_bool _ = false
fun unfold_conv _ =
SMT2_Util.if_true_conv (is_case_bool o Term.head_of)
(expand_head_conv (Conv.rewr_conv @{thm case_bool_if}))
in
fun rewrite_case_bool_conv ctxt =
SMT2_Util.if_exists_conv is_case_bool (Conv.top_conv unfold_conv ctxt)
val setup_case_bool = SMT2_Builtin.add_builtin_fun_ext'' @{const_name "bool.case_bool"}
end
(** unfold abs, min and max **)
val abs_min_max_table = [
(@{const_name min}, @{thm min_def_raw}),
(@{const_name max}, @{thm max_def_raw}),
(@{const_name abs}, @{thm abs_if_raw})]
local
fun abs_min_max ctxt (Const (n, Type (@{type_name fun}, [T, _]))) =
(case AList.lookup (op =) abs_min_max_table n of
NONE => NONE
| SOME thm => if SMT2_Builtin.is_builtin_typ_ext ctxt T then SOME thm else NONE)
| abs_min_max _ _ = NONE
fun unfold_amm_conv ctxt ct =
(case abs_min_max ctxt (Term.head_of (Thm.term_of ct)) of
SOME thm => expand_head_conv (Conv.rewr_conv thm)
| NONE => Conv.all_conv) ct
in
fun unfold_abs_min_max_conv ctxt =
SMT2_Util.if_exists_conv (is_some o abs_min_max ctxt) (Conv.top_conv unfold_amm_conv ctxt)
val setup_abs_min_max = fold (SMT2_Builtin.add_builtin_fun_ext'' o fst) abs_min_max_table
end
(** normalize numerals **)
local
(*
rewrite Numeral1 into 1
rewrite - 0 into 0
*)
fun is_irregular_number (Const (@{const_name numeral}, _) $ Const (@{const_name num.One}, _)) =
true
| is_irregular_number (Const (@{const_name uminus}, _) $ Const (@{const_name Groups.zero}, _)) =
true
| is_irregular_number _ = false
fun is_strange_number ctxt t = is_irregular_number t andalso SMT2_Builtin.is_builtin_num ctxt t
val proper_num_ss =
simpset_of (put_simpset HOL_ss @{context} addsimps @{thms Num.numeral_One minus_zero})
fun norm_num_conv ctxt =
SMT2_Util.if_conv (is_strange_number ctxt) (Simplifier.rewrite (put_simpset proper_num_ss ctxt))
Conv.no_conv
in
fun normalize_numerals_conv ctxt =
SMT2_Util.if_exists_conv (is_strange_number ctxt) (Conv.top_sweep_conv norm_num_conv ctxt)
end
(** combined unfoldings and rewritings **)
fun unfold_conv ctxt =
rewrite_case_bool_conv ctxt then_conv
unfold_abs_min_max_conv ctxt then_conv
Thm.beta_conversion true
fun unfold_polymorph ctxt = map (apsnd (Conv.fconv_rule (unfold_conv ctxt)))
fun unfold_monomorph ctxt = map (apsnd (Conv.fconv_rule (normalize_numerals_conv ctxt)))
(* overall normalization *)
fun burrow_ids f ithms =
let
val (is, thms) = split_list ithms
val (thms', extra_thms) = f thms
in (is ~~ thms') @ map (pair ~1) extra_thms end
type extra_norm = Proof.context -> thm list * thm list -> thm list * thm list
structure Extra_Norms = Generic_Data
(
type T = extra_norm SMT2_Util.dict
val empty = []
val extend = I
fun merge data = SMT2_Util.dict_merge fst data
)
fun add_extra_norm (cs, norm) = Extra_Norms.map (SMT2_Util.dict_update (cs, norm))
fun apply_extra_norms ctxt ithms =
let
val cs = SMT2_Config.solver_class_of ctxt
val es = SMT2_Util.dict_lookup (Extra_Norms.get (Context.Proof ctxt)) cs
in burrow_ids (fold (fn e => e ctxt) es o rpair []) ithms end
local
val ignored = member (op =) [@{const_name All}, @{const_name Ex},
@{const_name Let}, @{const_name If}, @{const_name HOL.eq}]
val schematic_consts_of =
let
fun collect (@{const trigger} $ p $ t) = collect_trigger p #> collect t
| collect (t $ u) = collect t #> collect u
| collect (Abs (_, _, t)) = collect t
| collect (t as Const (n, _)) =
if not (ignored n) then Monomorph.add_schematic_consts_of t else I
| collect _ = I
and collect_trigger t =
let val dest = these o try SMT2_Util.dest_symb_list
in fold (fold collect_pat o dest) (dest t) end
and collect_pat (Const (@{const_name pat}, _) $ t) = collect t
| collect_pat (Const (@{const_name nopat}, _) $ t) = collect t
| collect_pat _ = I
in (fn t => collect t Symtab.empty) end
in
fun monomorph ctxt xthms =
let val (xs, thms) = split_list xthms
in
map (pair 1) thms
|> Monomorph.monomorph schematic_consts_of ctxt
|> maps (uncurry (map o pair)) o map2 pair xs o map (map snd)
end
end
fun normalize ctxt wthms =
wthms
|> map_index I
|> gen_normalize ctxt
|> unfold_polymorph ctxt
|> monomorph ctxt
|> unfold_monomorph ctxt
|> apply_extra_norms ctxt
val _ = Theory.setup (Context.theory_map (
setup_atomize #>
setup_unfolded_quants #>
setup_trigger #>
setup_case_bool #>
setup_abs_min_max))
end;