be more precise: only treat constant 'distinct' applied to an explicit list as built-in
(* Title: HOL/Tools/SMT/smt_normalize.ML
Author: Sascha Boehme, TU Muenchen
Normalization steps on theorems required by SMT solvers:
* simplify trivial distincts (those with less than three elements),
* rewrite bool case expressions as if expressions,
* normalize numerals (e.g. replace negative numerals by negated positive
numerals),
* embed natural numbers into integers,
* add extra rules specifying types and constants which occur frequently,
* fully translate into object logic, add universal closure,
* monomorphize (create instances of schematic rules),
* lift lambda terms,
* make applications explicit for functions with varying number of arguments.
* add (hypothetical definitions for) missing datatype selectors,
*)
signature SMT_NORMALIZE =
sig
type extra_norm = bool -> (int * thm) list -> Proof.context ->
(int * thm) list * Proof.context
val normalize: extra_norm -> bool -> (int * thm) list -> Proof.context ->
(int * thm) list * Proof.context
val atomize_conv: Proof.context -> conv
val eta_expand_conv: (Proof.context -> conv) -> Proof.context -> conv
end
structure SMT_Normalize: SMT_NORMALIZE =
struct
structure U = SMT_Utils
infix 2 ??
fun (test ?? f) x = if test x then f x else x
(* simplification of trivial distincts (distinct should have at least
three elements in the argument list) *)
local
fun is_trivial_distinct (Const (@{const_name distinct}, _) $ t) =
(case try HOLogic.dest_list t of
SOME [] => true
| SOME [_] => true
| _ => false)
| is_trivial_distinct _ = false
val thms = map mk_meta_eq @{lemma
"distinct [] = True"
"distinct [x] = True"
"distinct [x, y] = (x ~= y)"
by simp_all}
fun distinct_conv _ =
U.if_true_conv is_trivial_distinct (Conv.rewrs_conv thms)
in
fun trivial_distinct ctxt =
map (apsnd ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ??
Conv.fconv_rule (Conv.top_conv distinct_conv ctxt)))
end
(* rewrite bool case expressions as if expressions *)
local
val is_bool_case = (fn
Const (@{const_name "bool.bool_case"}, _) $ _ $ _ $ _ => true
| _ => false)
val thm = mk_meta_eq @{lemma
"(case P of True => x | False => y) = (if P then x else y)" by simp}
val unfold_conv = U.if_true_conv is_bool_case (Conv.rewr_conv thm)
in
fun rewrite_bool_cases ctxt =
map (apsnd ((Term.exists_subterm is_bool_case o Thm.prop_of) ??
Conv.fconv_rule (Conv.top_conv (K unfold_conv) ctxt)))
end
(* normalization of numerals: rewriting of negative integer numerals into
positive numerals, Numeral0 into 0, Numeral1 into 1 *)
local
fun is_number_sort ctxt T =
Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring})
fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) $ _) =
(case try HOLogic.dest_number t of
SOME (T, i) => is_number_sort ctxt T andalso i < 2
| NONE => false)
| is_strange_number _ _ = false
val pos_numeral_ss = HOL_ss
addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}]
addsimps @{thms Int.pred_bin_simps}
addsimps @{thms Int.normalize_bin_simps}
addsimps @{lemma
"Int.Min = - Int.Bit1 Int.Pls"
"Int.Bit0 (- Int.Pls) = - Int.Pls"
"Int.Bit0 (- k) = - Int.Bit0 k"
"Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)"
by simp_all (simp add: pred_def)}
fun pos_conv ctxt = U.if_conv (is_strange_number ctxt)
(Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
Conv.no_conv
in
fun normalize_numerals ctxt =
map (apsnd ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ??
Conv.fconv_rule (Conv.top_sweep_conv pos_conv ctxt)))
end
(* embedding of standard natural number operations into integer operations *)
local
val nat_embedding = map (pair ~1) @{lemma
"nat (int n) = n"
"i >= 0 --> int (nat i) = i"
"i < 0 --> int (nat i) = 0"
by simp_all}
val nat_rewriting = @{lemma
"0 = nat 0"
"1 = nat 1"
"(number_of :: int => nat) = (%i. nat (number_of i))"
"int (nat 0) = 0"
"int (nat 1) = 1"
"op < = (%a b. int a < int b)"
"op <= = (%a b. int a <= int b)"
"Suc = (%a. nat (int a + 1))"
"op + = (%a b. nat (int a + int b))"
"op - = (%a b. nat (int a - int b))"
"op * = (%a b. nat (int a * int b))"
"op div = (%a b. nat (int a div int b))"
"op mod = (%a b. nat (int a mod int b))"
"min = (%a b. nat (min (int a) (int b)))"
"max = (%a b. nat (max (int a) (int b)))"
"int (nat (int a + int b)) = int a + int b"
"int (nat (int a + 1)) = int a + 1" (* special rule due to Suc above *)
"int (nat (int a * int b)) = int a * int b"
"int (nat (int a div int b)) = int a div int b"
"int (nat (int a mod int b)) = int a mod int b"
"int (nat (min (int a) (int b))) = min (int a) (int b)"
"int (nat (max (int a) (int b))) = max (int a) (int b)"
by (auto intro!: ext simp add: nat_mult_distrib nat_div_distrib
nat_mod_distrib int_mult[symmetric] zdiv_int[symmetric]
zmod_int[symmetric])}
fun on_positive num f x =
(case try HOLogic.dest_number (Thm.term_of num) of
SOME (_, i) => if i >= 0 then SOME (f x) else NONE
| NONE => NONE)
val cancel_int_nat_ss = HOL_ss
addsimps [@{thm Nat_Numeral.nat_number_of}]
addsimps [@{thm Nat_Numeral.int_nat_number_of}]
addsimps @{thms neg_simps}
val int_eq = Thm.cterm_of @{theory} @{const "==" (int)}
fun cancel_int_nat_simproc _ ss ct =
let
val num = Thm.dest_arg (Thm.dest_arg ct)
val goal = Thm.mk_binop int_eq ct num
val simpset = Simplifier.inherit_context ss cancel_int_nat_ss
fun tac _ = Simplifier.simp_tac simpset 1
in on_positive num (Goal.prove_internal [] goal) tac end
val nat_ss = HOL_ss
addsimps nat_rewriting
addsimprocs [
Simplifier.make_simproc {
name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}],
proc = cancel_int_nat_simproc, identifier = [] }]
fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss)
val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat}))
val uses_nat_int = Term.exists_subterm (member (op aconv)
[@{const of_nat (int)}, @{const nat}])
in
fun nat_as_int ctxt =
map (apsnd ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt))) #>
exists (uses_nat_int o Thm.prop_of o snd) ?? append nat_embedding
end
(* further normalizations: beta/eta, universal closure, atomize *)
val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)}
fun eta_expand_conv cv ctxt =
Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt
local
val eta_conv = eta_expand_conv
fun args_conv cv ct =
(case Thm.term_of ct of
_ $ _ => Conv.combination_conv (args_conv cv) cv
| _ => Conv.all_conv) ct
fun eta_args_conv cv 0 = args_conv o cv
| eta_args_conv cv i = eta_conv (eta_args_conv cv (i-1))
fun keep_conv ctxt = Conv.binder_conv (norm_conv o snd) ctxt
and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt)
and keep_let_conv ctxt = Conv.combination_conv
(Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt)
and unfold_let_conv ctxt = Conv.combination_conv
(Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt)
and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt
and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt
and unfold_ball_conv ctxt = unfold_conv (mk_meta_eq @{thm Ball_def}) ctxt
and unfold_bex_conv ctxt = unfold_conv (mk_meta_eq @{thm Bex_def}) ctxt
and norm_conv ctxt ct =
(case Thm.term_of ct of
Const (@{const_name All}, _) $ Abs _ => keep_conv
| Const (@{const_name All}, _) $ _ => eta_binder_conv
| Const (@{const_name All}, _) => eta_conv eta_binder_conv
| Const (@{const_name Ex}, _) $ Abs _ => keep_conv
| Const (@{const_name Ex}, _) $ _ => eta_binder_conv
| Const (@{const_name Ex}, _) => eta_conv eta_binder_conv
| Const (@{const_name Let}, _) $ _ $ Abs _ => keep_let_conv
| Const (@{const_name Let}, _) $ _ $ _ => unfold_let_conv
| Const (@{const_name Let}, _) $ _ => eta_conv unfold_let_conv
| Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv)
| Const (@{const_name Ex1}, _) $ _ => unfold_ex1_conv
| Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv
| Const (@{const_name Ball}, _) $ _ $ _ => unfold_ball_conv
| Const (@{const_name Ball}, _) $ _ => eta_conv unfold_ball_conv
| Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv)
| Const (@{const_name Bex}, _) $ _ $ _ => unfold_bex_conv
| Const (@{const_name Bex}, _) $ _ => eta_conv unfold_bex_conv
| Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv)
| Abs _ => Conv.abs_conv (norm_conv o snd)
| _ =>
(case Term.strip_comb (Thm.term_of ct) of
(Const (c as (_, T)), ts) =>
if SMT_Builtin.is_builtin ctxt c
then eta_args_conv norm_conv
(length (Term.binder_types T) - length ts)
else args_conv o norm_conv
| _ => args_conv o norm_conv)) ctxt ct
fun is_normed ctxt t =
(case t of
Const (@{const_name All}, _) $ Abs (_, _, u) => is_normed ctxt u
| Const (@{const_name All}, _) $ _ => false
| Const (@{const_name All}, _) => false
| Const (@{const_name Ex}, _) $ Abs (_, _, u) => is_normed ctxt u
| Const (@{const_name Ex}, _) $ _ => false
| Const (@{const_name Ex}, _) => false
| Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
is_normed ctxt u1 andalso is_normed ctxt u2
| Const (@{const_name Let}, _) $ _ $ _ => false
| Const (@{const_name Let}, _) $ _ => false
| Const (@{const_name Let}, _) => false
| Const (@{const_name Ex1}, _) $ _ => false
| Const (@{const_name Ex1}, _) => false
| Const (@{const_name Ball}, _) $ _ $ _ => false
| Const (@{const_name Ball}, _) $ _ => false
| Const (@{const_name Ball}, _) => false
| Const (@{const_name Bex}, _) $ _ $ _ => false
| Const (@{const_name Bex}, _) $ _ => false
| Const (@{const_name Bex}, _) => false
| Abs (_, _, u) => is_normed ctxt u
| _ =>
(case Term.strip_comb t of
(Const (c as (_, T)), ts) =>
if SMT_Builtin.is_builtin ctxt c
then length (Term.binder_types T) = length ts andalso
forall (is_normed ctxt) ts
else forall (is_normed ctxt) ts
| (_, ts) => forall (is_normed ctxt) ts))
in
fun norm_binder_conv ctxt =
U.if_conv (is_normed ctxt) Conv.all_conv (norm_conv ctxt)
end
fun norm_def ctxt thm =
(case Thm.prop_of thm of
@{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ _ $ Abs _) =>
norm_def ctxt (thm RS @{thm fun_cong})
| Const (@{const_name "=="}, _) $ _ $ Abs _ =>
norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq})
| _ => thm)
fun atomize_conv ctxt ct =
(case Thm.term_of ct of
@{const "==>"} $ _ $ _ =>
Conv.binop_conv (atomize_conv ctxt) then_conv
Conv.rewr_conv @{thm atomize_imp}
| Const (@{const_name "=="}, _) $ _ $ _ =>
Conv.binop_conv (atomize_conv ctxt) then_conv
Conv.rewr_conv @{thm atomize_eq}
| Const (@{const_name all}, _) $ Abs _ =>
Conv.binder_conv (atomize_conv o snd) ctxt then_conv
Conv.rewr_conv @{thm atomize_all}
| _ => Conv.all_conv) ct
fun normalize_rule ctxt =
Conv.fconv_rule (
(* reduce lambda abstractions, except at known binders: *)
Thm.beta_conversion true then_conv
Thm.eta_conversion then_conv
norm_binder_conv ctxt) #>
norm_def ctxt #>
Drule.forall_intr_vars #>
Conv.fconv_rule (atomize_conv ctxt)
(* lift lambda terms into additional rules *)
local
fun used_vars cvs ct =
let
val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs)
val add = (fn SOME ct => insert (op aconvc) ct | _ => I)
in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end
fun apply cv thm =
let val thm' = Thm.combination thm (Thm.reflexive cv)
in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end
fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq)
fun replace_lambda cvs ct (cx as (ctxt, defs)) =
let
val cvs' = used_vars cvs ct
val ct' = fold_rev Thm.cabs cvs' ct
in
(case Termtab.lookup defs (Thm.term_of ct') of
SOME eq => (apply_def cvs' eq, cx)
| NONE =>
let
val {T, ...} = Thm.rep_cterm ct' and n = Name.uu
val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
val cu = U.mk_cequals (U.certify ctxt (Free (n', T))) ct'
val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt'
val defs' = Termtab.update (Thm.term_of ct', eq) defs
in (apply_def cvs' eq, (ctxt'', defs')) end)
end
fun none ct cx = (Thm.reflexive ct, cx)
fun in_comb f g ct cx =
let val (cu1, cu2) = Thm.dest_comb ct
in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end
fun in_arg f = in_comb none f
fun in_abs f cvs ct (ctxt, defs) =
let
val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt
val (cv, cu) = Thm.dest_abs (SOME n) ct
in (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end
fun traverse cvs ct =
(case Thm.term_of ct of
Const (@{const_name All}, _) $ Abs _ => in_arg (in_abs traverse cvs)
| Const (@{const_name Ex}, _) $ Abs _ => in_arg (in_abs traverse cvs)
| Const (@{const_name Let}, _) $ _ $ Abs _ =>
in_comb (in_arg (traverse cvs)) (in_abs traverse cvs)
| Abs _ => at_lambda cvs
| _ $ _ => in_comb (traverse cvs) (traverse cvs)
| _ => none) ct
and at_lambda cvs ct =
in_abs traverse cvs ct #-> (fn thm =>
replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm)
fun has_free_lambdas t =
(case t of
Const (@{const_name All}, _) $ Abs (_, _, u) => has_free_lambdas u
| Const (@{const_name Ex}, _) $ Abs (_, _, u) => has_free_lambdas u
| Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
has_free_lambdas u1 orelse has_free_lambdas u2
| Abs _ => true
| u1 $ u2 => has_free_lambdas u1 orelse has_free_lambdas u2
| _ => false)
fun lift_lm f thm cx =
if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx)
else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm)
in
fun lift_lambdas irules ctxt =
let
val cx = (ctxt, Termtab.empty)
val (idxs, thms) = split_list irules
val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx
val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs []
in (map (pair ~1) eqs @ (idxs ~~ thms'), ctxt') end
end
(* make application explicit for functions with varying number of arguments *)
local
val const = prefix "c" and free = prefix "f"
fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e
fun add t i = Symtab.map_default (t, (false, i)) (min i)
fun traverse t =
(case Term.strip_comb t of
(Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts
| (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts
| (Abs (_, _, u), ts) => fold traverse (u :: ts)
| (_, ts) => fold traverse ts)
fun prune tab = Symtab.fold (fn (n, (true, i)) =>
Symtab.update (n, i) | _ => I) tab Symtab.empty
fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
fun nary_conv conv1 conv2 ct =
(Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct
fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) =>
let val n = fst (Term.dest_Free (Thm.term_of cv))
in conv (Symtab.update (free n, 0) tb) cx end)
val fun_app_rule = @{lemma "f x == fun_app f x" by (simp add: fun_app_def)}
in
fun explicit_application ctxt irules =
let
fun sub_conv tb ctxt ct =
(case Term.strip_comb (Thm.term_of ct) of
(Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt
| (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt
| (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
| (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct
and app_conv tb n i ctxt =
(case Symtab.lookup tb n of
NONE => nary_conv Conv.all_conv (sub_conv tb ctxt)
| SOME j => fun_app_conv tb ctxt (i - j))
and fun_app_conv tb ctxt i ct = (
if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
else
Conv.rewr_conv fun_app_rule then_conv
binop_conv (fun_app_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
fun needs_exp_app tab = Term.exists_subterm (fn
Bound _ $ _ => true
| Const (n, _) => Symtab.defined tab (const n)
| Free (n, _) => Symtab.defined tab (free n)
| _ => false)
fun rewrite tab ctxt thm =
if not (needs_exp_app tab (Thm.prop_of thm)) then thm
else Conv.fconv_rule (sub_conv tab ctxt) thm
val tab = prune (fold (traverse o Thm.prop_of o snd) irules Symtab.empty)
in map (apsnd (rewrite tab ctxt)) irules end
end
(* add missing datatype selectors via hypothetical definitions *)
local
val add = (fn Type (n, _) => Symtab.update (n, ()) | _ => I)
fun collect t =
(case Term.strip_comb t of
(Abs (_, T, t), _) => add T #> collect t
| (Const (_, T), ts) => collects T ts
| (Free (_, T), ts) => collects T ts
| _ => I)
and collects T ts =
let val ((Ts, Us), U) = Term.strip_type T |> apfst (chop (length ts))
in fold add Ts #> add (Us ---> U) #> fold collect ts end
fun add_constructors thy n =
(case Datatype.get_info thy n of
NONE => I
| SOME {descr, ...} => fold (fn (_, (_, _, cs)) => fold (fn (n, ds) =>
fold (insert (op =) o pair n) (1 upto length ds)) cs) descr)
fun add_selector (c as (n, i)) ctxt =
(case Datatype_Selectors.lookup_selector ctxt c of
SOME _ => ctxt
| NONE =>
let
val T = Sign.the_const_type (ProofContext.theory_of ctxt) n
val U = Term.body_type T --> nth (Term.binder_types T) (i-1)
in
ctxt
|> yield_singleton Variable.variant_fixes Name.uu
|>> pair ((n, T), i) o rpair U
|-> Context.proof_map o Datatype_Selectors.add_selector
end)
in
fun datatype_selectors irules ctxt =
let
val ns = Symtab.keys (fold (collect o Thm.prop_of o snd) irules Symtab.empty)
val cs = fold (add_constructors (ProofContext.theory_of ctxt)) ns []
in (irules, fold add_selector cs ctxt) end
(* FIXME: also generate hypothetical definitions for the selectors *)
end
(* combined normalization *)
type extra_norm = bool -> (int * thm) list -> Proof.context ->
(int * thm) list * Proof.context
fun with_context f irules ctxt = (f ctxt irules, ctxt)
fun normalize extra_norm with_datatypes irules ctxt =
let
fun norm f ctxt' (i, thm) =
if Config.get ctxt' SMT_Config.drop_bad_facts then
(case try (f ctxt') thm of
SOME thm' => SOME (i, thm')
| NONE => (SMT_Config.verbose_msg ctxt' (prefix ("Warning: " ^
"dropping assumption: ") o Display.string_of_thm ctxt') thm; NONE))
else SOME (i, f ctxt' thm)
in
irules
|> trivial_distinct ctxt
|> rewrite_bool_cases ctxt
|> normalize_numerals ctxt
|> nat_as_int ctxt
|> rpair ctxt
|-> extra_norm with_datatypes
|-> with_context (map_filter o norm normalize_rule)
|-> SMT_Monomorph.monomorph
|-> lift_lambdas
|-> with_context explicit_application
|-> (if with_datatypes then datatype_selectors else pair)
end
end