(* Title: HOL/SMT/Tools/smt_normalize.ML
Author: Sascha Boehme, TU Muenchen
Normalization steps on theorems required by SMT solvers:
* unfold trivial let expressions,
* replace negative numerals by negated positive numerals,
* embed natural numbers into integers,
* add extra rules specifying types and constants which occur frequently,
* lift lambda terms,
* make applications explicit for functions with varying number of arguments,
* fully translate into object logic, add universal closure.
*)
signature SMT_NORMALIZE =
sig
val normalize_rule: Proof.context -> thm -> thm
val instantiate_free: Thm.cterm * Thm.cterm -> thm -> thm
val discharge_definition: Thm.cterm -> thm -> thm
val trivial_let: Proof.context -> thm list -> thm list
val positive_numerals: Proof.context -> thm list -> thm list
val nat_as_int: Proof.context -> thm list -> thm list
val unfold_defs: bool Config.T
val add_pair_rules: Proof.context -> thm list -> thm list
val add_fun_upd_rules: Proof.context -> thm list -> thm list
val add_abs_min_max_rules: Proof.context -> thm list -> thm list
datatype config =
RewriteTrivialLets |
RewriteNegativeNumerals |
RewriteNaturalNumbers |
AddPairRules |
AddFunUpdRules |
AddAbsMinMaxRules
val normalize: config list -> Proof.context -> thm list ->
Thm.cterm list * thm list
val setup: theory -> theory
end
structure SMT_Normalize: SMT_NORMALIZE =
struct
val norm_binder_conv = Conv.try_conv (More_Conv.rewrs_conv [
@{lemma "All P == ALL x. P x" by (rule reflexive)},
@{lemma "Ex P == EX x. P x" by (rule reflexive)},
@{lemma "Let c P == let x = c in P x" by (rule reflexive)}])
fun cert ctxt = Thm.cterm_of (ProofContext.theory_of ctxt)
fun norm_meta_def 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 norm_def ctxt thm =
(case Thm.prop_of thm of
Const (@{const_name "=="}, _) $ _ $ Abs (_, T, _) =>
let val v = Var ((Name.uu, #maxidx (Thm.rep_thm thm) + 1), T)
in norm_def ctxt (norm_meta_def (cert ctxt v) thm) end
| @{term Trueprop} $ (Const (@{const_name "op ="}, _) $ _ $ Abs _) =>
norm_def ctxt (thm RS @{thm fun_cong})
| _ => thm)
fun normalize_rule ctxt =
Conv.fconv_rule (
Thm.beta_conversion true then_conv
Thm.eta_conversion then_conv
More_Conv.bottom_conv (K norm_binder_conv) ctxt) #>
norm_def ctxt #>
Drule.forall_intr_vars #>
Conv.fconv_rule ObjectLogic.atomize
fun instantiate_free (cv, ct) thm =
if Term.exists_subterm (equal (Thm.term_of cv)) (Thm.prop_of thm)
then Thm.forall_elim ct (Thm.forall_intr cv thm)
else thm
fun discharge_definition ct thm =
let val (cv, cu) = Thm.dest_equals ct
in
Thm.implies_intr ct thm
|> instantiate_free (cv, cu)
|> (fn thm => Thm.implies_elim thm (Thm.reflexive cu))
end
fun if_conv c cv1 cv2 ct = (if c (Thm.term_of ct) then cv1 else cv2) ct
fun if_true_conv c cv = if_conv c cv Conv.all_conv
(* simplification of trivial let expressions (whose bound variables occur at
most once) *)
local
fun count i (Bound j) = if j = i then 1 else 0
| count i (t $ u) = count i t + count i u
| count i (Abs (_, _, t)) = count (i + 1) t
| count _ _ = 0
fun is_trivial_let (Const (@{const_name Let}, _) $ _ $ Abs (_, _, t)) =
(count 0 t <= 1)
| is_trivial_let _ = false
fun let_conv _ = if_true_conv is_trivial_let (Conv.rewr_conv @{thm Let_def})
fun cond_let_conv ctxt = if_true_conv (Term.exists_subterm is_trivial_let)
(More_Conv.top_conv let_conv ctxt)
in
fun trivial_let ctxt = map (Conv.fconv_rule (cond_let_conv ctxt))
end
(* rewriting of negative integer numerals into positive numerals *)
local
fun neg_numeral @{term Int.Min} = true
| neg_numeral _ = false
fun is_number_sort thy T = Sign.of_sort thy (T, @{sort number_ring})
fun is_neg_number ctxt (Const (@{const_name number_of}, T) $ t) =
Term.exists_subterm neg_numeral t andalso
is_number_sort (ProofContext.theory_of ctxt) (Term.body_type T)
| is_neg_number _ _ = false
fun has_neg_number ctxt = Term.exists_subterm (is_neg_number ctxt)
val pos_numeral_ss = HOL_ss
addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
addsimps [@{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 = if_conv (is_neg_number ctxt)
(Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
Conv.no_conv
fun cond_pos_conv ctxt = if_true_conv (has_neg_number ctxt)
(More_Conv.top_sweep_conv pos_conv ctxt)
in
fun positive_numerals ctxt = map (Conv.fconv_rule (cond_pos_conv ctxt))
end
(* embedding of standard natural number operations into integer operations *)
local
val nat_embedding = @{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 i = nat (number_of i)"
"int (nat 0) = 0"
"int (nat 1) = 1"
"a < b = (int a < int b)"
"a <= b = (int a <= int b)"
"Suc a = nat (int a + 1)"
"a + b = nat (int a + int b)"
"a - b = nat (int a - int b)"
"a * b = nat (int a * int b)"
"a div b = nat (int a div int b)"
"a mod b = nat (int a mod int b)"
"int (nat (int a + int b)) = int a + int b"
"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"
by (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}
fun cancel_int_nat_simproc _ ss ct =
let
val num = Thm.dest_arg (Thm.dest_arg ct)
val goal = Thm.mk_binop @{cterm "op == :: int => _"} 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}))
in
fun nat_as_int ctxt thms =
let
fun norm thm uses_nat =
if not (uses_nat_type (Thm.prop_of thm)) then (thm, uses_nat)
else (Conv.fconv_rule (conv ctxt) thm, true)
val (thms', uses_nat) = fold_map norm thms false
in if uses_nat then nat_embedding @ thms' else thms' end
end
(* include additional rules *)
val (unfold_defs, unfold_defs_setup) =
Attrib.config_bool "smt_unfold_defs" true
local
val pair_rules = [@{thm fst_conv}, @{thm snd_conv}, @{thm pair_collapse}]
val pair_type = (fn Type (@{type_name "*"}, _) => true | _ => false)
val exists_pair_type = Term.exists_type (Term.exists_subtype pair_type)
val fun_upd_rules = [@{thm fun_upd_same}, @{thm fun_upd_apply}]
val is_fun_upd = (fn Const (@{const_name fun_upd}, _) => true | _ => false)
val exists_fun_upd = Term.exists_subterm is_fun_upd
in
fun add_pair_rules _ thms =
thms
|> exists (exists_pair_type o Thm.prop_of) thms ? append pair_rules
fun add_fun_upd_rules _ thms =
thms
|> exists (exists_fun_upd o Thm.prop_of) thms ? append fun_upd_rules
end
local
fun mk_entry t thm = (Term.head_of t, (thm, thm RS @{thm eq_reflection}))
fun prepare_def thm =
(case HOLogic.dest_Trueprop (Thm.prop_of thm) of
Const (@{const_name "op ="}, _) $ t $ _ => mk_entry t thm
| t => raise TERM ("prepare_def", [t]))
val defs = map prepare_def [
@{thm abs_if[where 'a = int]}, @{thm abs_if[where 'a = real]},
@{thm min_def[where 'a = int]}, @{thm min_def[where 'a = real]},
@{thm max_def[where 'a = int]}, @{thm max_def[where 'a = real]}]
fun add_sym t = if AList.defined (op =) defs t then insert (op =) t else I
fun add_syms thms = fold (Term.fold_aterms add_sym o Thm.prop_of) thms []
fun unfold_conv ctxt ct =
(case AList.lookup (op =) defs (Term.head_of (Thm.term_of ct)) of
SOME (_, eq) => Conv.rewr_conv eq
| NONE => Conv.all_conv) ct
in
fun add_abs_min_max_rules ctxt thms =
if Config.get ctxt unfold_defs
then map (Conv.fconv_rule (More_Conv.bottom_conv unfold_conv ctxt)) thms
else map fst (map_filter (AList.lookup (op =) defs) (add_syms thms)) @ thms
end
(* lift lambda terms into additional rules *)
local
val meta_eq = @{cpat "op =="}
val meta_eqT = hd (Thm.dest_ctyp (Thm.ctyp_of_term meta_eq))
fun inst_meta cT = Thm.instantiate_cterm ([(meta_eqT, cT)], []) meta_eq
fun mk_meta_eq ct cu = Thm.mk_binop (inst_meta (Thm.ctyp_of_term ct)) ct cu
fun lambda_conv conv =
let
fun sub_conv cvs ctxt ct =
(case Thm.term_of ct of
Const (@{const_name All}, _) $ Abs _ => quant_conv cvs ctxt
| Const (@{const_name Ex}, _) $ Abs _ => quant_conv cvs ctxt
| Const _ $ Abs _ => Conv.arg_conv (at_lambda_conv cvs ctxt)
| Const (@{const_name Let}, _) $ _ $ Abs _ => Conv.combination_conv
(Conv.arg_conv (sub_conv cvs ctxt)) (abs_conv cvs ctxt)
| Abs _ => at_lambda_conv cvs ctxt
| _ $ _ => Conv.comb_conv (sub_conv cvs ctxt)
| _ => Conv.all_conv) ct
and abs_conv cvs = Conv.abs_conv (fn (cv, cx) => sub_conv (cv::cvs) cx)
and quant_conv cvs ctxt = Conv.arg_conv (abs_conv cvs ctxt)
and at_lambda_conv cvs ctxt = abs_conv cvs ctxt then_conv conv cvs ctxt
in sub_conv [] end
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
val rev_int_fst_ord = rev_order o int_ord o pairself fst
fun ordered_values tab =
Termtab.fold (fn (_, x) => OrdList.insert rev_int_fst_ord x) tab []
|> map snd
in
fun lift_lambdas ctxt thms =
let
val declare_frees = fold (Thm.fold_terms Term.declare_term_frees)
val names = ref (declare_frees thms (Name.make_context []))
val fresh_name = change_result names o yield_singleton Name.variants
val defs = ref (Termtab.empty : (int * thm) Termtab.table)
fun add_def t thm = change defs (Termtab.update (t, (serial (), thm)))
fun make_def cvs eq = Thm.symmetric (fold norm_meta_def cvs eq)
fun def_conv cvs ctxt ct =
let
val cvs' = used_vars cvs ct
val ct' = fold Thm.cabs cvs' ct
in
(case Termtab.lookup (!defs) (Thm.term_of ct') of
SOME (_, eq) => make_def cvs' eq
| NONE =>
let
val {t, T, ...} = Thm.rep_cterm ct'
val eq = mk_meta_eq (cert ctxt (Free (fresh_name "", T))) ct'
val thm = Thm.assume eq
in (add_def t thm; make_def cvs' thm) end)
end
val thms' = map (Conv.fconv_rule (lambda_conv def_conv ctxt)) thms
val eqs = ordered_values (!defs)
in
(maps (#hyps o Thm.crep_thm) eqs, map (normalize_rule ctxt) eqs @ thms')
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)
val prune = (fn (n, (true, i)) => Symtab.update (n, i) | _ => I)
fun prune_tab tab = Symtab.fold prune 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 apply_rule = @{lemma "f x == apply f x" by (simp add: apply_def)}
in
fun explicit_application ctxt thms =
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 _, ts) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
| (_, ts) => 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 => apply_conv tb ctxt (i - j))
and apply_conv tb ctxt i ct = (
if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
else
Conv.rewr_conv apply_rule then_conv
binop_conv (apply_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
val tab = prune_tab (fold (traverse o Thm.prop_of) thms Symtab.empty)
in map (Conv.fconv_rule (sub_conv tab ctxt)) thms end
end
(* combined normalization *)
datatype config =
RewriteTrivialLets |
RewriteNegativeNumerals |
RewriteNaturalNumbers |
AddPairRules |
AddFunUpdRules |
AddAbsMinMaxRules
fun normalize config ctxt thms =
let fun if_enabled c f = member (op =) config c ? f ctxt
in
thms
|> if_enabled RewriteTrivialLets trivial_let
|> if_enabled RewriteNegativeNumerals positive_numerals
|> if_enabled RewriteNaturalNumbers nat_as_int
|> if_enabled AddPairRules add_pair_rules
|> if_enabled AddFunUpdRules add_fun_upd_rules
|> if_enabled AddAbsMinMaxRules add_abs_min_max_rules
|> map (normalize_rule ctxt)
|> lift_lambdas ctxt
|> apsnd (explicit_application ctxt)
end
val setup = unfold_defs_setup
end