boehmes@36898: (*  Title:      HOL/Tools/SMT/smt_normalize.ML
boehmes@36898:     Author:     Sascha Boehme, TU Muenchen
boehmes@36898: 
boehmes@36898: Normalization steps on theorems required by SMT solvers:
boehmes@36898:   * simplify trivial distincts (those with less than three elements),
boehmes@36898:   * rewrite bool case expressions as if expressions,
boehmes@36898:   * normalize numerals (e.g. replace negative numerals by negated positive
boehmes@36898:     numerals),
boehmes@36898:   * embed natural numbers into integers,
boehmes@36898:   * add extra rules specifying types and constants which occur frequently,
boehmes@36898:   * fully translate into object logic, add universal closure,
boehmes@39483:   * monomorphize (create instances of schematic rules),
boehmes@36898:   * lift lambda terms,
boehmes@36898:   * make applications explicit for functions with varying number of arguments.
boehmes@39483:   * add (hypothetical definitions for) missing datatype selectors,
boehmes@36898: *)
boehmes@36898: 
boehmes@36898: signature SMT_NORMALIZE =
boehmes@36898: sig
boehmes@40162:   type extra_norm = bool -> (int * thm) list -> Proof.context ->
boehmes@40161:     (int * thm) list * Proof.context
boehmes@40424:   val normalize: extra_norm -> bool -> (int * thm) list -> Proof.context ->
boehmes@40161:     (int * thm) list * Proof.context
boehmes@36899:   val atomize_conv: Proof.context -> conv
boehmes@36898:   val eta_expand_conv: (Proof.context -> conv) -> Proof.context -> conv
boehmes@36898: end
boehmes@36898: 
boehmes@36898: structure SMT_Normalize: SMT_NORMALIZE =
boehmes@36898: struct
boehmes@36898: 
boehmes@40663: structure U = SMT_Utils
boehmes@40663: 
boehmes@36898: infix 2 ??
boehmes@36898: fun (test ?? f) x = if test x then f x else x
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@40685: (* instantiate elimination rules *)
boehmes@40685:  
boehmes@40685: local
boehmes@40685:   val (cpfalse, cfalse) = `U.mk_cprop (Thm.cterm_of @{theory} @{const False})
boehmes@40685: 
boehmes@40685:   fun inst f ct thm =
boehmes@40685:     let val cv = f (Drule.strip_imp_concl (Thm.cprop_of thm))
boehmes@40685:     in Thm.instantiate ([], [(cv, ct)]) thm end
boehmes@40685: in
boehmes@40685: 
boehmes@40685: fun instantiate_elim thm =
boehmes@40685:   (case Thm.concl_of thm of
boehmes@40685:     @{const Trueprop} $ Var (_, @{typ bool}) => inst Thm.dest_arg cfalse thm
boehmes@40685:   | Var _ => inst I cpfalse thm
boehmes@40685:   | _ => thm)
boehmes@40685: 
boehmes@40685: end
boehmes@40685: 
boehmes@40685: 
boehmes@40685: 
boehmes@36898: (* simplification of trivial distincts (distinct should have at least
boehmes@36898:    three elements in the argument list) *)
boehmes@36898: 
boehmes@36898: local
boehmes@40681:   fun is_trivial_distinct (Const (@{const_name distinct}, _) $ t) =
boehmes@40681:         (case try HOLogic.dest_list t of
boehmes@40681:           SOME [] => true
boehmes@40681:         | SOME [_] => true
boehmes@40681:         | _ => false)
boehmes@36898:     | is_trivial_distinct _ = false
boehmes@36898: 
boehmes@37786:   val thms = map mk_meta_eq @{lemma
boehmes@40681:     "distinct [] = True"
boehmes@40681:     "distinct [x] = True"
boehmes@40681:     "distinct [x, y] = (x ~= y)"
boehmes@40681:     by simp_all}
boehmes@36898:   fun distinct_conv _ =
boehmes@40663:     U.if_true_conv is_trivial_distinct (Conv.rewrs_conv thms)
boehmes@36898: in
boehmes@36898: fun trivial_distinct ctxt =
boehmes@40161:   map (apsnd ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ??
boehmes@40161:     Conv.fconv_rule (Conv.top_conv distinct_conv ctxt)))
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* rewrite bool case expressions as if expressions *)
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   val is_bool_case = (fn
boehmes@36898:       Const (@{const_name "bool.bool_case"}, _) $ _ $ _ $ _ => true
boehmes@36898:     | _ => false)
boehmes@36898: 
boehmes@40275:   val thm = mk_meta_eq @{lemma
boehmes@40275:     "(case P of True => x | False => y) = (if P then x else y)" by simp}
boehmes@40663:   val unfold_conv = U.if_true_conv is_bool_case (Conv.rewr_conv thm)
boehmes@36898: in
boehmes@36898: fun rewrite_bool_cases ctxt =
boehmes@40161:   map (apsnd ((Term.exists_subterm is_bool_case o Thm.prop_of) ??
boehmes@40161:     Conv.fconv_rule (Conv.top_conv (K unfold_conv) ctxt)))
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* normalization of numerals: rewriting of negative integer numerals into
boehmes@36898:    positive numerals, Numeral0 into 0, Numeral1 into 1 *)
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   fun is_number_sort ctxt T =
boehmes@36898:     Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring})
boehmes@36898: 
boehmes@36898:   fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) $ _) =
boehmes@36898:         (case try HOLogic.dest_number t of
boehmes@36898:           SOME (T, i) => is_number_sort ctxt T andalso i < 2
boehmes@36898:         | NONE => false)
boehmes@36898:     | is_strange_number _ _ = false
boehmes@36898: 
boehmes@36898:   val pos_numeral_ss = HOL_ss
boehmes@36898:     addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
boehmes@36898:     addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}]
boehmes@36898:     addsimps @{thms Int.pred_bin_simps}
boehmes@36898:     addsimps @{thms Int.normalize_bin_simps}
boehmes@36898:     addsimps @{lemma
boehmes@36898:       "Int.Min = - Int.Bit1 Int.Pls"
boehmes@36898:       "Int.Bit0 (- Int.Pls) = - Int.Pls"
boehmes@36898:       "Int.Bit0 (- k) = - Int.Bit0 k"
boehmes@36898:       "Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)"
boehmes@36898:       by simp_all (simp add: pred_def)}
boehmes@36898: 
boehmes@40663:   fun pos_conv ctxt = U.if_conv (is_strange_number ctxt)
boehmes@36898:     (Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
boehmes@36898:     Conv.no_conv
boehmes@36898: in
boehmes@36898: fun normalize_numerals ctxt =
boehmes@40161:   map (apsnd ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ??
boehmes@40161:     Conv.fconv_rule (Conv.top_sweep_conv pos_conv ctxt)))
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* embedding of standard natural number operations into integer operations *)
boehmes@36898: 
boehmes@36898: local
boehmes@40161:   val nat_embedding = map (pair ~1) @{lemma
boehmes@36898:     "nat (int n) = n"
boehmes@36898:     "i >= 0 --> int (nat i) = i"
boehmes@36898:     "i < 0 --> int (nat i) = 0"
boehmes@36898:     by simp_all}
boehmes@36898: 
boehmes@36898:   val nat_rewriting = @{lemma
boehmes@36898:     "0 = nat 0"
boehmes@36898:     "1 = nat 1"
boehmes@40279:     "(number_of :: int => nat) = (%i. nat (number_of i))"
boehmes@36898:     "int (nat 0) = 0"
boehmes@36898:     "int (nat 1) = 1"
boehmes@40279:     "op < = (%a b. int a < int b)"
boehmes@40279:     "op <= = (%a b. int a <= int b)"
boehmes@40279:     "Suc = (%a. nat (int a + 1))"
boehmes@40279:     "op + = (%a b. nat (int a + int b))"
boehmes@40279:     "op - = (%a b. nat (int a - int b))"
boehmes@40279:     "op * = (%a b. nat (int a * int b))"
boehmes@40279:     "op div = (%a b. nat (int a div int b))"
boehmes@40279:     "op mod = (%a b. nat (int a mod int b))"
boehmes@40279:     "min = (%a b. nat (min (int a) (int b)))"
boehmes@40279:     "max = (%a b. nat (max (int a) (int b)))"
boehmes@36898:     "int (nat (int a + int b)) = int a + int b"
boehmes@40279:     "int (nat (int a + 1)) = int a + 1"  (* special rule due to Suc above *)
boehmes@36898:     "int (nat (int a * int b)) = int a * int b"
boehmes@36898:     "int (nat (int a div int b)) = int a div int b"
boehmes@36898:     "int (nat (int a mod int b)) = int a mod int b"
boehmes@36898:     "int (nat (min (int a) (int b))) = min (int a) (int b)"
boehmes@36898:     "int (nat (max (int a) (int b))) = max (int a) (int b)"
boehmes@40279:     by (auto intro!: ext simp add: nat_mult_distrib nat_div_distrib
boehmes@40279:       nat_mod_distrib int_mult[symmetric] zdiv_int[symmetric]
boehmes@40279:       zmod_int[symmetric])}
boehmes@36898: 
boehmes@36898:   fun on_positive num f x = 
boehmes@36898:     (case try HOLogic.dest_number (Thm.term_of num) of
boehmes@36898:       SOME (_, i) => if i >= 0 then SOME (f x) else NONE
boehmes@36898:     | NONE => NONE)
boehmes@36898: 
boehmes@36898:   val cancel_int_nat_ss = HOL_ss
boehmes@36898:     addsimps [@{thm Nat_Numeral.nat_number_of}]
boehmes@36898:     addsimps [@{thm Nat_Numeral.int_nat_number_of}]
boehmes@36898:     addsimps @{thms neg_simps}
boehmes@36898: 
boehmes@40579:   val int_eq = Thm.cterm_of @{theory} @{const "==" (int)}
boehmes@40579: 
boehmes@36898:   fun cancel_int_nat_simproc _ ss ct = 
boehmes@36898:     let
boehmes@36898:       val num = Thm.dest_arg (Thm.dest_arg ct)
boehmes@40579:       val goal = Thm.mk_binop int_eq ct num
boehmes@36898:       val simpset = Simplifier.inherit_context ss cancel_int_nat_ss
boehmes@36898:       fun tac _ = Simplifier.simp_tac simpset 1
boehmes@36898:     in on_positive num (Goal.prove_internal [] goal) tac end
boehmes@36898: 
boehmes@36898:   val nat_ss = HOL_ss
boehmes@36898:     addsimps nat_rewriting
boehmes@40279:     addsimprocs [
boehmes@40279:       Simplifier.make_simproc {
boehmes@40279:         name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}],
boehmes@40279:         proc = cancel_int_nat_simproc, identifier = [] }]
boehmes@36898: 
boehmes@36898:   fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss)
boehmes@36898: 
boehmes@36898:   val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat}))
boehmes@40579:   val uses_nat_int = Term.exists_subterm (member (op aconv)
boehmes@40579:     [@{const of_nat (int)}, @{const nat}])
boehmes@36898: in
boehmes@36898: fun nat_as_int ctxt =
boehmes@40161:   map (apsnd ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt))) #>
boehmes@40161:   exists (uses_nat_int o Thm.prop_of o snd) ?? append nat_embedding
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* further normalizations: beta/eta, universal closure, atomize *)
boehmes@36898: 
boehmes@36898: val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)}
boehmes@36898: 
boehmes@36898: fun eta_expand_conv cv ctxt =
boehmes@36898:   Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   val eta_conv = eta_expand_conv
boehmes@36898: 
boehmes@40279:   fun args_conv cv ct =
boehmes@40279:     (case Thm.term_of ct of
boehmes@40279:       _ $ _ => Conv.combination_conv (args_conv cv) cv
boehmes@40279:     | _ => Conv.all_conv) ct
boehmes@40279: 
boehmes@40279:   fun eta_args_conv cv 0 = args_conv o cv
boehmes@40279:     | eta_args_conv cv i = eta_conv (eta_args_conv cv (i-1))
boehmes@40279: 
wenzelm@36936:   fun keep_conv ctxt = Conv.binder_conv (norm_conv o snd) ctxt
boehmes@36898:   and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt)
boehmes@36898:   and keep_let_conv ctxt = Conv.combination_conv
boehmes@36898:     (Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt)
boehmes@36898:   and unfold_let_conv ctxt = Conv.combination_conv
boehmes@36898:     (Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt)
boehmes@36898:   and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt
boehmes@36898:   and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt
boehmes@37786:   and unfold_ball_conv ctxt = unfold_conv (mk_meta_eq @{thm Ball_def}) ctxt
boehmes@37786:   and unfold_bex_conv ctxt = unfold_conv (mk_meta_eq @{thm Bex_def}) ctxt
boehmes@36898:   and norm_conv ctxt ct =
boehmes@36898:     (case Thm.term_of ct of
boehmes@36898:       Const (@{const_name All}, _) $ Abs _ => keep_conv
boehmes@36898:     | Const (@{const_name All}, _) $ _ => eta_binder_conv
boehmes@36898:     | Const (@{const_name All}, _) => eta_conv eta_binder_conv
boehmes@36898:     | Const (@{const_name Ex}, _) $ Abs _ => keep_conv
boehmes@36898:     | Const (@{const_name Ex}, _) $ _ => eta_binder_conv
boehmes@36898:     | Const (@{const_name Ex}, _) => eta_conv eta_binder_conv
boehmes@36898:     | Const (@{const_name Let}, _) $ _ $ Abs _ => keep_let_conv
boehmes@36898:     | Const (@{const_name Let}, _) $ _ $ _ => unfold_let_conv
boehmes@36898:     | Const (@{const_name Let}, _) $ _ => eta_conv unfold_let_conv
boehmes@36898:     | Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv)
boehmes@36898:     | Const (@{const_name Ex1}, _) $ _ => unfold_ex1_conv
boehmes@36898:     | Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv 
boehmes@36898:     | Const (@{const_name Ball}, _) $ _ $ _ => unfold_ball_conv
boehmes@36898:     | Const (@{const_name Ball}, _) $ _ => eta_conv unfold_ball_conv
boehmes@36898:     | Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv)
boehmes@36898:     | Const (@{const_name Bex}, _) $ _ $ _ => unfold_bex_conv
boehmes@36898:     | Const (@{const_name Bex}, _) $ _ => eta_conv unfold_bex_conv
boehmes@36898:     | Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv)
boehmes@36898:     | Abs _ => Conv.abs_conv (norm_conv o snd)
boehmes@40279:     | _ =>
boehmes@40279:         (case Term.strip_comb (Thm.term_of ct) of
boehmes@40279:           (Const (c as (_, T)), ts) =>
boehmes@40279:             if SMT_Builtin.is_builtin ctxt c
boehmes@40279:             then eta_args_conv norm_conv
boehmes@40279:               (length (Term.binder_types T) - length ts)
boehmes@40279:             else args_conv o norm_conv
boehmes@40424:         | _ => args_conv o norm_conv)) ctxt ct
boehmes@36898: 
boehmes@40279:   fun is_normed ctxt t =
boehmes@36898:     (case t of
boehmes@40279:       Const (@{const_name All}, _) $ Abs (_, _, u) => is_normed ctxt u
boehmes@36898:     | Const (@{const_name All}, _) $ _ => false
boehmes@36898:     | Const (@{const_name All}, _) => false
boehmes@40279:     | Const (@{const_name Ex}, _) $ Abs (_, _, u) => is_normed ctxt u
boehmes@36898:     | Const (@{const_name Ex}, _) $ _ => false
boehmes@36898:     | Const (@{const_name Ex}, _) => false
boehmes@36898:     | Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
boehmes@40279:         is_normed ctxt u1 andalso is_normed ctxt u2
boehmes@36898:     | Const (@{const_name Let}, _) $ _ $ _ => false
boehmes@36898:     | Const (@{const_name Let}, _) $ _ => false
boehmes@36898:     | Const (@{const_name Let}, _) => false
boehmes@40279:     | Const (@{const_name Ex1}, _) $ _ => false
boehmes@36898:     | Const (@{const_name Ex1}, _) => false
boehmes@40279:     | Const (@{const_name Ball}, _) $ _ $ _ => false
boehmes@40279:     | Const (@{const_name Ball}, _) $ _ => false
boehmes@36898:     | Const (@{const_name Ball}, _) => false
boehmes@40279:     | Const (@{const_name Bex}, _) $ _ $ _ => false
boehmes@40279:     | Const (@{const_name Bex}, _) $ _ => false
boehmes@36898:     | Const (@{const_name Bex}, _) => false
boehmes@40279:     | Abs (_, _, u) => is_normed ctxt u
boehmes@40279:     | _ =>
boehmes@40279:         (case Term.strip_comb t of
boehmes@40279:           (Const (c as (_, T)), ts) =>
boehmes@40279:             if SMT_Builtin.is_builtin ctxt c
boehmes@40279:             then length (Term.binder_types T) = length ts andalso
boehmes@40279:               forall (is_normed ctxt) ts
boehmes@40279:             else forall (is_normed ctxt) ts
boehmes@40279:         | (_, ts) => forall (is_normed ctxt) ts))
boehmes@36898: in
boehmes@40279: fun norm_binder_conv ctxt =
boehmes@40663:   U.if_conv (is_normed ctxt) Conv.all_conv (norm_conv ctxt)
boehmes@36898: end
boehmes@36898: 
boehmes@36898: fun norm_def ctxt thm =
boehmes@36898:   (case Thm.prop_of thm of
boehmes@40579:     @{const Trueprop} $ (Const (@{const_name HOL.eq}, _) $ _ $ Abs _) =>
boehmes@36898:       norm_def ctxt (thm RS @{thm fun_cong})
boehmes@36898:   | Const (@{const_name "=="}, _) $ _ $ Abs _ =>
boehmes@36898:       norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq})
boehmes@36898:   | _ => thm)
boehmes@36898: 
boehmes@36898: fun atomize_conv ctxt ct =
boehmes@36898:   (case Thm.term_of ct of
boehmes@40579:     @{const "==>"} $ _ $ _ =>
boehmes@36898:       Conv.binop_conv (atomize_conv ctxt) then_conv
boehmes@36898:       Conv.rewr_conv @{thm atomize_imp}
boehmes@36898:   | Const (@{const_name "=="}, _) $ _ $ _ =>
boehmes@36898:       Conv.binop_conv (atomize_conv ctxt) then_conv
boehmes@36898:       Conv.rewr_conv @{thm atomize_eq}
boehmes@36898:   | Const (@{const_name all}, _) $ Abs _ =>
wenzelm@36936:       Conv.binder_conv (atomize_conv o snd) ctxt then_conv
boehmes@36898:       Conv.rewr_conv @{thm atomize_all}
boehmes@36898:   | _ => Conv.all_conv) ct
boehmes@36898: 
boehmes@36898: fun normalize_rule ctxt =
boehmes@36898:   Conv.fconv_rule (
boehmes@36898:     (* reduce lambda abstractions, except at known binders: *)
boehmes@36898:     Thm.beta_conversion true then_conv
boehmes@36898:     Thm.eta_conversion then_conv
boehmes@36898:     norm_binder_conv ctxt) #>
boehmes@36898:   norm_def ctxt #>
boehmes@36898:   Drule.forall_intr_vars #>
boehmes@36898:   Conv.fconv_rule (atomize_conv ctxt)
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* lift lambda terms into additional rules *)
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   fun used_vars cvs ct =
boehmes@36898:     let
boehmes@36898:       val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs)
boehmes@36898:       val add = (fn SOME ct => insert (op aconvc) ct | _ => I)
boehmes@36898:     in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end
boehmes@36898: 
boehmes@36898:   fun apply cv thm = 
boehmes@36898:     let val thm' = Thm.combination thm (Thm.reflexive cv)
boehmes@36898:     in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end
boehmes@36898:   fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq)
boehmes@36898: 
boehmes@36898:   fun replace_lambda cvs ct (cx as (ctxt, defs)) =
boehmes@36898:     let
boehmes@36898:       val cvs' = used_vars cvs ct
boehmes@36898:       val ct' = fold_rev Thm.cabs cvs' ct
boehmes@36898:     in
boehmes@36898:       (case Termtab.lookup defs (Thm.term_of ct') of
boehmes@36898:         SOME eq => (apply_def cvs' eq, cx)
boehmes@36898:       | NONE =>
boehmes@36898:           let
boehmes@36898:             val {T, ...} = Thm.rep_cterm ct' and n = Name.uu
boehmes@36898:             val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
boehmes@40663:             val cu = U.mk_cequals (U.certify ctxt (Free (n', T))) ct'
boehmes@36898:             val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt'
boehmes@36898:             val defs' = Termtab.update (Thm.term_of ct', eq) defs
boehmes@36898:           in (apply_def cvs' eq, (ctxt'', defs')) end)
boehmes@36898:     end
boehmes@36898: 
boehmes@36898:   fun none ct cx = (Thm.reflexive ct, cx)
boehmes@36898:   fun in_comb f g ct cx =
boehmes@36898:     let val (cu1, cu2) = Thm.dest_comb ct
boehmes@36898:     in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end
boehmes@36898:   fun in_arg f = in_comb none f
boehmes@36898:   fun in_abs f cvs ct (ctxt, defs) =
boehmes@36898:     let
boehmes@36898:       val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt
boehmes@36898:       val (cv, cu) = Thm.dest_abs (SOME n) ct
boehmes@36898:     in  (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end
boehmes@36898: 
boehmes@36898:   fun traverse cvs ct =
boehmes@36898:     (case Thm.term_of ct of
boehmes@36898:       Const (@{const_name All}, _) $ Abs _ => in_arg (in_abs traverse cvs)
boehmes@36898:     | Const (@{const_name Ex}, _) $ Abs _ => in_arg (in_abs traverse cvs)
boehmes@36898:     | Const (@{const_name Let}, _) $ _ $ Abs _ =>
boehmes@36898:         in_comb (in_arg (traverse cvs)) (in_abs traverse cvs)
boehmes@36898:     | Abs _ => at_lambda cvs
boehmes@36898:     | _ $ _ => in_comb (traverse cvs) (traverse cvs)
boehmes@36898:     | _ => none) ct
boehmes@36898: 
boehmes@36898:   and at_lambda cvs ct =
boehmes@36898:     in_abs traverse cvs ct #-> (fn thm =>
boehmes@36898:     replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm)
boehmes@36898: 
boehmes@36898:   fun has_free_lambdas t =
boehmes@36898:     (case t of
boehmes@36898:       Const (@{const_name All}, _) $ Abs (_, _, u) => has_free_lambdas u
boehmes@36898:     | Const (@{const_name Ex}, _) $ Abs (_, _, u) => has_free_lambdas u
boehmes@36898:     | Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
boehmes@36898:         has_free_lambdas u1 orelse has_free_lambdas u2
boehmes@36898:     | Abs _ => true
boehmes@36898:     | u1 $ u2 => has_free_lambdas u1 orelse has_free_lambdas u2
boehmes@36898:     | _ => false)
boehmes@36898: 
boehmes@36898:   fun lift_lm f thm cx =
boehmes@36898:     if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx)
boehmes@36898:     else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm)
boehmes@36898: in
boehmes@40161: fun lift_lambdas irules ctxt =
boehmes@36898:   let
boehmes@36898:     val cx = (ctxt, Termtab.empty)
boehmes@40161:     val (idxs, thms) = split_list irules
boehmes@36898:     val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx
boehmes@36898:     val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs []
boehmes@40161:   in (map (pair ~1) eqs @ (idxs ~~ thms'), ctxt') end
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@36898: (* make application explicit for functions with varying number of arguments *)
boehmes@36898: 
boehmes@36898: local
boehmes@36898:   val const = prefix "c" and free = prefix "f"
boehmes@36898:   fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e
boehmes@36898:   fun add t i = Symtab.map_default (t, (false, i)) (min i)
boehmes@36898:   fun traverse t =
boehmes@36898:     (case Term.strip_comb t of
boehmes@36898:       (Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts 
boehmes@36898:     | (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts
boehmes@36898:     | (Abs (_, _, u), ts) => fold traverse (u :: ts)
boehmes@36898:     | (_, ts) => fold traverse ts)
boehmes@40161:   fun prune tab = Symtab.fold (fn (n, (true, i)) =>
boehmes@40161:     Symtab.update (n, i) | _ => I) tab Symtab.empty
boehmes@36898: 
boehmes@36898:   fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
boehmes@36898:   fun nary_conv conv1 conv2 ct =
boehmes@36898:     (Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct
boehmes@36898:   fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) =>
boehmes@36898:     let val n = fst (Term.dest_Free (Thm.term_of cv))
boehmes@36898:     in conv (Symtab.update (free n, 0) tb) cx end)
boehmes@37153:   val fun_app_rule = @{lemma "f x == fun_app f x" by (simp add: fun_app_def)}
boehmes@36898: in
boehmes@40161: fun explicit_application ctxt irules =
boehmes@36898:   let
boehmes@36898:     fun sub_conv tb ctxt ct =
boehmes@36898:       (case Term.strip_comb (Thm.term_of ct) of
boehmes@36898:         (Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt
boehmes@36898:       | (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt
boehmes@36898:       | (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
boehmes@36898:       | (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct
boehmes@36898:     and app_conv tb n i ctxt =
boehmes@36898:       (case Symtab.lookup tb n of
boehmes@36898:         NONE => nary_conv Conv.all_conv (sub_conv tb ctxt)
boehmes@37153:       | SOME j => fun_app_conv tb ctxt (i - j))
boehmes@37153:     and fun_app_conv tb ctxt i ct = (
boehmes@36898:       if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
boehmes@36898:       else
boehmes@37153:         Conv.rewr_conv fun_app_rule then_conv
boehmes@37153:         binop_conv (fun_app_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
boehmes@36898: 
boehmes@36898:     fun needs_exp_app tab = Term.exists_subterm (fn
boehmes@36898:         Bound _ $ _ => true
boehmes@36898:       | Const (n, _) => Symtab.defined tab (const n)
boehmes@36898:       | Free (n, _) => Symtab.defined tab (free n)
boehmes@36898:       | _ => false)
boehmes@36898: 
boehmes@36898:     fun rewrite tab ctxt thm =
boehmes@36898:       if not (needs_exp_app tab (Thm.prop_of thm)) then thm
boehmes@36898:       else Conv.fconv_rule (sub_conv tab ctxt) thm
boehmes@36898: 
boehmes@40161:     val tab = prune (fold (traverse o Thm.prop_of o snd) irules Symtab.empty)
boehmes@40161:   in map (apsnd (rewrite tab ctxt)) irules end
boehmes@36898: end
boehmes@36898: 
boehmes@36898: 
boehmes@36898: 
boehmes@39483: (* add missing datatype selectors via hypothetical definitions *)
boehmes@39483: 
boehmes@39483: local
boehmes@39483:   val add = (fn Type (n, _) => Symtab.update (n, ()) | _ => I)
boehmes@39483: 
boehmes@39483:   fun collect t =
boehmes@39483:     (case Term.strip_comb t of
boehmes@39483:       (Abs (_, T, t), _) => add T #> collect t
boehmes@39483:     | (Const (_, T), ts) => collects T ts
boehmes@39483:     | (Free (_, T), ts) => collects T ts
boehmes@39483:     | _ => I)
boehmes@39483:   and collects T ts =
boehmes@39483:     let val ((Ts, Us), U) = Term.strip_type T |> apfst (chop (length ts))
boehmes@39483:     in fold add Ts #> add (Us ---> U) #> fold collect ts end
boehmes@39483: 
boehmes@39483:   fun add_constructors thy n =
boehmes@39483:     (case Datatype.get_info thy n of
boehmes@39483:       NONE => I
boehmes@39483:     | SOME {descr, ...} => fold (fn (_, (_, _, cs)) => fold (fn (n, ds) =>
boehmes@39483:         fold (insert (op =) o pair n) (1 upto length ds)) cs) descr)
boehmes@39483: 
boehmes@39483:   fun add_selector (c as (n, i)) ctxt =
boehmes@39483:     (case Datatype_Selectors.lookup_selector ctxt c of
boehmes@39483:       SOME _ => ctxt
boehmes@39483:     | NONE =>
boehmes@39483:         let
boehmes@39483:           val T = Sign.the_const_type (ProofContext.theory_of ctxt) n
boehmes@39483:           val U = Term.body_type T --> nth (Term.binder_types T) (i-1)
boehmes@39483:         in
boehmes@39483:           ctxt
boehmes@39483:           |> yield_singleton Variable.variant_fixes Name.uu
boehmes@39483:           |>> pair ((n, T), i) o rpair U
boehmes@39483:           |-> Context.proof_map o Datatype_Selectors.add_selector
boehmes@39483:         end)
boehmes@39483: in
boehmes@39483: 
boehmes@40161: fun datatype_selectors irules ctxt =
boehmes@39483:   let
boehmes@40161:     val ns = Symtab.keys (fold (collect o Thm.prop_of o snd) irules Symtab.empty)
boehmes@39483:     val cs = fold (add_constructors (ProofContext.theory_of ctxt)) ns []
boehmes@40161:   in (irules, fold add_selector cs ctxt) end
boehmes@39483:     (* FIXME: also generate hypothetical definitions for the selectors *)
boehmes@39483: 
boehmes@39483: end
boehmes@39483: 
boehmes@39483: 
boehmes@39483: 
boehmes@36898: (* combined normalization *)
boehmes@36898: 
boehmes@40162: type extra_norm = bool -> (int * thm) list -> Proof.context ->
boehmes@40161:   (int * thm) list * Proof.context
boehmes@36898: 
boehmes@40161: fun with_context f irules ctxt = (f ctxt irules, ctxt)
boehmes@36898: 
boehmes@40424: fun normalize extra_norm with_datatypes irules ctxt =
boehmes@40278:   let
boehmes@40278:     fun norm f ctxt' (i, thm) =
boehmes@40424:       if Config.get ctxt' SMT_Config.drop_bad_facts then
boehmes@40278:         (case try (f ctxt') thm of
boehmes@40278:           SOME thm' => SOME (i, thm')
boehmes@40424:         | NONE => (SMT_Config.verbose_msg ctxt' (prefix ("Warning: " ^
boehmes@40278:             "dropping assumption: ") o Display.string_of_thm ctxt') thm; NONE))
boehmes@40424:       else SOME (i, f ctxt' thm)
boehmes@40278:   in
boehmes@40278:     irules
boehmes@40685:     |> map (apsnd instantiate_elim)
boehmes@40278:     |> trivial_distinct ctxt
boehmes@40278:     |> rewrite_bool_cases ctxt
boehmes@40278:     |> normalize_numerals ctxt
boehmes@40278:     |> nat_as_int ctxt
boehmes@40278:     |> rpair ctxt
boehmes@40278:     |-> extra_norm with_datatypes
boehmes@40278:     |-> with_context (map_filter o norm normalize_rule)
boehmes@40278:     |-> SMT_Monomorph.monomorph
boehmes@40278:     |-> lift_lambdas
boehmes@40278:     |-> with_context explicit_application
boehmes@40278:     |-> (if with_datatypes then datatype_selectors else pair)
boehmes@40278:   end
boehmes@36898: 
boehmes@36898: end