src/HOL/Tools/SMT/smt_normalize.ML
author boehmes
Tue Jul 13 02:29:05 2010 +0200 (2010-07-13)
changeset 37786 4eb98849c5c0
parent 37153 8feed34275ce
child 38864 4abe644fcea5
permissions -rw-r--r--
fixed handling of Ball/Bex: turn equalities into meta-equalities for the rewriting conversions;
added tests for Ball/Bex
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(*  Title:      HOL/Tools/SMT/smt_normalize.ML
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    Author:     Sascha Boehme, TU Muenchen
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Normalization steps on theorems required by SMT solvers:
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  * simplify trivial distincts (those with less than three elements),
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  * rewrite bool case expressions as if expressions,
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  * normalize numerals (e.g. replace negative numerals by negated positive
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    numerals),
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  * embed natural numbers into integers,
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  * add extra rules specifying types and constants which occur frequently,
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  * fully translate into object logic, add universal closure,
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  * lift lambda terms,
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  * make applications explicit for functions with varying number of arguments.
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*)
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signature SMT_NORMALIZE =
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sig
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  type extra_norm = thm list -> Proof.context -> thm list * Proof.context
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  val normalize: extra_norm -> thm list -> Proof.context ->
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    thm list * Proof.context
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  val atomize_conv: Proof.context -> conv
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  val eta_expand_conv: (Proof.context -> conv) -> Proof.context -> conv
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end
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structure SMT_Normalize: SMT_NORMALIZE =
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struct
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infix 2 ??
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fun (test ?? f) x = if test x then f x else x
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fun if_conv c cv1 cv2 ct = (if c (Thm.term_of ct) then cv1 else cv2) ct
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fun if_true_conv c cv = if_conv c cv Conv.all_conv
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(* simplification of trivial distincts (distinct should have at least
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   three elements in the argument list) *)
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local
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  fun is_trivial_distinct (Const (@{const_name distinct}, _) $ t) =
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        length (HOLogic.dest_list t) <= 2
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    | is_trivial_distinct _ = false
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  val thms = map mk_meta_eq @{lemma
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    "distinct [] = True"
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    "distinct [x] = True"
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    "distinct [x, y] = (x ~= y)"
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    by simp_all}
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  fun distinct_conv _ =
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    if_true_conv is_trivial_distinct (Conv.rewrs_conv thms)
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in
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fun trivial_distinct ctxt =
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  map ((Term.exists_subterm is_trivial_distinct o Thm.prop_of) ??
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    Conv.fconv_rule (Conv.top_conv distinct_conv ctxt))
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end
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(* rewrite bool case expressions as if expressions *)
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local
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  val is_bool_case = (fn
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      Const (@{const_name "bool.bool_case"}, _) $ _ $ _ $ _ => true
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    | _ => false)
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  val thms = map mk_meta_eq @{lemma
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    "(case P of True => x | False => y) = (if P then x else y)"
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    "(case P of False => y | True => x) = (if P then x else y)"
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    by simp_all}
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  val unfold_conv = if_true_conv is_bool_case (Conv.rewrs_conv thms)
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in
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fun rewrite_bool_cases ctxt =
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  map ((Term.exists_subterm is_bool_case o Thm.prop_of) ??
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    Conv.fconv_rule (Conv.top_conv (K unfold_conv) ctxt))
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end
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(* normalization of numerals: rewriting of negative integer numerals into
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   positive numerals, Numeral0 into 0, Numeral1 into 1 *)
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local
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  fun is_number_sort ctxt T =
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    Sign.of_sort (ProofContext.theory_of ctxt) (T, @{sort number_ring})
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  fun is_strange_number ctxt (t as Const (@{const_name number_of}, _) $ _) =
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        (case try HOLogic.dest_number t of
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          SOME (T, i) => is_number_sort ctxt T andalso i < 2
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        | NONE => false)
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    | is_strange_number _ _ = false
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  val pos_numeral_ss = HOL_ss
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    addsimps [@{thm Int.number_of_minus}, @{thm Int.number_of_Min}]
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    addsimps [@{thm Int.number_of_Pls}, @{thm Int.numeral_1_eq_1}]
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    addsimps @{thms Int.pred_bin_simps}
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    addsimps @{thms Int.normalize_bin_simps}
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    addsimps @{lemma
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      "Int.Min = - Int.Bit1 Int.Pls"
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      "Int.Bit0 (- Int.Pls) = - Int.Pls"
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      "Int.Bit0 (- k) = - Int.Bit0 k"
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      "Int.Bit1 (- k) = - Int.Bit1 (Int.pred k)"
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      by simp_all (simp add: pred_def)}
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  fun pos_conv ctxt = if_conv (is_strange_number ctxt)
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    (Simplifier.rewrite (Simplifier.context ctxt pos_numeral_ss))
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    Conv.no_conv
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in
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fun normalize_numerals ctxt =
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  map ((Term.exists_subterm (is_strange_number ctxt) o Thm.prop_of) ??
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    Conv.fconv_rule (Conv.top_sweep_conv pos_conv ctxt))
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end
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(* embedding of standard natural number operations into integer operations *)
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local
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  val nat_embedding = @{lemma
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    "nat (int n) = n"
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    "i >= 0 --> int (nat i) = i"
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    "i < 0 --> int (nat i) = 0"
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    by simp_all}
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  val nat_rewriting = @{lemma
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    "0 = nat 0"
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    "1 = nat 1"
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    "number_of i = nat (number_of i)"
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    "int (nat 0) = 0"
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    "int (nat 1) = 1"
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    "a < b = (int a < int b)"
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    "a <= b = (int a <= int b)"
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    "Suc a = nat (int a + 1)"
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    "a + b = nat (int a + int b)"
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    "a - b = nat (int a - int b)"
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    "a * b = nat (int a * int b)"
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    "a div b = nat (int a div int b)"
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    "a mod b = nat (int a mod int b)"
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    "min a b = nat (min (int a) (int b))"
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    "max a b = nat (max (int a) (int b))"
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    "int (nat (int a + int b)) = int a + int b"
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    "int (nat (int a * int b)) = int a * int b"
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    "int (nat (int a div int b)) = int a div int b"
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    "int (nat (int a mod int b)) = int a mod int b"
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    "int (nat (min (int a) (int b))) = min (int a) (int b)"
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    "int (nat (max (int a) (int b))) = max (int a) (int b)"
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    by (simp_all add: nat_mult_distrib nat_div_distrib nat_mod_distrib
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      int_mult[symmetric] zdiv_int[symmetric] zmod_int[symmetric])}
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  fun on_positive num f x = 
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    (case try HOLogic.dest_number (Thm.term_of num) of
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      SOME (_, i) => if i >= 0 then SOME (f x) else NONE
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    | NONE => NONE)
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  val cancel_int_nat_ss = HOL_ss
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    addsimps [@{thm Nat_Numeral.nat_number_of}]
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    addsimps [@{thm Nat_Numeral.int_nat_number_of}]
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    addsimps @{thms neg_simps}
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  fun cancel_int_nat_simproc _ ss ct = 
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    let
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      val num = Thm.dest_arg (Thm.dest_arg ct)
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      val goal = Thm.mk_binop @{cterm "op == :: int => _"} ct num
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      val simpset = Simplifier.inherit_context ss cancel_int_nat_ss
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      fun tac _ = Simplifier.simp_tac simpset 1
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    in on_positive num (Goal.prove_internal [] goal) tac end
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  val nat_ss = HOL_ss
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    addsimps nat_rewriting
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    addsimprocs [Simplifier.make_simproc {
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      name = "cancel_int_nat_num", lhss = [@{cpat "int (nat _)"}],
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      proc = cancel_int_nat_simproc, identifier = [] }]
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  fun conv ctxt = Simplifier.rewrite (Simplifier.context ctxt nat_ss)
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  val uses_nat_type = Term.exists_type (Term.exists_subtype (equal @{typ nat}))
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  val uses_nat_int =
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    Term.exists_subterm (member (op aconv) [@{term int}, @{term nat}])
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in
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fun nat_as_int ctxt =
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  map ((uses_nat_type o Thm.prop_of) ?? Conv.fconv_rule (conv ctxt)) #>
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  exists (uses_nat_int o Thm.prop_of) ?? append nat_embedding
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end
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(* further normalizations: beta/eta, universal closure, atomize *)
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val eta_expand_eq = @{lemma "f == (%x. f x)" by (rule reflexive)}
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fun eta_expand_conv cv ctxt =
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  Conv.rewr_conv eta_expand_eq then_conv Conv.abs_conv (cv o snd) ctxt
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local
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  val eta_conv = eta_expand_conv
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  fun keep_conv ctxt = Conv.binder_conv (norm_conv o snd) ctxt
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  and eta_binder_conv ctxt = Conv.arg_conv (eta_conv norm_conv ctxt)
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  and keep_let_conv ctxt = Conv.combination_conv
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    (Conv.arg_conv (norm_conv ctxt)) (Conv.abs_conv (norm_conv o snd) ctxt)
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  and unfold_let_conv ctxt = Conv.combination_conv
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    (Conv.arg_conv (norm_conv ctxt)) (eta_conv norm_conv ctxt)
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  and unfold_conv thm ctxt = Conv.rewr_conv thm then_conv keep_conv ctxt
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  and unfold_ex1_conv ctxt = unfold_conv @{thm Ex1_def} ctxt
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  and unfold_ball_conv ctxt = unfold_conv (mk_meta_eq @{thm Ball_def}) ctxt
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  and unfold_bex_conv ctxt = unfold_conv (mk_meta_eq @{thm Bex_def}) ctxt
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  and norm_conv ctxt ct =
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    (case Thm.term_of ct of
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      Const (@{const_name All}, _) $ Abs _ => keep_conv
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    | Const (@{const_name All}, _) $ _ => eta_binder_conv
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    | Const (@{const_name All}, _) => eta_conv eta_binder_conv
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    | Const (@{const_name Ex}, _) $ Abs _ => keep_conv
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    | Const (@{const_name Ex}, _) $ _ => eta_binder_conv
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    | Const (@{const_name Ex}, _) => eta_conv eta_binder_conv
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    | Const (@{const_name Let}, _) $ _ $ Abs _ => keep_let_conv
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    | Const (@{const_name Let}, _) $ _ $ _ => unfold_let_conv
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    | Const (@{const_name Let}, _) $ _ => eta_conv unfold_let_conv
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    | Const (@{const_name Let}, _) => eta_conv (eta_conv unfold_let_conv)
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    | Const (@{const_name Ex1}, _) $ _ => unfold_ex1_conv
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    | Const (@{const_name Ex1}, _) => eta_conv unfold_ex1_conv 
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    | Const (@{const_name Ball}, _) $ _ $ _ => unfold_ball_conv
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    | Const (@{const_name Ball}, _) $ _ => eta_conv unfold_ball_conv
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    | Const (@{const_name Ball}, _) => eta_conv (eta_conv unfold_ball_conv)
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    | Const (@{const_name Bex}, _) $ _ $ _ => unfold_bex_conv
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    | Const (@{const_name Bex}, _) $ _ => eta_conv unfold_bex_conv
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    | Const (@{const_name Bex}, _) => eta_conv (eta_conv unfold_bex_conv)
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    | Abs _ => Conv.abs_conv (norm_conv o snd)
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    | _ $ _ => Conv.comb_conv o norm_conv
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    | _ => K Conv.all_conv) ctxt ct
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  fun is_normed t =
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    (case t of
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      Const (@{const_name All}, _) $ Abs (_, _, u) => is_normed u
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    | Const (@{const_name All}, _) $ _ => false
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    | Const (@{const_name All}, _) => false
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    | Const (@{const_name Ex}, _) $ Abs (_, _, u) => is_normed u
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    | Const (@{const_name Ex}, _) $ _ => false
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    | Const (@{const_name Ex}, _) => false
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    | Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
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        is_normed u1 andalso is_normed u2
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    | Const (@{const_name Let}, _) $ _ $ _ => false
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    | Const (@{const_name Let}, _) $ _ => false
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    | Const (@{const_name Let}, _) => false
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    | Const (@{const_name Ex1}, _) => false
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    | Const (@{const_name Ball}, _) => false
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    | Const (@{const_name Bex}, _) => false
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    | Abs (_, _, u) => is_normed u
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    | u1 $ u2 => is_normed u1 andalso is_normed u2
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    | _ => true)
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in
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fun norm_binder_conv ctxt = if_conv is_normed Conv.all_conv (norm_conv ctxt)
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end
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fun norm_def ctxt thm =
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  (case Thm.prop_of thm of
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    @{term Trueprop} $ (Const (@{const_name "op ="}, _) $ _ $ Abs _) =>
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      norm_def ctxt (thm RS @{thm fun_cong})
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  | Const (@{const_name "=="}, _) $ _ $ Abs _ =>
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      norm_def ctxt (thm RS @{thm meta_eq_to_obj_eq})
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  | _ => thm)
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fun atomize_conv ctxt ct =
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  (case Thm.term_of ct of
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    @{term "op ==>"} $ _ $ _ =>
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      Conv.binop_conv (atomize_conv ctxt) then_conv
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      Conv.rewr_conv @{thm atomize_imp}
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  | Const (@{const_name "=="}, _) $ _ $ _ =>
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      Conv.binop_conv (atomize_conv ctxt) then_conv
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      Conv.rewr_conv @{thm atomize_eq}
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  | Const (@{const_name all}, _) $ Abs _ =>
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      Conv.binder_conv (atomize_conv o snd) ctxt then_conv
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      Conv.rewr_conv @{thm atomize_all}
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  | _ => Conv.all_conv) ct
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fun normalize_rule ctxt =
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  Conv.fconv_rule (
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    (* reduce lambda abstractions, except at known binders: *)
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    Thm.beta_conversion true then_conv
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    Thm.eta_conversion then_conv
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    norm_binder_conv ctxt) #>
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  norm_def ctxt #>
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  Drule.forall_intr_vars #>
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  Conv.fconv_rule (atomize_conv ctxt)
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(* lift lambda terms into additional rules *)
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local
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  val meta_eq = @{cpat "op =="}
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  val meta_eqT = hd (Thm.dest_ctyp (Thm.ctyp_of_term meta_eq))
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  fun inst_meta cT = Thm.instantiate_cterm ([(meta_eqT, cT)], []) meta_eq
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  fun mk_meta_eq ct cu = Thm.mk_binop (inst_meta (Thm.ctyp_of_term ct)) ct cu
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  fun cert ctxt = Thm.cterm_of (ProofContext.theory_of ctxt)
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  fun used_vars cvs ct =
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    let
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      val lookup = AList.lookup (op aconv) (map (` Thm.term_of) cvs)
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      val add = (fn SOME ct => insert (op aconvc) ct | _ => I)
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    in Term.fold_aterms (add o lookup) (Thm.term_of ct) [] end
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  fun apply cv thm = 
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    let val thm' = Thm.combination thm (Thm.reflexive cv)
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    in Thm.transitive thm' (Thm.beta_conversion false (Thm.rhs_of thm')) end
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  fun apply_def cvs eq = Thm.symmetric (fold apply cvs eq)
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  fun replace_lambda cvs ct (cx as (ctxt, defs)) =
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    let
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      val cvs' = used_vars cvs ct
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      val ct' = fold_rev Thm.cabs cvs' ct
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    in
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      (case Termtab.lookup defs (Thm.term_of ct') of
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        SOME eq => (apply_def cvs' eq, cx)
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      | NONE =>
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          let
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            val {T, ...} = Thm.rep_cterm ct' and n = Name.uu
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            val (n', ctxt') = yield_singleton Variable.variant_fixes n ctxt
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            val cu = mk_meta_eq (cert ctxt (Free (n', T))) ct'
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            val (eq, ctxt'') = yield_singleton Assumption.add_assumes cu ctxt'
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            val defs' = Termtab.update (Thm.term_of ct', eq) defs
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          in (apply_def cvs' eq, (ctxt'', defs')) end)
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    end
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  fun none ct cx = (Thm.reflexive ct, cx)
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  fun in_comb f g ct cx =
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    let val (cu1, cu2) = Thm.dest_comb ct
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    in cx |> f cu1 ||>> g cu2 |>> uncurry Thm.combination end
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  fun in_arg f = in_comb none f
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  fun in_abs f cvs ct (ctxt, defs) =
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    let
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      val (n, ctxt') = yield_singleton Variable.variant_fixes Name.uu ctxt
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      val (cv, cu) = Thm.dest_abs (SOME n) ct
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    in  (ctxt', defs) |> f (cv :: cvs) cu |>> Thm.abstract_rule n cv end
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  fun traverse cvs ct =
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    (case Thm.term_of ct of
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      Const (@{const_name All}, _) $ Abs _ => in_arg (in_abs traverse cvs)
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    | Const (@{const_name Ex}, _) $ Abs _ => in_arg (in_abs traverse cvs)
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    | Const (@{const_name Let}, _) $ _ $ Abs _ =>
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        in_comb (in_arg (traverse cvs)) (in_abs traverse cvs)
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    | Abs _ => at_lambda cvs
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    | _ $ _ => in_comb (traverse cvs) (traverse cvs)
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    | _ => none) ct
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  and at_lambda cvs ct =
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    in_abs traverse cvs ct #-> (fn thm =>
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    replace_lambda cvs (Thm.rhs_of thm) #>> Thm.transitive thm)
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  fun has_free_lambdas t =
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    (case t of
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      Const (@{const_name All}, _) $ Abs (_, _, u) => has_free_lambdas u
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    | Const (@{const_name Ex}, _) $ Abs (_, _, u) => has_free_lambdas u
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    | Const (@{const_name Let}, _) $ u1 $ Abs (_, _, u2) =>
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        has_free_lambdas u1 orelse has_free_lambdas u2
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    | Abs _ => true
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    | u1 $ u2 => has_free_lambdas u1 orelse has_free_lambdas u2
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    | _ => false)
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  fun lift_lm f thm cx =
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    if not (has_free_lambdas (Thm.prop_of thm)) then (thm, cx)
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    else cx |> f (Thm.cprop_of thm) |>> (fn thm' => Thm.equal_elim thm' thm)
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in
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fun lift_lambdas thms ctxt =
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  let
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    val cx = (ctxt, Termtab.empty)
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    val (thms', (ctxt', defs)) = fold_map (lift_lm (traverse [])) thms cx
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    val eqs = Termtab.fold (cons o normalize_rule ctxt' o snd) defs []
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  in (eqs @ thms', ctxt') end
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end
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(* make application explicit for functions with varying number of arguments *)
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local
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  val const = prefix "c" and free = prefix "f"
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  fun min i (e as (_, j)) = if i <> j then (true, Int.min (i, j)) else e
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  fun add t i = Symtab.map_default (t, (false, i)) (min i)
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  fun traverse t =
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    (case Term.strip_comb t of
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      (Const (n, _), ts) => add (const n) (length ts) #> fold traverse ts 
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    | (Free (n, _), ts) => add (free n) (length ts) #> fold traverse ts
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    | (Abs (_, _, u), ts) => fold traverse (u :: ts)
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    | (_, ts) => fold traverse ts)
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  val prune = (fn (n, (true, i)) => Symtab.update (n, i) | _ => I)
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  fun prune_tab tab = Symtab.fold prune tab Symtab.empty
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  fun binop_conv cv1 cv2 = Conv.combination_conv (Conv.arg_conv cv1) cv2
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  fun nary_conv conv1 conv2 ct =
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    (Conv.combination_conv (nary_conv conv1 conv2) conv2 else_conv conv1) ct
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  fun abs_conv conv tb = Conv.abs_conv (fn (cv, cx) =>
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    let val n = fst (Term.dest_Free (Thm.term_of cv))
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    in conv (Symtab.update (free n, 0) tb) cx end)
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  val fun_app_rule = @{lemma "f x == fun_app f x" by (simp add: fun_app_def)}
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in
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fun explicit_application ctxt thms =
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  let
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    fun sub_conv tb ctxt ct =
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      (case Term.strip_comb (Thm.term_of ct) of
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        (Const (n, _), ts) => app_conv tb (const n) (length ts) ctxt
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      | (Free (n, _), ts) => app_conv tb (free n) (length ts) ctxt
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      | (Abs _, _) => nary_conv (abs_conv sub_conv tb ctxt) (sub_conv tb ctxt)
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      | (_, _) => nary_conv Conv.all_conv (sub_conv tb ctxt)) ct
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    and app_conv tb n i ctxt =
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      (case Symtab.lookup tb n of
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        NONE => nary_conv Conv.all_conv (sub_conv tb ctxt)
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      | SOME j => fun_app_conv tb ctxt (i - j))
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    and fun_app_conv tb ctxt i ct = (
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      if i = 0 then nary_conv Conv.all_conv (sub_conv tb ctxt)
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      else
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        Conv.rewr_conv fun_app_rule then_conv
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        binop_conv (fun_app_conv tb ctxt (i-1)) (sub_conv tb ctxt)) ct
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    fun needs_exp_app tab = Term.exists_subterm (fn
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        Bound _ $ _ => true
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      | Const (n, _) => Symtab.defined tab (const n)
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      | Free (n, _) => Symtab.defined tab (free n)
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      | _ => false)
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    fun rewrite tab ctxt thm =
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      if not (needs_exp_app tab (Thm.prop_of thm)) then thm
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      else Conv.fconv_rule (sub_conv tab ctxt) thm
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    val tab = prune_tab (fold (traverse o Thm.prop_of) thms Symtab.empty)
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  in map (rewrite tab ctxt) thms end
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end
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(* combined normalization *)
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type extra_norm = thm list -> Proof.context -> thm list * Proof.context
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fun with_context f thms ctxt = (f ctxt thms, ctxt)
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fun normalize extra_norm thms ctxt =
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  thms
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  |> trivial_distinct ctxt
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  |> rewrite_bool_cases ctxt
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  |> normalize_numerals ctxt
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  |> nat_as_int ctxt
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  |> rpair ctxt
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  |-> extra_norm
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  |-> with_context (fn cx => map (normalize_rule cx))
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  |-> SMT_Monomorph.monomorph
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  |-> lift_lambdas
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  |-> with_context explicit_application
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end