src/HOL/Tools/SMT/z3_proof_reconstruction.ML
author boehmes
Mon Dec 20 22:02:57 2010 +0100 (2010-12-20)
changeset 41328 6792a5c92a58
parent 41172 a17c2d669c40
child 41426 09615ed31f04
permissions -rw-r--r--
avoid ML structure aliases (especially single-letter abbreviations)
boehmes@36898
     1
(*  Title:      HOL/Tools/SMT/z3_proof_reconstruction.ML
boehmes@36898
     2
    Author:     Sascha Boehme, TU Muenchen
boehmes@36898
     3
boehmes@36898
     4
Proof reconstruction for proofs found by Z3.
boehmes@36898
     5
*)
boehmes@36898
     6
boehmes@36898
     7
signature Z3_PROOF_RECONSTRUCTION =
boehmes@36898
     8
sig
boehmes@36899
     9
  val add_z3_rule: thm -> Context.generic -> Context.generic
boehmes@40162
    10
  val reconstruct: Proof.context -> SMT_Translate.recon -> string list ->
boehmes@41127
    11
    int list * thm
boehmes@36898
    12
  val setup: theory -> theory
boehmes@36898
    13
end
boehmes@36898
    14
boehmes@36898
    15
structure Z3_Proof_Reconstruction: Z3_PROOF_RECONSTRUCTION =
boehmes@36898
    16
struct
boehmes@36898
    17
boehmes@36898
    18
boehmes@40424
    19
fun z3_exn msg = raise SMT_Failure.SMT (SMT_Failure.Other_Failure
boehmes@40162
    20
  ("Z3 proof reconstruction: " ^ msg))
boehmes@36898
    21
boehmes@36898
    22
boehmes@36898
    23
boehmes@41130
    24
(* net of schematic rules *)
boehmes@36898
    25
boehmes@36898
    26
val z3_ruleN = "z3_rule"
boehmes@36898
    27
boehmes@36898
    28
local
boehmes@36898
    29
  val description = "declaration of Z3 proof rules"
boehmes@36898
    30
boehmes@36898
    31
  val eq = Thm.eq_thm
boehmes@36898
    32
boehmes@36898
    33
  structure Z3_Rules = Generic_Data
boehmes@36898
    34
  (
boehmes@36898
    35
    type T = thm Net.net
boehmes@36898
    36
    val empty = Net.empty
boehmes@36898
    37
    val extend = I
boehmes@36898
    38
    val merge = Net.merge eq
boehmes@36898
    39
  )
boehmes@36898
    40
boehmes@41328
    41
  val prep =
boehmes@41328
    42
    `Thm.prop_of o Simplifier.rewrite_rule [Z3_Proof_Literals.rewrite_true]
boehmes@36898
    43
boehmes@36898
    44
  fun ins thm net = Net.insert_term eq (prep thm) net handle Net.INSERT => net
boehmes@36898
    45
  fun del thm net = Net.delete_term eq (prep thm) net handle Net.DELETE => net
boehmes@36898
    46
boehmes@36898
    47
  val add = Thm.declaration_attribute (Z3_Rules.map o ins)
boehmes@36898
    48
  val del = Thm.declaration_attribute (Z3_Rules.map o del)
boehmes@36898
    49
in
boehmes@36898
    50
boehmes@36899
    51
val add_z3_rule = Z3_Rules.map o ins
boehmes@36898
    52
boehmes@36898
    53
fun by_schematic_rule ctxt ct =
boehmes@41328
    54
  the (Z3_Proof_Tools.net_instance (Z3_Rules.get (Context.Proof ctxt)) ct)
boehmes@36898
    55
boehmes@36898
    56
val z3_rules_setup =
boehmes@36898
    57
  Attrib.setup (Binding.name z3_ruleN) (Attrib.add_del add del) description #>
wenzelm@39557
    58
  Global_Theory.add_thms_dynamic (Binding.name z3_ruleN, Net.content o Z3_Rules.get)
boehmes@36898
    59
boehmes@36898
    60
end
boehmes@36898
    61
boehmes@36898
    62
boehmes@36898
    63
boehmes@41130
    64
(* proof tools *)
boehmes@36898
    65
boehmes@36898
    66
fun named ctxt name prover ct =
boehmes@40424
    67
  let val _ = SMT_Config.trace_msg ctxt I ("Z3: trying " ^ name ^ " ...")
boehmes@36898
    68
  in prover ct end
boehmes@36898
    69
boehmes@36898
    70
fun NAMED ctxt name tac i st =
boehmes@40424
    71
  let val _ = SMT_Config.trace_msg ctxt I ("Z3: trying " ^ name ^ " ...")
boehmes@36898
    72
  in tac i st end
boehmes@36898
    73
boehmes@36898
    74
fun pretty_goal ctxt thms t =
boehmes@36898
    75
  [Pretty.block [Pretty.str "proposition: ", Syntax.pretty_term ctxt t]]
boehmes@36898
    76
  |> not (null thms) ? cons (Pretty.big_list "assumptions:"
boehmes@36898
    77
       (map (Display.pretty_thm ctxt) thms))
boehmes@36898
    78
boehmes@36898
    79
fun try_apply ctxt thms =
boehmes@36898
    80
  let
boehmes@36898
    81
    fun try_apply_err ct = Pretty.string_of (Pretty.chunks [
boehmes@36898
    82
      Pretty.big_list ("Z3 found a proof," ^
boehmes@36898
    83
        " but proof reconstruction failed at the following subgoal:")
boehmes@36898
    84
        (pretty_goal ctxt thms (Thm.term_of ct)),
boehmes@36898
    85
      Pretty.str ("Adding a rule to the lemma group " ^ quote z3_ruleN ^
boehmes@36898
    86
        " might solve this problem.")])
boehmes@36898
    87
boehmes@36898
    88
    fun apply [] ct = error (try_apply_err ct)
boehmes@36898
    89
      | apply (prover :: provers) ct =
boehmes@36898
    90
          (case try prover ct of
boehmes@40424
    91
            SOME thm => (SMT_Config.trace_msg ctxt I "Z3: succeeded"; thm)
boehmes@36898
    92
          | NONE => apply provers ct)
boehmes@36898
    93
boehmes@36898
    94
  in apply o cons (named ctxt "schematic rules" (by_schematic_rule ctxt)) end
boehmes@36898
    95
boehmes@36899
    96
local
boehmes@36899
    97
  val rewr_if =
boehmes@36899
    98
    @{lemma "(if P then Q1 else Q2) = ((P --> Q1) & (~P --> Q2))" by simp}
boehmes@36899
    99
in
boehmes@36899
   100
val simp_fast_tac =
boehmes@36899
   101
  Simplifier.simp_tac (HOL_ss addsimps [rewr_if])
boehmes@36899
   102
  THEN_ALL_NEW Classical.fast_tac HOL_cs
boehmes@36899
   103
end
boehmes@36899
   104
boehmes@36898
   105
boehmes@36898
   106
boehmes@41130
   107
(* theorems and proofs *)
boehmes@36898
   108
boehmes@41130
   109
(** theorem incarnations **)
boehmes@36898
   110
boehmes@36898
   111
datatype theorem =
boehmes@36898
   112
  Thm of thm | (* theorem without special features *)
boehmes@36898
   113
  MetaEq of thm | (* meta equality "t == s" *)
boehmes@41328
   114
  Literals of thm * Z3_Proof_Literals.littab
boehmes@36898
   115
    (* "P1 & ... & Pn" and table of all literals P1, ..., Pn *)
boehmes@36898
   116
boehmes@36898
   117
fun thm_of (Thm thm) = thm
boehmes@36898
   118
  | thm_of (MetaEq thm) = thm COMP @{thm meta_eq_to_obj_eq}
boehmes@36898
   119
  | thm_of (Literals (thm, _)) = thm
boehmes@36898
   120
boehmes@36898
   121
fun meta_eq_of (MetaEq thm) = thm
boehmes@36898
   122
  | meta_eq_of p = mk_meta_eq (thm_of p)
boehmes@36898
   123
boehmes@36898
   124
fun literals_of (Literals (_, lits)) = lits
boehmes@41328
   125
  | literals_of p = Z3_Proof_Literals.make_littab [thm_of p]
boehmes@36898
   126
boehmes@36898
   127
boehmes@36898
   128
boehmes@36898
   129
(** core proof rules **)
boehmes@36898
   130
boehmes@36898
   131
(* assumption *)
boehmes@41131
   132
boehmes@36898
   133
local
boehmes@41131
   134
  val remove_trigger = mk_meta_eq @{thm SMT.trigger_def}
boehmes@41131
   135
  val remove_weight = mk_meta_eq @{thm SMT.weight_def}
boehmes@41131
   136
  val remove_fun_app = mk_meta_eq @{thm SMT.fun_app_def}
boehmes@36898
   137
boehmes@36898
   138
  fun rewrite_conv ctxt eqs = Simplifier.full_rewrite
boehmes@36898
   139
    (Simplifier.context ctxt Simplifier.empty_ss addsimps eqs)
boehmes@36898
   140
boehmes@41131
   141
  val prep_rules = [@{thm Let_def}, remove_trigger, remove_weight,
boehmes@41328
   142
    remove_fun_app, Z3_Proof_Literals.rewrite_true]
boehmes@41131
   143
boehmes@41131
   144
  fun rewrite ctxt eqs = Conv.fconv_rule (rewrite_conv ctxt eqs)
boehmes@36898
   145
boehmes@40164
   146
  fun burrow_snd_option f (i, thm) = Option.map (pair i) (f thm)
boehmes@41131
   147
boehmes@41131
   148
  fun lookup_assm assms_net ct =
boehmes@41328
   149
    Z3_Proof_Tools.net_instance' burrow_snd_option assms_net ct
boehmes@41131
   150
    |> Option.map (fn ithm as (_, thm) => (ithm, Thm.cprop_of thm aconvc ct))
boehmes@36898
   151
in
boehmes@41131
   152
boehmes@41131
   153
fun add_asserted outer_ctxt rewrite_rules assms asserted ctxt =
boehmes@36898
   154
  let
boehmes@41328
   155
    val eqs = map (rewrite ctxt [Z3_Proof_Literals.rewrite_true]) rewrite_rules
boehmes@41131
   156
    val eqs' = union Thm.eq_thm eqs prep_rules
boehmes@41131
   157
boehmes@41131
   158
    val assms_net =
boehmes@41127
   159
      assms
boehmes@41131
   160
      |> map (apsnd (rewrite ctxt eqs'))
boehmes@41127
   161
      |> map (apsnd (Conv.fconv_rule Thm.eta_conversion))
boehmes@41328
   162
      |> Z3_Proof_Tools.thm_net_of snd 
boehmes@41131
   163
boehmes@41131
   164
    fun revert_conv ctxt = rewrite_conv ctxt eqs' then_conv Thm.eta_conversion
boehmes@41131
   165
boehmes@41131
   166
    fun assume thm ctxt =
boehmes@41131
   167
      let
boehmes@41131
   168
        val ct = Thm.cprem_of thm 1
boehmes@41131
   169
        val (thm', ctxt') = yield_singleton Assumption.add_assumes ct ctxt
boehmes@41131
   170
      in (Thm.implies_elim thm thm', ctxt') end
boehmes@36898
   171
boehmes@41131
   172
    fun add (idx, ct) ((is, thms), (ctxt, ptab)) =
boehmes@41131
   173
      let
boehmes@41131
   174
        val thm1 = 
boehmes@41131
   175
          Thm.trivial ct
boehmes@41131
   176
          |> Conv.fconv_rule (Conv.arg1_conv (revert_conv outer_ctxt))
boehmes@41131
   177
        val thm2 = singleton (Variable.export ctxt outer_ctxt) thm1
boehmes@41131
   178
      in
boehmes@41131
   179
        (case lookup_assm assms_net (Thm.cprem_of thm2 1) of
boehmes@41131
   180
          NONE =>
boehmes@41131
   181
            let val (thm, ctxt') = assume thm1 ctxt
boehmes@41131
   182
            in ((is, thms), (ctxt', Inttab.update (idx, Thm thm) ptab)) end
boehmes@41131
   183
        | SOME ((i, th), exact) =>
boehmes@41131
   184
            let
boehmes@41131
   185
              val (thm, ctxt') =
boehmes@41131
   186
                if exact then (Thm.implies_elim thm1 th, ctxt)
boehmes@41131
   187
                else assume thm1 ctxt
boehmes@41131
   188
              val thms' = if exact then thms else th :: thms
boehmes@41131
   189
            in 
boehmes@41131
   190
              ((insert (op =) i is, thms'),
boehmes@41131
   191
                (ctxt', Inttab.update (idx, Thm thm) ptab))
boehmes@41131
   192
            end)
boehmes@41131
   193
      end
boehmes@41131
   194
  in fold add asserted (([], []), (ctxt, Inttab.empty)) end
boehmes@40164
   195
boehmes@36898
   196
end
boehmes@36898
   197
boehmes@36898
   198
boehmes@36898
   199
(* P = Q ==> P ==> Q   or   P --> Q ==> P ==> Q *)
boehmes@36898
   200
local
boehmes@41328
   201
  val precomp = Z3_Proof_Tools.precompose2
boehmes@41328
   202
  val comp = Z3_Proof_Tools.compose
boehmes@36898
   203
boehmes@41328
   204
  val meta_iffD1 = @{lemma "P == Q ==> P ==> (Q::bool)" by simp}
boehmes@41328
   205
  val meta_iffD1_c = precomp Thm.dest_binop meta_iffD1
boehmes@41328
   206
boehmes@41328
   207
  val iffD1_c = precomp (Thm.dest_binop o Thm.dest_arg) @{thm iffD1}
boehmes@41328
   208
  val mp_c = precomp (Thm.dest_binop o Thm.dest_arg) @{thm mp}
boehmes@36898
   209
in
boehmes@41328
   210
fun mp (MetaEq thm) p = Thm (Thm.implies_elim (comp meta_iffD1_c thm) p)
boehmes@36898
   211
  | mp p_q p = 
boehmes@36898
   212
      let
boehmes@36898
   213
        val pq = thm_of p_q
boehmes@41328
   214
        val thm = comp iffD1_c pq handle THM _ => comp mp_c pq
boehmes@36898
   215
      in Thm (Thm.implies_elim thm p) end
boehmes@36898
   216
end
boehmes@36898
   217
boehmes@36898
   218
boehmes@36898
   219
(* and_elim:     P1 & ... & Pn ==> Pi *)
boehmes@36898
   220
(* not_or_elim:  ~(P1 | ... | Pn) ==> ~Pi *)
boehmes@36898
   221
local
boehmes@41328
   222
  fun is_sublit conj t = Z3_Proof_Literals.exists_lit conj (fn u => u aconv t)
boehmes@36898
   223
boehmes@36898
   224
  fun derive conj t lits idx ptab =
boehmes@36898
   225
    let
boehmes@41328
   226
      val lit = the (Z3_Proof_Literals.get_first_lit (is_sublit conj t) lits)
boehmes@41328
   227
      val ls = Z3_Proof_Literals.explode conj false false [t] lit
boehmes@41328
   228
      val lits' = fold Z3_Proof_Literals.insert_lit ls
boehmes@41328
   229
        (Z3_Proof_Literals.delete_lit lit lits)
boehmes@36898
   230
boehmes@41130
   231
      fun upd thm = Literals (thm_of thm, lits')
boehmes@41328
   232
      val ptab' = Inttab.map_entry idx upd ptab
boehmes@41328
   233
    in (the (Z3_Proof_Literals.lookup_lit lits' t), ptab') end
boehmes@36898
   234
boehmes@36898
   235
  fun lit_elim conj (p, idx) ct ptab =
boehmes@36898
   236
    let val lits = literals_of p
boehmes@36898
   237
    in
boehmes@41328
   238
      (case Z3_Proof_Literals.lookup_lit lits (SMT_Utils.term_of ct) of
boehmes@36898
   239
        SOME lit => (Thm lit, ptab)
boehmes@41328
   240
      | NONE => apfst Thm (derive conj (SMT_Utils.term_of ct) lits idx ptab))
boehmes@36898
   241
    end
boehmes@36898
   242
in
boehmes@36898
   243
val and_elim = lit_elim true
boehmes@36898
   244
val not_or_elim = lit_elim false
boehmes@36898
   245
end
boehmes@36898
   246
boehmes@36898
   247
boehmes@36898
   248
(* P1, ..., Pn |- False ==> |- ~P1 | ... | ~Pn *)
boehmes@36898
   249
local
boehmes@36898
   250
  fun step lit thm =
boehmes@36898
   251
    Thm.implies_elim (Thm.implies_intr (Thm.cprop_of lit) thm) lit
boehmes@41328
   252
  val explode_disj = Z3_Proof_Literals.explode false false false
boehmes@36898
   253
  fun intro hyps thm th = fold step (explode_disj hyps th) thm
boehmes@36898
   254
boehmes@36898
   255
  fun dest_ccontr ct = [Thm.dest_arg (Thm.dest_arg (Thm.dest_arg1 ct))]
boehmes@41328
   256
  val ccontr = Z3_Proof_Tools.precompose dest_ccontr @{thm ccontr}
boehmes@36898
   257
in
boehmes@36898
   258
fun lemma thm ct =
boehmes@36898
   259
  let
boehmes@41328
   260
    val cu = Z3_Proof_Literals.negate (Thm.dest_arg ct)
boehmes@36898
   261
    val hyps = map_filter (try HOLogic.dest_Trueprop) (#hyps (Thm.rep_thm thm))
boehmes@41328
   262
    val th = Z3_Proof_Tools.under_assumption (intro hyps thm) cu
boehmes@41328
   263
  in Thm (Z3_Proof_Tools.compose ccontr th) end
boehmes@36898
   264
end
boehmes@36898
   265
boehmes@36898
   266
boehmes@36898
   267
(* \/{P1, ..., Pn, Q1, ..., Qn}, ~P1, ..., ~Pn ==> \/{Q1, ..., Qn} *)
boehmes@36898
   268
local
boehmes@41328
   269
  val explode_disj = Z3_Proof_Literals.explode false true false
boehmes@41328
   270
  val join_disj = Z3_Proof_Literals.join false
boehmes@36898
   271
  fun unit thm thms th =
boehmes@41328
   272
    let
boehmes@41328
   273
      val t = @{const Not} $ SMT_Utils.prop_of thm
boehmes@41328
   274
      val ts = map SMT_Utils.prop_of thms
boehmes@41328
   275
    in
boehmes@41328
   276
      join_disj (Z3_Proof_Literals.make_littab (thms @ explode_disj ts th)) t
boehmes@41328
   277
    end
boehmes@36898
   278
boehmes@36898
   279
  fun dest_arg2 ct = Thm.dest_arg (Thm.dest_arg ct)
boehmes@36898
   280
  fun dest ct = pairself dest_arg2 (Thm.dest_binop ct)
boehmes@41328
   281
  val contrapos =
boehmes@41328
   282
    Z3_Proof_Tools.precompose2 dest @{lemma "(~P ==> ~Q) ==> Q ==> P" by fast}
boehmes@36898
   283
in
boehmes@36898
   284
fun unit_resolution thm thms ct =
boehmes@41328
   285
  Z3_Proof_Literals.negate (Thm.dest_arg ct)
boehmes@41328
   286
  |> Z3_Proof_Tools.under_assumption (unit thm thms)
boehmes@41328
   287
  |> Thm o Z3_Proof_Tools.discharge thm o Z3_Proof_Tools.compose contrapos
boehmes@36898
   288
end
boehmes@36898
   289
boehmes@36898
   290
boehmes@36898
   291
(* P ==> P == True   or   P ==> P == False *)
boehmes@36898
   292
local
boehmes@36898
   293
  val iff1 = @{lemma "P ==> P == (~ False)" by simp}
boehmes@36898
   294
  val iff2 = @{lemma "~P ==> P == False" by simp}
boehmes@36898
   295
in
boehmes@36898
   296
fun iff_true thm = MetaEq (thm COMP iff1)
boehmes@36898
   297
fun iff_false thm = MetaEq (thm COMP iff2)
boehmes@36898
   298
end
boehmes@36898
   299
boehmes@36898
   300
boehmes@36898
   301
(* distributivity of | over & *)
boehmes@36898
   302
fun distributivity ctxt = Thm o try_apply ctxt [] [
boehmes@41328
   303
  named ctxt "fast" (Z3_Proof_Tools.by_tac (Classical.fast_tac HOL_cs))]
boehmes@36898
   304
    (* FIXME: not very well tested *)
boehmes@36898
   305
boehmes@36898
   306
boehmes@36898
   307
(* Tseitin-like axioms *)
boehmes@36898
   308
local
boehmes@36898
   309
  val disjI1 = @{lemma "(P ==> Q) ==> ~P | Q" by fast}
boehmes@36898
   310
  val disjI2 = @{lemma "(~P ==> Q) ==> P | Q" by fast}
boehmes@36898
   311
  val disjI3 = @{lemma "(~Q ==> P) ==> P | Q" by fast}
boehmes@36898
   312
  val disjI4 = @{lemma "(Q ==> P) ==> P | ~Q" by fast}
boehmes@36898
   313
boehmes@36898
   314
  fun prove' conj1 conj2 ct2 thm =
boehmes@41328
   315
    let
boehmes@41328
   316
      val littab =
boehmes@41328
   317
        Z3_Proof_Literals.explode conj1 true (conj1 <> conj2) [] thm
boehmes@41328
   318
        |> cons Z3_Proof_Literals.true_thm
boehmes@41328
   319
        |> Z3_Proof_Literals.make_littab
boehmes@41328
   320
    in Z3_Proof_Literals.join conj2 littab (Thm.term_of ct2) end
boehmes@36898
   321
boehmes@36898
   322
  fun prove rule (ct1, conj1) (ct2, conj2) =
boehmes@41328
   323
    Z3_Proof_Tools.under_assumption (prove' conj1 conj2 ct2) ct1 COMP rule
boehmes@36898
   324
boehmes@36898
   325
  fun prove_def_axiom ct =
boehmes@36898
   326
    let val (ct1, ct2) = Thm.dest_binop (Thm.dest_arg ct)
boehmes@36898
   327
    in
boehmes@36898
   328
      (case Thm.term_of ct1 of
boehmes@40579
   329
        @{const Not} $ (@{const HOL.conj} $ _ $ _) =>
boehmes@36898
   330
          prove disjI1 (Thm.dest_arg ct1, true) (ct2, true)
boehmes@40579
   331
      | @{const HOL.conj} $ _ $ _ =>
boehmes@41328
   332
          prove disjI3 (Z3_Proof_Literals.negate ct2, false) (ct1, true)
boehmes@40579
   333
      | @{const Not} $ (@{const HOL.disj} $ _ $ _) =>
boehmes@41328
   334
          prove disjI3 (Z3_Proof_Literals.negate ct2, false) (ct1, false)
boehmes@40579
   335
      | @{const HOL.disj} $ _ $ _ =>
boehmes@41328
   336
          prove disjI2 (Z3_Proof_Literals.negate ct1, false) (ct2, true)
boehmes@40681
   337
      | Const (@{const_name distinct}, _) $ _ =>
boehmes@36898
   338
          let
boehmes@36898
   339
            fun dis_conv cv = Conv.arg_conv (Conv.arg1_conv cv)
boehmes@41328
   340
            val unfold_dis_conv = dis_conv Z3_Proof_Tools.unfold_distinct_conv
boehmes@36898
   341
            fun prv cu =
boehmes@36898
   342
              let val (cu1, cu2) = Thm.dest_binop (Thm.dest_arg cu)
boehmes@36898
   343
              in prove disjI4 (Thm.dest_arg cu2, true) (cu1, true) end
boehmes@41328
   344
          in Z3_Proof_Tools.with_conv unfold_dis_conv prv ct end
boehmes@40681
   345
      | @{const Not} $ (Const (@{const_name distinct}, _) $ _) =>
boehmes@36898
   346
          let
boehmes@36898
   347
            fun dis_conv cv = Conv.arg_conv (Conv.arg1_conv (Conv.arg_conv cv))
boehmes@41328
   348
            val unfold_dis_conv = dis_conv Z3_Proof_Tools.unfold_distinct_conv
boehmes@36898
   349
            fun prv cu =
boehmes@36898
   350
              let val (cu1, cu2) = Thm.dest_binop (Thm.dest_arg cu)
boehmes@36898
   351
              in prove disjI1 (Thm.dest_arg cu1, true) (cu2, true) end
boehmes@41328
   352
          in Z3_Proof_Tools.with_conv unfold_dis_conv prv ct end
boehmes@36898
   353
      | _ => raise CTERM ("prove_def_axiom", [ct]))
boehmes@36898
   354
    end
boehmes@36898
   355
in
boehmes@36898
   356
fun def_axiom ctxt = Thm o try_apply ctxt [] [
boehmes@36898
   357
  named ctxt "conj/disj/distinct" prove_def_axiom,
boehmes@41328
   358
  Z3_Proof_Tools.by_abstraction (true, false) ctxt [] (fn ctxt' =>
boehmes@41328
   359
    named ctxt' "simp+fast" (Z3_Proof_Tools.by_tac simp_fast_tac))]
boehmes@36898
   360
end
boehmes@36898
   361
boehmes@36898
   362
boehmes@36898
   363
(* local definitions *)
boehmes@36898
   364
local
boehmes@36898
   365
  val intro_rules = [
boehmes@36898
   366
    @{lemma "n == P ==> (~n | P) & (n | ~P)" by simp},
boehmes@36898
   367
    @{lemma "n == (if P then s else t) ==> (~P | n = s) & (P | n = t)"
boehmes@36898
   368
      by simp},
boehmes@36898
   369
    @{lemma "n == P ==> n = P" by (rule meta_eq_to_obj_eq)} ]
boehmes@36898
   370
boehmes@36898
   371
  val apply_rules = [
boehmes@36898
   372
    @{lemma "(~n | P) & (n | ~P) ==> P == n" by (atomize(full)) fast},
boehmes@36898
   373
    @{lemma "(~P | n = s) & (P | n = t) ==> (if P then s else t) == n"
boehmes@36898
   374
      by (atomize(full)) fastsimp} ]
boehmes@36898
   375
boehmes@41328
   376
  val inst_rule = Z3_Proof_Tools.match_instantiate Thm.dest_arg
boehmes@36898
   377
boehmes@36898
   378
  fun apply_rule ct =
boehmes@36898
   379
    (case get_first (try (inst_rule ct)) intro_rules of
boehmes@36898
   380
      SOME thm => thm
boehmes@36898
   381
    | NONE => raise CTERM ("intro_def", [ct]))
boehmes@36898
   382
in
boehmes@41328
   383
fun intro_def ct = Z3_Proof_Tools.make_hyp_def (apply_rule ct) #>> Thm
boehmes@36898
   384
boehmes@36898
   385
fun apply_def thm =
boehmes@36898
   386
  get_first (try (fn rule => MetaEq (thm COMP rule))) apply_rules
boehmes@36898
   387
  |> the_default (Thm thm)
boehmes@36898
   388
end
boehmes@36898
   389
boehmes@36898
   390
boehmes@36898
   391
(* negation normal form *)
boehmes@36898
   392
local
boehmes@36898
   393
  val quant_rules1 = ([
boehmes@36898
   394
    @{lemma "(!!x. P x == Q) ==> ALL x. P x == Q" by simp},
boehmes@36898
   395
    @{lemma "(!!x. P x == Q) ==> EX x. P x == Q" by simp}], [
boehmes@36898
   396
    @{lemma "(!!x. P x == Q x) ==> ALL x. P x == ALL x. Q x" by simp},
boehmes@36898
   397
    @{lemma "(!!x. P x == Q x) ==> EX x. P x == EX x. Q x" by simp}])
boehmes@36898
   398
boehmes@36898
   399
  val quant_rules2 = ([
boehmes@36898
   400
    @{lemma "(!!x. ~P x == Q) ==> ~(ALL x. P x) == Q" by simp},
boehmes@36898
   401
    @{lemma "(!!x. ~P x == Q) ==> ~(EX x. P x) == Q" by simp}], [
boehmes@36898
   402
    @{lemma "(!!x. ~P x == Q x) ==> ~(ALL x. P x) == EX x. Q x" by simp},
boehmes@36898
   403
    @{lemma "(!!x. ~P x == Q x) ==> ~(EX x. P x) == ALL x. Q x" by simp}])
boehmes@36898
   404
boehmes@36898
   405
  fun nnf_quant_tac thm (qs as (qs1, qs2)) i st = (
boehmes@36898
   406
    Tactic.rtac thm ORELSE'
boehmes@36898
   407
    (Tactic.match_tac qs1 THEN' nnf_quant_tac thm qs) ORELSE'
boehmes@36898
   408
    (Tactic.match_tac qs2 THEN' nnf_quant_tac thm qs)) i st
boehmes@36898
   409
boehmes@41328
   410
  fun nnf_quant_tac_varified vars eq =
boehmes@41328
   411
    nnf_quant_tac (Z3_Proof_Tools.varify vars eq)
boehmes@41328
   412
boehmes@36898
   413
  fun nnf_quant vars qs p ct =
boehmes@41328
   414
    Z3_Proof_Tools.as_meta_eq ct
boehmes@41328
   415
    |> Z3_Proof_Tools.by_tac (nnf_quant_tac_varified vars (meta_eq_of p) qs)
boehmes@36898
   416
boehmes@36898
   417
  fun prove_nnf ctxt = try_apply ctxt [] [
boehmes@41328
   418
    named ctxt "conj/disj" Z3_Proof_Literals.prove_conj_disj_eq,
boehmes@41328
   419
    Z3_Proof_Tools.by_abstraction (true, false) ctxt [] (fn ctxt' =>
boehmes@41328
   420
      named ctxt' "simp+fast" (Z3_Proof_Tools.by_tac simp_fast_tac))]
boehmes@36898
   421
in
boehmes@36898
   422
fun nnf ctxt vars ps ct =
boehmes@41328
   423
  (case SMT_Utils.term_of ct of
boehmes@36898
   424
    _ $ (l as Const _ $ Abs _) $ (r as Const _ $ Abs _) =>
boehmes@36898
   425
      if l aconv r
boehmes@36898
   426
      then MetaEq (Thm.reflexive (Thm.dest_arg (Thm.dest_arg ct)))
boehmes@36898
   427
      else MetaEq (nnf_quant vars quant_rules1 (hd ps) ct)
boehmes@40579
   428
  | _ $ (@{const Not} $ (Const _ $ Abs _)) $ (Const _ $ Abs _) =>
boehmes@36898
   429
      MetaEq (nnf_quant vars quant_rules2 (hd ps) ct)
boehmes@36898
   430
  | _ =>
boehmes@36898
   431
      let
boehmes@36898
   432
        val nnf_rewr_conv = Conv.arg_conv (Conv.arg_conv
boehmes@41328
   433
          (Z3_Proof_Tools.unfold_eqs ctxt
boehmes@41328
   434
            (map (Thm.symmetric o meta_eq_of) ps)))
boehmes@41328
   435
      in Thm (Z3_Proof_Tools.with_conv nnf_rewr_conv (prove_nnf ctxt) ct) end)
boehmes@36898
   436
end
boehmes@36898
   437
boehmes@36898
   438
boehmes@36898
   439
boehmes@36898
   440
(** equality proof rules **)
boehmes@36898
   441
boehmes@36898
   442
(* |- t = t *)
boehmes@36898
   443
fun refl ct = MetaEq (Thm.reflexive (Thm.dest_arg (Thm.dest_arg ct)))
boehmes@36898
   444
boehmes@36898
   445
boehmes@36898
   446
(* s = t ==> t = s *)
boehmes@36898
   447
local
boehmes@36898
   448
  val symm_rule = @{lemma "s = t ==> t == s" by simp}
boehmes@36898
   449
in
boehmes@36898
   450
fun symm (MetaEq thm) = MetaEq (Thm.symmetric thm)
boehmes@36898
   451
  | symm p = MetaEq (thm_of p COMP symm_rule)
boehmes@36898
   452
end
boehmes@36898
   453
boehmes@36898
   454
boehmes@36898
   455
(* s = t ==> t = u ==> s = u *)
boehmes@36898
   456
local
boehmes@36898
   457
  val trans1 = @{lemma "s == t ==> t =  u ==> s == u" by simp}
boehmes@36898
   458
  val trans2 = @{lemma "s =  t ==> t == u ==> s == u" by simp}
boehmes@36898
   459
  val trans3 = @{lemma "s =  t ==> t =  u ==> s == u" by simp}
boehmes@36898
   460
in
boehmes@36898
   461
fun trans (MetaEq thm1) (MetaEq thm2) = MetaEq (Thm.transitive thm1 thm2)
boehmes@36898
   462
  | trans (MetaEq thm) q = MetaEq (thm_of q COMP (thm COMP trans1))
boehmes@36898
   463
  | trans p (MetaEq thm) = MetaEq (thm COMP (thm_of p COMP trans2))
boehmes@36898
   464
  | trans p q = MetaEq (thm_of q COMP (thm_of p COMP trans3))
boehmes@36898
   465
end
boehmes@36898
   466
boehmes@36898
   467
boehmes@36898
   468
(* t1 = s1 ==> ... ==> tn = sn ==> f t1 ... tn = f s1 .. sn
boehmes@36898
   469
   (reflexive antecendents are droppped) *)
boehmes@36898
   470
local
boehmes@36898
   471
  exception MONO
boehmes@36898
   472
boehmes@36898
   473
  fun prove_refl (ct, _) = Thm.reflexive ct
boehmes@36898
   474
  fun prove_comb f g cp =
boehmes@36898
   475
    let val ((ct1, ct2), (cu1, cu2)) = pairself Thm.dest_comb cp
boehmes@36898
   476
    in Thm.combination (f (ct1, cu1)) (g (ct2, cu2)) end
boehmes@36898
   477
  fun prove_arg f = prove_comb prove_refl f
boehmes@36898
   478
boehmes@36898
   479
  fun prove f cp = prove_comb (prove f) f cp handle CTERM _ => prove_refl cp
boehmes@36898
   480
boehmes@36898
   481
  fun prove_nary is_comb f =
boehmes@36898
   482
    let
boehmes@36898
   483
      fun prove (cp as (ct, _)) = f cp handle MONO =>
boehmes@36898
   484
        if is_comb (Thm.term_of ct)
boehmes@36898
   485
        then prove_comb (prove_arg prove) prove cp
boehmes@36898
   486
        else prove_refl cp
boehmes@36898
   487
    in prove end
boehmes@36898
   488
boehmes@36898
   489
  fun prove_list f n cp =
boehmes@36898
   490
    if n = 0 then prove_refl cp
boehmes@36898
   491
    else prove_comb (prove_arg f) (prove_list f (n-1)) cp
boehmes@36898
   492
boehmes@36898
   493
  fun with_length f (cp as (cl, _)) =
boehmes@36898
   494
    f (length (HOLogic.dest_list (Thm.term_of cl))) cp
boehmes@36898
   495
boehmes@36898
   496
  fun prove_distinct f = prove_arg (with_length (prove_list f))
boehmes@36898
   497
boehmes@36898
   498
  fun prove_eq exn lookup cp =
boehmes@36898
   499
    (case lookup (Logic.mk_equals (pairself Thm.term_of cp)) of
boehmes@36898
   500
      SOME eq => eq
boehmes@36898
   501
    | NONE => if exn then raise MONO else prove_refl cp)
boehmes@36898
   502
  
boehmes@41328
   503
  val prove_exn = prove_eq true
boehmes@41328
   504
  and prove_safe = prove_eq false
boehmes@36898
   505
boehmes@36898
   506
  fun mono f (cp as (cl, _)) =
boehmes@36898
   507
    (case Term.head_of (Thm.term_of cl) of
boehmes@41328
   508
      @{const HOL.conj} => prove_nary Z3_Proof_Literals.is_conj (prove_exn f)
boehmes@41328
   509
    | @{const HOL.disj} => prove_nary Z3_Proof_Literals.is_disj (prove_exn f)
boehmes@41328
   510
    | Const (@{const_name distinct}, _) => prove_distinct (prove_safe f)
boehmes@41328
   511
    | _ => prove (prove_safe f)) cp
boehmes@36898
   512
in
boehmes@36898
   513
fun monotonicity eqs ct =
boehmes@36898
   514
  let
boehmes@40680
   515
    fun and_symmetric (t, thm) = [(t, thm), (t, Thm.symmetric thm)]
boehmes@40680
   516
    val teqs = maps (and_symmetric o `Thm.prop_of o meta_eq_of) eqs
boehmes@40680
   517
    val lookup = AList.lookup (op aconv) teqs
boehmes@36898
   518
    val cp = Thm.dest_binop (Thm.dest_arg ct)
boehmes@41328
   519
  in MetaEq (prove_exn lookup cp handle MONO => mono lookup cp) end
boehmes@36898
   520
end
boehmes@36898
   521
boehmes@36898
   522
boehmes@36898
   523
(* |- f a b = f b a (where f is equality) *)
boehmes@36898
   524
local
boehmes@36898
   525
  val rule = @{lemma "a = b == b = a" by (atomize(full)) (rule eq_commute)}
boehmes@36898
   526
in
boehmes@41328
   527
fun commutativity ct =
boehmes@41328
   528
  MetaEq (Z3_Proof_Tools.match_instantiate I
boehmes@41328
   529
    (Z3_Proof_Tools.as_meta_eq ct) rule)
boehmes@36898
   530
end
boehmes@36898
   531
boehmes@36898
   532
boehmes@36898
   533
boehmes@36898
   534
(** quantifier proof rules **)
boehmes@36898
   535
boehmes@36898
   536
(* P ?x = Q ?x ==> (ALL x. P x) = (ALL x. Q x)
boehmes@36898
   537
   P ?x = Q ?x ==> (EX x. P x) = (EX x. Q x)    *)
boehmes@36898
   538
local
boehmes@36898
   539
  val rules = [
boehmes@36898
   540
    @{lemma "(!!x. P x == Q x) ==> (ALL x. P x) == (ALL x. Q x)" by simp},
boehmes@36898
   541
    @{lemma "(!!x. P x == Q x) ==> (EX x. P x) == (EX x. Q x)" by simp}]
boehmes@36898
   542
in
boehmes@36898
   543
fun quant_intro vars p ct =
boehmes@36898
   544
  let
boehmes@36898
   545
    val thm = meta_eq_of p
boehmes@41328
   546
    val rules' = Z3_Proof_Tools.varify vars thm :: rules
boehmes@41328
   547
    val cu = Z3_Proof_Tools.as_meta_eq ct
boehmes@41328
   548
    val tac = REPEAT_ALL_NEW (Tactic.match_tac rules')
boehmes@41328
   549
  in MetaEq (Z3_Proof_Tools.by_tac tac cu) end
boehmes@36898
   550
end
boehmes@36898
   551
boehmes@36898
   552
boehmes@36898
   553
(* |- ((ALL x. P x) | Q) = (ALL x. P x | Q) *)
boehmes@36898
   554
fun pull_quant ctxt = Thm o try_apply ctxt [] [
boehmes@41328
   555
  named ctxt "fast" (Z3_Proof_Tools.by_tac (Classical.fast_tac HOL_cs))]
boehmes@36898
   556
    (* FIXME: not very well tested *)
boehmes@36898
   557
boehmes@36898
   558
boehmes@36898
   559
(* |- (ALL x. P x & Q x) = ((ALL x. P x) & (ALL x. Q x)) *)
boehmes@36898
   560
fun push_quant ctxt = Thm o try_apply ctxt [] [
boehmes@41328
   561
  named ctxt "fast" (Z3_Proof_Tools.by_tac (Classical.fast_tac HOL_cs))]
boehmes@36898
   562
    (* FIXME: not very well tested *)
boehmes@36898
   563
boehmes@36898
   564
boehmes@36898
   565
(* |- (ALL x1 ... xn y1 ... yn. P x1 ... xn) = (ALL x1 ... xn. P x1 ... xn) *)
boehmes@36898
   566
local
boehmes@36898
   567
  val elim_all = @{lemma "(ALL x. P) == P" by simp}
boehmes@36898
   568
  val elim_ex = @{lemma "(EX x. P) == P" by simp}
boehmes@36898
   569
boehmes@36898
   570
  fun elim_unused_conv ctxt =
boehmes@36898
   571
    Conv.params_conv ~1 (K (Conv.arg_conv (Conv.arg1_conv
wenzelm@36936
   572
      (Conv.rewrs_conv [elim_all, elim_ex])))) ctxt
boehmes@36898
   573
boehmes@36898
   574
  fun elim_unused_tac ctxt =
boehmes@36898
   575
    REPEAT_ALL_NEW (
boehmes@36898
   576
      Tactic.match_tac [@{thm refl}, @{thm iff_allI}, @{thm iff_exI}]
boehmes@36898
   577
      ORELSE' CONVERSION (elim_unused_conv ctxt))
boehmes@36898
   578
in
boehmes@41328
   579
fun elim_unused_vars ctxt = Thm o Z3_Proof_Tools.by_tac (elim_unused_tac ctxt)
boehmes@36898
   580
end
boehmes@36898
   581
boehmes@36898
   582
boehmes@36898
   583
(* |- (ALL x1 ... xn. ~(x1 = t1 & ... xn = tn) | P x1 ... xn) = P t1 ... tn *)
boehmes@36898
   584
fun dest_eq_res ctxt = Thm o try_apply ctxt [] [
boehmes@41328
   585
  named ctxt "fast" (Z3_Proof_Tools.by_tac (Classical.fast_tac HOL_cs))]
boehmes@36898
   586
    (* FIXME: not very well tested *)
boehmes@36898
   587
boehmes@36898
   588
boehmes@36898
   589
(* |- ~(ALL x1...xn. P x1...xn) | P a1...an *)
boehmes@36898
   590
local
boehmes@36898
   591
  val rule = @{lemma "~ P x | Q ==> ~(ALL x. P x) | Q" by fast}
boehmes@36898
   592
in
boehmes@41328
   593
val quant_inst = Thm o Z3_Proof_Tools.by_tac (
boehmes@36898
   594
  REPEAT_ALL_NEW (Tactic.match_tac [rule])
boehmes@36898
   595
  THEN' Tactic.rtac @{thm excluded_middle})
boehmes@36898
   596
end
boehmes@36898
   597
boehmes@36898
   598
boehmes@36898
   599
(* c = SOME x. P x |- (EX x. P x) = P c
boehmes@36898
   600
   c = SOME x. ~ P x |- ~(ALL x. P x) = ~ P c *)
boehmes@36898
   601
local
boehmes@36898
   602
  val elim_ex = @{lemma "EX x. P == P" by simp}
boehmes@36898
   603
  val elim_all = @{lemma "~ (ALL x. P) == ~P" by simp}
boehmes@36898
   604
  val sk_ex = @{lemma "c == SOME x. P x ==> EX x. P x == P c"
boehmes@36898
   605
    by simp (intro eq_reflection some_eq_ex[symmetric])}
boehmes@36898
   606
  val sk_all = @{lemma "c == SOME x. ~ P x ==> ~(ALL x. P x) == ~ P c"
boehmes@36898
   607
    by (simp only: not_all) (intro eq_reflection some_eq_ex[symmetric])}
boehmes@36898
   608
  val sk_ex_rule = ((sk_ex, I), elim_ex)
boehmes@36898
   609
  and sk_all_rule = ((sk_all, Thm.dest_arg), elim_all)
boehmes@36898
   610
boehmes@36898
   611
  fun dest f sk_rule = 
boehmes@36898
   612
    Thm.dest_comb (f (Thm.dest_arg (Thm.dest_arg (Thm.cprop_of sk_rule))))
boehmes@36898
   613
  fun type_of f sk_rule = Thm.ctyp_of_term (snd (dest f sk_rule))
boehmes@36898
   614
  fun pair2 (a, b) (c, d) = [(a, c), (b, d)]
boehmes@36898
   615
  fun inst_sk (sk_rule, f) p c =
boehmes@36898
   616
    Thm.instantiate ([(type_of f sk_rule, Thm.ctyp_of_term c)], []) sk_rule
boehmes@36898
   617
    |> (fn sk' => Thm.instantiate ([], (pair2 (dest f sk') (p, c))) sk')
boehmes@36898
   618
    |> Conv.fconv_rule (Thm.beta_conversion true)
boehmes@36898
   619
boehmes@36898
   620
  fun kind (Const (@{const_name Ex}, _) $ _) = (sk_ex_rule, I, I)
boehmes@40579
   621
    | kind (@{const Not} $ (Const (@{const_name All}, _) $ _)) =
boehmes@41328
   622
        (sk_all_rule, Thm.dest_arg, Z3_Proof_Literals.negate)
boehmes@36898
   623
    | kind t = raise TERM ("skolemize", [t])
boehmes@36898
   624
boehmes@36898
   625
  fun dest_abs_type (Abs (_, T, _)) = T
boehmes@36898
   626
    | dest_abs_type t = raise TERM ("dest_abs_type", [t])
boehmes@36898
   627
boehmes@36898
   628
  fun bodies_of thy lhs rhs =
boehmes@36898
   629
    let
boehmes@36898
   630
      val (rule, dest, make) = kind (Thm.term_of lhs)
boehmes@36898
   631
boehmes@36898
   632
      fun dest_body idx cbs ct =
boehmes@36898
   633
        let
boehmes@36898
   634
          val cb = Thm.dest_arg (dest ct)
boehmes@36898
   635
          val T = dest_abs_type (Thm.term_of cb)
boehmes@36898
   636
          val cv = Thm.cterm_of thy (Var (("x", idx), T))
boehmes@36898
   637
          val cu = make (Drule.beta_conv cb cv)
boehmes@36898
   638
          val cbs' = (cv, cb) :: cbs
boehmes@36898
   639
        in
boehmes@36898
   640
          (snd (Thm.first_order_match (cu, rhs)), rev cbs')
boehmes@36898
   641
          handle Pattern.MATCH => dest_body (idx+1) cbs' cu
boehmes@36898
   642
        end
boehmes@36898
   643
    in (rule, dest_body 1 [] lhs) end
boehmes@36898
   644
boehmes@36898
   645
  fun transitive f thm = Thm.transitive thm (f (Thm.rhs_of thm))
boehmes@36898
   646
boehmes@36898
   647
  fun sk_step (rule, elim) (cv, mct, cb) ((is, thm), ctxt) =
boehmes@36898
   648
    (case mct of
boehmes@36898
   649
      SOME ct =>
boehmes@36898
   650
        ctxt
boehmes@41328
   651
        |> Z3_Proof_Tools.make_hyp_def
boehmes@41328
   652
             (inst_sk rule (Thm.instantiate_cterm ([], is) cb) ct)
boehmes@36898
   653
        |>> pair ((cv, ct) :: is) o Thm.transitive thm
boehmes@36898
   654
    | NONE => ((is, transitive (Conv.rewr_conv elim) thm), ctxt))
boehmes@36898
   655
in
boehmes@36898
   656
fun skolemize ct ctxt =
boehmes@36898
   657
  let
boehmes@36898
   658
    val (lhs, rhs) = Thm.dest_binop (Thm.dest_arg ct)
boehmes@36898
   659
    val (rule, (ctab, cbs)) = bodies_of (ProofContext.theory_of ctxt) lhs rhs
boehmes@36898
   660
    fun lookup_var (cv, cb) = (cv, AList.lookup (op aconvc) ctab cv, cb)
boehmes@36898
   661
  in
boehmes@36898
   662
    (([], Thm.reflexive lhs), ctxt)
boehmes@36898
   663
    |> fold (sk_step rule) (map lookup_var cbs)
boehmes@36898
   664
    |>> MetaEq o snd
boehmes@36898
   665
  end
boehmes@36898
   666
end
boehmes@36898
   667
boehmes@36898
   668
boehmes@36898
   669
(** theory proof rules **)
boehmes@36898
   670
boehmes@36898
   671
(* theory lemmas: linear arithmetic, arrays *)
boehmes@36898
   672
fun th_lemma ctxt simpset thms = Thm o try_apply ctxt thms [
boehmes@41328
   673
  Z3_Proof_Tools.by_abstraction (false, true) ctxt thms (fn ctxt' =>
boehmes@41328
   674
    Z3_Proof_Tools.by_tac (
boehmes@41328
   675
      NAMED ctxt' "arith" (Arith_Data.arith_tac ctxt')
boehmes@41328
   676
      ORELSE' NAMED ctxt' "simp+arith" (
boehmes@41328
   677
        Simplifier.simp_tac simpset
boehmes@41328
   678
        THEN_ALL_NEW Arith_Data.arith_tac ctxt')))]
boehmes@36898
   679
boehmes@36898
   680
boehmes@36898
   681
(* rewriting: prove equalities:
boehmes@36898
   682
     * ACI of conjunction/disjunction
boehmes@36898
   683
     * contradiction, excluded middle
boehmes@36898
   684
     * logical rewriting rules (for negation, implication, equivalence,
boehmes@36898
   685
         distinct)
boehmes@36898
   686
     * normal forms for polynoms (integer/real arithmetic)
boehmes@36898
   687
     * quantifier elimination over linear arithmetic
boehmes@36898
   688
     * ... ? **)
boehmes@36898
   689
structure Z3_Simps = Named_Thms
boehmes@36898
   690
(
boehmes@36898
   691
  val name = "z3_simp"
boehmes@36898
   692
  val description = "simplification rules for Z3 proof reconstruction"
boehmes@36898
   693
)
boehmes@36898
   694
boehmes@36898
   695
local
boehmes@36898
   696
  fun spec_meta_eq_of thm =
boehmes@36898
   697
    (case try (fn th => th RS @{thm spec}) thm of
boehmes@36898
   698
      SOME thm' => spec_meta_eq_of thm'
boehmes@36898
   699
    | NONE => mk_meta_eq thm)
boehmes@36898
   700
boehmes@36898
   701
  fun prep (Thm thm) = spec_meta_eq_of thm
boehmes@36898
   702
    | prep (MetaEq thm) = thm
boehmes@36898
   703
    | prep (Literals (thm, _)) = spec_meta_eq_of thm
boehmes@36898
   704
boehmes@36898
   705
  fun unfold_conv ctxt ths =
boehmes@41328
   706
    Conv.arg_conv (Conv.binop_conv (Z3_Proof_Tools.unfold_eqs ctxt
boehmes@41328
   707
      (map prep ths)))
boehmes@36898
   708
boehmes@36898
   709
  fun with_conv _ [] prv = prv
boehmes@41328
   710
    | with_conv ctxt ths prv =
boehmes@41328
   711
        Z3_Proof_Tools.with_conv (unfold_conv ctxt ths) prv
boehmes@36898
   712
boehmes@36898
   713
  val unfold_conv =
boehmes@41328
   714
    Conv.arg_conv (Conv.binop_conv
boehmes@41328
   715
      (Conv.try_conv Z3_Proof_Tools.unfold_distinct_conv))
boehmes@41328
   716
  val prove_conj_disj_eq =
boehmes@41328
   717
    Z3_Proof_Tools.with_conv unfold_conv Z3_Proof_Literals.prove_conj_disj_eq
boehmes@40663
   718
boehmes@40663
   719
  fun assume_prems ctxt thm =
boehmes@40663
   720
    Assumption.add_assumes (Drule.cprems_of thm) ctxt
boehmes@40663
   721
    |>> (fn thms => fold Thm.elim_implies thms thm)
boehmes@36898
   722
in
boehmes@36898
   723
boehmes@40663
   724
fun rewrite simpset ths ct ctxt =
boehmes@40663
   725
  apfst Thm (assume_prems ctxt (with_conv ctxt ths (try_apply ctxt [] [
boehmes@40663
   726
    named ctxt "conj/disj/distinct" prove_conj_disj_eq,
boehmes@41328
   727
    Z3_Proof_Tools.by_abstraction (true, false) ctxt [] (fn ctxt' =>
boehmes@41328
   728
      Z3_Proof_Tools.by_tac (
boehmes@41328
   729
        NAMED ctxt' "simp (logic)" (Simplifier.simp_tac simpset)
boehmes@41328
   730
        THEN_ALL_NEW NAMED ctxt' "fast (logic)" (Classical.fast_tac HOL_cs))),
boehmes@41328
   731
    Z3_Proof_Tools.by_abstraction (false, true) ctxt [] (fn ctxt' =>
boehmes@41328
   732
      Z3_Proof_Tools.by_tac (
boehmes@41328
   733
        NAMED ctxt' "simp (theory)" (Simplifier.simp_tac simpset)
boehmes@41328
   734
        THEN_ALL_NEW (
boehmes@41328
   735
          NAMED ctxt' "fast (theory)" (Classical.fast_tac HOL_cs)
boehmes@41328
   736
          ORELSE' NAMED ctxt' "arith (theory)" (Arith_Data.arith_tac ctxt')))),
boehmes@41328
   737
    Z3_Proof_Tools.by_abstraction (true, true) ctxt [] (fn ctxt' =>
boehmes@41328
   738
      Z3_Proof_Tools.by_tac (
boehmes@41328
   739
        NAMED ctxt' "simp (full)" (Simplifier.simp_tac simpset)
boehmes@41328
   740
        THEN_ALL_NEW (
boehmes@41328
   741
          NAMED ctxt' "fast (full)" (Classical.fast_tac HOL_cs)
boehmes@41328
   742
          ORELSE' NAMED ctxt' "arith (full)" (Arith_Data.arith_tac ctxt')))),
boehmes@41328
   743
    named ctxt "injectivity" (Z3_Proof_Methods.prove_injectivity ctxt)]) ct))
boehmes@36898
   744
boehmes@36898
   745
end
boehmes@36898
   746
boehmes@36898
   747
boehmes@36898
   748
boehmes@41130
   749
(* proof reconstruction *)
boehmes@36898
   750
boehmes@41130
   751
(** tracing and checking **)
boehmes@36898
   752
boehmes@41130
   753
fun trace_before ctxt idx = SMT_Config.trace_msg ctxt (fn r =>
boehmes@41328
   754
  "Z3: #" ^ string_of_int idx ^ ": " ^ Z3_Proof_Parser.string_of_rule r)
boehmes@36898
   755
boehmes@41130
   756
fun check_after idx r ps ct (p, (ctxt, _)) =
boehmes@41130
   757
  if not (Config.get ctxt SMT_Config.trace) then ()
boehmes@41130
   758
  else
boehmes@36898
   759
    let val thm = thm_of p |> tap (Thm.join_proofs o single)
boehmes@36898
   760
    in
boehmes@36898
   761
      if (Thm.cprop_of thm) aconvc ct then ()
boehmes@41328
   762
      else
boehmes@41328
   763
        z3_exn (Pretty.string_of (Pretty.big_list
boehmes@41328
   764
          ("proof step failed: " ^ quote (Z3_Proof_Parser.string_of_rule r) ^
boehmes@41328
   765
            " (#" ^ string_of_int idx ^ ")")
boehmes@36898
   766
          (pretty_goal ctxt (map (thm_of o fst) ps) (Thm.prop_of thm) @
boehmes@41328
   767
            [Pretty.block [Pretty.str "expected: ",
boehmes@41328
   768
              Syntax.pretty_term ctxt (Thm.term_of ct)]])))
boehmes@36898
   769
    end
boehmes@36898
   770
boehmes@36898
   771
boehmes@41130
   772
(** overall reconstruction procedure **)
boehmes@36898
   773
boehmes@40164
   774
local
boehmes@40164
   775
  fun not_supported r = raise Fail ("Z3: proof rule not implemented: " ^
boehmes@41328
   776
    quote (Z3_Proof_Parser.string_of_rule r))
boehmes@36898
   777
boehmes@41131
   778
  fun prove_step simpset vars r ps ct (cxp as (cx, ptab)) =
boehmes@40164
   779
    (case (r, ps) of
boehmes@40164
   780
      (* core rules *)
boehmes@41328
   781
      (Z3_Proof_Parser.True_Axiom, _) => (Thm Z3_Proof_Literals.true_thm, cxp)
boehmes@41328
   782
    | (Z3_Proof_Parser.Asserted, _) => raise Fail "bad assertion"
boehmes@41328
   783
    | (Z3_Proof_Parser.Goal, _) => raise Fail "bad assertion"
boehmes@41328
   784
    | (Z3_Proof_Parser.Modus_Ponens, [(p, _), (q, _)]) =>
boehmes@41328
   785
        (mp q (thm_of p), cxp)
boehmes@41328
   786
    | (Z3_Proof_Parser.Modus_Ponens_Oeq, [(p, _), (q, _)]) =>
boehmes@41328
   787
        (mp q (thm_of p), cxp)
boehmes@41328
   788
    | (Z3_Proof_Parser.And_Elim, [(p, i)]) =>
boehmes@41328
   789
        and_elim (p, i) ct ptab ||> pair cx
boehmes@41328
   790
    | (Z3_Proof_Parser.Not_Or_Elim, [(p, i)]) =>
boehmes@41328
   791
        not_or_elim (p, i) ct ptab ||> pair cx
boehmes@41328
   792
    | (Z3_Proof_Parser.Hypothesis, _) => (Thm (Thm.assume ct), cxp)
boehmes@41328
   793
    | (Z3_Proof_Parser.Lemma, [(p, _)]) => (lemma (thm_of p) ct, cxp)
boehmes@41328
   794
    | (Z3_Proof_Parser.Unit_Resolution, (p, _) :: ps) =>
boehmes@40164
   795
        (unit_resolution (thm_of p) (map (thm_of o fst) ps) ct, cxp)
boehmes@41328
   796
    | (Z3_Proof_Parser.Iff_True, [(p, _)]) => (iff_true (thm_of p), cxp)
boehmes@41328
   797
    | (Z3_Proof_Parser.Iff_False, [(p, _)]) => (iff_false (thm_of p), cxp)
boehmes@41328
   798
    | (Z3_Proof_Parser.Distributivity, _) => (distributivity cx ct, cxp)
boehmes@41328
   799
    | (Z3_Proof_Parser.Def_Axiom, _) => (def_axiom cx ct, cxp)
boehmes@41328
   800
    | (Z3_Proof_Parser.Intro_Def, _) => intro_def ct cx ||> rpair ptab
boehmes@41328
   801
    | (Z3_Proof_Parser.Apply_Def, [(p, _)]) => (apply_def (thm_of p), cxp)
boehmes@41328
   802
    | (Z3_Proof_Parser.Iff_Oeq, [(p, _)]) => (p, cxp)
boehmes@41328
   803
    | (Z3_Proof_Parser.Nnf_Pos, _) => (nnf cx vars (map fst ps) ct, cxp)
boehmes@41328
   804
    | (Z3_Proof_Parser.Nnf_Neg, _) => (nnf cx vars (map fst ps) ct, cxp)
boehmes@36898
   805
boehmes@40164
   806
      (* equality rules *)
boehmes@41328
   807
    | (Z3_Proof_Parser.Reflexivity, _) => (refl ct, cxp)
boehmes@41328
   808
    | (Z3_Proof_Parser.Symmetry, [(p, _)]) => (symm p, cxp)
boehmes@41328
   809
    | (Z3_Proof_Parser.Transitivity, [(p, _), (q, _)]) => (trans p q, cxp)
boehmes@41328
   810
    | (Z3_Proof_Parser.Monotonicity, _) => (monotonicity (map fst ps) ct, cxp)
boehmes@41328
   811
    | (Z3_Proof_Parser.Commutativity, _) => (commutativity ct, cxp)
boehmes@40164
   812
boehmes@40164
   813
      (* quantifier rules *)
boehmes@41328
   814
    | (Z3_Proof_Parser.Quant_Intro, [(p, _)]) => (quant_intro vars p ct, cxp)
boehmes@41328
   815
    | (Z3_Proof_Parser.Pull_Quant, _) => (pull_quant cx ct, cxp)
boehmes@41328
   816
    | (Z3_Proof_Parser.Push_Quant, _) => (push_quant cx ct, cxp)
boehmes@41328
   817
    | (Z3_Proof_Parser.Elim_Unused_Vars, _) => (elim_unused_vars cx ct, cxp)
boehmes@41328
   818
    | (Z3_Proof_Parser.Dest_Eq_Res, _) => (dest_eq_res cx ct, cxp)
boehmes@41328
   819
    | (Z3_Proof_Parser.Quant_Inst, _) => (quant_inst ct, cxp)
boehmes@41328
   820
    | (Z3_Proof_Parser.Skolemize, _) => skolemize ct cx ||> rpair ptab
boehmes@40164
   821
boehmes@40164
   822
      (* theory rules *)
boehmes@41328
   823
    | (Z3_Proof_Parser.Th_Lemma _, _) =>  (* FIXME: use arguments *)
boehmes@40164
   824
        (th_lemma cx simpset (map (thm_of o fst) ps) ct, cxp)
boehmes@41328
   825
    | (Z3_Proof_Parser.Rewrite, _) => rewrite simpset [] ct cx ||> rpair ptab
boehmes@41328
   826
    | (Z3_Proof_Parser.Rewrite_Star, ps) =>
boehmes@41328
   827
        rewrite simpset (map fst ps) ct cx ||> rpair ptab
boehmes@36898
   828
boehmes@41328
   829
    | (Z3_Proof_Parser.Nnf_Star, _) => not_supported r
boehmes@41328
   830
    | (Z3_Proof_Parser.Cnf_Star, _) => not_supported r
boehmes@41328
   831
    | (Z3_Proof_Parser.Transitivity_Star, _) => not_supported r
boehmes@41328
   832
    | (Z3_Proof_Parser.Pull_Quant_Star, _) => not_supported r
boehmes@36898
   833
boehmes@41328
   834
    | _ => raise Fail ("Z3: proof rule " ^
boehmes@41328
   835
        quote (Z3_Proof_Parser.string_of_rule r) ^
boehmes@41328
   836
        " has an unexpected number of arguments."))
boehmes@36898
   837
boehmes@41130
   838
  fun lookup_proof ptab idx =
boehmes@41130
   839
    (case Inttab.lookup ptab idx of
boehmes@41130
   840
      SOME p => (p, idx)
boehmes@41130
   841
    | NONE => z3_exn ("unknown proof id: " ^ quote (string_of_int idx)))
boehmes@41130
   842
boehmes@41131
   843
  fun prove simpset vars (idx, step) (_, cxp as (ctxt, ptab)) =
boehmes@40164
   844
    let
boehmes@41328
   845
      val Z3_Proof_Parser.Proof_Step {rule=r, prems, prop, ...} = step
boehmes@41130
   846
      val ps = map (lookup_proof ptab) prems
boehmes@41130
   847
      val _ = trace_before ctxt idx r
boehmes@41130
   848
      val (thm, (ctxt', ptab')) =
boehmes@41130
   849
        cxp
boehmes@41131
   850
        |> prove_step simpset vars r ps prop
boehmes@41130
   851
        |> tap (check_after idx r ps prop)
boehmes@41130
   852
    in (thm, (ctxt', Inttab.update (idx, thm) ptab')) end
boehmes@36898
   853
boehmes@41131
   854
  val disch_rules = [@{thm allI}, @{thm refl}, @{thm reflexive}]
boehmes@41131
   855
  fun all_disch_rules rules = map (pair false) (disch_rules @ rules)
boehmes@41127
   856
boehmes@41131
   857
  fun disch_assm rules thm =
boehmes@41127
   858
    if Thm.nprems_of thm = 0 then Drule.flexflex_unique thm
boehmes@41127
   859
    else
boehmes@41131
   860
      (case Seq.pull (Thm.biresolution false rules 1 thm) of
boehmes@41131
   861
        SOME (thm', _) => disch_assm rules thm'
boehmes@41127
   862
      | NONE => raise THM ("failed to discharge premise", 1, [thm]))
boehmes@41127
   863
boehmes@41131
   864
  fun discharge rules outer_ctxt (p, (inner_ctxt, _)) =
boehmes@41130
   865
    thm_of p
boehmes@41127
   866
    |> singleton (ProofContext.export inner_ctxt outer_ctxt)
boehmes@41131
   867
    |> disch_assm rules
boehmes@40164
   868
in
boehmes@40164
   869
boehmes@41127
   870
fun reconstruct outer_ctxt recon output =
boehmes@40164
   871
  let
boehmes@41127
   872
    val {context=ctxt, typs, terms, rewrite_rules, assms} = recon
boehmes@41328
   873
    val (asserted, steps, vars, ctxt1) =
boehmes@41328
   874
      Z3_Proof_Parser.parse ctxt typs terms output
boehmes@41131
   875
boehmes@41328
   876
    val simpset = Z3_Proof_Tools.make_simpset ctxt1 (Z3_Simps.get ctxt1)
boehmes@41131
   877
boehmes@41131
   878
    val ((is, rules), cxp as (ctxt2, _)) =
boehmes@41131
   879
      add_asserted outer_ctxt rewrite_rules assms asserted ctxt1
boehmes@36898
   880
  in
boehmes@41131
   881
    if Config.get ctxt2 SMT_Config.filter_only_facts then (is, @{thm TrueI})
boehmes@41127
   882
    else
boehmes@41131
   883
      (Thm @{thm TrueI}, cxp)
boehmes@41131
   884
      |> fold (prove simpset vars) steps 
boehmes@41131
   885
      |> discharge (all_disch_rules rules) outer_ctxt
boehmes@41127
   886
      |> pair []
boehmes@36898
   887
  end
boehmes@36898
   888
boehmes@40164
   889
end
boehmes@36898
   890
boehmes@40164
   891
val setup = z3_rules_setup #> Z3_Simps.setup
boehmes@36898
   892
boehmes@36898
   893
end