src/Doc/IsarRef/Generic.thy
author wenzelm
Sat Nov 10 20:16:16 2012 +0100 (2012-11-10)
changeset 50079 5c36db9db335
parent 50077 1edd0db7b6c4
child 50080 200f749c96db
permissions -rw-r--r--
updated subgoaler/solver/looper;
wenzelm@26782
     1
theory Generic
wenzelm@42651
     2
imports Base Main
wenzelm@26782
     3
begin
wenzelm@26782
     4
wenzelm@26782
     5
chapter {* Generic tools and packages \label{ch:gen-tools} *}
wenzelm@26782
     6
wenzelm@42655
     7
section {* Configuration options \label{sec:config} *}
wenzelm@26782
     8
wenzelm@40291
     9
text {* Isabelle/Pure maintains a record of named configuration
wenzelm@40291
    10
  options within the theory or proof context, with values of type
wenzelm@40291
    11
  @{ML_type bool}, @{ML_type int}, @{ML_type real}, or @{ML_type
wenzelm@40291
    12
  string}.  Tools may declare options in ML, and then refer to these
wenzelm@40291
    13
  values (relative to the context).  Thus global reference variables
wenzelm@40291
    14
  are easily avoided.  The user may change the value of a
wenzelm@40291
    15
  configuration option by means of an associated attribute of the same
wenzelm@40291
    16
  name.  This form of context declaration works particularly well with
wenzelm@42655
    17
  commands such as @{command "declare"} or @{command "using"} like
wenzelm@42655
    18
  this:
wenzelm@42655
    19
*}
wenzelm@42655
    20
wenzelm@42655
    21
declare [[show_main_goal = false]]
wenzelm@26782
    22
wenzelm@42655
    23
notepad
wenzelm@42655
    24
begin
wenzelm@42655
    25
  note [[show_main_goal = true]]
wenzelm@42655
    26
end
wenzelm@42655
    27
wenzelm@42655
    28
text {* For historical reasons, some tools cannot take the full proof
wenzelm@26782
    29
  context into account and merely refer to the background theory.
wenzelm@26782
    30
  This is accommodated by configuration options being declared as
wenzelm@26782
    31
  ``global'', which may not be changed within a local context.
wenzelm@26782
    32
wenzelm@26782
    33
  \begin{matharray}{rcll}
wenzelm@28761
    34
    @{command_def "print_configs"} & : & @{text "context \<rightarrow>"} \\
wenzelm@26782
    35
  \end{matharray}
wenzelm@26782
    36
wenzelm@42596
    37
  @{rail "
wenzelm@42596
    38
    @{syntax name} ('=' ('true' | 'false' | @{syntax int} | @{syntax float} | @{syntax name}))?
wenzelm@42596
    39
  "}
wenzelm@26782
    40
wenzelm@28760
    41
  \begin{description}
wenzelm@26782
    42
  
wenzelm@28760
    43
  \item @{command "print_configs"} prints the available configuration
wenzelm@28760
    44
  options, with names, types, and current values.
wenzelm@26782
    45
  
wenzelm@28760
    46
  \item @{text "name = value"} as an attribute expression modifies the
wenzelm@28760
    47
  named option, with the syntax of the value depending on the option's
wenzelm@28760
    48
  type.  For @{ML_type bool} the default value is @{text true}.  Any
wenzelm@28760
    49
  attempt to change a global option in a local context is ignored.
wenzelm@26782
    50
wenzelm@28760
    51
  \end{description}
wenzelm@26782
    52
*}
wenzelm@26782
    53
wenzelm@26782
    54
wenzelm@27040
    55
section {* Basic proof tools *}
wenzelm@26782
    56
wenzelm@26782
    57
subsection {* Miscellaneous methods and attributes \label{sec:misc-meth-att} *}
wenzelm@26782
    58
wenzelm@26782
    59
text {*
wenzelm@26782
    60
  \begin{matharray}{rcl}
wenzelm@28761
    61
    @{method_def unfold} & : & @{text method} \\
wenzelm@28761
    62
    @{method_def fold} & : & @{text method} \\
wenzelm@28761
    63
    @{method_def insert} & : & @{text method} \\[0.5ex]
wenzelm@28761
    64
    @{method_def erule}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
    65
    @{method_def drule}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
    66
    @{method_def frule}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@43365
    67
    @{method_def intro} & : & @{text method} \\
wenzelm@43365
    68
    @{method_def elim} & : & @{text method} \\
wenzelm@28761
    69
    @{method_def succeed} & : & @{text method} \\
wenzelm@28761
    70
    @{method_def fail} & : & @{text method} \\
wenzelm@26782
    71
  \end{matharray}
wenzelm@26782
    72
wenzelm@42596
    73
  @{rail "
wenzelm@42596
    74
    (@@{method fold} | @@{method unfold} | @@{method insert}) @{syntax thmrefs}
wenzelm@26782
    75
    ;
wenzelm@42596
    76
    (@@{method erule} | @@{method drule} | @@{method frule})
wenzelm@42596
    77
      ('(' @{syntax nat} ')')? @{syntax thmrefs}
wenzelm@43365
    78
    ;
wenzelm@43365
    79
    (@@{method intro} | @@{method elim}) @{syntax thmrefs}?
wenzelm@42596
    80
  "}
wenzelm@26782
    81
wenzelm@28760
    82
  \begin{description}
wenzelm@26782
    83
  
wenzelm@28760
    84
  \item @{method unfold}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{method fold}~@{text
wenzelm@28760
    85
  "a\<^sub>1 \<dots> a\<^sub>n"} expand (or fold back) the given definitions throughout
wenzelm@28760
    86
  all goals; any chained facts provided are inserted into the goal and
wenzelm@28760
    87
  subject to rewriting as well.
wenzelm@26782
    88
wenzelm@28760
    89
  \item @{method insert}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} inserts theorems as facts
wenzelm@28760
    90
  into all goals of the proof state.  Note that current facts
wenzelm@28760
    91
  indicated for forward chaining are ignored.
wenzelm@26782
    92
wenzelm@30397
    93
  \item @{method erule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, @{method
wenzelm@30397
    94
  drule}~@{text "a\<^sub>1 \<dots> a\<^sub>n"}, and @{method frule}~@{text
wenzelm@30397
    95
  "a\<^sub>1 \<dots> a\<^sub>n"} are similar to the basic @{method rule}
wenzelm@30397
    96
  method (see \secref{sec:pure-meth-att}), but apply rules by
wenzelm@30397
    97
  elim-resolution, destruct-resolution, and forward-resolution,
wenzelm@30397
    98
  respectively \cite{isabelle-implementation}.  The optional natural
wenzelm@30397
    99
  number argument (default 0) specifies additional assumption steps to
wenzelm@30397
   100
  be performed here.
wenzelm@26782
   101
wenzelm@26782
   102
  Note that these methods are improper ones, mainly serving for
wenzelm@26782
   103
  experimentation and tactic script emulation.  Different modes of
wenzelm@26782
   104
  basic rule application are usually expressed in Isar at the proof
wenzelm@26782
   105
  language level, rather than via implicit proof state manipulations.
wenzelm@26782
   106
  For example, a proper single-step elimination would be done using
wenzelm@26782
   107
  the plain @{method rule} method, with forward chaining of current
wenzelm@26782
   108
  facts.
wenzelm@26782
   109
wenzelm@43365
   110
  \item @{method intro} and @{method elim} repeatedly refine some goal
wenzelm@43365
   111
  by intro- or elim-resolution, after having inserted any chained
wenzelm@43365
   112
  facts.  Exactly the rules given as arguments are taken into account;
wenzelm@43365
   113
  this allows fine-tuned decomposition of a proof problem, in contrast
wenzelm@43365
   114
  to common automated tools.
wenzelm@43365
   115
wenzelm@28760
   116
  \item @{method succeed} yields a single (unchanged) result; it is
wenzelm@26782
   117
  the identity of the ``@{text ","}'' method combinator (cf.\
wenzelm@28754
   118
  \secref{sec:proof-meth}).
wenzelm@26782
   119
wenzelm@28760
   120
  \item @{method fail} yields an empty result sequence; it is the
wenzelm@26782
   121
  identity of the ``@{text "|"}'' method combinator (cf.\
wenzelm@28754
   122
  \secref{sec:proof-meth}).
wenzelm@26782
   123
wenzelm@28760
   124
  \end{description}
wenzelm@26782
   125
wenzelm@26782
   126
  \begin{matharray}{rcl}
wenzelm@28761
   127
    @{attribute_def tagged} & : & @{text attribute} \\
wenzelm@28761
   128
    @{attribute_def untagged} & : & @{text attribute} \\[0.5ex]
wenzelm@28761
   129
    @{attribute_def THEN} & : & @{text attribute} \\
wenzelm@28761
   130
    @{attribute_def unfolded} & : & @{text attribute} \\
wenzelm@47497
   131
    @{attribute_def folded} & : & @{text attribute} \\
wenzelm@47497
   132
    @{attribute_def abs_def} & : & @{text attribute} \\[0.5ex]
wenzelm@28761
   133
    @{attribute_def rotated} & : & @{text attribute} \\
wenzelm@28761
   134
    @{attribute_def (Pure) elim_format} & : & @{text attribute} \\
wenzelm@28761
   135
    @{attribute_def standard}@{text "\<^sup>*"} & : & @{text attribute} \\
wenzelm@28761
   136
    @{attribute_def no_vars}@{text "\<^sup>*"} & : & @{text attribute} \\
wenzelm@26782
   137
  \end{matharray}
wenzelm@26782
   138
wenzelm@42596
   139
  @{rail "
wenzelm@42596
   140
    @@{attribute tagged} @{syntax name} @{syntax name}
wenzelm@26782
   141
    ;
wenzelm@42596
   142
    @@{attribute untagged} @{syntax name}
wenzelm@26782
   143
    ;
wenzelm@48205
   144
    @@{attribute THEN} ('[' @{syntax nat} ']')? @{syntax thmref}
wenzelm@26782
   145
    ;
wenzelm@42596
   146
    (@@{attribute unfolded} | @@{attribute folded}) @{syntax thmrefs}
wenzelm@26782
   147
    ;
wenzelm@42596
   148
    @@{attribute rotated} @{syntax int}?
wenzelm@42596
   149
  "}
wenzelm@26782
   150
wenzelm@28760
   151
  \begin{description}
wenzelm@26782
   152
wenzelm@28764
   153
  \item @{attribute tagged}~@{text "name value"} and @{attribute
wenzelm@28760
   154
  untagged}~@{text name} add and remove \emph{tags} of some theorem.
wenzelm@26782
   155
  Tags may be any list of string pairs that serve as formal comment.
wenzelm@28764
   156
  The first string is considered the tag name, the second its value.
wenzelm@28764
   157
  Note that @{attribute untagged} removes any tags of the same name.
wenzelm@26782
   158
wenzelm@48205
   159
  \item @{attribute THEN}~@{text a} composes rules by resolution; it
wenzelm@48205
   160
  resolves with the first premise of @{text a} (an alternative
wenzelm@48205
   161
  position may be also specified).  See also @{ML_op "RS"} in
wenzelm@48205
   162
  \cite{isabelle-implementation}.
wenzelm@26782
   163
  
wenzelm@28760
   164
  \item @{attribute unfolded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} and @{attribute
wenzelm@28760
   165
  folded}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} expand and fold back again the given
wenzelm@28760
   166
  definitions throughout a rule.
wenzelm@26782
   167
wenzelm@47497
   168
  \item @{attribute abs_def} turns an equation of the form @{prop "f x
wenzelm@47497
   169
  y \<equiv> t"} into @{prop "f \<equiv> \<lambda>x y. t"}, which ensures that @{method
wenzelm@47497
   170
  simp} or @{method unfold} steps always expand it.  This also works
wenzelm@47497
   171
  for object-logic equality.
wenzelm@47497
   172
wenzelm@28760
   173
  \item @{attribute rotated}~@{text n} rotate the premises of a
wenzelm@26782
   174
  theorem by @{text n} (default 1).
wenzelm@26782
   175
wenzelm@28760
   176
  \item @{attribute (Pure) elim_format} turns a destruction rule into
wenzelm@28760
   177
  elimination rule format, by resolving with the rule @{prop "PROP A \<Longrightarrow>
wenzelm@28760
   178
  (PROP A \<Longrightarrow> PROP B) \<Longrightarrow> PROP B"}.
wenzelm@26782
   179
  
wenzelm@26782
   180
  Note that the Classical Reasoner (\secref{sec:classical}) provides
wenzelm@26782
   181
  its own version of this operation.
wenzelm@26782
   182
wenzelm@28760
   183
  \item @{attribute standard} puts a theorem into the standard form of
wenzelm@28760
   184
  object-rules at the outermost theory level.  Note that this
wenzelm@26782
   185
  operation violates the local proof context (including active
wenzelm@26782
   186
  locales).
wenzelm@26782
   187
wenzelm@28760
   188
  \item @{attribute no_vars} replaces schematic variables by free
wenzelm@26782
   189
  ones; this is mainly for tuning output of pretty printed theorems.
wenzelm@26782
   190
wenzelm@28760
   191
  \end{description}
wenzelm@26782
   192
*}
wenzelm@26782
   193
wenzelm@26782
   194
wenzelm@27044
   195
subsection {* Low-level equational reasoning *}
wenzelm@27044
   196
wenzelm@27044
   197
text {*
wenzelm@27044
   198
  \begin{matharray}{rcl}
wenzelm@28761
   199
    @{method_def subst} & : & @{text method} \\
wenzelm@28761
   200
    @{method_def hypsubst} & : & @{text method} \\
wenzelm@28761
   201
    @{method_def split} & : & @{text method} \\
wenzelm@27044
   202
  \end{matharray}
wenzelm@27044
   203
wenzelm@42596
   204
  @{rail "
wenzelm@42704
   205
    @@{method subst} ('(' 'asm' ')')? \\ ('(' (@{syntax nat}+) ')')? @{syntax thmref}
wenzelm@27044
   206
    ;
wenzelm@44094
   207
    @@{method split} @{syntax thmrefs}
wenzelm@42596
   208
  "}
wenzelm@27044
   209
wenzelm@27044
   210
  These methods provide low-level facilities for equational reasoning
wenzelm@27044
   211
  that are intended for specialized applications only.  Normally,
wenzelm@27044
   212
  single step calculations would be performed in a structured text
wenzelm@27044
   213
  (see also \secref{sec:calculation}), while the Simplifier methods
wenzelm@27044
   214
  provide the canonical way for automated normalization (see
wenzelm@27044
   215
  \secref{sec:simplifier}).
wenzelm@27044
   216
wenzelm@28760
   217
  \begin{description}
wenzelm@27044
   218
wenzelm@28760
   219
  \item @{method subst}~@{text eq} performs a single substitution step
wenzelm@28760
   220
  using rule @{text eq}, which may be either a meta or object
wenzelm@27044
   221
  equality.
wenzelm@27044
   222
wenzelm@28760
   223
  \item @{method subst}~@{text "(asm) eq"} substitutes in an
wenzelm@27044
   224
  assumption.
wenzelm@27044
   225
wenzelm@28760
   226
  \item @{method subst}~@{text "(i \<dots> j) eq"} performs several
wenzelm@27044
   227
  substitutions in the conclusion. The numbers @{text i} to @{text j}
wenzelm@27044
   228
  indicate the positions to substitute at.  Positions are ordered from
wenzelm@27044
   229
  the top of the term tree moving down from left to right. For
wenzelm@27044
   230
  example, in @{text "(a + b) + (c + d)"} there are three positions
wenzelm@28760
   231
  where commutativity of @{text "+"} is applicable: 1 refers to @{text
wenzelm@28760
   232
  "a + b"}, 2 to the whole term, and 3 to @{text "c + d"}.
wenzelm@27044
   233
wenzelm@27044
   234
  If the positions in the list @{text "(i \<dots> j)"} are non-overlapping
wenzelm@27044
   235
  (e.g.\ @{text "(2 3)"} in @{text "(a + b) + (c + d)"}) you may
wenzelm@27044
   236
  assume all substitutions are performed simultaneously.  Otherwise
wenzelm@27044
   237
  the behaviour of @{text subst} is not specified.
wenzelm@27044
   238
wenzelm@28760
   239
  \item @{method subst}~@{text "(asm) (i \<dots> j) eq"} performs the
wenzelm@27071
   240
  substitutions in the assumptions. The positions refer to the
wenzelm@27071
   241
  assumptions in order from left to right.  For example, given in a
wenzelm@27071
   242
  goal of the form @{text "P (a + b) \<Longrightarrow> P (c + d) \<Longrightarrow> \<dots>"}, position 1 of
wenzelm@27071
   243
  commutativity of @{text "+"} is the subterm @{text "a + b"} and
wenzelm@27071
   244
  position 2 is the subterm @{text "c + d"}.
wenzelm@27044
   245
wenzelm@28760
   246
  \item @{method hypsubst} performs substitution using some
wenzelm@27044
   247
  assumption; this only works for equations of the form @{text "x =
wenzelm@27044
   248
  t"} where @{text x} is a free or bound variable.
wenzelm@27044
   249
wenzelm@28760
   250
  \item @{method split}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} performs single-step case
wenzelm@44094
   251
  splitting using the given rules.  Splitting is performed in the
wenzelm@44094
   252
  conclusion or some assumption of the subgoal, depending of the
wenzelm@44094
   253
  structure of the rule.
wenzelm@27044
   254
  
wenzelm@27044
   255
  Note that the @{method simp} method already involves repeated
wenzelm@44094
   256
  application of split rules as declared in the current context, using
wenzelm@44094
   257
  @{attribute split}, for example.
wenzelm@27044
   258
wenzelm@28760
   259
  \end{description}
wenzelm@27044
   260
*}
wenzelm@27044
   261
wenzelm@27044
   262
wenzelm@26782
   263
subsection {* Further tactic emulations \label{sec:tactics} *}
wenzelm@26782
   264
wenzelm@26782
   265
text {*
wenzelm@26782
   266
  The following improper proof methods emulate traditional tactics.
wenzelm@26782
   267
  These admit direct access to the goal state, which is normally
wenzelm@26782
   268
  considered harmful!  In particular, this may involve both numbered
wenzelm@26782
   269
  goal addressing (default 1), and dynamic instantiation within the
wenzelm@26782
   270
  scope of some subgoal.
wenzelm@26782
   271
wenzelm@26782
   272
  \begin{warn}
wenzelm@26782
   273
    Dynamic instantiations refer to universally quantified parameters
wenzelm@26782
   274
    of a subgoal (the dynamic context) rather than fixed variables and
wenzelm@26782
   275
    term abbreviations of a (static) Isar context.
wenzelm@26782
   276
  \end{warn}
wenzelm@26782
   277
wenzelm@26782
   278
  Tactic emulation methods, unlike their ML counterparts, admit
wenzelm@26782
   279
  simultaneous instantiation from both dynamic and static contexts.
wenzelm@26782
   280
  If names occur in both contexts goal parameters hide locally fixed
wenzelm@26782
   281
  variables.  Likewise, schematic variables refer to term
wenzelm@26782
   282
  abbreviations, if present in the static context.  Otherwise the
wenzelm@26782
   283
  schematic variable is interpreted as a schematic variable and left
wenzelm@26782
   284
  to be solved by unification with certain parts of the subgoal.
wenzelm@26782
   285
wenzelm@26782
   286
  Note that the tactic emulation proof methods in Isabelle/Isar are
wenzelm@26782
   287
  consistently named @{text foo_tac}.  Note also that variable names
wenzelm@26782
   288
  occurring on left hand sides of instantiations must be preceded by a
wenzelm@26782
   289
  question mark if they coincide with a keyword or contain dots.  This
wenzelm@26782
   290
  is consistent with the attribute @{attribute "where"} (see
wenzelm@26782
   291
  \secref{sec:pure-meth-att}).
wenzelm@26782
   292
wenzelm@26782
   293
  \begin{matharray}{rcl}
wenzelm@28761
   294
    @{method_def rule_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   295
    @{method_def erule_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   296
    @{method_def drule_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   297
    @{method_def frule_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   298
    @{method_def cut_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   299
    @{method_def thin_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   300
    @{method_def subgoal_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   301
    @{method_def rename_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   302
    @{method_def rotate_tac}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   303
    @{method_def tactic}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@28761
   304
    @{method_def raw_tactic}@{text "\<^sup>*"} & : & @{text method} \\
wenzelm@26782
   305
  \end{matharray}
wenzelm@26782
   306
wenzelm@42596
   307
  @{rail "
wenzelm@42596
   308
    (@@{method rule_tac} | @@{method erule_tac} | @@{method drule_tac} |
wenzelm@42705
   309
      @@{method frule_tac} | @@{method cut_tac} | @@{method thin_tac}) @{syntax goal_spec}? \\
wenzelm@42617
   310
    ( dynamic_insts @'in' @{syntax thmref} | @{syntax thmrefs} )
wenzelm@26782
   311
    ;
wenzelm@42705
   312
    @@{method subgoal_tac} @{syntax goal_spec}? (@{syntax prop} +)
wenzelm@42596
   313
    ;
wenzelm@42705
   314
    @@{method rename_tac} @{syntax goal_spec}? (@{syntax name} +)
wenzelm@26782
   315
    ;
wenzelm@42705
   316
    @@{method rotate_tac} @{syntax goal_spec}? @{syntax int}?
wenzelm@26782
   317
    ;
wenzelm@42596
   318
    (@@{method tactic} | @@{method raw_tactic}) @{syntax text}
wenzelm@26782
   319
    ;
wenzelm@26782
   320
wenzelm@42617
   321
    dynamic_insts: ((@{syntax name} '=' @{syntax term}) + @'and')
wenzelm@42617
   322
  "}
wenzelm@26782
   323
wenzelm@28760
   324
\begin{description}
wenzelm@26782
   325
wenzelm@28760
   326
  \item @{method rule_tac} etc. do resolution of rules with explicit
wenzelm@26782
   327
  instantiation.  This works the same way as the ML tactics @{ML
wenzelm@30397
   328
  res_inst_tac} etc. (see \cite{isabelle-implementation})
wenzelm@26782
   329
wenzelm@26782
   330
  Multiple rules may be only given if there is no instantiation; then
wenzelm@26782
   331
  @{method rule_tac} is the same as @{ML resolve_tac} in ML (see
wenzelm@30397
   332
  \cite{isabelle-implementation}).
wenzelm@26782
   333
wenzelm@28760
   334
  \item @{method cut_tac} inserts facts into the proof state as
wenzelm@46706
   335
  assumption of a subgoal; instantiations may be given as well.  Note
wenzelm@46706
   336
  that the scope of schematic variables is spread over the main goal
wenzelm@46706
   337
  statement and rule premises are turned into new subgoals.  This is
wenzelm@46706
   338
  in contrast to the regular method @{method insert} which inserts
wenzelm@46706
   339
  closed rule statements.
wenzelm@26782
   340
wenzelm@46277
   341
  \item @{method thin_tac}~@{text \<phi>} deletes the specified premise
wenzelm@46277
   342
  from a subgoal.  Note that @{text \<phi>} may contain schematic
wenzelm@46277
   343
  variables, to abbreviate the intended proposition; the first
wenzelm@46277
   344
  matching subgoal premise will be deleted.  Removing useless premises
wenzelm@46277
   345
  from a subgoal increases its readability and can make search tactics
wenzelm@46277
   346
  run faster.
wenzelm@28760
   347
wenzelm@46271
   348
  \item @{method subgoal_tac}~@{text "\<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} adds the propositions
wenzelm@46271
   349
  @{text "\<phi>\<^sub>1 \<dots> \<phi>\<^sub>n"} as local premises to a subgoal, and poses the same
wenzelm@46271
   350
  as new subgoals (in the original context).
wenzelm@26782
   351
wenzelm@28760
   352
  \item @{method rename_tac}~@{text "x\<^sub>1 \<dots> x\<^sub>n"} renames parameters of a
wenzelm@28760
   353
  goal according to the list @{text "x\<^sub>1, \<dots>, x\<^sub>n"}, which refers to the
wenzelm@28760
   354
  \emph{suffix} of variables.
wenzelm@26782
   355
wenzelm@46274
   356
  \item @{method rotate_tac}~@{text n} rotates the premises of a
wenzelm@46274
   357
  subgoal by @{text n} positions: from right to left if @{text n} is
wenzelm@26782
   358
  positive, and from left to right if @{text n} is negative; the
wenzelm@46274
   359
  default value is 1.
wenzelm@26782
   360
wenzelm@28760
   361
  \item @{method tactic}~@{text "text"} produces a proof method from
wenzelm@26782
   362
  any ML text of type @{ML_type tactic}.  Apart from the usual ML
wenzelm@27223
   363
  environment and the current proof context, the ML code may refer to
wenzelm@27223
   364
  the locally bound values @{ML_text facts}, which indicates any
wenzelm@27223
   365
  current facts used for forward-chaining.
wenzelm@26782
   366
wenzelm@28760
   367
  \item @{method raw_tactic} is similar to @{method tactic}, but
wenzelm@27223
   368
  presents the goal state in its raw internal form, where simultaneous
wenzelm@27223
   369
  subgoals appear as conjunction of the logical framework instead of
wenzelm@27223
   370
  the usual split into several subgoals.  While feature this is useful
wenzelm@27223
   371
  for debugging of complex method definitions, it should not never
wenzelm@27223
   372
  appear in production theories.
wenzelm@26782
   373
wenzelm@28760
   374
  \end{description}
wenzelm@26782
   375
*}
wenzelm@26782
   376
wenzelm@26782
   377
wenzelm@27040
   378
section {* The Simplifier \label{sec:simplifier} *}
wenzelm@26782
   379
wenzelm@50063
   380
text {* The Simplifier performs conditional and unconditional
wenzelm@50063
   381
  rewriting and uses contextual information: rule declarations in the
wenzelm@50063
   382
  background theory or local proof context are taken into account, as
wenzelm@50063
   383
  well as chained facts and subgoal premises (``local assumptions'').
wenzelm@50063
   384
  There are several general hooks that allow to modify the
wenzelm@50063
   385
  simplification strategy, or incorporate other proof tools that solve
wenzelm@50063
   386
  sub-problems, produce rewrite rules on demand etc.
wenzelm@50063
   387
wenzelm@50075
   388
  The rewriting strategy is always strictly bottom up, except for
wenzelm@50075
   389
  congruence rules, which are applied while descending into a term.
wenzelm@50075
   390
  Conditions in conditional rewrite rules are solved recursively
wenzelm@50075
   391
  before the rewrite rule is applied.
wenzelm@50075
   392
wenzelm@50063
   393
  The default Simplifier setup of major object logics (HOL, HOLCF,
wenzelm@50063
   394
  FOL, ZF) makes the Simplifier ready for immediate use, without
wenzelm@50063
   395
  engaging into the internal structures.  Thus it serves as
wenzelm@50063
   396
  general-purpose proof tool with the main focus on equational
wenzelm@50075
   397
  reasoning, and a bit more than that.
wenzelm@50075
   398
*}
wenzelm@50063
   399
wenzelm@50063
   400
wenzelm@50063
   401
subsection {* Simplification methods \label{sec:simp-meth} *}
wenzelm@26782
   402
wenzelm@26782
   403
text {*
wenzelm@26782
   404
  \begin{matharray}{rcl}
wenzelm@28761
   405
    @{method_def simp} & : & @{text method} \\
wenzelm@28761
   406
    @{method_def simp_all} & : & @{text method} \\
wenzelm@26782
   407
  \end{matharray}
wenzelm@26782
   408
wenzelm@42596
   409
  @{rail "
wenzelm@42596
   410
    (@@{method simp} | @@{method simp_all}) opt? (@{syntax simpmod} * )
wenzelm@26782
   411
    ;
wenzelm@26782
   412
wenzelm@40255
   413
    opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use' | 'asm_lr' ) ')'
wenzelm@26782
   414
    ;
wenzelm@50063
   415
    @{syntax_def simpmod}: ('add' | 'del' | 'only' | 'split' (() | 'add' | 'del') |
wenzelm@50063
   416
      'cong' (() | 'add' | 'del')) ':' @{syntax thmrefs}
wenzelm@42596
   417
  "}
wenzelm@26782
   418
wenzelm@28760
   419
  \begin{description}
wenzelm@26782
   420
wenzelm@50063
   421
  \item @{method simp} invokes the Simplifier on the first subgoal,
wenzelm@50063
   422
  after inserting chained facts as additional goal premises; further
wenzelm@50063
   423
  rule declarations may be included via @{text "(simp add: facts)"}.
wenzelm@50063
   424
  The proof method fails if the subgoal remains unchanged after
wenzelm@50063
   425
  simplification.
wenzelm@26782
   426
wenzelm@50063
   427
  Note that the original goal premises and chained facts are subject
wenzelm@50063
   428
  to simplification themselves, while declarations via @{text
wenzelm@50063
   429
  "add"}/@{text "del"} merely follow the policies of the object-logic
wenzelm@50063
   430
  to extract rewrite rules from theorems, without further
wenzelm@50063
   431
  simplification.  This may lead to slightly different behavior in
wenzelm@50063
   432
  either case, which might be required precisely like that in some
wenzelm@50063
   433
  boundary situations to perform the intended simplification step!
wenzelm@50063
   434
wenzelm@50063
   435
  \medskip The @{text only} modifier first removes all other rewrite
wenzelm@50063
   436
  rules, looper tactics (including split rules), congruence rules, and
wenzelm@50063
   437
  then behaves like @{text add}.  Implicit solvers remain, which means
wenzelm@50063
   438
  that trivial rules like reflexivity or introduction of @{text
wenzelm@50063
   439
  "True"} are available to solve the simplified subgoals, but also
wenzelm@50063
   440
  non-trivial tools like linear arithmetic in HOL.  The latter may
wenzelm@50063
   441
  lead to some surprise of the meaning of ``only'' in Isabelle/HOL
wenzelm@50063
   442
  compared to English!
wenzelm@26782
   443
wenzelm@42596
   444
  \medskip The @{text split} modifiers add or delete rules for the
wenzelm@50079
   445
  Splitter (see also \secref{sec:simp-strategies} on the looper).
wenzelm@26782
   446
  This works only if the Simplifier method has been properly setup to
wenzelm@26782
   447
  include the Splitter (all major object logics such HOL, HOLCF, FOL,
wenzelm@26782
   448
  ZF do this already).
wenzelm@26782
   449
wenzelm@50065
   450
  There is also a separate @{method_ref split} method available for
wenzelm@50065
   451
  single-step case splitting.  The effect of repeatedly applying
wenzelm@50065
   452
  @{text "(split thms)"} can be imitated by ``@{text "(simp only:
wenzelm@50065
   453
  split: thms)"}''.
wenzelm@50065
   454
wenzelm@50063
   455
  \medskip The @{text cong} modifiers add or delete Simplifier
wenzelm@50063
   456
  congruence rules (see also \secref{sec:simp-rules}); the default is
wenzelm@50063
   457
  to add.
wenzelm@50063
   458
wenzelm@28760
   459
  \item @{method simp_all} is similar to @{method simp}, but acts on
wenzelm@50063
   460
  all goals, working backwards from the last to the first one as usual
wenzelm@50063
   461
  in Isabelle.\footnote{The order is irrelevant for goals without
wenzelm@50063
   462
  schematic variables, so simplification might actually be performed
wenzelm@50063
   463
  in parallel here.}
wenzelm@50063
   464
wenzelm@50063
   465
  Chained facts are inserted into all subgoals, before the
wenzelm@50063
   466
  simplification process starts.  Further rule declarations are the
wenzelm@50063
   467
  same as for @{method simp}.
wenzelm@50063
   468
wenzelm@50063
   469
  The proof method fails if all subgoals remain unchanged after
wenzelm@50063
   470
  simplification.
wenzelm@26782
   471
wenzelm@28760
   472
  \end{description}
wenzelm@26782
   473
wenzelm@50063
   474
  By default the Simplifier methods above take local assumptions fully
wenzelm@50063
   475
  into account, using equational assumptions in the subsequent
wenzelm@50063
   476
  normalization process, or simplifying assumptions themselves.
wenzelm@50063
   477
  Further options allow to fine-tune the behavior of the Simplifier
wenzelm@50063
   478
  in this respect, corresponding to a variety of ML tactics as
wenzelm@50063
   479
  follows.\footnote{Unlike the corresponding Isar proof methods, the
wenzelm@50063
   480
  ML tactics do not insist in changing the goal state.}
wenzelm@50063
   481
wenzelm@50063
   482
  \begin{center}
wenzelm@50063
   483
  \small
wenzelm@50065
   484
  \begin{supertabular}{|l|l|p{0.3\textwidth}|}
wenzelm@50063
   485
  \hline
wenzelm@50063
   486
  Isar method & ML tactic & behavior \\\hline
wenzelm@50063
   487
wenzelm@50063
   488
  @{text "(simp (no_asm))"} & @{ML simp_tac} & assumptions are ignored
wenzelm@50063
   489
  completely \\\hline
wenzelm@26782
   490
wenzelm@50063
   491
  @{text "(simp (no_asm_simp))"} & @{ML asm_simp_tac} & assumptions
wenzelm@50063
   492
  are used in the simplification of the conclusion but are not
wenzelm@50063
   493
  themselves simplified \\\hline
wenzelm@50063
   494
wenzelm@50063
   495
  @{text "(simp (no_asm_use))"} & @{ML full_simp_tac} & assumptions
wenzelm@50063
   496
  are simplified but are not used in the simplification of each other
wenzelm@50063
   497
  or the conclusion \\\hline
wenzelm@26782
   498
wenzelm@50063
   499
  @{text "(simp)"} & @{ML asm_full_simp_tac} & assumptions are used in
wenzelm@50063
   500
  the simplification of the conclusion and to simplify other
wenzelm@50063
   501
  assumptions \\\hline
wenzelm@50063
   502
wenzelm@50063
   503
  @{text "(simp (asm_lr))"} & @{ML asm_lr_simp_tac} & compatibility
wenzelm@50063
   504
  mode: an assumption is only used for simplifying assumptions which
wenzelm@50063
   505
  are to the right of it \\\hline
wenzelm@50063
   506
wenzelm@50065
   507
  \end{supertabular}
wenzelm@50063
   508
  \end{center}
wenzelm@26782
   509
*}
wenzelm@26782
   510
wenzelm@26782
   511
wenzelm@50064
   512
subsubsection {* Examples *}
wenzelm@50064
   513
wenzelm@50064
   514
text {* We consider basic algebraic simplifications in Isabelle/HOL.
wenzelm@50064
   515
  The rather trivial goal @{prop "0 + (x + 0) = x + 0 + 0"} looks like
wenzelm@50064
   516
  a good candidate to be solved by a single call of @{method simp}:
wenzelm@50064
   517
*}
wenzelm@50064
   518
wenzelm@50064
   519
lemma "0 + (x + 0) = x + 0 + 0" apply simp? oops
wenzelm@50064
   520
wenzelm@50064
   521
text {* The above attempt \emph{fails}, because @{term "0"} and @{term
wenzelm@50064
   522
  "op +"} in the HOL library are declared as generic type class
wenzelm@50064
   523
  operations, without stating any algebraic laws yet.  More specific
wenzelm@50064
   524
  types are required to get access to certain standard simplifications
wenzelm@50064
   525
  of the theory context, e.g.\ like this: *}
wenzelm@50064
   526
wenzelm@50064
   527
lemma fixes x :: nat shows "0 + (x + 0) = x + 0 + 0" by simp
wenzelm@50064
   528
lemma fixes x :: int shows "0 + (x + 0) = x + 0 + 0" by simp
wenzelm@50064
   529
lemma fixes x :: "'a :: monoid_add" shows "0 + (x + 0) = x + 0 + 0" by simp
wenzelm@50064
   530
wenzelm@50064
   531
text {*
wenzelm@50064
   532
  \medskip In many cases, assumptions of a subgoal are also needed in
wenzelm@50064
   533
  the simplification process.  For example:
wenzelm@50064
   534
*}
wenzelm@50064
   535
wenzelm@50064
   536
lemma fixes x :: nat shows "x = 0 \<Longrightarrow> x + x = 0" by simp
wenzelm@50064
   537
lemma fixes x :: nat assumes "x = 0" shows "x + x = 0" apply simp oops
wenzelm@50064
   538
lemma fixes x :: nat assumes "x = 0" shows "x + x = 0" using assms by simp
wenzelm@50064
   539
wenzelm@50064
   540
text {* As seen above, local assumptions that shall contribute to
wenzelm@50064
   541
  simplification need to be part of the subgoal already, or indicated
wenzelm@50064
   542
  explicitly for use by the subsequent method invocation.  Both too
wenzelm@50064
   543
  little or too much information can make simplification fail, for
wenzelm@50064
   544
  different reasons.
wenzelm@50064
   545
wenzelm@50064
   546
  In the next example the malicious assumption @{prop "\<And>x::nat. f x =
wenzelm@50064
   547
  g (f (g x))"} does not contribute to solve the problem, but makes
wenzelm@50064
   548
  the default @{method simp} method loop: the rewrite rule @{text "f
wenzelm@50064
   549
  ?x \<equiv> g (f (g ?x))"} extracted from the assumption does not
wenzelm@50064
   550
  terminate.  The Simplifier notices certain simple forms of
wenzelm@50064
   551
  nontermination, but not this one.  The problem can be solved
wenzelm@50064
   552
  nonetheless, by ignoring assumptions via special options as
wenzelm@50064
   553
  explained before:
wenzelm@50064
   554
*}
wenzelm@50064
   555
wenzelm@50064
   556
lemma "(\<And>x::nat. f x = g (f (g x))) \<Longrightarrow> f 0 = f 0 + 0"
wenzelm@50064
   557
  by (simp (no_asm))
wenzelm@50064
   558
wenzelm@50064
   559
text {* The latter form is typical for long unstructured proof
wenzelm@50064
   560
  scripts, where the control over the goal content is limited.  In
wenzelm@50064
   561
  structured proofs it is usually better to avoid pushing too many
wenzelm@50064
   562
  facts into the goal state in the first place.  Assumptions in the
wenzelm@50064
   563
  Isar proof context do not intrude the reasoning if not used
wenzelm@50064
   564
  explicitly.  This is illustrated for a toplevel statement and a
wenzelm@50064
   565
  local proof body as follows:
wenzelm@50064
   566
*}
wenzelm@50064
   567
wenzelm@50064
   568
lemma
wenzelm@50064
   569
  assumes "\<And>x::nat. f x = g (f (g x))"
wenzelm@50064
   570
  shows "f 0 = f 0 + 0" by simp
wenzelm@50064
   571
wenzelm@50064
   572
notepad
wenzelm@50064
   573
begin
wenzelm@50064
   574
  assume "\<And>x::nat. f x = g (f (g x))"
wenzelm@50064
   575
  have "f 0 = f 0 + 0" by simp
wenzelm@50064
   576
end
wenzelm@50064
   577
wenzelm@50064
   578
text {* \medskip Because assumptions may simplify each other, there
wenzelm@50064
   579
  can be very subtle cases of nontermination. For example, the regular
wenzelm@50064
   580
  @{method simp} method applied to @{prop "P (f x) \<Longrightarrow> y = x \<Longrightarrow> f x = f y
wenzelm@50064
   581
  \<Longrightarrow> Q"} gives rise to the infinite reduction sequence
wenzelm@50064
   582
  \[
wenzelm@50064
   583
  @{text "P (f x)"} \stackrel{@{text "f x \<equiv> f y"}}{\longmapsto}
wenzelm@50064
   584
  @{text "P (f y)"} \stackrel{@{text "y \<equiv> x"}}{\longmapsto}
wenzelm@50064
   585
  @{text "P (f x)"} \stackrel{@{text "f x \<equiv> f y"}}{\longmapsto} \cdots
wenzelm@50064
   586
  \]
wenzelm@50064
   587
  whereas applying the same to @{prop "y = x \<Longrightarrow> f x = f y \<Longrightarrow> P (f x) \<Longrightarrow>
wenzelm@50064
   588
  Q"} terminates (without solving the goal):
wenzelm@50064
   589
*}
wenzelm@50064
   590
wenzelm@50064
   591
lemma "y = x \<Longrightarrow> f x = f y \<Longrightarrow> P (f x) \<Longrightarrow> Q"
wenzelm@50064
   592
  apply simp
wenzelm@50064
   593
  oops
wenzelm@50064
   594
wenzelm@50064
   595
text {* See also \secref{sec:simp-config} for options to enable
wenzelm@50064
   596
  Simplifier trace mode, which often helps to diagnose problems with
wenzelm@50064
   597
  rewrite systems.
wenzelm@50064
   598
*}
wenzelm@50064
   599
wenzelm@50064
   600
wenzelm@50063
   601
subsection {* Declaring rules \label{sec:simp-rules} *}
wenzelm@26782
   602
wenzelm@26782
   603
text {*
wenzelm@26782
   604
  \begin{matharray}{rcl}
wenzelm@28761
   605
    @{attribute_def simp} & : & @{text attribute} \\
wenzelm@28761
   606
    @{attribute_def split} & : & @{text attribute} \\
wenzelm@50063
   607
    @{attribute_def cong} & : & @{text attribute} \\
wenzelm@50077
   608
    @{command_def "print_simpset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@26782
   609
  \end{matharray}
wenzelm@26782
   610
wenzelm@42596
   611
  @{rail "
wenzelm@50063
   612
    (@@{attribute simp} | @@{attribute split} | @@{attribute cong})
wenzelm@50063
   613
      (() | 'add' | 'del')
wenzelm@42596
   614
  "}
wenzelm@26782
   615
wenzelm@28760
   616
  \begin{description}
wenzelm@26782
   617
wenzelm@50076
   618
  \item @{attribute simp} declares rewrite rules, by adding or
wenzelm@50065
   619
  deleting them from the simpset within the theory or proof context.
wenzelm@50076
   620
  Rewrite rules are theorems expressing some form of equality, for
wenzelm@50076
   621
  example:
wenzelm@50076
   622
wenzelm@50076
   623
  @{text "Suc ?m + ?n = ?m + Suc ?n"} \\
wenzelm@50076
   624
  @{text "?P \<and> ?P \<longleftrightarrow> ?P"} \\
wenzelm@50076
   625
  @{text "?A \<union> ?B \<equiv> {x. x \<in> ?A \<or> x \<in> ?B}"}
wenzelm@50076
   626
wenzelm@50076
   627
  \smallskip
wenzelm@50076
   628
  Conditional rewrites such as @{text "?m < ?n \<Longrightarrow> ?m div ?n = 0"} are
wenzelm@50076
   629
  also permitted; the conditions can be arbitrary formulas.
wenzelm@50076
   630
wenzelm@50076
   631
  \medskip Internally, all rewrite rules are translated into Pure
wenzelm@50076
   632
  equalities, theorems with conclusion @{text "lhs \<equiv> rhs"}. The
wenzelm@50076
   633
  simpset contains a function for extracting equalities from arbitrary
wenzelm@50076
   634
  theorems, which is usually installed when the object-logic is
wenzelm@50076
   635
  configured initially. For example, @{text "\<not> ?x \<in> {}"} could be
wenzelm@50076
   636
  turned into @{text "?x \<in> {} \<equiv> False"}. Theorems that are declared as
wenzelm@50076
   637
  @{attribute simp} and local assumptions within a goal are treated
wenzelm@50076
   638
  uniformly in this respect.
wenzelm@50076
   639
wenzelm@50076
   640
  The Simplifier accepts the following formats for the @{text "lhs"}
wenzelm@50076
   641
  term:
wenzelm@50076
   642
wenzelm@50076
   643
  \begin{enumerate}
wenzelm@50065
   644
wenzelm@50076
   645
  \item First-order patterns, considering the sublanguage of
wenzelm@50076
   646
  application of constant operators to variable operands, without
wenzelm@50076
   647
  @{text "\<lambda>"}-abstractions or functional variables.
wenzelm@50076
   648
  For example:
wenzelm@50076
   649
wenzelm@50076
   650
  @{text "(?x + ?y) + ?z \<equiv> ?x + (?y + ?z)"} \\
wenzelm@50076
   651
  @{text "f (f ?x ?y) ?z \<equiv> f ?x (f ?y ?z)"}
wenzelm@50076
   652
wenzelm@50076
   653
  \item Higher-order patterns in the sense of \cite{nipkow-patterns}.
wenzelm@50076
   654
  These are terms in @{text "\<beta>"}-normal form (this will always be the
wenzelm@50076
   655
  case unless you have done something strange) where each occurrence
wenzelm@50076
   656
  of an unknown is of the form @{text "?F x\<^sub>1 \<dots> x\<^sub>n"}, where the
wenzelm@50076
   657
  @{text "x\<^sub>i"} are distinct bound variables.
wenzelm@50076
   658
wenzelm@50076
   659
  For example, @{text "(\<forall>x. ?P x \<and> ?Q x) \<equiv> (\<forall>x. ?P x) \<and> (\<forall>x. ?Q x)"}
wenzelm@50076
   660
  or its symmetric form, since the @{text "rhs"} is also a
wenzelm@50076
   661
  higher-order pattern.
wenzelm@50076
   662
wenzelm@50076
   663
  \item Physical first-order patterns over raw @{text "\<lambda>"}-term
wenzelm@50076
   664
  structure without @{text "\<alpha>\<beta>\<eta>"}-equality; abstractions and bound
wenzelm@50076
   665
  variables are treated like quasi-constant term material.
wenzelm@50076
   666
wenzelm@50076
   667
  For example, the rule @{text "?f ?x \<in> range ?f = True"} rewrites the
wenzelm@50076
   668
  term @{text "g a \<in> range g"} to @{text "True"}, but will fail to
wenzelm@50076
   669
  match @{text "g (h b) \<in> range (\<lambda>x. g (h x))"}. However, offending
wenzelm@50076
   670
  subterms (in our case @{text "?f ?x"}, which is not a pattern) can
wenzelm@50076
   671
  be replaced by adding new variables and conditions like this: @{text
wenzelm@50076
   672
  "?y = ?f ?x \<Longrightarrow> ?y \<in> range ?f = True"} is acceptable as a conditional
wenzelm@50076
   673
  rewrite rule of the second category since conditions can be
wenzelm@50076
   674
  arbitrary terms.
wenzelm@50076
   675
wenzelm@50076
   676
  \end{enumerate}
wenzelm@26782
   677
wenzelm@28760
   678
  \item @{attribute split} declares case split rules.
wenzelm@26782
   679
wenzelm@45645
   680
  \item @{attribute cong} declares congruence rules to the Simplifier
wenzelm@45645
   681
  context.
wenzelm@45645
   682
wenzelm@45645
   683
  Congruence rules are equalities of the form @{text [display]
wenzelm@45645
   684
  "\<dots> \<Longrightarrow> f ?x\<^sub>1 \<dots> ?x\<^sub>n = f ?y\<^sub>1 \<dots> ?y\<^sub>n"}
wenzelm@45645
   685
wenzelm@45645
   686
  This controls the simplification of the arguments of @{text f}.  For
wenzelm@45645
   687
  example, some arguments can be simplified under additional
wenzelm@45645
   688
  assumptions: @{text [display] "?P\<^sub>1 \<longleftrightarrow> ?Q\<^sub>1 \<Longrightarrow> (?Q\<^sub>1 \<Longrightarrow> ?P\<^sub>2 \<longleftrightarrow> ?Q\<^sub>2) \<Longrightarrow>
wenzelm@45645
   689
  (?P\<^sub>1 \<longrightarrow> ?P\<^sub>2) \<longleftrightarrow> (?Q\<^sub>1 \<longrightarrow> ?Q\<^sub>2)"}
wenzelm@45645
   690
wenzelm@45645
   691
  Given this rule, the simplifier assumes @{text "?Q\<^sub>1"} and extracts
wenzelm@45645
   692
  rewrite rules from it when simplifying @{text "?P\<^sub>2"}.  Such local
wenzelm@45645
   693
  assumptions are effective for rewriting formulae such as @{text "x =
wenzelm@45645
   694
  0 \<longrightarrow> y + x = y"}.
wenzelm@45645
   695
wenzelm@45645
   696
  %FIXME
wenzelm@45645
   697
  %The local assumptions are also provided as theorems to the solver;
wenzelm@45645
   698
  %see \secref{sec:simp-solver} below.
wenzelm@45645
   699
wenzelm@45645
   700
  \medskip The following congruence rule for bounded quantifiers also
wenzelm@45645
   701
  supplies contextual information --- about the bound variable:
wenzelm@45645
   702
  @{text [display] "(?A = ?B) \<Longrightarrow> (\<And>x. x \<in> ?B \<Longrightarrow> ?P x \<longleftrightarrow> ?Q x) \<Longrightarrow>
wenzelm@45645
   703
    (\<forall>x \<in> ?A. ?P x) \<longleftrightarrow> (\<forall>x \<in> ?B. ?Q x)"}
wenzelm@45645
   704
wenzelm@45645
   705
  \medskip This congruence rule for conditional expressions can
wenzelm@45645
   706
  supply contextual information for simplifying the arms:
wenzelm@45645
   707
  @{text [display] "?p = ?q \<Longrightarrow> (?q \<Longrightarrow> ?a = ?c) \<Longrightarrow> (\<not> ?q \<Longrightarrow> ?b = ?d) \<Longrightarrow>
wenzelm@45645
   708
    (if ?p then ?a else ?b) = (if ?q then ?c else ?d)"}
wenzelm@45645
   709
wenzelm@45645
   710
  A congruence rule can also \emph{prevent} simplification of some
wenzelm@45645
   711
  arguments.  Here is an alternative congruence rule for conditional
wenzelm@45645
   712
  expressions that conforms to non-strict functional evaluation:
wenzelm@45645
   713
  @{text [display] "?p = ?q \<Longrightarrow> (if ?p then ?a else ?b) = (if ?q then ?a else ?b)"}
wenzelm@45645
   714
wenzelm@45645
   715
  Only the first argument is simplified; the others remain unchanged.
wenzelm@45645
   716
  This can make simplification much faster, but may require an extra
wenzelm@45645
   717
  case split over the condition @{text "?q"} to prove the goal.
wenzelm@50063
   718
wenzelm@50077
   719
  \item @{command "print_simpset"} prints the collection of rules
wenzelm@50077
   720
  declared to the Simplifier, which is also known as ``simpset''
wenzelm@50077
   721
  internally.
wenzelm@50077
   722
wenzelm@50077
   723
  For historical reasons, simpsets may occur independently from the
wenzelm@50077
   724
  current context, but are conceptually dependent on it.  When the
wenzelm@50077
   725
  Simplifier is invoked via one of its main entry points in the Isar
wenzelm@50077
   726
  source language (as proof method \secref{sec:simp-meth} or rule
wenzelm@50077
   727
  attribute \secref{sec:simp-meth}), its simpset is derived from the
wenzelm@50077
   728
  current proof context, and carries a back-reference to that for
wenzelm@50077
   729
  other tools that might get invoked internally (e.g.\ simplification
wenzelm@50077
   730
  procedures \secref{sec:simproc}).  A mismatch of the context of the
wenzelm@50077
   731
  simpset and the context of the problem being simplified may lead to
wenzelm@50077
   732
  unexpected results.
wenzelm@50077
   733
wenzelm@50063
   734
  \end{description}
wenzelm@50065
   735
wenzelm@50065
   736
  The implicit simpset of the theory context is propagated
wenzelm@50065
   737
  monotonically through the theory hierarchy: forming a new theory,
wenzelm@50065
   738
  the union of the simpsets of its imports are taken as starting
wenzelm@50065
   739
  point.  Also note that definitional packages like @{command
wenzelm@50065
   740
  "datatype"}, @{command "primrec"}, @{command "fun"} routinely
wenzelm@50065
   741
  declare Simplifier rules to the target context, while plain
wenzelm@50065
   742
  @{command "definition"} is an exception in \emph{not} declaring
wenzelm@50065
   743
  anything.
wenzelm@50065
   744
wenzelm@50065
   745
  \medskip It is up the user to manipulate the current simpset further
wenzelm@50065
   746
  by explicitly adding or deleting theorems as simplification rules,
wenzelm@50065
   747
  or installing other tools via simplification procedures
wenzelm@50065
   748
  (\secref{sec:simproc}).  Good simpsets are hard to design.  Rules
wenzelm@50065
   749
  that obviously simplify, like @{text "?n + 0 \<equiv> ?n"} are good
wenzelm@50065
   750
  candidates for the implicit simpset, unless a special
wenzelm@50065
   751
  non-normalizing behavior of certain operations is intended.  More
wenzelm@50065
   752
  specific rules (such as distributive laws, which duplicate subterms)
wenzelm@50065
   753
  should be added only for specific proof steps.  Conversely,
wenzelm@50065
   754
  sometimes a rule needs to be deleted just for some part of a proof.
wenzelm@50065
   755
  The need of frequent additions or deletions may indicate a poorly
wenzelm@50065
   756
  designed simpset.
wenzelm@50065
   757
wenzelm@50065
   758
  \begin{warn}
wenzelm@50065
   759
  The union of simpsets from theory imports (as described above) is
wenzelm@50065
   760
  not always a good starting point for the new theory.  If some
wenzelm@50065
   761
  ancestors have deleted simplification rules because they are no
wenzelm@50065
   762
  longer wanted, while others have left those rules in, then the union
wenzelm@50065
   763
  will contain the unwanted rules, and thus have to be deleted again
wenzelm@50065
   764
  in the theory body.
wenzelm@50065
   765
  \end{warn}
wenzelm@45645
   766
*}
wenzelm@45645
   767
wenzelm@45645
   768
wenzelm@50063
   769
subsection {* Configuration options \label{sec:simp-config} *}
wenzelm@50063
   770
wenzelm@50063
   771
text {*
wenzelm@50063
   772
  \begin{tabular}{rcll}
wenzelm@50063
   773
    @{attribute_def simp_depth_limit} & : & @{text attribute} & default @{text 100} \\
wenzelm@50063
   774
    @{attribute_def simp_trace} & : & @{text attribute} & default @{text false} \\
wenzelm@50063
   775
    @{attribute_def simp_trace_depth_limit} & : & @{text attribute} & default @{text 1} \\
wenzelm@50063
   776
    @{attribute_def simp_debug} & : & @{text attribute} & default @{text false} \\
wenzelm@50063
   777
  \end{tabular}
wenzelm@50063
   778
  \medskip
wenzelm@50063
   779
wenzelm@50063
   780
  These configurations options control further aspects of the Simplifier.
wenzelm@50063
   781
  See also \secref{sec:config}.
wenzelm@50063
   782
wenzelm@50063
   783
  \begin{description}
wenzelm@50063
   784
wenzelm@50063
   785
  \item @{attribute simp_depth_limit} limits the number of recursive
wenzelm@50063
   786
  invocations of the Simplifier during conditional rewriting.
wenzelm@50063
   787
wenzelm@50063
   788
  \item @{attribute simp_trace} makes the Simplifier output internal
wenzelm@50063
   789
  operations.  This includes rewrite steps, but also bookkeeping like
wenzelm@50063
   790
  modifications of the simpset.
wenzelm@50063
   791
wenzelm@50063
   792
  \item @{attribute simp_trace_depth_limit} limits the effect of
wenzelm@50063
   793
  @{attribute simp_trace} to the given depth of recursive Simplifier
wenzelm@50063
   794
  invocations (when solving conditions of rewrite rules).
wenzelm@50063
   795
wenzelm@50063
   796
  \item @{attribute simp_debug} makes the Simplifier output some extra
wenzelm@50063
   797
  information about internal operations.  This includes any attempted
wenzelm@50063
   798
  invocation of simplification procedures.
wenzelm@50063
   799
wenzelm@50063
   800
  \end{description}
wenzelm@50063
   801
*}
wenzelm@50063
   802
wenzelm@50063
   803
wenzelm@50063
   804
subsection {* Simplification procedures \label{sec:simproc} *}
wenzelm@26782
   805
wenzelm@42925
   806
text {* Simplification procedures are ML functions that produce proven
wenzelm@42925
   807
  rewrite rules on demand.  They are associated with higher-order
wenzelm@42925
   808
  patterns that approximate the left-hand sides of equations.  The
wenzelm@42925
   809
  Simplifier first matches the current redex against one of the LHS
wenzelm@42925
   810
  patterns; if this succeeds, the corresponding ML function is
wenzelm@42925
   811
  invoked, passing the Simplifier context and redex term.  Thus rules
wenzelm@42925
   812
  may be specifically fashioned for particular situations, resulting
wenzelm@42925
   813
  in a more powerful mechanism than term rewriting by a fixed set of
wenzelm@42925
   814
  rules.
wenzelm@42925
   815
wenzelm@42925
   816
  Any successful result needs to be a (possibly conditional) rewrite
wenzelm@42925
   817
  rule @{text "t \<equiv> u"} that is applicable to the current redex.  The
wenzelm@42925
   818
  rule will be applied just as any ordinary rewrite rule.  It is
wenzelm@42925
   819
  expected to be already in \emph{internal form}, bypassing the
wenzelm@42925
   820
  automatic preprocessing of object-level equivalences.
wenzelm@42925
   821
wenzelm@26782
   822
  \begin{matharray}{rcl}
wenzelm@28761
   823
    @{command_def "simproc_setup"} & : & @{text "local_theory \<rightarrow> local_theory"} \\
wenzelm@28761
   824
    simproc & : & @{text attribute} \\
wenzelm@26782
   825
  \end{matharray}
wenzelm@26782
   826
wenzelm@42596
   827
  @{rail "
wenzelm@42596
   828
    @@{command simproc_setup} @{syntax name} '(' (@{syntax term} + '|') ')' '='
wenzelm@42596
   829
      @{syntax text} \\ (@'identifier' (@{syntax nameref}+))?
wenzelm@26782
   830
    ;
wenzelm@26782
   831
wenzelm@42596
   832
    @@{attribute simproc} (('add' ':')? | 'del' ':') (@{syntax name}+)
wenzelm@42596
   833
  "}
wenzelm@26782
   834
wenzelm@28760
   835
  \begin{description}
wenzelm@26782
   836
wenzelm@28760
   837
  \item @{command "simproc_setup"} defines a named simplification
wenzelm@26782
   838
  procedure that is invoked by the Simplifier whenever any of the
wenzelm@26782
   839
  given term patterns match the current redex.  The implementation,
wenzelm@26782
   840
  which is provided as ML source text, needs to be of type @{ML_type
wenzelm@26782
   841
  "morphism -> simpset -> cterm -> thm option"}, where the @{ML_type
wenzelm@26782
   842
  cterm} represents the current redex @{text r} and the result is
wenzelm@26782
   843
  supposed to be some proven rewrite rule @{text "r \<equiv> r'"} (or a
wenzelm@26782
   844
  generalized version), or @{ML NONE} to indicate failure.  The
wenzelm@26782
   845
  @{ML_type simpset} argument holds the full context of the current
wenzelm@26782
   846
  Simplifier invocation, including the actual Isar proof context.  The
wenzelm@26782
   847
  @{ML_type morphism} informs about the difference of the original
wenzelm@26782
   848
  compilation context wrt.\ the one of the actual application later
wenzelm@26782
   849
  on.  The optional @{keyword "identifier"} specifies theorems that
wenzelm@26782
   850
  represent the logical content of the abstract theory of this
wenzelm@26782
   851
  simproc.
wenzelm@26782
   852
wenzelm@26782
   853
  Morphisms and identifiers are only relevant for simprocs that are
wenzelm@26782
   854
  defined within a local target context, e.g.\ in a locale.
wenzelm@26782
   855
wenzelm@28760
   856
  \item @{text "simproc add: name"} and @{text "simproc del: name"}
wenzelm@26782
   857
  add or delete named simprocs to the current Simplifier context.  The
wenzelm@26782
   858
  default is to add a simproc.  Note that @{command "simproc_setup"}
wenzelm@26782
   859
  already adds the new simproc to the subsequent context.
wenzelm@26782
   860
wenzelm@28760
   861
  \end{description}
wenzelm@26782
   862
*}
wenzelm@26782
   863
wenzelm@26782
   864
wenzelm@42925
   865
subsubsection {* Example *}
wenzelm@42925
   866
wenzelm@42925
   867
text {* The following simplification procedure for @{thm
wenzelm@42925
   868
  [source=false, show_types] unit_eq} in HOL performs fine-grained
wenzelm@42925
   869
  control over rule application, beyond higher-order pattern matching.
wenzelm@42925
   870
  Declaring @{thm unit_eq} as @{attribute simp} directly would make
wenzelm@42925
   871
  the simplifier loop!  Note that a version of this simplification
wenzelm@42925
   872
  procedure is already active in Isabelle/HOL.  *}
wenzelm@42925
   873
wenzelm@42925
   874
simproc_setup unit ("x::unit") = {*
wenzelm@42925
   875
  fn _ => fn _ => fn ct =>
wenzelm@42925
   876
    if HOLogic.is_unit (term_of ct) then NONE
wenzelm@42925
   877
    else SOME (mk_meta_eq @{thm unit_eq})
wenzelm@42925
   878
*}
wenzelm@42925
   879
wenzelm@42925
   880
text {* Since the Simplifier applies simplification procedures
wenzelm@42925
   881
  frequently, it is important to make the failure check in ML
wenzelm@42925
   882
  reasonably fast. *}
wenzelm@42925
   883
wenzelm@42925
   884
wenzelm@50079
   885
subsection {* Configurable Simplifier strategies \label{sec:simp-strategies} *}
wenzelm@50079
   886
wenzelm@50079
   887
text {* The core term-rewriting engine of the Simplifier is normally
wenzelm@50079
   888
  used in combination with some add-on components that modify the
wenzelm@50079
   889
  strategy and allow to integrate other non-Simplifier proof tools.
wenzelm@50079
   890
  These may be reconfigured in ML as explained below.  Even if the
wenzelm@50079
   891
  default strategies of object-logics like Isabelle/HOL are used
wenzelm@50079
   892
  unchanged, it helps to understand how the standard Simplifier
wenzelm@50079
   893
  strategies work. *}
wenzelm@50079
   894
wenzelm@50079
   895
wenzelm@50079
   896
subsubsection {* The subgoaler *}
wenzelm@50079
   897
wenzelm@50079
   898
text {*
wenzelm@50079
   899
  \begin{mldecls}
wenzelm@50079
   900
  @{index_ML Simplifier.set_subgoaler: "(simpset -> int -> tactic) ->
wenzelm@50079
   901
  simpset -> simpset"} \\
wenzelm@50079
   902
  @{index_ML Simplifier.prems_of: "simpset -> thm list"} \\
wenzelm@50079
   903
  \end{mldecls}
wenzelm@50079
   904
wenzelm@50079
   905
  The subgoaler is the tactic used to solve subgoals arising out of
wenzelm@50079
   906
  conditional rewrite rules or congruence rules.  The default should
wenzelm@50079
   907
  be simplification itself.  In rare situations, this strategy may
wenzelm@50079
   908
  need to be changed.  For example, if the premise of a conditional
wenzelm@50079
   909
  rule is an instance of its conclusion, as in @{text "Suc ?m < ?n \<Longrightarrow>
wenzelm@50079
   910
  ?m < ?n"}, the default strategy could loop.  % FIXME !??
wenzelm@50079
   911
wenzelm@50079
   912
  \begin{description}
wenzelm@50079
   913
wenzelm@50079
   914
  \item @{ML Simplifier.set_subgoaler}~@{text "ss tac"} sets the
wenzelm@50079
   915
  subgoaler of simpset @{text "ss"} to @{text "tac"}.  The tactic will
wenzelm@50079
   916
  be applied to the context of the running Simplifier instance,
wenzelm@50079
   917
  expressed as a simpset.
wenzelm@50079
   918
wenzelm@50079
   919
  \item @{ML Simplifier.prems_of}~@{text "ss"} retrieves the current
wenzelm@50079
   920
  set of premises from simpset @{text "ss"} that represents the
wenzelm@50079
   921
  context of the running Simplifier.  This may be non-empty only if
wenzelm@50079
   922
  the Simplifier has been told to utilize local assumptions in the
wenzelm@50079
   923
  first place (cf.\ the options in \secref{sec:simp-meth}).
wenzelm@50079
   924
wenzelm@50079
   925
  \end{description}
wenzelm@50079
   926
wenzelm@50079
   927
  As an example, consider the following alternative subgoaler:
wenzelm@50079
   928
*}
wenzelm@50079
   929
wenzelm@50079
   930
ML {*
wenzelm@50079
   931
  fun subgoaler_tac ss =
wenzelm@50079
   932
    assume_tac ORELSE'
wenzelm@50079
   933
    resolve_tac (Simplifier.prems_of ss) ORELSE'
wenzelm@50079
   934
    asm_simp_tac ss
wenzelm@50079
   935
*}
wenzelm@50079
   936
wenzelm@50079
   937
text {* This tactic first tries to solve the subgoal by assumption or
wenzelm@50079
   938
  by resolving with with one of the premises, calling simplification
wenzelm@50079
   939
  only if that fails. *}
wenzelm@50079
   940
wenzelm@50079
   941
wenzelm@50079
   942
subsubsection {* The solver *}
wenzelm@50079
   943
wenzelm@50079
   944
text {*
wenzelm@50079
   945
  \begin{mldecls}
wenzelm@50079
   946
  @{index_ML_type solver} \\
wenzelm@50079
   947
  @{index_ML Simplifier.mk_solver: "string -> (simpset -> int -> tactic) ->
wenzelm@50079
   948
  solver"} \\
wenzelm@50079
   949
  @{index_ML_op setSolver: "simpset * solver -> simpset"} \\
wenzelm@50079
   950
  @{index_ML_op addSolver: "simpset * solver -> simpset"} \\
wenzelm@50079
   951
  @{index_ML_op setSSolver: "simpset * solver -> simpset"} \\
wenzelm@50079
   952
  @{index_ML_op addSSolver: "simpset * solver -> simpset"} \\
wenzelm@50079
   953
  \end{mldecls}
wenzelm@50079
   954
wenzelm@50079
   955
  A solver is a tactic that attempts to solve a subgoal after
wenzelm@50079
   956
  simplification.  Its core functionality is to prove trivial subgoals
wenzelm@50079
   957
  such as @{prop "True"} and @{text "t = t"}, but object-logics might
wenzelm@50079
   958
  be more ambitious.  For example, Isabelle/HOL performs a restricted
wenzelm@50079
   959
  version of linear arithmetic here.
wenzelm@50079
   960
wenzelm@50079
   961
  Solvers are packaged up in abstract type @{ML_type solver}, with
wenzelm@50079
   962
  @{ML Simplifier.mk_solver} as the only operation to create a solver.
wenzelm@50079
   963
wenzelm@50079
   964
  \medskip Rewriting does not instantiate unknowns.  For example,
wenzelm@50079
   965
  rewriting alone cannot prove @{text "a \<in> ?A"} since this requires
wenzelm@50079
   966
  instantiating @{text "?A"}.  The solver, however, is an arbitrary
wenzelm@50079
   967
  tactic and may instantiate unknowns as it pleases.  This is the only
wenzelm@50079
   968
  way the Simplifier can handle a conditional rewrite rule whose
wenzelm@50079
   969
  condition contains extra variables.  When a simplification tactic is
wenzelm@50079
   970
  to be combined with other provers, especially with the Classical
wenzelm@50079
   971
  Reasoner, it is important whether it can be considered safe or not.
wenzelm@50079
   972
  For this reason a simpset contains two solvers: safe and unsafe.
wenzelm@50079
   973
wenzelm@50079
   974
  The standard simplification strategy solely uses the unsafe solver,
wenzelm@50079
   975
  which is appropriate in most cases.  For special applications where
wenzelm@50079
   976
  the simplification process is not allowed to instantiate unknowns
wenzelm@50079
   977
  within the goal, simplification starts with the safe solver, but may
wenzelm@50079
   978
  still apply the ordinary unsafe one in nested simplifications for
wenzelm@50079
   979
  conditional rules or congruences. Note that in this way the overall
wenzelm@50079
   980
  tactic is not totally safe: it may instantiate unknowns that appear
wenzelm@50079
   981
  also in other subgoals.
wenzelm@50079
   982
wenzelm@50079
   983
  \begin{description}
wenzelm@50079
   984
wenzelm@50079
   985
  \item @{ML Simplifier.mk_solver}~@{text "name tac"} turns @{text
wenzelm@50079
   986
  "tac"} into a solver; the @{text "name"} is only attached as a
wenzelm@50079
   987
  comment and has no further significance.
wenzelm@50079
   988
wenzelm@50079
   989
  \item @{text "ss setSSolver solver"} installs @{text "solver"} as
wenzelm@50079
   990
  the safe solver of @{text "ss"}.
wenzelm@50079
   991
wenzelm@50079
   992
  \item @{text "ss addSSolver solver"} adds @{text "solver"} as an
wenzelm@50079
   993
  additional safe solver; it will be tried after the solvers which had
wenzelm@50079
   994
  already been present in @{text "ss"}.
wenzelm@50079
   995
wenzelm@50079
   996
  \item @{text "ss setSolver solver"} installs @{text "solver"} as the
wenzelm@50079
   997
  unsafe solver of @{text "ss"}.
wenzelm@50079
   998
wenzelm@50079
   999
  \item @{text "ss addSolver solver"} adds @{text "solver"} as an
wenzelm@50079
  1000
  additional unsafe solver; it will be tried after the solvers which
wenzelm@50079
  1001
  had already been present in @{text "ss"}.
wenzelm@50079
  1002
wenzelm@50079
  1003
  \end{description}
wenzelm@50079
  1004
wenzelm@50079
  1005
  \medskip The solver tactic is invoked with a simpset that represents
wenzelm@50079
  1006
  the context of the running Simplifier.  Further simpset operations
wenzelm@50079
  1007
  may be used to retrieve relevant information, such as the list of
wenzelm@50079
  1008
  local Simplifier premises via @{ML Simplifier.prems_of} --- this
wenzelm@50079
  1009
  list may be non-empty only if the Simplifier runs in a mode that
wenzelm@50079
  1010
  utilizes local assumptions (see also \secref{sec:simp-meth}).  The
wenzelm@50079
  1011
  solver is also presented the full goal including its assumptions in
wenzelm@50079
  1012
  any case.  Thus it can use these (e.g.\ by calling @{ML
wenzelm@50079
  1013
  assume_tac}), even if the Simplifier proper happens to ignore local
wenzelm@50079
  1014
  premises at the moment.
wenzelm@50079
  1015
wenzelm@50079
  1016
  \medskip As explained before, the subgoaler is also used to solve
wenzelm@50079
  1017
  the premises of congruence rules.  These are usually of the form
wenzelm@50079
  1018
  @{text "s = ?x"}, where @{text "s"} needs to be simplified and
wenzelm@50079
  1019
  @{text "?x"} needs to be instantiated with the result.  Typically,
wenzelm@50079
  1020
  the subgoaler will invoke the Simplifier at some point, which will
wenzelm@50079
  1021
  eventually call the solver.  For this reason, solver tactics must be
wenzelm@50079
  1022
  prepared to solve goals of the form @{text "t = ?x"}, usually by
wenzelm@50079
  1023
  reflexivity.  In particular, reflexivity should be tried before any
wenzelm@50079
  1024
  of the fancy automated proof tools.
wenzelm@50079
  1025
wenzelm@50079
  1026
  It may even happen that due to simplification the subgoal is no
wenzelm@50079
  1027
  longer an equality.  For example, @{text "False \<longleftrightarrow> ?Q"} could be
wenzelm@50079
  1028
  rewritten to @{text "\<not> ?Q"}.  To cover this case, the solver could
wenzelm@50079
  1029
  try resolving with the theorem @{text "\<not> False"} of the
wenzelm@50079
  1030
  object-logic.
wenzelm@50079
  1031
wenzelm@50079
  1032
  \medskip
wenzelm@50079
  1033
wenzelm@50079
  1034
  \begin{warn}
wenzelm@50079
  1035
  If a premise of a congruence rule cannot be proved, then the
wenzelm@50079
  1036
  congruence is ignored.  This should only happen if the rule is
wenzelm@50079
  1037
  \emph{conditional} --- that is, contains premises not of the form
wenzelm@50079
  1038
  @{text "t = ?x"}.  Otherwise it indicates that some congruence rule,
wenzelm@50079
  1039
  or possibly the subgoaler or solver, is faulty.
wenzelm@50079
  1040
  \end{warn}
wenzelm@50079
  1041
*}
wenzelm@50079
  1042
wenzelm@50079
  1043
wenzelm@50079
  1044
subsubsection {* The looper *}
wenzelm@50079
  1045
wenzelm@50079
  1046
text {*
wenzelm@50079
  1047
  \begin{mldecls}
wenzelm@50079
  1048
  @{index_ML_op setloop: "simpset * (int -> tactic) -> simpset"} \\
wenzelm@50079
  1049
  @{index_ML_op setloop': "simpset * (simpset -> int -> tactic) -> simpset"} \\
wenzelm@50079
  1050
  @{index_ML_op addloop: "simpset * (string * (int -> tactic)) -> simpset"} \\
wenzelm@50079
  1051
  @{index_ML_op addloop': "simpset * (string * (simpset -> int -> tactic))
wenzelm@50079
  1052
  -> simpset"} \\
wenzelm@50079
  1053
  @{index_ML_op delloop: "simpset * string -> simpset"} \\
wenzelm@50079
  1054
  @{index_ML_op Splitter.add_split: "thm -> simpset -> simpset"} \\
wenzelm@50079
  1055
  @{index_ML_op Splitter.del_split: "thm -> simpset -> simpset"} \\
wenzelm@50079
  1056
  \end{mldecls}
wenzelm@50079
  1057
wenzelm@50079
  1058
  The looper is a list of tactics that are applied after
wenzelm@50079
  1059
  simplification, in case the solver failed to solve the simplified
wenzelm@50079
  1060
  goal.  If the looper succeeds, the simplification process is started
wenzelm@50079
  1061
  all over again.  Each of the subgoals generated by the looper is
wenzelm@50079
  1062
  attacked in turn, in reverse order.
wenzelm@50079
  1063
wenzelm@50079
  1064
  A typical looper is \emph{case splitting}: the expansion of a
wenzelm@50079
  1065
  conditional.  Another possibility is to apply an elimination rule on
wenzelm@50079
  1066
  the assumptions.  More adventurous loopers could start an induction.
wenzelm@50079
  1067
wenzelm@50079
  1068
  \begin{description}
wenzelm@50079
  1069
wenzelm@50079
  1070
  \item @{text "ss setloop tac"} installs @{text "tac"} as the only
wenzelm@50079
  1071
  looper tactic of @{text "ss"}.  The variant @{text "setloop'"}
wenzelm@50079
  1072
  allows the tactic to depend on the running Simplifier context, which
wenzelm@50079
  1073
  is represented as simpset.
wenzelm@50079
  1074
wenzelm@50079
  1075
  \item @{text "ss addloop (name, tac)"} adds @{text "tac"} as an
wenzelm@50079
  1076
  additional looper tactic with name @{text "name"}, which is
wenzelm@50079
  1077
  significant for managing the collection of loopers.  The tactic will
wenzelm@50079
  1078
  be tried after the looper tactics that had already been present in
wenzelm@50079
  1079
  @{text "ss"}.  The variant @{text "addloop'"} allows the tactic to
wenzelm@50079
  1080
  depend on the running Simplifier context, which is represented as
wenzelm@50079
  1081
  simpset.
wenzelm@50079
  1082
wenzelm@50079
  1083
  \item @{text "ss delloop name"} deletes the looper tactic that was
wenzelm@50079
  1084
  associated with @{text "name"} from @{text "ss"}.
wenzelm@50079
  1085
wenzelm@50079
  1086
  \item @{ML Splitter.add_split}~@{text "thm ss"} adds split tactics
wenzelm@50079
  1087
  for @{text "thm"} as additional looper tactics of @{text "ss"}.
wenzelm@50079
  1088
wenzelm@50079
  1089
  \item @{ML Splitter.del_split}~@{text "thm ss"} deletes the split
wenzelm@50079
  1090
  tactic corresponding to @{text thm} from the looper tactics of
wenzelm@50079
  1091
  @{text "ss"}.
wenzelm@50079
  1092
wenzelm@50079
  1093
  \end{description}
wenzelm@50079
  1094
wenzelm@50079
  1095
  The splitter replaces applications of a given function; the
wenzelm@50079
  1096
  right-hand side of the replacement can be anything.  For example,
wenzelm@50079
  1097
  here is a splitting rule for conditional expressions:
wenzelm@50079
  1098
wenzelm@50079
  1099
  @{text [display] "?P (if ?Q ?x ?y) \<longleftrightarrow> (?Q \<longrightarrow> ?P ?x) \<and> (\<not> ?Q \<longrightarrow> ?P ?y)"}
wenzelm@50079
  1100
wenzelm@50079
  1101
  Another example is the elimination operator for Cartesian products
wenzelm@50079
  1102
  (which happens to be called @{text split} in Isabelle/HOL:
wenzelm@50079
  1103
wenzelm@50079
  1104
  @{text [display] "?P (split ?f ?p) \<longleftrightarrow> (\<forall>a b. ?p = (a, b) \<longrightarrow> ?P (f a b))"}
wenzelm@50079
  1105
wenzelm@50079
  1106
  For technical reasons, there is a distinction between case splitting
wenzelm@50079
  1107
  in the conclusion and in the premises of a subgoal.  The former is
wenzelm@50079
  1108
  done by @{ML Splitter.split_tac} with rules like @{thm [source]
wenzelm@50079
  1109
  split_if} or @{thm [source] option.split}, which do not split the
wenzelm@50079
  1110
  subgoal, while the latter is done by @{ML Splitter.split_asm_tac}
wenzelm@50079
  1111
  with rules like @{thm [source] split_if_asm} or @{thm [source]
wenzelm@50079
  1112
  option.split_asm}, which split the subgoal.  The function @{ML
wenzelm@50079
  1113
  Splitter.add_split} automatically takes care of which tactic to
wenzelm@50079
  1114
  call, analyzing the form of the rules given as argument; it is the
wenzelm@50079
  1115
  same operation behind @{text "split"} attribute or method modifier
wenzelm@50079
  1116
  syntax in the Isar source language.
wenzelm@50079
  1117
wenzelm@50079
  1118
  Case splits should be allowed only when necessary; they are
wenzelm@50079
  1119
  expensive and hard to control.  Case-splitting on if-expressions in
wenzelm@50079
  1120
  the conclusion is usually beneficial, so it is enabled by default in
wenzelm@50079
  1121
  Isabelle/HOL and Isabelle/FOL/ZF.
wenzelm@50079
  1122
wenzelm@50079
  1123
  \begin{warn}
wenzelm@50079
  1124
  With @{ML Splitter.split_asm_tac} as looper component, the
wenzelm@50079
  1125
  Simplifier may split subgoals!  This might cause unexpected problems
wenzelm@50079
  1126
  in tactic expressions that silently assume 0 or 1 subgoals after
wenzelm@50079
  1127
  simplification.
wenzelm@50079
  1128
  \end{warn}
wenzelm@50079
  1129
*}
wenzelm@50079
  1130
wenzelm@50079
  1131
wenzelm@50063
  1132
subsection {* Forward simplification \label{sec:simp-forward} *}
wenzelm@26782
  1133
wenzelm@26782
  1134
text {*
wenzelm@26782
  1135
  \begin{matharray}{rcl}
wenzelm@28761
  1136
    @{attribute_def simplified} & : & @{text attribute} \\
wenzelm@26782
  1137
  \end{matharray}
wenzelm@26782
  1138
wenzelm@42596
  1139
  @{rail "
wenzelm@42596
  1140
    @@{attribute simplified} opt? @{syntax thmrefs}?
wenzelm@26782
  1141
    ;
wenzelm@26782
  1142
wenzelm@40255
  1143
    opt: '(' ('no_asm' | 'no_asm_simp' | 'no_asm_use') ')'
wenzelm@42596
  1144
  "}
wenzelm@26782
  1145
wenzelm@28760
  1146
  \begin{description}
wenzelm@26782
  1147
  
wenzelm@28760
  1148
  \item @{attribute simplified}~@{text "a\<^sub>1 \<dots> a\<^sub>n"} causes a theorem to
wenzelm@28760
  1149
  be simplified, either by exactly the specified rules @{text "a\<^sub>1, \<dots>,
wenzelm@28760
  1150
  a\<^sub>n"}, or the implicit Simplifier context if no arguments are given.
wenzelm@28760
  1151
  The result is fully simplified by default, including assumptions and
wenzelm@28760
  1152
  conclusion; the options @{text no_asm} etc.\ tune the Simplifier in
wenzelm@28760
  1153
  the same way as the for the @{text simp} method.
wenzelm@26782
  1154
wenzelm@26782
  1155
  Note that forward simplification restricts the simplifier to its
wenzelm@26782
  1156
  most basic operation of term rewriting; solver and looper tactics
wenzelm@50079
  1157
  (\secref{sec:simp-strategies}) are \emph{not} involved here.  The
wenzelm@50079
  1158
  @{attribute simplified} attribute should be only rarely required
wenzelm@50079
  1159
  under normal circumstances.
wenzelm@26782
  1160
wenzelm@28760
  1161
  \end{description}
wenzelm@26782
  1162
*}
wenzelm@26782
  1163
wenzelm@26782
  1164
wenzelm@27040
  1165
section {* The Classical Reasoner \label{sec:classical} *}
wenzelm@26782
  1166
wenzelm@42930
  1167
subsection {* Basic concepts *}
wenzelm@42927
  1168
wenzelm@42927
  1169
text {* Although Isabelle is generic, many users will be working in
wenzelm@42927
  1170
  some extension of classical first-order logic.  Isabelle/ZF is built
wenzelm@42927
  1171
  upon theory FOL, while Isabelle/HOL conceptually contains
wenzelm@42927
  1172
  first-order logic as a fragment.  Theorem-proving in predicate logic
wenzelm@42927
  1173
  is undecidable, but many automated strategies have been developed to
wenzelm@42927
  1174
  assist in this task.
wenzelm@42927
  1175
wenzelm@42927
  1176
  Isabelle's classical reasoner is a generic package that accepts
wenzelm@42927
  1177
  certain information about a logic and delivers a suite of automatic
wenzelm@42927
  1178
  proof tools, based on rules that are classified and declared in the
wenzelm@42927
  1179
  context.  These proof procedures are slow and simplistic compared
wenzelm@42927
  1180
  with high-end automated theorem provers, but they can save
wenzelm@42927
  1181
  considerable time and effort in practice.  They can prove theorems
wenzelm@42927
  1182
  such as Pelletier's \cite{pelletier86} problems 40 and 41 in a few
wenzelm@42927
  1183
  milliseconds (including full proof reconstruction): *}
wenzelm@42927
  1184
wenzelm@42927
  1185
lemma "(\<exists>y. \<forall>x. F x y \<longleftrightarrow> F x x) \<longrightarrow> \<not> (\<forall>x. \<exists>y. \<forall>z. F z y \<longleftrightarrow> \<not> F z x)"
wenzelm@42927
  1186
  by blast
wenzelm@42927
  1187
wenzelm@42927
  1188
lemma "(\<forall>z. \<exists>y. \<forall>x. f x y \<longleftrightarrow> f x z \<and> \<not> f x x) \<longrightarrow> \<not> (\<exists>z. \<forall>x. f x z)"
wenzelm@42927
  1189
  by blast
wenzelm@42927
  1190
wenzelm@42927
  1191
text {* The proof tools are generic.  They are not restricted to
wenzelm@42927
  1192
  first-order logic, and have been heavily used in the development of
wenzelm@42927
  1193
  the Isabelle/HOL library and applications.  The tactics can be
wenzelm@42927
  1194
  traced, and their components can be called directly; in this manner,
wenzelm@42927
  1195
  any proof can be viewed interactively.  *}
wenzelm@42927
  1196
wenzelm@42927
  1197
wenzelm@42927
  1198
subsubsection {* The sequent calculus *}
wenzelm@42927
  1199
wenzelm@42927
  1200
text {* Isabelle supports natural deduction, which is easy to use for
wenzelm@42927
  1201
  interactive proof.  But natural deduction does not easily lend
wenzelm@42927
  1202
  itself to automation, and has a bias towards intuitionism.  For
wenzelm@42927
  1203
  certain proofs in classical logic, it can not be called natural.
wenzelm@42927
  1204
  The \emph{sequent calculus}, a generalization of natural deduction,
wenzelm@42927
  1205
  is easier to automate.
wenzelm@42927
  1206
wenzelm@42927
  1207
  A \textbf{sequent} has the form @{text "\<Gamma> \<turnstile> \<Delta>"}, where @{text "\<Gamma>"}
wenzelm@42927
  1208
  and @{text "\<Delta>"} are sets of formulae.\footnote{For first-order
wenzelm@42927
  1209
  logic, sequents can equivalently be made from lists or multisets of
wenzelm@42927
  1210
  formulae.} The sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} is
wenzelm@42927
  1211
  \textbf{valid} if @{text "P\<^sub>1 \<and> \<dots> \<and> P\<^sub>m"} implies @{text "Q\<^sub>1 \<or> \<dots> \<or>
wenzelm@42927
  1212
  Q\<^sub>n"}.  Thus @{text "P\<^sub>1, \<dots>, P\<^sub>m"} represent assumptions, each of which
wenzelm@42927
  1213
  is true, while @{text "Q\<^sub>1, \<dots>, Q\<^sub>n"} represent alternative goals.  A
wenzelm@42927
  1214
  sequent is \textbf{basic} if its left and right sides have a common
wenzelm@42927
  1215
  formula, as in @{text "P, Q \<turnstile> Q, R"}; basic sequents are trivially
wenzelm@42927
  1216
  valid.
wenzelm@42927
  1217
wenzelm@42927
  1218
  Sequent rules are classified as \textbf{right} or \textbf{left},
wenzelm@42927
  1219
  indicating which side of the @{text "\<turnstile>"} symbol they operate on.
wenzelm@42927
  1220
  Rules that operate on the right side are analogous to natural
wenzelm@42927
  1221
  deduction's introduction rules, and left rules are analogous to
wenzelm@42927
  1222
  elimination rules.  The sequent calculus analogue of @{text "(\<longrightarrow>I)"}
wenzelm@42927
  1223
  is the rule
wenzelm@42927
  1224
  \[
wenzelm@42927
  1225
  \infer[@{text "(\<longrightarrow>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<longrightarrow> Q"}}{@{text "P, \<Gamma> \<turnstile> \<Delta>, Q"}}
wenzelm@42927
  1226
  \]
wenzelm@42927
  1227
  Applying the rule backwards, this breaks down some implication on
wenzelm@42927
  1228
  the right side of a sequent; @{text "\<Gamma>"} and @{text "\<Delta>"} stand for
wenzelm@42927
  1229
  the sets of formulae that are unaffected by the inference.  The
wenzelm@42927
  1230
  analogue of the pair @{text "(\<or>I1)"} and @{text "(\<or>I2)"} is the
wenzelm@42927
  1231
  single rule
wenzelm@42927
  1232
  \[
wenzelm@42927
  1233
  \infer[@{text "(\<or>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, P \<or> Q"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P, Q"}}
wenzelm@42927
  1234
  \]
wenzelm@42927
  1235
  This breaks down some disjunction on the right side, replacing it by
wenzelm@42927
  1236
  both disjuncts.  Thus, the sequent calculus is a kind of
wenzelm@42927
  1237
  multiple-conclusion logic.
wenzelm@42927
  1238
wenzelm@42927
  1239
  To illustrate the use of multiple formulae on the right, let us
wenzelm@42927
  1240
  prove the classical theorem @{text "(P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}.  Working
wenzelm@42927
  1241
  backwards, we reduce this formula to a basic sequent:
wenzelm@42927
  1242
  \[
wenzelm@42927
  1243
  \infer[@{text "(\<or>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)"}}
wenzelm@42927
  1244
    {\infer[@{text "(\<longrightarrow>R)"}]{@{text "\<turnstile> (P \<longrightarrow> Q), (Q \<longrightarrow> P)"}}
wenzelm@42927
  1245
      {\infer[@{text "(\<longrightarrow>R)"}]{@{text "P \<turnstile> Q, (Q \<longrightarrow> P)"}}
wenzelm@42927
  1246
        {@{text "P, Q \<turnstile> Q, P"}}}}
wenzelm@42927
  1247
  \]
wenzelm@42927
  1248
wenzelm@42927
  1249
  This example is typical of the sequent calculus: start with the
wenzelm@42927
  1250
  desired theorem and apply rules backwards in a fairly arbitrary
wenzelm@42927
  1251
  manner.  This yields a surprisingly effective proof procedure.
wenzelm@42927
  1252
  Quantifiers add only few complications, since Isabelle handles
wenzelm@42927
  1253
  parameters and schematic variables.  See \cite[Chapter
wenzelm@42927
  1254
  10]{paulson-ml2} for further discussion.  *}
wenzelm@42927
  1255
wenzelm@42927
  1256
wenzelm@42927
  1257
subsubsection {* Simulating sequents by natural deduction *}
wenzelm@42927
  1258
wenzelm@42927
  1259
text {* Isabelle can represent sequents directly, as in the
wenzelm@42927
  1260
  object-logic LK.  But natural deduction is easier to work with, and
wenzelm@42927
  1261
  most object-logics employ it.  Fortunately, we can simulate the
wenzelm@42927
  1262
  sequent @{text "P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n"} by the Isabelle formula
wenzelm@42927
  1263
  @{text "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>2 \<Longrightarrow> ... \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> Q\<^sub>1"} where the order of
wenzelm@42927
  1264
  the assumptions and the choice of @{text "Q\<^sub>1"} are arbitrary.
wenzelm@42927
  1265
  Elim-resolution plays a key role in simulating sequent proofs.
wenzelm@42927
  1266
wenzelm@42927
  1267
  We can easily handle reasoning on the left.  Elim-resolution with
wenzelm@42927
  1268
  the rules @{text "(\<or>E)"}, @{text "(\<bottom>E)"} and @{text "(\<exists>E)"} achieves
wenzelm@42927
  1269
  a similar effect as the corresponding sequent rules.  For the other
wenzelm@42927
  1270
  connectives, we use sequent-style elimination rules instead of
wenzelm@42927
  1271
  destruction rules such as @{text "(\<and>E1, 2)"} and @{text "(\<forall>E)"}.
wenzelm@42927
  1272
  But note that the rule @{text "(\<not>L)"} has no effect under our
wenzelm@42927
  1273
  representation of sequents!
wenzelm@42927
  1274
  \[
wenzelm@42927
  1275
  \infer[@{text "(\<not>L)"}]{@{text "\<not> P, \<Gamma> \<turnstile> \<Delta>"}}{@{text "\<Gamma> \<turnstile> \<Delta>, P"}}
wenzelm@42927
  1276
  \]
wenzelm@42927
  1277
wenzelm@42927
  1278
  What about reasoning on the right?  Introduction rules can only
wenzelm@42927
  1279
  affect the formula in the conclusion, namely @{text "Q\<^sub>1"}.  The
wenzelm@42927
  1280
  other right-side formulae are represented as negated assumptions,
wenzelm@42927
  1281
  @{text "\<not> Q\<^sub>2, \<dots>, \<not> Q\<^sub>n"}.  In order to operate on one of these, it
wenzelm@42927
  1282
  must first be exchanged with @{text "Q\<^sub>1"}.  Elim-resolution with the
wenzelm@42927
  1283
  @{text swap} rule has this effect: @{text "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"}
wenzelm@42927
  1284
wenzelm@42927
  1285
  To ensure that swaps occur only when necessary, each introduction
wenzelm@42927
  1286
  rule is converted into a swapped form: it is resolved with the
wenzelm@42927
  1287
  second premise of @{text "(swap)"}.  The swapped form of @{text
wenzelm@42927
  1288
  "(\<and>I)"}, which might be called @{text "(\<not>\<and>E)"}, is
wenzelm@42927
  1289
  @{text [display] "\<not> (P \<and> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> (\<not> R \<Longrightarrow> Q) \<Longrightarrow> R"}
wenzelm@42927
  1290
wenzelm@42927
  1291
  Similarly, the swapped form of @{text "(\<longrightarrow>I)"} is
wenzelm@42927
  1292
  @{text [display] "\<not> (P \<longrightarrow> Q) \<Longrightarrow> (\<not> R \<Longrightarrow> P \<Longrightarrow> Q) \<Longrightarrow> R"}
wenzelm@42927
  1293
wenzelm@42927
  1294
  Swapped introduction rules are applied using elim-resolution, which
wenzelm@42927
  1295
  deletes the negated formula.  Our representation of sequents also
wenzelm@42927
  1296
  requires the use of ordinary introduction rules.  If we had no
wenzelm@42927
  1297
  regard for readability of intermediate goal states, we could treat
wenzelm@42927
  1298
  the right side more uniformly by representing sequents as @{text
wenzelm@42927
  1299
  [display] "P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> \<bottom>"}
wenzelm@42927
  1300
*}
wenzelm@42927
  1301
wenzelm@42927
  1302
wenzelm@42927
  1303
subsubsection {* Extra rules for the sequent calculus *}
wenzelm@42927
  1304
wenzelm@42927
  1305
text {* As mentioned, destruction rules such as @{text "(\<and>E1, 2)"} and
wenzelm@42927
  1306
  @{text "(\<forall>E)"} must be replaced by sequent-style elimination rules.
wenzelm@42927
  1307
  In addition, we need rules to embody the classical equivalence
wenzelm@42927
  1308
  between @{text "P \<longrightarrow> Q"} and @{text "\<not> P \<or> Q"}.  The introduction
wenzelm@42927
  1309
  rules @{text "(\<or>I1, 2)"} are replaced by a rule that simulates
wenzelm@42927
  1310
  @{text "(\<or>R)"}: @{text [display] "(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q"}
wenzelm@42927
  1311
wenzelm@42927
  1312
  The destruction rule @{text "(\<longrightarrow>E)"} is replaced by @{text [display]
wenzelm@42927
  1313
  "(P \<longrightarrow> Q) \<Longrightarrow> (\<not> P \<Longrightarrow> R) \<Longrightarrow> (Q \<Longrightarrow> R) \<Longrightarrow> R"}
wenzelm@42927
  1314
wenzelm@42927
  1315
  Quantifier replication also requires special rules.  In classical
wenzelm@42927
  1316
  logic, @{text "\<exists>x. P x"} is equivalent to @{text "\<not> (\<forall>x. \<not> P x)"};
wenzelm@42927
  1317
  the rules @{text "(\<exists>R)"} and @{text "(\<forall>L)"} are dual:
wenzelm@42927
  1318
  \[
wenzelm@42927
  1319
  \infer[@{text "(\<exists>R)"}]{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x"}}{@{text "\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x, P t"}}
wenzelm@42927
  1320
  \qquad
wenzelm@42927
  1321
  \infer[@{text "(\<forall>L)"}]{@{text "\<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}{@{text "P t, \<forall>x. P x, \<Gamma> \<turnstile> \<Delta>"}}
wenzelm@42927
  1322
  \]
wenzelm@42927
  1323
  Thus both kinds of quantifier may be replicated.  Theorems requiring
wenzelm@42927
  1324
  multiple uses of a universal formula are easy to invent; consider
wenzelm@42927
  1325
  @{text [display] "(\<forall>x. P x \<longrightarrow> P (f x)) \<and> P a \<longrightarrow> P (f\<^sup>n a)"} for any
wenzelm@42927
  1326
  @{text "n > 1"}.  Natural examples of the multiple use of an
wenzelm@42927
  1327
  existential formula are rare; a standard one is @{text "\<exists>x. \<forall>y. P x
wenzelm@42927
  1328
  \<longrightarrow> P y"}.
wenzelm@42927
  1329
wenzelm@42927
  1330
  Forgoing quantifier replication loses completeness, but gains
wenzelm@42927
  1331
  decidability, since the search space becomes finite.  Many useful
wenzelm@42927
  1332
  theorems can be proved without replication, and the search generally
wenzelm@42927
  1333
  delivers its verdict in a reasonable time.  To adopt this approach,
wenzelm@42927
  1334
  represent the sequent rules @{text "(\<exists>R)"}, @{text "(\<exists>L)"} and
wenzelm@42927
  1335
  @{text "(\<forall>R)"} by @{text "(\<exists>I)"}, @{text "(\<exists>E)"} and @{text "(\<forall>I)"},
wenzelm@42927
  1336
  respectively, and put @{text "(\<forall>E)"} into elimination form: @{text
wenzelm@42927
  1337
  [display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"}
wenzelm@42927
  1338
wenzelm@42927
  1339
  Elim-resolution with this rule will delete the universal formula
wenzelm@42927
  1340
  after a single use.  To replicate universal quantifiers, replace the
wenzelm@42927
  1341
  rule by @{text [display] "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"}
wenzelm@42927
  1342
wenzelm@42927
  1343
  To replicate existential quantifiers, replace @{text "(\<exists>I)"} by
wenzelm@42927
  1344
  @{text [display] "(\<not> (\<exists>x. P x) \<Longrightarrow> P t) \<Longrightarrow> \<exists>x. P x"}
wenzelm@42927
  1345
wenzelm@42927
  1346
  All introduction rules mentioned above are also useful in swapped
wenzelm@42927
  1347
  form.
wenzelm@42927
  1348
wenzelm@42927
  1349
  Replication makes the search space infinite; we must apply the rules
wenzelm@42927
  1350
  with care.  The classical reasoner distinguishes between safe and
wenzelm@42927
  1351
  unsafe rules, applying the latter only when there is no alternative.
wenzelm@42927
  1352
  Depth-first search may well go down a blind alley; best-first search
wenzelm@42927
  1353
  is better behaved in an infinite search space.  However, quantifier
wenzelm@42927
  1354
  replication is too expensive to prove any but the simplest theorems.
wenzelm@42927
  1355
*}
wenzelm@42927
  1356
wenzelm@42927
  1357
wenzelm@42928
  1358
subsection {* Rule declarations *}
wenzelm@42928
  1359
wenzelm@42928
  1360
text {* The proof tools of the Classical Reasoner depend on
wenzelm@42928
  1361
  collections of rules declared in the context, which are classified
wenzelm@42928
  1362
  as introduction, elimination or destruction and as \emph{safe} or
wenzelm@42928
  1363
  \emph{unsafe}.  In general, safe rules can be attempted blindly,
wenzelm@42928
  1364
  while unsafe rules must be used with care.  A safe rule must never
wenzelm@42928
  1365
  reduce a provable goal to an unprovable set of subgoals.
wenzelm@42928
  1366
wenzelm@42928
  1367
  The rule @{text "P \<Longrightarrow> P \<or> Q"} is unsafe because it reduces @{text "P
wenzelm@42928
  1368
  \<or> Q"} to @{text "P"}, which might turn out as premature choice of an
wenzelm@42928
  1369
  unprovable subgoal.  Any rule is unsafe whose premises contain new
wenzelm@42928
  1370
  unknowns.  The elimination rule @{text "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> Q) \<Longrightarrow> Q"} is
wenzelm@42928
  1371
  unsafe, since it is applied via elim-resolution, which discards the
wenzelm@42928
  1372
  assumption @{text "\<forall>x. P x"} and replaces it by the weaker
wenzelm@42928
  1373
  assumption @{text "P t"}.  The rule @{text "P t \<Longrightarrow> \<exists>x. P x"} is
wenzelm@42928
  1374
  unsafe for similar reasons.  The quantifier duplication rule @{text
wenzelm@42928
  1375
  "\<forall>x. P x \<Longrightarrow> (P t \<Longrightarrow> \<forall>x. P x \<Longrightarrow> Q) \<Longrightarrow> Q"} is unsafe in a different sense:
wenzelm@42928
  1376
  since it keeps the assumption @{text "\<forall>x. P x"}, it is prone to
wenzelm@42928
  1377
  looping.  In classical first-order logic, all rules are safe except
wenzelm@42928
  1378
  those mentioned above.
wenzelm@42928
  1379
wenzelm@42928
  1380
  The safe~/ unsafe distinction is vague, and may be regarded merely
wenzelm@42928
  1381
  as a way of giving some rules priority over others.  One could argue
wenzelm@42928
  1382
  that @{text "(\<or>E)"} is unsafe, because repeated application of it
wenzelm@42928
  1383
  could generate exponentially many subgoals.  Induction rules are
wenzelm@42928
  1384
  unsafe because inductive proofs are difficult to set up
wenzelm@42928
  1385
  automatically.  Any inference is unsafe that instantiates an unknown
wenzelm@42928
  1386
  in the proof state --- thus matching must be used, rather than
wenzelm@42928
  1387
  unification.  Even proof by assumption is unsafe if it instantiates
wenzelm@42928
  1388
  unknowns shared with other subgoals.
wenzelm@42928
  1389
wenzelm@42928
  1390
  \begin{matharray}{rcl}
wenzelm@42928
  1391
    @{command_def "print_claset"}@{text "\<^sup>*"} & : & @{text "context \<rightarrow>"} \\
wenzelm@42928
  1392
    @{attribute_def intro} & : & @{text attribute} \\
wenzelm@42928
  1393
    @{attribute_def elim} & : & @{text attribute} \\
wenzelm@42928
  1394
    @{attribute_def dest} & : & @{text attribute} \\
wenzelm@42928
  1395
    @{attribute_def rule} & : & @{text attribute} \\
wenzelm@42928
  1396
    @{attribute_def iff} & : & @{text attribute} \\
wenzelm@42928
  1397
    @{attribute_def swapped} & : & @{text attribute} \\
wenzelm@42928
  1398
  \end{matharray}
wenzelm@42928
  1399
wenzelm@42928
  1400
  @{rail "
wenzelm@42928
  1401
    (@@{attribute intro} | @@{attribute elim} | @@{attribute dest}) ('!' | () | '?') @{syntax nat}?
wenzelm@42928
  1402
    ;
wenzelm@42928
  1403
    @@{attribute rule} 'del'
wenzelm@42928
  1404
    ;
wenzelm@42928
  1405
    @@{attribute iff} (((() | 'add') '?'?) | 'del')
wenzelm@42928
  1406
  "}
wenzelm@42928
  1407
wenzelm@42928
  1408
  \begin{description}
wenzelm@42928
  1409
wenzelm@42928
  1410
  \item @{command "print_claset"} prints the collection of rules
wenzelm@42928
  1411
  declared to the Classical Reasoner, i.e.\ the @{ML_type claset}
wenzelm@42928
  1412
  within the context.
wenzelm@42928
  1413
wenzelm@42928
  1414
  \item @{attribute intro}, @{attribute elim}, and @{attribute dest}
wenzelm@42928
  1415
  declare introduction, elimination, and destruction rules,
wenzelm@42928
  1416
  respectively.  By default, rules are considered as \emph{unsafe}
wenzelm@42928
  1417
  (i.e.\ not applied blindly without backtracking), while ``@{text
wenzelm@42928
  1418
  "!"}'' classifies as \emph{safe}.  Rule declarations marked by
wenzelm@42928
  1419
  ``@{text "?"}'' coincide with those of Isabelle/Pure, cf.\
wenzelm@42928
  1420
  \secref{sec:pure-meth-att} (i.e.\ are only applied in single steps
wenzelm@42928
  1421
  of the @{method rule} method).  The optional natural number
wenzelm@42928
  1422
  specifies an explicit weight argument, which is ignored by the
wenzelm@42928
  1423
  automated reasoning tools, but determines the search order of single
wenzelm@42928
  1424
  rule steps.
wenzelm@42928
  1425
wenzelm@42928
  1426
  Introduction rules are those that can be applied using ordinary
wenzelm@42928
  1427
  resolution.  Their swapped forms are generated internally, which
wenzelm@42928
  1428
  will be applied using elim-resolution.  Elimination rules are
wenzelm@42928
  1429
  applied using elim-resolution.  Rules are sorted by the number of
wenzelm@42928
  1430
  new subgoals they will yield; rules that generate the fewest
wenzelm@42928
  1431
  subgoals will be tried first.  Otherwise, later declarations take
wenzelm@42928
  1432
  precedence over earlier ones.
wenzelm@42928
  1433
wenzelm@42928
  1434
  Rules already present in the context with the same classification
wenzelm@42928
  1435
  are ignored.  A warning is printed if the rule has already been
wenzelm@42928
  1436
  added with some other classification, but the rule is added anyway
wenzelm@42928
  1437
  as requested.
wenzelm@42928
  1438
wenzelm@42928
  1439
  \item @{attribute rule}~@{text del} deletes all occurrences of a
wenzelm@42928
  1440
  rule from the classical context, regardless of its classification as
wenzelm@42928
  1441
  introduction~/ elimination~/ destruction and safe~/ unsafe.
wenzelm@42928
  1442
wenzelm@42928
  1443
  \item @{attribute iff} declares logical equivalences to the
wenzelm@42928
  1444
  Simplifier and the Classical reasoner at the same time.
wenzelm@42928
  1445
  Non-conditional rules result in a safe introduction and elimination
wenzelm@42928
  1446
  pair; conditional ones are considered unsafe.  Rules with negative
wenzelm@42928
  1447
  conclusion are automatically inverted (using @{text "\<not>"}-elimination
wenzelm@42928
  1448
  internally).
wenzelm@42928
  1449
wenzelm@42928
  1450
  The ``@{text "?"}'' version of @{attribute iff} declares rules to
wenzelm@42928
  1451
  the Isabelle/Pure context only, and omits the Simplifier
wenzelm@42928
  1452
  declaration.
wenzelm@42928
  1453
wenzelm@42928
  1454
  \item @{attribute swapped} turns an introduction rule into an
wenzelm@42928
  1455
  elimination, by resolving with the classical swap principle @{text
wenzelm@42928
  1456
  "\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R"} in the second position.  This is mainly for
wenzelm@42928
  1457
  illustrative purposes: the Classical Reasoner already swaps rules
wenzelm@42928
  1458
  internally as explained above.
wenzelm@42928
  1459
wenzelm@28760
  1460
  \end{description}
wenzelm@26782
  1461
*}
wenzelm@26782
  1462
wenzelm@26782
  1463
wenzelm@43365
  1464
subsection {* Structured methods *}
wenzelm@43365
  1465
wenzelm@43365
  1466
text {*
wenzelm@43365
  1467
  \begin{matharray}{rcl}
wenzelm@43365
  1468
    @{method_def rule} & : & @{text method} \\
wenzelm@43365
  1469
    @{method_def contradiction} & : & @{text method} \\
wenzelm@43365
  1470
  \end{matharray}
wenzelm@43365
  1471
wenzelm@43365
  1472
  @{rail "
wenzelm@43365
  1473
    @@{method rule} @{syntax thmrefs}?
wenzelm@43365
  1474
  "}
wenzelm@43365
  1475
wenzelm@43365
  1476
  \begin{description}
wenzelm@43365
  1477
wenzelm@43365
  1478
  \item @{method rule} as offered by the Classical Reasoner is a
wenzelm@43365
  1479
  refinement over the Pure one (see \secref{sec:pure-meth-att}).  Both
wenzelm@43365
  1480
  versions work the same, but the classical version observes the
wenzelm@43365
  1481
  classical rule context in addition to that of Isabelle/Pure.
wenzelm@43365
  1482
wenzelm@43365
  1483
  Common object logics (HOL, ZF, etc.) declare a rich collection of
wenzelm@43365
  1484
  classical rules (even if these would qualify as intuitionistic
wenzelm@43365
  1485
  ones), but only few declarations to the rule context of
wenzelm@43365
  1486
  Isabelle/Pure (\secref{sec:pure-meth-att}).
wenzelm@43365
  1487
wenzelm@43365
  1488
  \item @{method contradiction} solves some goal by contradiction,
wenzelm@43365
  1489
  deriving any result from both @{text "\<not> A"} and @{text A}.  Chained
wenzelm@43365
  1490
  facts, which are guaranteed to participate, may appear in either
wenzelm@43365
  1491
  order.
wenzelm@43365
  1492
wenzelm@43365
  1493
  \end{description}
wenzelm@43365
  1494
*}
wenzelm@43365
  1495
wenzelm@43365
  1496
wenzelm@50070
  1497
subsection {* Fully automated methods *}
wenzelm@26782
  1498
wenzelm@26782
  1499
text {*
wenzelm@26782
  1500
  \begin{matharray}{rcl}
wenzelm@28761
  1501
    @{method_def blast} & : & @{text method} \\
wenzelm@42930
  1502
    @{method_def auto} & : & @{text method} \\
wenzelm@42930
  1503
    @{method_def force} & : & @{text method} \\
wenzelm@28761
  1504
    @{method_def fast} & : & @{text method} \\
wenzelm@28761
  1505
    @{method_def slow} & : & @{text method} \\
wenzelm@28761
  1506
    @{method_def best} & : & @{text method} \\
nipkow@44911
  1507
    @{method_def fastforce} & : & @{text method} \\
wenzelm@28761
  1508
    @{method_def slowsimp} & : & @{text method} \\
wenzelm@28761
  1509
    @{method_def bestsimp} & : & @{text method} \\
wenzelm@43367
  1510
    @{method_def deepen} & : & @{text method} \\
wenzelm@26782
  1511
  \end{matharray}
wenzelm@26782
  1512
wenzelm@42596
  1513
  @{rail "
wenzelm@42930
  1514
    @@{method blast} @{syntax nat}? (@{syntax clamod} * )
wenzelm@42930
  1515
    ;
wenzelm@42596
  1516
    @@{method auto} (@{syntax nat} @{syntax nat})? (@{syntax clasimpmod} * )
wenzelm@26782
  1517
    ;
wenzelm@42930
  1518
    @@{method force} (@{syntax clasimpmod} * )
wenzelm@42930
  1519
    ;
wenzelm@42930
  1520
    (@@{method fast} | @@{method slow} | @@{method best}) (@{syntax clamod} * )
wenzelm@26782
  1521
    ;
nipkow@44911
  1522
    (@@{method fastforce} | @@{method slowsimp} | @@{method bestsimp})
wenzelm@42930
  1523
      (@{syntax clasimpmod} * )
wenzelm@42930
  1524
    ;
wenzelm@43367
  1525
    @@{method deepen} (@{syntax nat} ?) (@{syntax clamod} * )
wenzelm@43367
  1526
    ;
wenzelm@42930
  1527
    @{syntax_def clamod}:
wenzelm@42930
  1528
      (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del') ':' @{syntax thmrefs}
wenzelm@42930
  1529
    ;
wenzelm@42596
  1530
    @{syntax_def clasimpmod}: ('simp' (() | 'add' | 'del' | 'only') |
wenzelm@26782
  1531
      ('cong' | 'split') (() | 'add' | 'del') |
wenzelm@26782
  1532
      'iff' (((() | 'add') '?'?) | 'del') |
wenzelm@42596
  1533
      (('intro' | 'elim' | 'dest') ('!' | () | '?') | 'del')) ':' @{syntax thmrefs}
wenzelm@42596
  1534
  "}
wenzelm@26782
  1535
wenzelm@28760
  1536
  \begin{description}
wenzelm@26782
  1537
wenzelm@42930
  1538
  \item @{method blast} is a separate classical tableau prover that
wenzelm@42930
  1539
  uses the same classical rule declarations as explained before.
wenzelm@42930
  1540
wenzelm@42930
  1541
  Proof search is coded directly in ML using special data structures.
wenzelm@42930
  1542
  A successful proof is then reconstructed using regular Isabelle
wenzelm@42930
  1543
  inferences.  It is faster and more powerful than the other classical
wenzelm@42930
  1544
  reasoning tools, but has major limitations too.
wenzelm@42930
  1545
wenzelm@42930
  1546
  \begin{itemize}
wenzelm@42930
  1547
wenzelm@42930
  1548
  \item It does not use the classical wrapper tacticals, such as the
nipkow@44911
  1549
  integration with the Simplifier of @{method fastforce}.
wenzelm@42930
  1550
wenzelm@42930
  1551
  \item It does not perform higher-order unification, as needed by the
wenzelm@42930
  1552
  rule @{thm [source=false] rangeI} in HOL.  There are often
wenzelm@42930
  1553
  alternatives to such rules, for example @{thm [source=false]
wenzelm@42930
  1554
  range_eqI}.
wenzelm@42930
  1555
wenzelm@42930
  1556
  \item Function variables may only be applied to parameters of the
wenzelm@42930
  1557
  subgoal.  (This restriction arises because the prover does not use
wenzelm@42930
  1558
  higher-order unification.)  If other function variables are present
wenzelm@42930
  1559
  then the prover will fail with the message \texttt{Function Var's
wenzelm@42930
  1560
  argument not a bound variable}.
wenzelm@42930
  1561
wenzelm@42930
  1562
  \item Its proof strategy is more general than @{method fast} but can
wenzelm@42930
  1563
  be slower.  If @{method blast} fails or seems to be running forever,
wenzelm@42930
  1564
  try @{method fast} and the other proof tools described below.
wenzelm@42930
  1565
wenzelm@42930
  1566
  \end{itemize}
wenzelm@42930
  1567
wenzelm@42930
  1568
  The optional integer argument specifies a bound for the number of
wenzelm@42930
  1569
  unsafe steps used in a proof.  By default, @{method blast} starts
wenzelm@42930
  1570
  with a bound of 0 and increases it successively to 20.  In contrast,
wenzelm@42930
  1571
  @{text "(blast lim)"} tries to prove the goal using a search bound
wenzelm@42930
  1572
  of @{text "lim"}.  Sometimes a slow proof using @{method blast} can
wenzelm@42930
  1573
  be made much faster by supplying the successful search bound to this
wenzelm@42930
  1574
  proof method instead.
wenzelm@42930
  1575
wenzelm@42930
  1576
  \item @{method auto} combines classical reasoning with
wenzelm@42930
  1577
  simplification.  It is intended for situations where there are a lot
wenzelm@42930
  1578
  of mostly trivial subgoals; it proves all the easy ones, leaving the
wenzelm@42930
  1579
  ones it cannot prove.  Occasionally, attempting to prove the hard
wenzelm@42930
  1580
  ones may take a long time.
wenzelm@42930
  1581
wenzelm@43332
  1582
  The optional depth arguments in @{text "(auto m n)"} refer to its
wenzelm@43332
  1583
  builtin classical reasoning procedures: @{text m} (default 4) is for
wenzelm@43332
  1584
  @{method blast}, which is tried first, and @{text n} (default 2) is
wenzelm@43332
  1585
  for a slower but more general alternative that also takes wrappers
wenzelm@43332
  1586
  into account.
wenzelm@42930
  1587
wenzelm@42930
  1588
  \item @{method force} is intended to prove the first subgoal
wenzelm@42930
  1589
  completely, using many fancy proof tools and performing a rather
wenzelm@42930
  1590
  exhaustive search.  As a result, proof attempts may take rather long
wenzelm@42930
  1591
  or diverge easily.
wenzelm@42930
  1592
wenzelm@42930
  1593
  \item @{method fast}, @{method best}, @{method slow} attempt to
wenzelm@42930
  1594
  prove the first subgoal using sequent-style reasoning as explained
wenzelm@42930
  1595
  before.  Unlike @{method blast}, they construct proofs directly in
wenzelm@42930
  1596
  Isabelle.
wenzelm@26782
  1597
wenzelm@42930
  1598
  There is a difference in search strategy and back-tracking: @{method
wenzelm@42930
  1599
  fast} uses depth-first search and @{method best} uses best-first
wenzelm@42930
  1600
  search (guided by a heuristic function: normally the total size of
wenzelm@42930
  1601
  the proof state).
wenzelm@42930
  1602
wenzelm@42930
  1603
  Method @{method slow} is like @{method fast}, but conducts a broader
wenzelm@42930
  1604
  search: it may, when backtracking from a failed proof attempt, undo
wenzelm@42930
  1605
  even the step of proving a subgoal by assumption.
wenzelm@42930
  1606
wenzelm@47967
  1607
  \item @{method fastforce}, @{method slowsimp}, @{method bestsimp}
wenzelm@47967
  1608
  are like @{method fast}, @{method slow}, @{method best},
wenzelm@47967
  1609
  respectively, but use the Simplifier as additional wrapper. The name
wenzelm@47967
  1610
  @{method fastforce}, reflects the behaviour of this popular method
wenzelm@47967
  1611
  better without requiring an understanding of its implementation.
wenzelm@42930
  1612
wenzelm@43367
  1613
  \item @{method deepen} works by exhaustive search up to a certain
wenzelm@43367
  1614
  depth.  The start depth is 4 (unless specified explicitly), and the
wenzelm@43367
  1615
  depth is increased iteratively up to 10.  Unsafe rules are modified
wenzelm@43367
  1616
  to preserve the formula they act on, so that it be used repeatedly.
wenzelm@43367
  1617
  This method can prove more goals than @{method fast}, but is much
wenzelm@43367
  1618
  slower, for example if the assumptions have many universal
wenzelm@43367
  1619
  quantifiers.
wenzelm@43367
  1620
wenzelm@42930
  1621
  \end{description}
wenzelm@42930
  1622
wenzelm@42930
  1623
  Any of the above methods support additional modifiers of the context
wenzelm@42930
  1624
  of classical (and simplifier) rules, but the ones related to the
wenzelm@42930
  1625
  Simplifier are explicitly prefixed by @{text simp} here.  The
wenzelm@42930
  1626
  semantics of these ad-hoc rule declarations is analogous to the
wenzelm@42930
  1627
  attributes given before.  Facts provided by forward chaining are
wenzelm@42930
  1628
  inserted into the goal before commencing proof search.
wenzelm@42930
  1629
*}
wenzelm@42930
  1630
wenzelm@42930
  1631
wenzelm@50070
  1632
subsection {* Partially automated methods *}
wenzelm@42930
  1633
wenzelm@42930
  1634
text {* These proof methods may help in situations when the
wenzelm@42930
  1635
  fully-automated tools fail.  The result is a simpler subgoal that
wenzelm@42930
  1636
  can be tackled by other means, such as by manual instantiation of
wenzelm@42930
  1637
  quantifiers.
wenzelm@42930
  1638
wenzelm@42930
  1639
  \begin{matharray}{rcl}
wenzelm@42930
  1640
    @{method_def safe} & : & @{text method} \\
wenzelm@42930
  1641
    @{method_def clarify} & : & @{text method} \\
wenzelm@42930
  1642
    @{method_def clarsimp} & : & @{text method} \\
wenzelm@42930
  1643
  \end{matharray}
wenzelm@42930
  1644
wenzelm@42930
  1645
  @{rail "
wenzelm@42930
  1646
    (@@{method safe} | @@{method clarify}) (@{syntax clamod} * )
wenzelm@42930
  1647
    ;
wenzelm@42930
  1648
    @@{method clarsimp} (@{syntax clasimpmod} * )
wenzelm@42930
  1649
  "}
wenzelm@42930
  1650
wenzelm@42930
  1651
  \begin{description}
wenzelm@42930
  1652
wenzelm@42930
  1653
  \item @{method safe} repeatedly performs safe steps on all subgoals.
wenzelm@42930
  1654
  It is deterministic, with at most one outcome.
wenzelm@42930
  1655
wenzelm@43366
  1656
  \item @{method clarify} performs a series of safe steps without
wenzelm@43366
  1657
  splitting subgoals; see also @{ML clarify_step_tac}.
wenzelm@42930
  1658
wenzelm@42930
  1659
  \item @{method clarsimp} acts like @{method clarify}, but also does
wenzelm@42930
  1660
  simplification.  Note that if the Simplifier context includes a
wenzelm@42930
  1661
  splitter for the premises, the subgoal may still be split.
wenzelm@26782
  1662
wenzelm@28760
  1663
  \end{description}
wenzelm@26782
  1664
*}
wenzelm@26782
  1665
wenzelm@26782
  1666
wenzelm@43366
  1667
subsection {* Single-step tactics *}
wenzelm@43366
  1668
wenzelm@43366
  1669
text {*
wenzelm@50071
  1670
  \begin{mldecls}
wenzelm@43366
  1671
    @{index_ML safe_step_tac: "Proof.context -> int -> tactic"} \\
wenzelm@43366
  1672
    @{index_ML inst_step_tac: "Proof.context -> int -> tactic"} \\
wenzelm@43366
  1673
    @{index_ML step_tac: "Proof.context -> int -> tactic"} \\
wenzelm@43366
  1674
    @{index_ML slow_step_tac: "Proof.context -> int -> tactic"} \\
wenzelm@43366
  1675
    @{index_ML clarify_step_tac: "Proof.context -> int -> tactic"} \\
wenzelm@50071
  1676
  \end{mldecls}
wenzelm@43366
  1677
wenzelm@50070
  1678
  These are the primitive tactics behind the automated proof methods
wenzelm@50070
  1679
  of the Classical Reasoner.  By calling them yourself, you can
wenzelm@50070
  1680
  execute these procedures one step at a time.
wenzelm@43366
  1681
wenzelm@43366
  1682
  \begin{description}
wenzelm@43366
  1683
wenzelm@43366
  1684
  \item @{ML safe_step_tac}~@{text "ctxt i"} performs a safe step on
wenzelm@43366
  1685
  subgoal @{text i}.  The safe wrapper tacticals are applied to a
wenzelm@43366
  1686
  tactic that may include proof by assumption or Modus Ponens (taking
wenzelm@43366
  1687
  care not to instantiate unknowns), or substitution.
wenzelm@43366
  1688
wenzelm@43366
  1689
  \item @{ML inst_step_tac} is like @{ML safe_step_tac}, but allows
wenzelm@43366
  1690
  unknowns to be instantiated.
wenzelm@43366
  1691
wenzelm@43366
  1692
  \item @{ML step_tac}~@{text "ctxt i"} is the basic step of the proof
wenzelm@43366
  1693
  procedure.  The unsafe wrapper tacticals are applied to a tactic
wenzelm@43366
  1694
  that tries @{ML safe_tac}, @{ML inst_step_tac}, or applies an unsafe
wenzelm@43366
  1695
  rule from the context.
wenzelm@43366
  1696
wenzelm@43366
  1697
  \item @{ML slow_step_tac} resembles @{ML step_tac}, but allows
wenzelm@43366
  1698
  backtracking between using safe rules with instantiation (@{ML
wenzelm@43366
  1699
  inst_step_tac}) and using unsafe rules.  The resulting search space
wenzelm@43366
  1700
  is larger.
wenzelm@43366
  1701
wenzelm@43366
  1702
  \item @{ML clarify_step_tac}~@{text "ctxt i"} performs a safe step
wenzelm@43366
  1703
  on subgoal @{text i}.  No splitting step is applied; for example,
wenzelm@43366
  1704
  the subgoal @{text "A \<and> B"} is left as a conjunction.  Proof by
wenzelm@43366
  1705
  assumption, Modus Ponens, etc., may be performed provided they do
wenzelm@43366
  1706
  not instantiate unknowns.  Assumptions of the form @{text "x = t"}
wenzelm@43366
  1707
  may be eliminated.  The safe wrapper tactical is applied.
wenzelm@43366
  1708
wenzelm@43366
  1709
  \end{description}
wenzelm@43366
  1710
*}
wenzelm@43366
  1711
wenzelm@43366
  1712
wenzelm@50071
  1713
subsection {* Modifying the search step *}
wenzelm@50071
  1714
wenzelm@50071
  1715
text {*
wenzelm@50071
  1716
  \begin{mldecls}
wenzelm@50071
  1717
    @{index_ML_type wrapper: "(int -> tactic) -> (int -> tactic)"} \\[0.5ex]
wenzelm@50071
  1718
    @{index_ML_op addSWrapper: "claset * (string * (Proof.context -> wrapper))
wenzelm@50071
  1719
  -> claset"} \\
wenzelm@50071
  1720
    @{index_ML_op addSbefore: "claset * (string * (int -> tactic)) -> claset"} \\
wenzelm@50071
  1721
    @{index_ML_op addSafter: "claset * (string * (int -> tactic)) -> claset"} \\
wenzelm@50071
  1722
    @{index_ML_op delSWrapper: "claset * string -> claset"} \\[0.5ex]
wenzelm@50071
  1723
    @{index_ML_op addWrapper: "claset * (string * (Proof.context -> wrapper))
wenzelm@50071
  1724
  -> claset"} \\
wenzelm@50071
  1725
    @{index_ML_op addbefore: "claset * (string * (int -> tactic)) -> claset"} \\
wenzelm@50071
  1726
    @{index_ML_op addafter: "claset * (string * (int -> tactic)) -> claset"} \\
wenzelm@50071
  1727
    @{index_ML_op delWrapper: "claset * string -> claset"} \\[0.5ex]
wenzelm@50071
  1728
    @{index_ML addSss: "Proof.context -> Proof.context"} \\
wenzelm@50071
  1729
    @{index_ML addss: "Proof.context -> Proof.context"} \\
wenzelm@50071
  1730
  \end{mldecls}
wenzelm@50071
  1731
wenzelm@50071
  1732
  The proof strategy of the Classical Reasoner is simple.  Perform as
wenzelm@50071
  1733
  many safe inferences as possible; or else, apply certain safe rules,
wenzelm@50071
  1734
  allowing instantiation of unknowns; or else, apply an unsafe rule.
wenzelm@50071
  1735
  The tactics also eliminate assumptions of the form @{text "x = t"}
wenzelm@50071
  1736
  by substitution if they have been set up to do so.  They may perform
wenzelm@50071
  1737
  a form of Modus Ponens: if there are assumptions @{text "P \<longrightarrow> Q"} and
wenzelm@50071
  1738
  @{text "P"}, then replace @{text "P \<longrightarrow> Q"} by @{text "Q"}.
wenzelm@50071
  1739
wenzelm@50071
  1740
  The classical reasoning tools --- except @{method blast} --- allow
wenzelm@50071
  1741
  to modify this basic proof strategy by applying two lists of
wenzelm@50071
  1742
  arbitrary \emph{wrapper tacticals} to it.  The first wrapper list,
wenzelm@50071
  1743
  which is considered to contain safe wrappers only, affects @{ML
wenzelm@50071
  1744
  safe_step_tac} and all the tactics that call it.  The second one,
wenzelm@50071
  1745
  which may contain unsafe wrappers, affects the unsafe parts of @{ML
wenzelm@50071
  1746
  step_tac}, @{ML slow_step_tac}, and the tactics that call them.  A
wenzelm@50071
  1747
  wrapper transforms each step of the search, for example by
wenzelm@50071
  1748
  attempting other tactics before or after the original step tactic.
wenzelm@50071
  1749
  All members of a wrapper list are applied in turn to the respective
wenzelm@50071
  1750
  step tactic.
wenzelm@50071
  1751
wenzelm@50071
  1752
  Initially the two wrapper lists are empty, which means no
wenzelm@50071
  1753
  modification of the step tactics. Safe and unsafe wrappers are added
wenzelm@50071
  1754
  to a claset with the functions given below, supplying them with
wenzelm@50071
  1755
  wrapper names.  These names may be used to selectively delete
wenzelm@50071
  1756
  wrappers.
wenzelm@50071
  1757
wenzelm@50071
  1758
  \begin{description}
wenzelm@50071
  1759
wenzelm@50071
  1760
  \item @{text "cs addSWrapper (name, wrapper)"} adds a new wrapper,
wenzelm@50071
  1761
  which should yield a safe tactic, to modify the existing safe step
wenzelm@50071
  1762
  tactic.
wenzelm@50071
  1763
wenzelm@50071
  1764
  \item @{text "cs addSbefore (name, tac)"} adds the given tactic as a
wenzelm@50071
  1765
  safe wrapper, such that it is tried \emph{before} each safe step of
wenzelm@50071
  1766
  the search.
wenzelm@50071
  1767
wenzelm@50071
  1768
  \item @{text "cs addSafter (name, tac)"} adds the given tactic as a
wenzelm@50071
  1769
  safe wrapper, such that it is tried when a safe step of the search
wenzelm@50071
  1770
  would fail.
wenzelm@50071
  1771
wenzelm@50071
  1772
  \item @{text "cs delSWrapper name"} deletes the safe wrapper with
wenzelm@50071
  1773
  the given name.
wenzelm@50071
  1774
wenzelm@50071
  1775
  \item @{text "cs addWrapper (name, wrapper)"} adds a new wrapper to
wenzelm@50071
  1776
  modify the existing (unsafe) step tactic.
wenzelm@50071
  1777
wenzelm@50071
  1778
  \item @{text "cs addbefore (name, tac)"} adds the given tactic as an
wenzelm@50071
  1779
  unsafe wrapper, such that it its result is concatenated
wenzelm@50071
  1780
  \emph{before} the result of each unsafe step.
wenzelm@50071
  1781
wenzelm@50071
  1782
  \item @{text "cs addafter (name, tac)"} adds the given tactic as an
wenzelm@50071
  1783
  unsafe wrapper, such that it its result is concatenated \emph{after}
wenzelm@50071
  1784
  the result of each unsafe step.
wenzelm@50071
  1785
wenzelm@50071
  1786
  \item @{text "cs delWrapper name"} deletes the unsafe wrapper with
wenzelm@50071
  1787
  the given name.
wenzelm@50071
  1788
wenzelm@50071
  1789
  \item @{text "addSss"} adds the simpset of the context to its
wenzelm@50071
  1790
  classical set. The assumptions and goal will be simplified, in a
wenzelm@50071
  1791
  rather safe way, after each safe step of the search.
wenzelm@50071
  1792
wenzelm@50071
  1793
  \item @{text "addss"} adds the simpset of the context to its
wenzelm@50071
  1794
  classical set. The assumptions and goal will be simplified, before
wenzelm@50071
  1795
  the each unsafe step of the search.
wenzelm@50071
  1796
wenzelm@50071
  1797
  \end{description}
wenzelm@50071
  1798
*}
wenzelm@50071
  1799
wenzelm@50071
  1800
wenzelm@27044
  1801
section {* Object-logic setup \label{sec:object-logic} *}
wenzelm@26790
  1802
wenzelm@26790
  1803
text {*
wenzelm@26790
  1804
  \begin{matharray}{rcl}
wenzelm@28761
  1805
    @{command_def "judgment"} & : & @{text "theory \<rightarrow> theory"} \\
wenzelm@28761
  1806
    @{method_def atomize} & : & @{text method} \\
wenzelm@28761
  1807
    @{attribute_def atomize} & : & @{text attribute} \\
wenzelm@28761
  1808
    @{attribute_def rule_format} & : & @{text attribute} \\
wenzelm@28761
  1809
    @{attribute_def rulify} & : & @{text attribute} \\
wenzelm@26790
  1810
  \end{matharray}
wenzelm@26790
  1811
wenzelm@26790
  1812
  The very starting point for any Isabelle object-logic is a ``truth
wenzelm@26790
  1813
  judgment'' that links object-level statements to the meta-logic
wenzelm@26790
  1814
  (with its minimal language of @{text prop} that covers universal
wenzelm@26790
  1815
  quantification @{text "\<And>"} and implication @{text "\<Longrightarrow>"}).
wenzelm@26790
  1816
wenzelm@26790
  1817
  Common object-logics are sufficiently expressive to internalize rule
wenzelm@26790
  1818
  statements over @{text "\<And>"} and @{text "\<Longrightarrow>"} within their own
wenzelm@26790
  1819
  language.  This is useful in certain situations where a rule needs
wenzelm@26790
  1820
  to be viewed as an atomic statement from the meta-level perspective,
wenzelm@26790
  1821
  e.g.\ @{text "\<And>x. x \<in> A \<Longrightarrow> P x"} versus @{text "\<forall>x \<in> A. P x"}.
wenzelm@26790
  1822
wenzelm@26790
  1823
  From the following language elements, only the @{method atomize}
wenzelm@26790
  1824
  method and @{attribute rule_format} attribute are occasionally
wenzelm@26790
  1825
  required by end-users, the rest is for those who need to setup their
wenzelm@26790
  1826
  own object-logic.  In the latter case existing formulations of
wenzelm@26790
  1827
  Isabelle/FOL or Isabelle/HOL may be taken as realistic examples.
wenzelm@26790
  1828
wenzelm@26790
  1829
  Generic tools may refer to the information provided by object-logic
wenzelm@26790
  1830
  declarations internally.
wenzelm@26790
  1831
wenzelm@42596
  1832
  @{rail "
wenzelm@46494
  1833
    @@{command judgment} @{syntax name} '::' @{syntax type} @{syntax mixfix}?
wenzelm@26790
  1834
    ;
wenzelm@42596
  1835
    @@{attribute atomize} ('(' 'full' ')')?
wenzelm@26790
  1836
    ;
wenzelm@42596
  1837
    @@{attribute rule_format} ('(' 'noasm' ')')?
wenzelm@42596
  1838
  "}
wenzelm@26790
  1839
wenzelm@28760
  1840
  \begin{description}
wenzelm@26790
  1841
  
wenzelm@28760
  1842
  \item @{command "judgment"}~@{text "c :: \<sigma> (mx)"} declares constant
wenzelm@28760
  1843
  @{text c} as the truth judgment of the current object-logic.  Its
wenzelm@28760
  1844
  type @{text \<sigma>} should specify a coercion of the category of
wenzelm@28760
  1845
  object-level propositions to @{text prop} of the Pure meta-logic;
wenzelm@28760
  1846
  the mixfix annotation @{text "(mx)"} would typically just link the
wenzelm@28760
  1847
  object language (internally of syntactic category @{text logic})
wenzelm@28760
  1848
  with that of @{text prop}.  Only one @{command "judgment"}
wenzelm@28760
  1849
  declaration may be given in any theory development.
wenzelm@26790
  1850
  
wenzelm@28760
  1851
  \item @{method atomize} (as a method) rewrites any non-atomic
wenzelm@26790
  1852
  premises of a sub-goal, using the meta-level equations declared via
wenzelm@26790
  1853
  @{attribute atomize} (as an attribute) beforehand.  As a result,
wenzelm@26790
  1854
  heavily nested goals become amenable to fundamental operations such
wenzelm@42626
  1855
  as resolution (cf.\ the @{method (Pure) rule} method).  Giving the ``@{text
wenzelm@26790
  1856
  "(full)"}'' option here means to turn the whole subgoal into an
wenzelm@26790
  1857
  object-statement (if possible), including the outermost parameters
wenzelm@26790
  1858
  and assumptions as well.
wenzelm@26790
  1859
wenzelm@26790
  1860
  A typical collection of @{attribute atomize} rules for a particular
wenzelm@26790
  1861
  object-logic would provide an internalization for each of the
wenzelm@26790
  1862
  connectives of @{text "\<And>"}, @{text "\<Longrightarrow>"}, and @{text "\<equiv>"}.
wenzelm@26790
  1863
  Meta-level conjunction should be covered as well (this is
wenzelm@26790
  1864
  particularly important for locales, see \secref{sec:locale}).
wenzelm@26790
  1865
wenzelm@28760
  1866
  \item @{attribute rule_format} rewrites a theorem by the equalities
wenzelm@28760
  1867
  declared as @{attribute rulify} rules in the current object-logic.
wenzelm@28760
  1868
  By default, the result is fully normalized, including assumptions
wenzelm@28760
  1869
  and conclusions at any depth.  The @{text "(no_asm)"} option
wenzelm@28760
  1870
  restricts the transformation to the conclusion of a rule.
wenzelm@26790
  1871
wenzelm@26790
  1872
  In common object-logics (HOL, FOL, ZF), the effect of @{attribute
wenzelm@26790
  1873
  rule_format} is to replace (bounded) universal quantification
wenzelm@26790
  1874
  (@{text "\<forall>"}) and implication (@{text "\<longrightarrow>"}) by the corresponding
wenzelm@26790
  1875
  rule statements over @{text "\<And>"} and @{text "\<Longrightarrow>"}.
wenzelm@26790
  1876
wenzelm@28760
  1877
  \end{description}
wenzelm@26790
  1878
*}
wenzelm@26790
  1879
wenzelm@26782
  1880
end