theory Tactical

imports Base FirstSteps

begin

chapter {* Tactical Reasoning\label{chp:tactical} *}

text {*

  The main reason for descending to the ML-level of Isabelle is to be able to

  implement automatic proof procedures. Such proof procedures usually lessen

  considerably the burden of manual reasoning, for example, when introducing

  new definitions. These proof procedures are centred around refining a goal

  state using tactics. This is similar to the \isacommand{apply}-style

  reasoning at the user level, where goals are modified in a sequence of proof

  steps until all of them are solved. However, there are also more structured

  operations available on the ML-level that help with the handling of

  variables and assumptions.

*}

section {* Basics of Reasoning with Tactics*}

text {*

  To see how tactics work, let us first transcribe a simple \isacommand{apply}-style proof 

  into ML. Suppose the following proof.

*}

lemma disj_swap: "P \<or> Q \<Longrightarrow> Q \<or> P"

apply(erule disjE)

apply(rule disjI2)

apply(assumption)

apply(rule disjI1)

apply(assumption)

done

text {*

  This proof translates to the following ML-code.

@{ML_response_fake [display,gray]

"let

  val ctxt = @{context}

  val goal = @{prop \"P \<or> Q \<Longrightarrow> Q \<or> P\"}

in

  Goal.prove ctxt [\"P\", \"Q\"] [] goal 

   (fn _ => 

      etac @{thm disjE} 1

      THEN rtac @{thm disjI2} 1

      THEN atac 1

      THEN rtac @{thm disjI1} 1

      THEN atac 1)

end" "?P \<or> ?Q \<Longrightarrow> ?Q \<or> ?P"}

  To start the proof, the function @{ML "Goal.prove"}~@{text "ctxt xs As C

  tac"} sets up a goal state for proving the goal @{text C} 

  (that is @{prop "P \<or> Q \<Longrightarrow> Q \<or> P"} in the proof at hand) under the

  assumptions @{text As} (happens to be empty) with the variables

  @{text xs} that will be generalised once the goal is proved (in our case

  @{text P} and @{text Q}). The @{text "tac"} is the tactic that proves the goal;

  it can make use of the local assumptions (there are none in this example). 

  The functions @{ML etac}, @{ML rtac} and @{ML atac} correspond to 

  @{text erule}, @{text rule} and @{text assumption}, respectively. 

  The operator @{ML THEN} strings the tactics together. 

  \begin{readmore}

  To learn more about the function @{ML Goal.prove} see \isccite{sec:results}

  and the file @{ML_file "Pure/goal.ML"}.  See @{ML_file

  "Pure/tactic.ML"} and @{ML_file "Pure/tctical.ML"} for the code of basic

  tactics and tactic combinators; see also Chapters 3 and 4 in the old

  Isabelle Reference Manual.

  \end{readmore}

  Note that in the code above we used antiquotations for referencing the theorems. Many theorems

  also have ML-bindings with the same name. Therefore, we could also just have

  written @{ML "etac disjE 1"}, or in case there are no ML-binding obtained

  the theorem dynamically using the function @{ML thm}; for example 

  \mbox{@{ML  "etac (thm \"disjE\") 1"}}. Both ways however are considered bad style! 

  The reason

  is that the binding for @{ML disjE} can be re-assigned by the user and thus

  one does not have complete control over which theorem is actually

  applied. This problem is nicely prevented by using antiquotations, because

  then the theorems are fixed statically at compile-time.

  During the development of automatic proof procedures, you will often find it 

  necessary to test a tactic on examples. This can be conveniently 

  done with the command \isacommand{apply}@{text "(tactic \<verbopen> \<dots> \<verbclose>)"}. 

  Consider the following sequence of tactics

*}

ML{*val foo_tac = 

      (etac @{thm disjE} 1

       THEN rtac @{thm disjI2} 1

       THEN atac 1

       THEN rtac @{thm disjI1} 1

       THEN atac 1)*}

text {* and the Isabelle proof: *}

lemma "P \<or> Q \<Longrightarrow> Q \<or> P"

apply(tactic {* foo_tac *})

done

text {*

  By using @{text "tactic \<verbopen> \<dots> \<verbclose>"} you can call from the 

  user level of Isabelle the tactic @{ML foo_tac} or 

  any other function that returns a tactic.

  The tactic @{ML foo_tac} is just a sequence of simple tactics stringed

  together by @{ML THEN}. As can be seen, each simple tactic in @{ML foo_tac}

  has a hard-coded number that stands for the subgoal analysed by the

  tactic (@{text "1"} stands for the first, or top-most, subgoal). This hard-coding

  of goals is sometimes wanted, but usually it is not. To avoid the explicit numbering, 

  you can write\label{tac:footacprime}

*}

ML{*val foo_tac' = 

      (etac @{thm disjE} 

       THEN' rtac @{thm disjI2} 

       THEN' atac 

       THEN' rtac @{thm disjI1} 

       THEN' atac)*}

text {* 

  and then give the number for the subgoal explicitly when the tactic is

  called. So in the next proof you can first discharge the second subgoal, and

  subsequently the first.

*}

lemma "P1 \<or> Q1 \<Longrightarrow> Q1 \<or> P1"

   and "P2 \<or> Q2 \<Longrightarrow> Q2 \<or> P2"

apply(tactic {* foo_tac' 2 *})

apply(tactic {* foo_tac' 1 *})

done

text {*

  This kind of addressing is more difficult to achieve when the goal is 

  hard-coded inside the tactic. For most operators that combine tactics 

  (@{ML THEN} is only one such operator) a ``primed'' version exists.

  The tactics @{ML foo_tac} and @{ML foo_tac'} are very specific for

  analysing goals being only of the form @{prop "P \<or> Q \<Longrightarrow> Q \<or> P"}. If the goal is not

  of this form, then they return the error message:

  \begin{isabelle}

  @{text "*** empty result sequence -- proof command failed"}\\

  @{text "*** At command \"apply\"."}

  \end{isabelle}

  This means the tactics failed. The reason for this error message is that tactics 

  are functions mapping a goal state to a (lazy) sequence of successor states. 

  Hence the type of a tactic is:

*}

ML{*type tactic = thm -> thm Seq.seq*}

text {*

  By convention, if a tactic fails, then it should return the empty sequence. 

  Therefore, if you write your own tactics, they  should not raise exceptions 

  willy-nilly; only in very grave failure situations should a tactic raise the 

  exception @{text THM}.

  The simplest tactics are @{ML no_tac} and @{ML all_tac}. The first returns

  the empty sequence and is defined as

*}

ML{*fun no_tac thm = Seq.empty*}

text {*

  which means @{ML no_tac} always fails. The second returns the given theorem wrapped 

  up in a single member sequence; it is defined as

*}

ML{*fun all_tac thm = Seq.single thm*}

text {*

  which means @{ML all_tac} always succeeds, but also does not make any progress 

  with the proof.

  The lazy list of possible successor goal states shows through at the user-level 

  of Isabelle when using the command \isacommand{back}. For instance in the 

  following proof there are two possibilities for how to apply 

  @{ML foo_tac'}: either using the first assumption or the second.

*}

lemma "\<lbrakk>P \<or> Q; P \<or> Q\<rbrakk> \<Longrightarrow> Q \<or> P"

apply(tactic {* foo_tac' 1 *})

back

done

text {*

  By using \isacommand{back}, we construct the proof that uses the

  second assumption. While in the proof above, it does not really matter which 

  assumption is used, in more interesting cases provability might depend

  on exploring different possibilities.

  \begin{readmore}

  See @{ML_file "Pure/General/seq.ML"} for the implementation of lazy

  sequences. In day-to-day Isabelle programming, however, one rarely 

  constructs sequences explicitly, but uses the predefined tactics and 

  tactic combinators instead.

  \end{readmore}

  It might be surprising that tactics, which transform

  one goal state to the next, are functions from theorems to theorem 

  (sequences). The surprise resolves by knowing that every 

  goal state is indeed a theorem. To shed more light on this,

  let us modify the code of @{ML all_tac} to obtain the following

  tactic

*}

ML{*fun my_print_tac ctxt thm =

 let

   val _ = warning (str_of_thm ctxt thm)

 in 

   Seq.single thm

 end*}

text_raw {*

  \begin{figure}[p]

  \begin{isabelle}

*}

lemma shows "\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B" 

apply(tactic {* my_print_tac @{context} *})

txt{* \begin{minipage}{\textwidth}

      @{subgoals [display]} 

      \end{minipage}\medskip      

      \begin{minipage}{\textwidth}

      \small\colorbox{gray!20}{

      \begin{tabular}{@ {}l@ {}}

      internal goal state:\\

      @{text "(\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B) \<Longrightarrow> (\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B)"}

      \end{tabular}}

      \end{minipage}\medskip

*}

apply(rule conjI)

apply(tactic {* my_print_tac @{context} *})

txt{* \begin{minipage}{\textwidth}

      @{subgoals [display]} 

      \end{minipage}\medskip

      \begin{minipage}{\textwidth}

      \small\colorbox{gray!20}{

      \begin{tabular}{@ {}l@ {}}

      internal goal state:\\

      @{text "(\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A) \<Longrightarrow> (\<lbrakk>A; B\<rbrakk> \<Longrightarrow> B) \<Longrightarrow> (\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B)"}

      \end{tabular}}

      \end{minipage}\medskip

*}

apply(assumption)

apply(tactic {* my_print_tac @{context} *})

txt{* \begin{minipage}{\textwidth}

      @{subgoals [display]} 

      \end{minipage}\medskip

      \begin{minipage}{\textwidth}

      \small\colorbox{gray!20}{

      \begin{tabular}{@ {}l@ {}}

      internal goal state:\\

      @{text "(\<lbrakk>A; B\<rbrakk> \<Longrightarrow> B) \<Longrightarrow> (\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B)"}

      \end{tabular}}

      \end{minipage}\medskip

*}

apply(assumption)

apply(tactic {* my_print_tac @{context} *})

txt{* \begin{minipage}{\textwidth}

      @{subgoals [display]} 

      \end{minipage}\medskip 

      \begin{minipage}{\textwidth}

      \small\colorbox{gray!20}{

      \begin{tabular}{@ {}l@ {}}

      internal goal state:\\

      @{text "\<lbrakk>A; B\<rbrakk> \<Longrightarrow> A \<and> B"}

      \end{tabular}}

      \end{minipage}\medskip   

   *}

done

text_raw {*  

  \end{isabelle}

  \caption{A proof where we show the goal state as printed 

  by the Isabelle system and as represented internally 

  (highlighted boxes).\label{fig:goalstates}}

  \end{figure}

*}

text {*

  which prints out the given theorem (using the string-function defined in

  Section~\ref{sec:printing}) and then behaves like @{ML all_tac}. With this

  tactic we are in the position to inspect every goal state in a

  proof. Consider now the proof in Figure~\ref{fig:goalstates}: as can be seen, 

  internally every goal state is an implication of the form

  @{text[display] "A\<^isub>1 \<Longrightarrow> \<dots> \<Longrightarrow> A\<^isub>n \<Longrightarrow> (C)"}

  where @{term C} is the goal to be proved and the @{term "A\<^isub>i"} are

  the subgoals. So after setting up the lemma, the goal state is always of the

  form @{text "C \<Longrightarrow> (C)"}; when the proof is finished we are left with @{text

  "(C)"}. Since the goal @{term C} can potentially be an implication, there is

  a ``protector'' wrapped around it (in from of an outermost constant @{text

  "Const (\"prop\", bool \<Rightarrow> bool)"} applied to each goal; however this constant

  is invisible in the figure). This prevents that premises of @{text C} are

  mis-interpreted as open subgoals. While tactics can operate on the subgoals

  (the @{text "A\<^isub>i"} above), they are expected to leave the conclusion

  @{term C} intact, with the exception of possibly instantiating schematic

  variables. If you use the predefined tactics, which we describe in the next

  section, this will always be the case.

  \begin{readmore}

  For more information about the internals of goals see \isccite{sec:tactical-goals}.

  \end{readmore}

*}

section {* Simple Tactics *}

text {*

  Let us start with the tactic @{ML print_tac}, which is quite useful for low-level 

  debugging of tactics. It just prints out a message and the current goal state. 

  Processing the proof

*}

lemma shows "False \<Longrightarrow> True"

apply(tactic {* print_tac "foo message" *})

txt{*gives:\medskip

     \begin{minipage}{\textwidth}\small

     @{text "foo message"}\\[3mm] 

     @{prop "False \<Longrightarrow> True"}\\

     @{text " 1. False \<Longrightarrow> True"}\\

     \end{minipage}

   *}

(*<*)oops(*>*)

text {*

  Another simple tactic is the function @{ML atac}, which, as shown in the previous

  section, corresponds to the assumption command.

*}

lemma shows "P \<Longrightarrow> P"

apply(tactic {* atac 1 *})

txt{*\begin{minipage}{\textwidth}

     @{subgoals [display]}

     \end{minipage}*}

(*<*)oops(*>*)

text {*

  Similarly, @{ML rtac}, @{ML dtac}, @{ML etac} and @{ML ftac} correspond

  to @{text rule}, @{text drule}, @{text erule} and @{text frule}, 

  respectively. Each of them takes a theorem as argument and attempts to 

  apply it to a goal. Below are three self-explanatory examples.

*}

lemma shows "P \<and> Q"

apply(tactic {* rtac @{thm conjI} 1 *})

txt{*\begin{minipage}{\textwidth}

     @{subgoals [display]}

     \end{minipage}*}

(*<*)oops(*>*)

lemma shows "P \<and> Q \<Longrightarrow> False"

apply(tactic {* etac @{thm conjE} 1 *})

txt{*\begin{minipage}{\textwidth}

     @{subgoals [display]}

     \end{minipage}*}

(*<*)oops(*>*)

lemma shows "False \<and> True \<Longrightarrow> False"

apply(tactic {* dtac @{thm conjunct2} 1 *})

txt{*\begin{minipage}{\textwidth}

     @{subgoals [display]}

     \end{minipage}*}

(*<*)oops(*>*)

text {*

  Note the number in each tactic call. Also as mentioned in the previous section, most 

  basic tactics take such an argument; it addresses the subgoal they are analysing. 

  In the proof below, we first split up the conjunction in  the second subgoal by 

  focusing on this subgoal first.

*}

lemma shows "Foo" and "P \<and> Q"

apply(tactic {* rtac @{thm conjI} 2 *})

txt {*\begin{minipage}{\textwidth}

      @{subgoals [display]}

      \end{minipage}*}

(*<*)oops(*>*)

text {*

  (FIXME: is it important to get the number of subgoals?)

  The function @{ML resolve_tac} is similar to @{ML rtac}, except that it

  expects a list of theorems as arguments. From this list it will apply the

  first applicable theorem (later theorems that are also applicable can be

  explored via the lazy sequences mechanism). Given the code

*}

ML{*val resolve_tac_xmp = resolve_tac [@{thm impI}, @{thm conjI}]*}

text {*

  an example for @{ML resolve_tac} is the following proof where first an outermost 

  implication is analysed and then an outermost conjunction.

*}

lemma shows "C \<longrightarrow> (A \<and> B)" and "(A \<longrightarrow> B) \<and> C"

apply(tactic {* resolve_tac_xmp 1 *})

apply(tactic {* resolve_tac_xmp 2 *})

txt{*\begin{minipage}{\textwidth}

     @{subgoals [display]} 

     \end{minipage}*}

(*<*)oops(*>*)

text {* 

  Similarly versions taking a list of theorems exist for the tactics 

  @{ML dtac} (@{ML dresolve_tac}), @{ML etac} (@{ML eresolve_tac}) and so on.

  Another simple tactic is @{ML cut_facts_tac}. It inserts a list of theorems

  into the assumptions of the current goal state. For example

*}

lemma shows "True \<noteq> False"

apply(tactic {* cut_facts_tac [@{thm True_def}, @{thm False_def}] 1 *})

txt{*produces the goal state\medskip

     \begin{minipage}{\textwidth}

     @{subgoals [display]} 

     \end{minipage}*}

(*<*)oops(*>*)

text {*

  Since rules are applied using higher-order unification, an automatic proof

  procedure might become too fragile, if it just applies inference rules as 

  shown above. The reason is that a number of rules introduce meta-variables 

  into the goal state. Consider for example the proof

*}

lemma shows "\<forall>x\<in>A. P x \<Longrightarrow> Q x"

apply(drule bspec)

txt{*\begin{minipage}{\textwidth}

     @{subgoals [display]} 

     \end{minipage}*}

(*<*)oops(*>*)

text {*

  where the application of Rule @{text bspec} generates two subgoals involving the

  meta-variable @{text "?x"}. Now, if you are not careful, tactics 

  applied to the first subgoal might instantiate this meta-variable in such a 

  way that the second subgoal becomes unprovable. If it is clear what the @{text "?x"}

  should be, then this situation can be avoided by introducing a more

  constraint version of the @{text bspec}-rule. Such constraints can be given by

  pre-instantiating theorems with other theorems. One function to do this is

  @{ML RS}

  @{ML_response_fake [display,gray]

  "@{thm disjI1} RS @{thm conjI}" "\<lbrakk>?P1; ?Q\<rbrakk> \<Longrightarrow> (?P1 \<or> ?Q1) \<and> ?Q"}

  which in the example instantiates the first premise of the @{text conjI}-rule 

  with the rule @{text disjI1}. If the instantiation is impossible, as in the 

  case of

  @{ML_response_fake_both [display,gray]

  "@{thm conjI} RS @{thm mp}" 

"*** Exception- THM (\"RSN: no unifiers\", 1, 

[\"\<lbrakk>?P; ?Q\<rbrakk> \<Longrightarrow> ?P \<and> ?Q\", \"\<lbrakk>?P \<longrightarrow> ?Q; ?P\<rbrakk> \<Longrightarrow> ?Q\"]) raised"}

  then the function raises an exception. The function @{ML RSN} is similar to @{ML RS}, but 

  takes an additional number as argument that makes explicit which premise 

  should be instantiated. 

  To improve readability of the theorems we produce below, we shall use 

  the following function

*}

ML{*fun no_vars ctxt thm =

let 

  val ((_, [thm']), _) = Variable.import_thms true [thm] ctxt

in

  thm'

end*}

text {*

  that transform the schematic variables of a theorem into free variables.

  Using this function for the first @{ML RS}-expression above would produce

  the more readable result: