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 @{text

  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. Consider 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. A difference

  between the \isacommand{apply}-script and the ML-code is that the

  former causes the lemma to be stored under the name @{text "disj_swap"}, 

  whereas the latter does not include any code for this.

  \begin{readmore}

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

  and the file @{ML_file "Pure/goal.ML"}. For more information about the

  internals of goals see \isccite{sec:tactical-goals}.  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 we used antiquotations for referencing the theorems. We could also

  just have written @{ML "etac disjE 1"} and so on, but this is 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 {*

  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 is 

  sometimes wanted, but usually not. To avoid the explicit numbering in 

  the tactic, you can write

*}

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. For every operator that combines tactics (@{ML THEN} is only one 

  such operator), a primed version exists. So in the next proof you 

  can first discharge the second subgoal, and after that 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.

  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 @{ML foo_tac} throws the error message:

  \begin{isabelle}

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

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

  \end{isabelle}

  Meaning the tactic failed. The reason for this error message is that tactics 

  are functions that map a goal state to a (lazy) sequence of successor states, 

  hence the type of a tactic is:

  @{text [display, gray] "type tactic = thm -> thm Seq.seq"}

  It is custom that if a tactic fails, it should return the empty sequence: 

  your own tactics should not raise exceptions willy-nilly.

  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 

  as 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 states shows through to 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'}.

*}

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. In more interesting situations, different possibilities 

  can lead to different proofs and even often need to be explored when

  a first proof attempt is unsuccessful.

  \begin{readmore}

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

  sequences. However in day-to-day Isabelle programming, one rarely 

  constructs sequences explicitly, but uses the predefined functions

  instead.

  \end{readmore}

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

  (sequences). The surprise resolves by knowing that every 

  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 {*

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

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

  now can inspect every proof state in the follwing proof. On the left-hand

  side we show the goal state as shown by the system; on the right-hand

*}

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

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

txt{* \small 

      \begin{tabular}{@ {}l@ {}p{0.7\textwidth}@ {}}

      \begin{minipage}[t]{0.3\textwidth}

      @{subgoals [display]} 

      \end{minipage} &

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

      \end{tabular}

*}

apply(rule conjI)

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

txt{* \small 

      \begin{tabular}{@ {}l@ {}p{0.76\textwidth}@ {}}

      \begin{minipage}[t]{0.26\textwidth}

      @{subgoals [display]} 

      \end{minipage} &

      \hfill@{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}

*}

apply(assumption)

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

txt{* \small 

      \begin{tabular}{@ {}l@ {}p{0.7\textwidth}@ {}}

      \begin{minipage}[t]{0.3\textwidth}

      @{subgoals [display]} 

      \end{minipage} &

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

      \end{tabular}

*}

apply(assumption)

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

txt{* \small 

      \begin{tabular}{@ {}l@ {}p{0.7\textwidth}@ {}}

      \begin{minipage}[t]{0.3\textwidth}

      @{subgoals [display]} 

      \end{minipage} &

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

      \end{tabular}

*}

done

text {*

  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  

  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. 

*}

section {* Simple Tactics *}

text {*

  As seen above, the function @{ML atac} corresponds to the assumption tactic.

*}

lemma shows "P \<Longrightarrow> P"

apply(tactic {* atac 1 *})

done

text {*

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

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

*}

lemma shows "P \<and> Q"

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

(*<*)oops(*>*)

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

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

(*<*)oops(*>*)

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

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

(*<*)oops(*>*)

text {*

  As mentioned above, most basic tactics take a number as argument, which

  addresses to subgoal they are analysing.

*}

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

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

(*<*)oops(*>*)

text {* 

  Corresponding to @{ML rtac}, there is also the tactic @{ML resolve_tac}, which

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

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

  explored via the lazy sequences mechanism). Given the abbreviation

*}

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 *})

(*<*)oops(*>*)

text {* 

  Similarly versions exists for @{ML atac} (@{ML assume_tac}), @{ML etac} 

  (@{ML eresolve_tac}) and so on.

  The tactic @{ML print_tac} is useful for low-level debugging of tactics: it

  prints out a message and the current goal state.

*}

lemma shows "False \<Longrightarrow> True"

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

(*<*)oops(*>*)

text {*

  (FIXME explain RS MRS)

  safety is provided by the tactic @{ML SUBPROOF}. This tactic fixes the parameters 

  and binds the various components of a proof state into a record. 

*}

text_raw{*

\begin{figure}

\begin{isabelle}

*}

ML{*fun sp_tac {prems, params, asms, concl, context, schematics} = 

  let 

    val str_of_params = str_of_cterms context params

    val str_of_asms = str_of_cterms context asms

    val str_of_concl = str_of_cterm context concl

    val str_of_prems = str_of_thms context prems   

    val str_of_schms = str_of_cterms context (snd schematics)    

    val _ = (warning ("params: " ^ str_of_params);

             warning ("schematics: " ^ str_of_schms);

             warning ("assumptions: " ^ str_of_asms);

             warning ("conclusion: " ^ str_of_concl);

             warning ("premises: " ^ str_of_prems))

  in

    no_tac 

  end*}

text_raw{*

\end{isabelle}

\caption{A function that prints out the various parameters provided by the tactic

 @{ML SUBPROOF}. It uses the functions extracting strings from @{ML_type cterm}s 

  and @{ML_type thm}s, which are defined in Section~\ref{sec:printing}.\label{fig:sptac}}

\end{figure}

*}

text {*

  To see what happens, assume the function defined in Figure~\ref{fig:sptac}, which

  takes a record as argument and just prints out the content of this record (using the 

  string transformation functions defined in Section~\ref{sec:printing}). Consider

  now the proof

*}

lemma shows "\<And>x y. A x y \<Longrightarrow> B y x \<longrightarrow> C (?z y) x"

apply(tactic {* SUBPROOF sp_tac @{context} 1 *})?

txt {*

  which yields the printout:

  \begin{quote}\small

  \begin{tabular}{ll}

  params:      & @{term x}, @{term y}\\

  schematics:  & @{term z}\\

  assumptions: & @{term "A x y"}\\

  conclusion:  & @{term "B y x \<longrightarrow> C (z y) x"}\\

  premises:    & @{term "A x y"}

  \end{tabular}

  \end{quote}

  Note in the actual output the brown colour of the variables @{term x} and 

  @{term y}. Although parameters in the original goal, they are fixed inside

  the subproof. Similarly the schematic variable @{term z}. The assumption 

  Notice also that we had to append @{text "?"} to \isacommand{apply}. The 

  reason is that @{ML SUBPROOF} normally expects that the subgoal is solved completely.

  Since in the function @{ML sp_tac} we returned the tactic @{ML no_tac}, the subproof

  obviously fails. The question-mark allows us to recover from this failure

  in a graceful manner so that the warning messages are not overwritten

  If we continue the proof script by applying the @{text impI}-rule

*}

apply(rule impI)

apply(tactic {* SUBPROOF sp_tac @{context} 1 *})?

txt {*

  then @{ML SUBPROOF} prints out 

  \begin{quote}\small

  \begin{tabular}{ll}

  params:      & @{term x}, @{term y}\\

  schematics:  & @{term z}\\

  assumptions: & @{term "A x y"}, @{term "B y x"}\\

  conclusion:  & @{term "C (z y) x"}\\

  premises:    & @{term "A x y"}, @{term "B y x"}

  \end{tabular}

  \end{quote}

*}

(*<*)oops(*>*)

text {*

  One convenience of @{ML SUBPROOF} is that we can apply assumption

  using the usual tactics, because the parameter @{text prems} 

  contains the assumptions as theorems. With this we can easily 

  implement a tactic that almost behaves like @{ML atac}:

*}

lemma shows "\<And>x y. \<lbrakk>B x y; A x y; C x y\<rbrakk> \<Longrightarrow> A x y"

txt{* yields

      @{subgoals [display]} *}

(*<*)oops(*>*)