text\<open>
Isabelle/Pure maintains a record of named configuration options within the theory or proofcontext, with values of type \<^ML_type>\<open>bool\<close>, \<^ML_type>\<open>int\<close>, \<^ML_type>\<open>real\<close>, or \<^ML_type>\<open>string\<close>. Tools may declare options in
ML, andthen refer to these values (relative to the context). Thusglobal
reference variables are easily avoided. The user may change the value of a
configuration option by means of an associated attribute of the same name.
This form of contextdeclaration works particularly well with commands such
as @{command "declare"} or @{command "using"} like this: \<close>
\<^descr> @{command "print_options"} prints the available configuration options, with names, types, and current values; the ``\<open>!\<close>'' option indicates extra
verbosity.
\<^descr> \<open>name = value\<close> as an attribute expression modifies the named option, with
the syntax of the value depending on the option's type. For \<^ML_type>\bool\
the default valueis\<open>true\<close>. Any attempt to change a global option in a localcontextis ignored. \<close>
section \<open>Basic proof tools\<close>
subsection \<open>Miscellaneous methods and attributes \label{sec:misc-meth-att}\<close>
\<^descr> @{method unfold}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> and @{method fold}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> expand (or
fold back) the given definitions throughout all goals; any chained facts
provided are inserted into the goal and subject to rewriting as well.
Unfolding works in two stages: first, the given equations are used directly for rewriting; second, the equations are passed through the attribute
@{attribute_ref abs_def} before rewriting --- to ensure that definitions are
fully expanded, regardless of the actual parameters that are provided.
\<^descr> @{method insert}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> inserts theorems as facts into all goals of
the proof state. Note that current facts indicated for forward chaining are
ignored.
\<^descr> @{method erule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close>, @{method drule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close>, and @{method
frule}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> are similar to the basic @{method rule} method (see \secref{sec:pure-meth-att}), but apply rules by elim-resolution,
destruct-resolution, and forward-resolution, respectively \<^cite>\<open>"isabelle-implementation"\<close>. The optional natural number argument (default 0)
specifies additional assumption steps to be performed here.
Note that these methods are improper ones, mainly serving for
experimentation and tactic script emulation. Different modes of basic rule
application are usually expressed in Isar at the proof language level,
rather than via implicit proof state manipulations. For example, a proper
single-step elimination would be doneusing the plain @{method rule} method, with forward chaining of current facts.
\<^descr> @{method intro} and @{method elim} repeatedly refine some goal by intro-
or elim-resolution, after having inserted any chained facts. Exactly the
rules given as arguments are taken into account; this allows fine-tuned
decomposition of a proof problem, in contrast to common automated tools.
\<^descr> @{method fail} yields an empty result sequence; it is the identity of the
``\<open>|\<close>'' method combinator (cf.\ \secref{sec:proof-meth}).
\<^descr> @{method succeed} yields a single (unchanged) result; it is the identity
of the ``\<open>,\<close>'' method combinator (cf.\ \secref{sec:proof-meth}).
\<^descr> @{method sleep}~\<open>s\<close> succeeds after a real-time delay of \<open>s\<close> seconds. This is occasionally useful for demonstration and testing purposes.
\<^descr> @{attribute tagged}~\<open>name value\<close> and @{attribute untagged}~\<open>name\<close> add and
remove \<^emph>\<open>tags\<close> of some theorem. Tags may be any list of string pairs that
serve as formal comment. The first string is considered the tag name, the
second its value. Note that @{attribute untagged} removes any tags of the
same name.
\<^descr> @{attribute THEN}~\<open>a\<close> composes rules by resolution; it resolves with the
first premise of \<open>a\<close> (an alternative position may be also specified). See also\<^ML_infix>\<open>RS\<close> in \<^cite>\<open>"isabelle-implementation"\<close>.
\<^descr> @{attribute unfolded}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> and @{attribute folded}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close>
expand and fold back again the given definitions throughout a rule.
\<^descr> @{attribute abs_def} turns an equation of the form \<^prop>\<open>f x y \<equiv> t\<close>
into \<^prop>\<open>f \<equiv> \<lambda>x y. t\<close>, which ensures that @{method simp} steps always
expand it. This also works for object-logic equality.
\<^descr> @{attribute rotated}~\<open>n\<close> rotate the premises of a theorem by \<open>n\<close> (default
1).
\<^descr> @{attribute (Pure) elim_format} turns a destruction rule into elimination
rule format, by resolving with the rule \<^prop>\<open>PROP A \<Longrightarrow> (PROP A \<Longrightarrow> PROP B) \<Longrightarrow> PROP B\<close>.
Note that the Classical Reasoner (\secref{sec:classical}) provides its own
version of this operation.
\<^descr> @{attribute no_vars} replaces schematic variables by free ones; this is
mainly for tuning output of pretty printed theorems. \<close>
These methods provide low-level facilities for equational reasoning that are
intended for specialized applications only. Normally, single step
calculations would be performed in a structured text (see also \secref{sec:calculation}), while the Simplifier methods provide the
canonical way for automated normalization (see \secref{sec:simplifier}).
\<^descr> @{method subst}~\<open>eq\<close> performs a single substitution step using rule \<open>eq\<close>,
which may be either a meta or object equality.
\<^descr> @{method subst}~\<open>(asm) eq\<close> substitutes in an assumption.
\<^descr> @{method subst}~\<open>(i \<dots> j) eq\<close> performs several substitutions in the
conclusion. The numbers \<open>i\<close> to \<open>j\<close> indicate the positions to substitute at.
Positions are ordered from the top of the term tree moving down from left to
right. For example, in\<open>(a + b) + (c + d)\<close> there are three positions where
commutativity of \<open>+\<close> is applicable: 1 refers to \<open>a + b\<close>, 2 to the whole term, and 3 to\<open>c + d\<close>.
If the positions in the list \<open>(i \<dots> j)\<close> are non-overlapping (e.g.\ \<open>(2 3)\<close> in \<open>(a + b) + (c + d)\<close>) you may assume all substitutions are performed
simultaneously. Otherwise the behaviour of \<open>subst\<close> is not specified.
\<^descr> @{method subst}~\<open>(asm) (i \<dots> j) eq\<close> performs the substitutions in the
assumptions. The positions refer to the assumptions in order from left to
right. For example, given in a goal of the form \<open>P (a + b) \<Longrightarrow> P (c + d) \<Longrightarrow> \<dots>\<close>,
position 1 of commutativity of \<open>+\<close> is the subterm \<open>a + b\<close> and position 2 is
the subterm \<open>c + d\<close>.
\<^descr> @{method hypsubst} performs substitution using some assumption; this only
works for equations of the form \<open>x = t\<close> where \<open>x\<close> is a free or bound
variable.
\<^descr> @{method split}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> performs single-step case splitting using the
given rules. Splitting is performed in the conclusion or some assumption of
the subgoal, depending of the structure of the rule.
Note that the @{method simp} method already involves repeated application of
split rules as declared in the current context, using @{attribute split}, for example. \<close>
text\<open>
The Simplifier performs conditional and unconditional rewriting anduses
contextual information: rule declarations in the background theory or local proofcontext are taken into account, as well as chained facts and subgoal
premises (``local assumptions''). There are several general hooks that allow to modify the simplification strategy, or incorporate other proof tools that
solve sub-problems, produce rewrite rules on demand etc.
The rewriting strategy is always strictly bottom up, except for congruence
rules, which are applied while descending into a term. Conditions in
conditional rewrite rules are solved recursively before the rewrite rule is
applied.
The default Simplifier setup of major object logics (HOL, HOLCF, FOL, ZF)
makes the Simplifier ready for immediate use, without engaging into the
internal structures. Thus it serves as general-purpose proof tool with the
main focus on equational reasoning, and a bit more than that. \<close>
\<^descr> @{method simp} invokes the Simplifier on the first subgoal, after
inserting chained facts as additional goal premises; further rule
declarations may be included via \<open>(simp add: facts)\<close>. The proof method fails if the subgoal remains unchanged after simplification.
Note that the original goal premises and chained facts are subject to
simplification themselves, while declarations via \<open>add\<close>/\<open>del\<close> merely follow
the policies of the object-logic toextract rewrite rules fromtheorems,
without further simplification. This may lead to slightly different behavior in either case, which might be required precisely like that in some boundary
situations to perform the intended simplification step!
\<^medskip>
Modifier \<open>flip\<close> deletes the following theorems from the simpset and adds
their symmetric version (i.e.\ lhs and rhs exchanged). No warning is shown if the original theorem was not present.
\<^medskip>
The \<open>only\<close> modifier first removes all other rewrite rules, looper tactics
(including split rules), congruence rules, andthen behaves like \<open>add\<close>.
Implicit solvers remain, which means that trivial rules like reflexivity or
introduction of \<open>True\<close> are available to solve the simplified subgoals, but also non-trivial tools like linear arithmetic in HOL. The latter may lead to
some surprise of the meaning of ``only''in Isabelle/HOL compared to
English!
\<^medskip>
The \<open>split\<close> modifiers add or delete rules for the Splitter (see also \secref{sec:simp-strategies} on the looper). This works only if the
Simplifier method has been properly setupto include the Splitter (all major
object logics such HOL, HOLCF, FOL, ZF do this already).
The \<open>!\<close> option causes the split rules to be used aggressively:
after each application of a split rule in the conclusion, the \<open>safe\<close>
tactic of the classical reasoner (see \secref{sec:classical:partial}) is applied to the new goal. The net effect is that the goal is split into
the different cases. This option can speed up simplification of goals with many nested conditional or case expressions significantly.
There isalso a separate @{method_ref split} method available for
single-step case splitting. The effect of repeatedly applying \<open>(split thms)\<close>
can be imitated by ``\<open>(simp only: split: thms)\<close>''.
\<^medskip>
The \<open>cong\<close> modifiers add or delete Simplifier congruence rules (see also \secref{sec:simp-rules}); the default is to add.
\<^descr> @{method simp_all} is similar to @{method simp}, but acts on all goals,
working backwards from the last to the first one as usual in Isabelle.\<^footnote>\<open>The
order is irrelevant for goals without schematic variables, so simplification
might actually be performed in parallel here.\<close>
Chained facts are inserted into all subgoals, before the simplification
process starts. Further rule declarations are the same as for @{method
simp}.
The proof method fails if all subgoals remain unchanged after
simplification.
\<^descr> @{attribute simp_depth_limit} limits the number of recursive invocations
of the Simplifier during conditional rewriting.
By default the Simplifier methods above take local assumptions fully into
account, using equational assumptions in the subsequent normalization
process, or simplifying assumptions themselves. Further options allow to
fine-tune the behavior of the Simplifier in this respect, corresponding to a
variety of ML tactics as follows.\<^footnote>\<open>Unlike the corresponding Isar proof
methods, the ML tactics do not insist in changing the goal state.\<close>
\<open>(simp (no_asm))\<close> & \<^ML>\<open>simp_tac\<close> & assumptions are ignored completely \\\hline
\<open>(simp (no_asm_simp))\<close> & \<^ML>\<open>asm_simp_tac\<close> & assumptions are used in the
simplification of the conclusion but are not themselves simplified \\\hline
\<open>(simp (no_asm_use))\<close> & \<^ML>\<open>full_simp_tac\<close> & assumptions are simplified but
are not used in the simplification of each other or the conclusion \\\hline
\<open>(simp)\<close> & \<^ML>\<open>asm_full_simp_tac\<close> & assumptions are used in the
simplification of the conclusion andto simplify other assumptions \\\hline
\<open>(simp (asm_lr))\<close> & \<^ML>\<open>asm_lr_simp_tac\<close> & compatibility mode: an
assumption is only used for simplifying assumptions which are to the right
of it \\\hline
\end{tabular} \end{center}
\<^medskip> In Isabelle/Pure, proof methods @{method (Pure) simp} and @{method (Pure)
simp_all} only know about meta-equality \<open>\<equiv>\<close>. Any new object-logic needs to
re-define these methods via \<^ML>\<open>Simplifier.method_setup\<close> in ML:
Isabelle/FOL or Isabelle/HOL may serve as blue-prints. \<close>
subsubsection \<open>Examples\<close>
text\<open>
We consider basic algebraic simplifications in Isabelle/HOL. The rather
trivial goal \<^prop>\<open>0 + (x + 0) = x + 0 + 0\<close> looks like a good candidate to be solved by a single call of @{method simp}: \<close>
lemma"0 + (x + 0) = x + 0 + 0"apply simp? oops
text\<open>
The above attempt \<^emph>\<open>fails\<close>, because \<^term>\<open>0\<close> and \<^term>\<open>(+)\<close> in the
HOL library are declared as generic type class operations, without stating
any algebraic laws yet. More specific types are required to get access to
certain standard simplifications of the theorycontext, e.g.\ like this:\<close>
lemmafixes x :: nat shows"0 + (x + 0) = x + 0 + 0"by simp lemmafixes x :: int shows"0 + (x + 0) = x + 0 + 0"by simp lemmafixes x :: "'a :: monoid_add"shows"0 + (x + 0) = x + 0 + 0"by simp
text\<open> \<^medskip> In many cases, assumptions of a subgoal are also needed in the
simplification process. For example: \<close>
lemmafixes x :: nat shows"x = 0 \ x + x = 0" by simp lemmafixes x :: nat assumes"x = 0"shows"x + x = 0"apply simp oops lemmafixes x :: nat assumes"x = 0"shows"x + x = 0"using assms by simp
text\<open>
As seen above, local assumptions that shall contribute to simplification
need to be part of the subgoal already, or indicated explicitly foruseby
the subsequent method invocation. Both too little or too much information
can make simplification fail, for different reasons.
In the next example the malicious assumption \<^prop>\<open>\<And>x::nat. f x = g (f (g
x))\<close> does not contribute to solve the problem, but makes the default
@{method simp} method loop: the rewrite rule \<open>f ?x \<equiv> g (f (g ?x))\<close> extracted from the assumption does not terminate. The Simplifier notices certain
simple forms of nontermination, but not this one. The problem can be solved
nonetheless, by ignoring assumptions via special options as explained
before: \<close>
lemma"(\x::nat. f x = g (f (g x))) \ f 0 = f 0 + 0" by (simp (no_asm))
text\<open>
The latter form is typical for long unstructured proof scripts, where the
control over the goal content is limited. In structured proofs it is usually
better to avoid pushing too many facts into the goal state in the first
place. Assumptions in the Isar proofcontext do not intrude the reasoning if
not used explicitly. This is illustrated for a toplevel statement and a localproof body as follows: \<close>
lemma assumes"\x::nat. f x = g (f (g x))" shows"f 0 = f 0 + 0"by simp
notepad begin assume"\x::nat. f x = g (f (g x))" have"f 0 = f 0 + 0"by simp end
text\<open> \<^medskip>
Because assumptions may simplify each other, there can be very subtle cases
of nontermination. For example, the regular @{method simp} method applied to \<^prop>\<open>P (f x) \<Longrightarrow> y = x \<Longrightarrow> f x = f y \<Longrightarrow> Q\<close> gives rise to the infinite
reduction sequence \[ \<open>P (f x)\<close> \stackrel{\<open>f x \<equiv> f y\<close>}{\longmapsto} \<open>P (f y)\<close> \stackrel{\<open>y \<equiv> x\<close>}{\longmapsto} \<open>P (f x)\<close> \stackrel{\<open>f x \<equiv> f y\<close>}{\longmapsto} \cdots \]
whereas applying the same to\<^prop>\<open>y = x \<Longrightarrow> f x = f y \<Longrightarrow> P (f x) \<Longrightarrow> Q\<close>
terminates (without solving the goal): \<close>
lemma"y = x \ f x = f y \ P (f x) \ Q" apply simp oops
text\<open>
See also\secref{sec:simp-trace} for options to enable Simplifier trace
mode, which often helps to diagnose problems with rewrite systems. \<close>
\<^descr> @{attribute simp} declares rewrite rules, by adding or deleting them from
the simpset within the theory or proofcontext. Rewrite rules are theorems
expressing some form of equality, for example:
\<^medskip>
Conditional rewrites such as \<open>?m < ?n \<Longrightarrow> ?m div ?n = 0\<close> are also permitted;
the conditions can be arbitrary formulas.
\<^medskip>
Internally, all rewrite rules are translated into Pure equalities, theorems with conclusion \<open>lhs \<equiv> rhs\<close>. The simpset contains a function for extracting
equalities from arbitrary theorems, which is usually installed when the
object-logic is configured initially. For example, \<open>\<not> ?x \<in> {}\<close> could be
turned into \<open>?x \<in> {} \<equiv> False\<close>. Theorems that are declared as @{attribute
simp} andlocal assumptions within a goal are treated uniformly in this
respect.
The Simplifier accepts the following formats for the \<open>lhs\<close> term:
\<^enum> First-order patterns, considering the sublanguage of application of
constant operators to variable operands, without \<open>\<lambda>\<close>-abstractions or
functional variables. For example:
\<^enum> Higher-order patterns in the sense of \<^cite>\<open>"nipkow-patterns"\<close>. These
are terms in\<open>\<beta>\<close>-normal form (this will always be the case unless you have done something strange) where each occurrence of an unknown is of the form \<open>?F x\<^sub>1 \<dots> x\<^sub>n\<close>, where the \<open>x\<^sub>i\<close> are distinct bound variables.
For example, \<open>(\<forall>x. ?P x \<and> ?Q x) \<equiv> (\<forall>x. ?P x) \<and> (\<forall>x. ?Q x)\<close> or its
symmetric form, since the \<open>rhs\<close> is also a higher-order pattern.
\<^enum> Physical first-order patterns over raw \<open>\<lambda>\<close>-term structure without \<open>\<alpha>\<beta>\<eta>\<close>-equality; abstractions and bound variables are treated like
quasi-constant term material.
For example, the rule \<open>?f ?x \<in> range ?f = True\<close> rewrites the term \<open>g a \<in>
range g\<close> to \<open>True\<close>, but will fail to match \<open>g (h b) \<in> range (\<lambda>x. g (h
x))\<close>. However, offending subterms (in our case \<open>?f ?x\<close>, which is not a
pattern) can be replaced by adding new variables and conditions like this: \<open>?y = ?f ?x \<Longrightarrow> ?y \<in> range ?f = True\<close> is acceptable as a conditional rewrite
rule of the second category since conditions can be arbitrary terms.
\<^descr> @{attribute split} declares case split rules.
\<^descr> @{attribute cong} declares congruence rules to the Simplifier context.
Congruence rules are equalities of the form @{text [display] "\ \ f ?x\<^sub>1 \ ?x\<^sub>n = f ?y\<^sub>1 \ ?y\<^sub>n"}
This controls the simplification of the arguments of \<open>f\<close>. For example, some
arguments can be simplified under additional assumptions:
@{text [display] "?P\<^sub>1 \ ?Q\<^sub>1 \
(?Q\<^sub>1 \<Longrightarrow> ?P\<^sub>2 \<longleftrightarrow> ?Q\<^sub>2) \<Longrightarrow>
(?P\<^sub>1 \<longrightarrow> ?P\<^sub>2) \<longleftrightarrow> (?Q\<^sub>1 \<longrightarrow> ?Q\<^sub>2)"}
Given this rule, the Simplifier assumes\<open>?Q\<^sub>1\<close> and extracts rewrite rules from it when simplifying \<open>?P\<^sub>2\<close>. Such local assumptions are effective for
rewriting formulae such as \<open>x = 0 \<longrightarrow> y + x = y\<close>.
%FIXME
%The local assumptions are also provided as theoremsto the solver;
%see \secref{sec:simp-solver} below.
\<^medskip>
The following congruence rule for bounded quantifiers also supplies
contextual information --- about the bound variable: @{text [display] "(?A = ?B) \
(\<And>x. x \<in> ?B \<Longrightarrow> ?P x \<longleftrightarrow> ?Q x) \<Longrightarrow>
(\<forall>x \<in> ?A. ?P x) \<longleftrightarrow> (\<forall>x \<in> ?B. ?Q x)"}
\<^medskip>
This congruence rule for conditional expressions can supply contextual
information for simplifying the arms: @{text [display] "?p = ?q \
(?q \<Longrightarrow> ?a = ?c) \<Longrightarrow>
(\<not> ?q \<Longrightarrow> ?b = ?d) \<Longrightarrow>
(if ?p then ?a else ?b) = (if ?q then ?c else ?d)"}
A congruence rule can also\<^emph>\<open>prevent\<close> simplification of some arguments. Here is an alternative congruence rule for conditional expressions that conforms to non-strict functional evaluation: @{text [display] "?p = ?q \
(if ?p then ?a else ?b) = (if ?q then ?a else ?b)"}
Only the first argument is simplified; the others remain unchanged. This can
make simplification much faster, but may require an extra case split over
the condition \<open>?q\<close> to prove the goal.
\<^descr> @{command "print_simpset"} prints the collection of rules declared to the
Simplifier, which isalso known as ``simpset'' internally; the ``\<open>!\<close>''
option indicates extra verbosity.
The implicit simpset of the theorycontextis propagated monotonically
through the theory hierarchy: forming a new theory, the union of the
simpsets of its imports are taken as starting point. Alsonote that
definitional packages like @{command "datatype"}, @{command "primrec"},
@{command "fun"} routinely declare Simplifier rules to the target context,
while plain @{command "definition"} is an exception in\<^emph>\<open>not\<close> declaring
anything.
\<^medskip>
It is up the user to manipulate the current simpset further by explicitly
adding or deleting theorems as simplification rules, or installing other
tools via simplification procedures (\secref{sec:simproc}). Good simpsets
are hard to design. Rules that obviously simplify, like \<open>?n + 0 \<equiv> ?n\<close> are
good candidates for the implicit simpset, unless a special non-normalizing
behavior of certain operations is intended. More specific rules (such as
distributive laws, which duplicate subterms) should be added only for
specific proof steps. Conversely, sometimes a rule needs to be deleted just for some part of a proof. The need of frequent additions or deletions may
indicate a poorly designed simpset.
\begin{warn}
The union of simpsets fromtheoryimports (as described above) is not always
a good starting point for the new theory. If some ancestors have deleted
simplification rules because they are no longer wanted, while others have
left those rules in, then the union will contain the unwanted rules, and thushaveto be deleted again in the theory body. \end{warn} \<close>
subsection \<open>Ordered rewriting with permutative rules\<close>
text\<open>
A rewrite rule is\<^emph>\<open>permutative\<close> if the left-hand side and right-hand side
are the equal up to renaming of variables. The most common permutative rule is commutativity: \<open>?x + ?y = ?y + ?x\<close>. Other examples include \<open>(?x - ?y) -
?z = (?x - ?z) - ?y\<close> in arithmetic and \<open>insert ?x (insert ?y ?A) = insert ?y
(insert ?x ?A)\<close> for sets. Such rules are common enough to merit special
attention.
Because ordinary rewriting loops given such rules, the Simplifier employs a
special strategy, called \<^emph>\<open>ordered rewriting\<close>. Permutative rules are
detected and only applied if the rewriting step decreases the redex wrt.\ a
given term ordering. For example, commutativity rewrites \<open>b + a\<close> to \<open>a + b\<close>,
but then stops, because the redex cannot be decreased further in the sense
of the term ordering.
The default is lexicographic ordering of termstructure, but this could be also changed locally for special applications via @{define_ML
Simplifier.set_term_ord} in Isabelle/ML.
\<^medskip>
Permutative rewrite rules are declared to the Simplifier just like other
rewrite rules. Their special status is recognized automatically, and their
application is guarded by the term ordering accordingly. \<close>
subsubsection \<open>Rewriting with AC operators\<close>
text\<open>
Ordered rewriting is particularly effective in the case of
associative-commutative operators. (Associativity by itself is not
permutative.) When dealing with an AC-operator \<open>f\<close>, keep the following
points in mind:
\<^item> The associative law must always be oriented from left to right, namely \<open>f (f x y) z = f x (f y z)\<close>. The opposite orientation, if used with
commutativity, leads to looping in conjunction with the standard term
order.
\<^item> To complete your set of rewrite rules, you must add not just
associativity (A) and commutativity (C) but also a derived rule \<^emph>\<open>left-commutativity\<close> (LC): \<open>f x (f y z) = f y (f x z)\<close>.
Ordered rewriting with the combination of A, C, and LC sorts a term
lexicographically --- the rewriting engine imitates bubble-sort. \<close>
experiment fixes f :: "'a \ 'a \ 'a" (infix \\\ 60) assumes assoc: "(x \ y) \ z = x \ (y \ z)" assumes commute: "x \ y = y \ x" begin
lemma left_commute: "x \ (y \ z) = y \ (x \ z)" proof - have"(x \ y) \ z = (y \ x) \ z" by (simp only: commute) thenshow ?thesis by (simp only: assoc) qed
lemmas AC_rules = assoc commute left_commute
text\<open> Thus the Simplifier is able to establish equalities with arbitrary
permutations of subterms, by normalizing to a common standard form. For
example: \<close>
lemma"(b \ c) \ a = xxx" apply (simp only: AC_rules) txt\<open>\<^subgoals>\<close> oops
lemma"(b \ c) \ a = a \ (b \ c)" by (simp only: AC_rules) lemma"(b \ c) \ a = c \ (b \ a)" by (simp only: AC_rules) lemma"(b \ c) \ a = (c \ b) \ a" by (simp only: AC_rules)
end
text\<open>
Martin and Nipkow \<^cite>\<open>"martin-nipkow"\<close> discuss the theory and give many
examples; other algebraic structures are amenable to ordered rewriting, such
as Boolean rings. The Boyer-Moore theorem prover \<^cite>\<open>bm88book\<close> also
employs ordered rewriting. \<close>
text\<open>Another application of ordered rewriting uses the derived rule
@{thm [source] eq_commute}: @{thm [source = false] eq_commute} to
reverse equations.
This is occasionally useful to re-orient local assumptions according to the term ordering, when other built-in mechanisms of
reorientation and mutual simplification fail toapply.\<close>
subsection \<open>Simplifier tracing and debugging \label{sec:simp-trace}\<close>
These attributes and configurations options control various aspects of
Simplifier tracing and debugging.
\<^descr> @{attribute simp_trace} makes the Simplifier output internal operations.
This includes rewrite steps (but not traces from simproc calls),
but also bookkeeping like modifications of the simpset.
\<^descr> @{attribute simp_trace_depth_limit} limits the effect of @{attribute
simp_trace} to the given depth of recursive Simplifier invocations (when
solving conditions of rewrite rules).
\<^descr> @{attribute simp_debug} makes the Simplifier output some extra information
about internal operations. This includes any attempted invocation of
simplification procedures and the corresponding traces.
\<^descr> @{attribute simp_trace_new} controls Simplifier tracing within
Isabelle/PIDE applications, notably Isabelle/jEdit \<^cite>\<open>"isabelle-jedit"\<close>.
This provides a hierarchical representation of the rewriting steps performed by the Simplifier.
Users can configure the behaviour by specifying breakpoints, verbosity and
enabling or disabling the interactive mode. In normal verbosity (the
default), only rule applications matching a breakpoint will be shown to the
user. In full verbosity, all rule applications will be logged. Interactive
mode interrupts the normal flow of the Simplifier and defers the decision
how to continue to the user via some GUI dialog.
\<^descr> @{attribute simp_break} declares term or theorem breakpoints for
@{attribute simp_trace_new} as described above. Term breakpoints are
patterns which are checked for matches on the redex of a rule application. Theorem breakpoints trigger when the corresponding theoremis applied in a
rewrite step. For example: \<close>
text\<open>
A \<^emph>\<open>simplification procedure\<close> or \<^emph>\<open>simproc\<close> is an ML function that produces
proven rewrite rules on demand. Simprocs are guarded by multiple \<^emph>\<open>patterns\<close> for the left-hand sides of equations. The Simplifier first matches the
current redex against one of the LHS patterns; if this succeeds, the
corresponding ML functionis invoked, passing the Simplifier contextand
redex term. The function may choose to succeed with a specific result for
the redex, or fail.
The successful result of a simproc needs to be a (possibly conditional)
rewrite rule \<open>t \<equiv> u\<close> that is applicable to the current redex. The rule will
be applied just as any ordinary rewrite rule. It is expected to be already in\<^emph>\<open>internal form\<close> of the Pure logic, bypassing the automatic preprocessing
of object-level equivalences.
\<^descr> Command @{command "simproc_setup"} defines a named simplification
procedure that is invoked by the Simplifier whenever any of the given term
patterns match the current redex. The implementation, which is provided as
embedded ML source, needs to be of type \<^ML_type>\<open>morphism -> Proof.context -> cterm -> thm option\<close>, where the \<^ML_type>\<open>cterm\<close> represents the current redex \<open>r\<close> and the result is supposed to be \<^ML>\<open>SOME\<close> proven rewrite rule \<open>r \<equiv> r'\<close> (or a
generalized version); \<^ML>\<open>NONE\<close> indicates failure. The \<^ML_type>\<open>Proof.context\<close> argument holds the full context of the current
Simplifier invocation.
The \<^ML_type>\<open>morphism\<close> tells how to move from the abstract context of the
original definition into the concrete context of applications. This is only
relevant for simprocs that are defined ``\<^theory_text>\<open>in\<close>'' a local theory context
(e.g.\ @{command "locale"} with later @{command "interpretation"}).
By default, the simproc is declared to the current Simplifier contextand thus\<^emph>\<open>active\<close>. The keyword \<^theory_text>\<open>passive\<close> avoids that: it merely defines a
simproc that can be activated in a different context later on.
Regular simprocs produce rewrite rules on the fly, but it isalso possible to congruence rules via the @{syntax proc_kind} keywords: \<^theory_text>\<open>congproc\<close> or \<^theory_text>\<open>weak_congproc\<close>. See also \<^file>\<open>~~/src/HOL/Imperative_HOL/ex/Congproc_Ex.thy\<close> for further explanations and examples.
\<^descr> ML antiquotation @{ML_antiquotation_ref simproc_setup} is like command
@{command simproc_setup}, with slightly extended syntax following @{syntax
simproc_setup_id}. It allows to introduce a new simproc conveniently within
an ML module, and refer directly to its ML value. For example, see various usesin @{file"~~/src/HOL/Tools/record.ML"}.
The optional \<^theory_text>\<open>identifier\<close> specifies characteristic theorems to distinguish
simproc instances after application of morphisms, e.g.\ @{command locale} with multiple @{command interpretation}. See also the minimal example below.
\<^descr> Attributes \<open>[simproc add: name]\<close> and \<open>[simproc del: name]\<close> add or delete
named simprocs to the current Simplifier context. The default isto add a
simproc. Note that @{command "simproc_setup"} already adds the new simproc by default, unless it specified as \<^theory_text>\<open>passive\<close>. \<close>
subsubsection \<open>Examples\<close>
text\<open>
The following simplification procedure for @{thm [source = false,
show_types] unit_eq} in HOL performs fine-grained control over rule
application, beyond higher-order pattern matching. Declaring @{thm unit_eq}
as @{attribute simp} directly would make the Simplifier loop! Note that a
version of this simplification procedure is already active in Isabelle/HOL. \<close>
(*<*)experiment begin(*>*) simproc_setup unit ("x::unit") = \<open>K (K (fn ct => if HOLogic.is_unit (Thm.term_of ct) then NONE
else SOME (mk_meta_eq @{thm unit_eq})))\<close> (*<*)end(*>*)
text\<open>
Since the Simplifier applies simplification procedures frequently, it is
important to make the failure check in ML reasonably fast.
\<^medskip> The subsequent example shows how to define a local simproc with extra identifierto distinguish its instances after interpretation: \<close>
locale loc = fixes x y :: 'a assumes eq: "x = y" begin
text\<open>
The core term-rewriting engine of the Simplifier is normally used in
combination with some add-on components that modify the strategy and allow to integrate other non-Simplifier proof tools. These may be reconfigured in
ML as explained below. Even if the default strategies of object-logics like
Isabelle/HOL are used unchanged, it helps to understand how the standard
Simplifier strategies work.\<close>
The subgoaler is the tactic used to solve subgoals arising out of
conditional rewrite rules or congruence rules. The default should be
simplification itself. In rare situations, this strategy may need to be
changed. For example, if the premise of a conditional rule is an instance of
its conclusion, as in\<open>Suc ?m < ?n \<Longrightarrow> ?m < ?n\<close>, the default strategy could
loop. % FIXME !??
\<^descr> \<^ML>\<open>Simplifier.set_subgoaler\<close>~\<open>tac ctxt\<close> sets the subgoaler of the contextto\<open>tac\<close>. The tactic will be applied to the context of the running
Simplifier instance.
\<^descr> \<^ML>\<open>Simplifier.prems_of\<close>~\<open>ctxt\<close> retrieves the current set of premises from the context. This may be non-empty only if the Simplifier has been
told to utilize local assumptions in the first place (cf.\ the options in \secref{sec:simp-meth}).
As an example, consider the following alternative subgoaler: \<close>
text\<open>
This tactic first tries to solve the subgoal by assumption or by resolving withwith one of the premises, calling simplification only if that fails.\<close>
A solver is a tactic that attempts to solve a subgoal after simplification.
Its core functionality isto prove trivial subgoals such as \<^prop>\<open>True\<close> and\<open>t = t\<close>, but object-logics might be more ambitious. For example,
Isabelle/HOL performs a restricted version of linear arithmetic here.
Solvers are packaged up in abstract type \<^ML_type>\<open>solver\<close>, with \<^ML>\<open>Simplifier.mk_solver\<close> as the only operation to create a solver.
\<^medskip>
Rewriting does not instantiate unknowns. For example, rewriting alone cannot
prove \<open>a \<in> ?A\<close> since this requires instantiating \<open>?A\<close>. The solver, however, is an arbitrary tactic and may instantiate unknowns as it pleases. This is
the only way the Simplifier can handle a conditional rewrite rule whose
condition contains extra variables. When a simplification tactic isto be
combined with other provers, especially with the Classical Reasoner, it is
important whether it can be considered safe or not. For this reason a
simpset contains two solvers: safe and unsafe.
The standard simplification strategy solely uses the unsafe solver, which is
appropriate in most cases. For special applications where the simplification
process is not allowed to instantiate unknowns within the goal,
simplification starts with the safe solver, but may still apply the ordinary
unsafe one in nested simplifications for conditional rules or congruences. Note that in this way the overall tactic is not totally safe: it may
instantiate unknowns that appear alsoin other subgoals.
\<^descr> \<^ML>\<open>Simplifier.mk_solver\<close>~\<open>name tac\<close> turns \<open>tac\<close> into a solver; the \<open>name\<close> is only attached as a comment and has no further significance.
\<^descr> \<^ML>\<open>Simplifier.set_safe_solver\<close>~\<open>solver ctxt\<close> installs \<open>solver\<close> as the safe solver of \<open>ctxt\<close>.
\<^descr> \<^ML>\<open>Simplifier.add_safe_solver\<close>~\<open>solver ctxt\<close> adds \<open>solver\<close> as an additional safe solver; it
will be tried after the solvers which had already been present in\<open>ctxt\<close>.
\<^descr> \<^ML>\<open>Simplifier.set_unsafe_solver\<close>~\<open>solver ctxt\<close> installs \<open>solver\<close> as the unsafe solver of \<open>ctxt\<close>.
\<^descr> \<^ML>\<open>Simplifier.add_unsafe_solver\<close>~\<open>solver ctxt\<close> adds \<open>solver\<close> as an additional unsafe solver; it
will be tried after the solvers which had already been present in\<open>ctxt\<close>.
\<^medskip>
The solver tactic is invoked with the context of the running Simplifier.
Further operations may be used to retrieve relevant information, such as the
list of local Simplifier premises via \<^ML>\<open>Simplifier.prems_of\<close> --- this
list may be non-empty only if the Simplifier runs in a mode that utilizes local assumptions (see also\secref{sec:simp-meth}). The solver is also
presented the full goal including its assumptions in any case. Thus it can use these (e.g.\ by calling \<^ML>\<open>assume_tac\<close>), even if the Simplifier proper
happens to ignore local premises at the moment.
\<^medskip>
As explained before, the subgoaler isalso used to solve the premises of
congruence rules. These are usually of the form \<open>s = ?x\<close>, where \<open>s\<close> needs to
be simplified and\<open>?x\<close> needs to be instantiated with the result. Typically,
the subgoaler will invoke the Simplifier at some point, which will
eventually call the solver. For this reason, solver tactics must be prepared to solve goals of the form \<open>t = ?x\<close>, usually by reflexivity. In particular,
reflexivity should be tried before any of the fancy automated proof tools.
It may even happen that due to simplification the subgoal is no longer an
equality. For example, \<open>False \<longleftrightarrow> ?Q\<close> could be rewritten to \<open>\<not> ?Q\<close>. To cover
this case, the solver could try resolving with the theorem\<open>\<not> False\<close> of the
object-logic.
\<^medskip> \begin{warn} If a premise of a congruence rule cannot be proved, then the congruence is
ignored. This should only happen if the rule is\<^emph>\<open>conditional\<close> --- that is, contains premises not of the form \<open>t = ?x\<close>. Otherwise it indicates that some
congruence rule, or possibly the subgoaler or solver, is faulty. \end{warn} \<close>
The looper is a list of tactics that are applied after simplification, in case the solver failed to solve the simplified goal. If the looper succeeds,
the simplification process is started all over again. Each of the subgoals
generated by the looper is attacked in turn, in reverse order.
A typical looper is\<^emph>\<open>case splitting\<close>: the expansion of a conditional.
Another possibility istoapply an elimination rule on the assumptions. More
adventurous loopers could start an induction.
\<^descr> \<^ML>\<open>Simplifier.set_loop\<close>~\<open>tac ctxt\<close> installs \<open>tac\<close> as the only looper tactic of \<open>ctxt\<close>.
\<^descr> \<^ML>\<open>Simplifier.add_loop\<close>~\<open>(name, tac) ctxt\<close> adds \<open>tac\<close> as an additional looper tactic with name \<open>name\<close>, which is significant for managing the collection of
loopers. The tactic will be tried after the looper tactics that had
already been present in\<open>ctxt\<close>.
\<^descr> \<^ML>\<open>Simplifier.del_loop\<close>~\<open>name ctxt\<close> deletes the looper tactic that was associated with \<open>name\<close> from \<open>ctxt\<close>.
\<^descr> \<^ML>\<open>Splitter.add_split\<close>~\<open>thm ctxt\<close> adds split tactic for\<open>thm\<close> as additional looper tactic of \<open>ctxt\<close>
(overwriting previous split tactic for the same constant).
\<^descr> \<^ML>\<open>Splitter.add_split_bang\<close>~\<open>thm ctxt\<close> adds aggressive
(see \S\ref{sec:simp-meth})
split tactic for\<open>thm\<close> as additional looper tactic of \<open>ctxt\<close>
(overwriting previous split tactic for the same constant).
\<^descr> \<^ML>\<open>Splitter.del_split\<close>~\<open>thm ctxt\<close> deletes the split tactic
corresponding to\<open>thm\<close> from the looper tactics of \<open>ctxt\<close>.
The splitter replaces applications of a given function; the right-hand side
of the replacement can be anything. For example, here is a splitting rule for conditional expressions:
Another example is the elimination operator for Cartesian products (which
happens to be called \<^const>\<open>case_prod\<close> in Isabelle/HOL:
@{text [display] "?P (case_prod ?f ?p) \ (\a b. ?p = (a, b) \ ?P (f a b))"}
For technical reasons, there is a distinction between case splitting in the
conclusion andin the premises of a subgoal. The former isdoneby\<^ML>\<open>Splitter.split_tac\<close> with rules like @{thm [source] if_split} or @{thm
[source] option.split}, which do not split the subgoal, while the latter is doneby\<^ML>\<open>Splitter.split_asm_tac\<close> with rules like @{thm [source]
if_split_asm} or @{thm [source] option.split_asm}, which split the subgoal.
The function\<^ML>\<open>Splitter.add_split\<close> automatically takes care of which
tactic to call, analyzing the form of the rules given as argument; it is the
same operation behind \<open>split\<close> attribute or method modifier syntax in the
Isar source language.
Case splits should be allowed only when necessary; they are expensive and
hard to control. Case-splitting on if-expressions in the conclusion is
usually beneficial, so it is enabled by default in Isabelle/HOL and
Isabelle/FOL/ZF.
\begin{warn} With\<^ML>\<open>Splitter.split_asm_tac\<close> as looper component, the Simplifier may
split subgoals! This might cause unexpected problems in tactic expressions
that silently assume 0 or 1 subgoals after simplification. \end{warn} \<close>
\<^descr> @{attribute simplified}~\<open>a\<^sub>1 \<dots> a\<^sub>n\<close> causes a theorem to be simplified,
either by exactly the specified rules \<open>a\<^sub>1, \<dots>, a\<^sub>n\<close>, or the implicit
Simplifier contextif no arguments are given. The result is fully simplified by default, including assumptions and conclusion; the options \<open>no_asm\<close> etc.\
tune the Simplifier in the same way as the for the \<open>simp\<close> method.
Note that forward simplification restricts the Simplifier to its most basic
operation of term rewriting; solver and looper tactics
(\secref{sec:simp-strategies}) are \<^emph>\<open>not\<close> involved here. The @{attribute
simplified} attribute should be only rarely required under normal
circumstances. \<close>
text\<open>Although Isabelle is generic, many users will be working in
some extension of classical first-order logic. Isabelle/ZF is built
upon theory FOL, while Isabelle/HOL conceptually contains
first-order logic as a fragment. Theorem-proving in predicate logic is undecidable, but many automated strategies have been developed to
assist in this task.
Isabelle's classical reasoner is a generic package that accepts
certain information about a logic and delivers a suite of automatic proof tools, based on rules that are classified and declared in the context. These proof procedures are slow and simplistic compared with high-end automated theorem provers, but they can save
considerable time and effort in practice. They can prove theorems
such as Pelletier's \<^cite>\pelletier86\ problems 40 and 41 in a few
milliseconds (including full proof reconstruction):\<close>
lemma"(\y. \x. F x y \ F x x) \ \ (\x. \y. \z. F z y \ \ F z x)" by blast
lemma"(\z. \y. \x. f x y \ f x z \ \ f x x) \ \ (\z. \x. f x z)" by blast
text\<open>The proof tools are generic. They are not restricted to
first-order logic, andhave been heavily used in the development of
the Isabelle/HOL library and applications. The tactics can be
traced, and their components can be called directly; in this manner,
any proof can be viewed interactively.\<close>
subsubsection \<open>The sequent calculus\<close>
text\<open>Isabelle supports natural deduction, which is easy to use for
interactive proof. But natural deduction does not easily lend
itself to automation, and has a bias towards intuitionism. For
certain proofs in classical logic, it can not be called natural.
The \<^emph>\<open>sequent calculus\<close>, a generalization of natural deduction, is easier to automate.
A \<^bold>\<open>sequent\<close> has the form \<open>\<Gamma> \<turnstile> \<Delta>\<close>, where \<open>\<Gamma>\<close> and\<open>\<Delta>\<close> are sets of formulae.\<^footnote>\<open>For first-order
logic, sequents can equivalently be made from lists or multisets of
formulae.\<close> The sequent \<open>P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n\<close> is \<^bold>\<open>valid\<close> if \<open>P\<^sub>1 \<and> \<dots> \<and> P\<^sub>m\<close> implies \<open>Q\<^sub>1 \<or> \<dots> \<or>
Q\<^sub>n\<close>. Thus \<open>P\<^sub>1, \<dots>, P\<^sub>m\<close> represent assumptions, each of which is true, while \<open>Q\<^sub>1, \<dots>, Q\<^sub>n\<close> represent alternative goals. A
sequent is\<^bold>\<open>basic\<close> if its left and right sides have a common
formula, as in\<open>P, Q \<turnstile> Q, R\<close>; basic sequents are trivially
valid.
Sequent rules are classified as \<^bold>\<open>right\<close> or \<^bold>\<open>left\<close>,
indicating which side of the \<open>\<turnstile>\<close> symbol they operate on.
Rules that operate on the right side are analogous to natural
deduction's introduction rules, and left rules are analogous to
elimination rules. The sequent calculus analogue of \<open>(\<longrightarrow>I)\<close> is the rule \[ \infer[\<open>(\<longrightarrow>R)\<close>]{\<open>\<Gamma> \<turnstile> \<Delta>, P \<longrightarrow> Q\<close>}{\<open>P, \<Gamma> \<turnstile> \<Delta>, Q\<close>} \]
Applying the rule backwards, this breaks down some implication on
the right side of a sequent; \<open>\<Gamma>\<close> and \<open>\<Delta>\<close> stand for
the sets of formulae that are unaffected by the inference. The
analogue of the pair \<open>(\<or>I1)\<close> and \<open>(\<or>I2)\<close> is the
single rule \[ \infer[\<open>(\<or>R)\<close>]{\<open>\<Gamma> \<turnstile> \<Delta>, P \<or> Q\<close>}{\<open>\<Gamma> \<turnstile> \<Delta>, P, Q\<close>} \]
This breaks down some disjunction on the right side, replacing it by
both disjuncts. Thus, the sequent calculus is a kind of
multiple-conclusion logic.
To illustrate the use of multiple formulae on the right, let us
prove the classical theorem\<open>(P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)\<close>. Working
backwards, we reduce this formula to a basic sequent: \[ \infer[\<open>(\<or>R)\<close>]{\<open>\<turnstile> (P \<longrightarrow> Q) \<or> (Q \<longrightarrow> P)\<close>}
{\infer[\<open>(\<longrightarrow>R)\<close>]{\<open>\<turnstile> (P \<longrightarrow> Q), (Q \<longrightarrow> P)\<close>}
{\infer[\<open>(\<longrightarrow>R)\<close>]{\<open>P \<turnstile> Q, (Q \<longrightarrow> P)\<close>}
{\<open>P, Q \<turnstile> Q, P\<close>}}} \]
This example is typical of the sequent calculus: start with the
desired theoremandapply rules backwards in a fairly arbitrary
manner. This yields a surprisingly effective proof procedure.
Quantifiers add only few complications, since Isabelle handles
parameters and schematic variables. See \<^cite>\<open>\<open>Chapter 10\<close> in "paulson-ml2"\<close> for further discussion.\<close>
subsubsection \<open>Simulating sequents by natural deduction\<close>
text\<open>Isabelle can represent sequents directly, as in the
object-logic LK. But natural deduction is easier to work with, and
most object-logics employ it. Fortunately, we can simulate the
sequent \<open>P\<^sub>1, \<dots>, P\<^sub>m \<turnstile> Q\<^sub>1, \<dots>, Q\<^sub>n\<close> by the Isabelle formula \<open>P\<^sub>1 \<Longrightarrow> \<dots> \<Longrightarrow> P\<^sub>m \<Longrightarrow> \<not> Q\<^sub>2 \<Longrightarrow> ... \<Longrightarrow> \<not> Q\<^sub>n \<Longrightarrow> Q\<^sub>1\<close> where the order of
the assumptions and the choice of \<open>Q\<^sub>1\<close> are arbitrary.
Elim-resolution plays a key role in simulating sequent proofs.
We can easily handle reasoning on the left. Elim-resolution with
the rules \<open>(\<or>E)\<close>, \<open>(\<bottom>E)\<close> and \<open>(\<exists>E)\<close> achieves
a similar effect as the corresponding sequent rules. For the other
connectives, we use sequent-style elimination rules instead of
destruction rules such as \<open>(\<and>E1, 2)\<close> and \<open>(\<forall>E)\<close>.
But note that the rule \<open>(\<not>L)\<close> has no effect under our
representation of sequents! \[ \infer[\<open>(\<not>L)\<close>]{\<open>\<not> P, \<Gamma> \<turnstile> \<Delta>\<close>}{\<open>\<Gamma> \<turnstile> \<Delta>, P\<close>} \]
What about reasoning on the right? Introduction rules can only
affect the formula in the conclusion, namely \<open>Q\<^sub>1\<close>. The
other right-side formulae are represented as negated assumptions, \<open>\<not> Q\<^sub>2, \<dots>, \<not> Q\<^sub>n\<close>. In order to operate on one of these, it
must first be exchanged with\<open>Q\<^sub>1\<close>. Elim-resolution with the \<open>swap\<close> rule has this effect: \<open>\<not> P \<Longrightarrow> (\<not> R \<Longrightarrow> P) \<Longrightarrow> R\<close>
To ensure that swaps occur only when necessary, each introduction
rule is converted into a swapped form: it is resolved with the
second premise of \<open>(swap)\<close>. The swapped form of \<open>(\<and>I)\<close>, which might be called \<open>(\<not>\<and>E)\<close>, is
@{text [display] "\ (P \ Q) \ (\ R \ P) \ (\ R \ Q) \ R"}
Similarly, the swapped form of \<open>(\<longrightarrow>I)\<close> is
@{text [display] "\ (P \ Q) \ (\ R \ P \ Q) \ R"}
Swapped introduction rules are applied using elim-resolution, which
deletes the negated formula. Our representation of sequents also
requires the use of ordinary introduction rules. If we had no
regard for readability of intermediate goal states, we could treat
the right side more uniformly by representing sequents as @{text
[display] "P\<^sub>1 \ \ \ P\<^sub>m \ \ Q\<^sub>1 \ \ \ \ Q\<^sub>n \ \"} \<close>
subsubsection \<open>Extra rules for the sequent calculus\<close>
text\<open>As mentioned, destruction rules such as \<open>(\<and>E1, 2)\<close> and \<open>(\<forall>E)\<close> must be replaced by sequent-style elimination rules. In addition, we need rules to embody the classical equivalence
between \<open>P \<longrightarrow> Q\<close> and \<open>\<not> P \<or> Q\<close>. The introduction
rules \<open>(\<or>I1, 2)\<close> are replaced by a rule that simulates \<open>(\<or>R)\<close>: @{text [display] "(\<not> Q \<Longrightarrow> P) \<Longrightarrow> P \<or> Q"}
The destruction rule \<open>(\<longrightarrow>E)\<close> is replaced by @{text [display] "(P \ Q) \ (\ P \ R) \ (Q \ R) \ R"}
Quantifier replication also requires special rules. In classical
logic, \<open>\<exists>x. P x\<close> is equivalent to \<open>\<not> (\<forall>x. \<not> P x)\<close>;
the rules \<open>(\<exists>R)\<close> and \<open>(\<forall>L)\<close> are dual: \[ \infer[\<open>(\<exists>R)\<close>]{\<open>\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x\<close>}{\<open>\<Gamma> \<turnstile> \<Delta>, \<exists>x. P x, P t\<close>} \qquad \infer[\<open>(\<forall>L)\<close>]{\<open>\<forall>x. P x, \<Gamma> \<turnstile> \<Delta>\<close>}{\<open>P t, \<forall>x. P x, \<Gamma> \<turnstile> \<Delta>\<close>} \] Thus both kinds of quantifier may be replicated. Theorems requiring
multiple uses of a universal formula are easy to invent; consider
@{text [display] "(\x. P x \ P (f x)) \ P a \ P (f\<^sup>n a)"} for any \<open>n > 1\<close>. Natural examples of the multiple use of an
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