(*:maxLineLen=78:*)
theory Framework
imports Base Main
begin
chapter \<open>The Isabelle/Isar Framework \label{ch:isar-framework}\<close>
text \<open>
Isabelle/Isar @{cite "Wenzel:1999:TPHOL" and "Wenzel-PhD" and
"Nipkow-TYPES02" and "Wenzel-Paulson:2006" and "Wenzel:2006:Festschrift"}
is intended as a generic framework for developing formal
mathematical documents with full proof checking. Definitions and
proofs are organized as theories. An assembly of theory sources may
be presented as a printed document; see also
\chref{ch:document-prep}.
The main objective of Isar is the design of a human-readable
structured proof language, which is called the ``primary proof
format'' in Isar terminology. Such a primary proof language is
somewhere in the middle between the extremes of primitive proof
objects and actual natural language. In this respect, Isar is a bit
more formalistic than Mizar @{cite "Trybulec:1993:MizarFeatures" and
"Rudnicki:1992:MizarOverview" and "Wiedijk:1999:Mizar"},
using logical symbols for certain reasoning schemes where Mizar
would prefer English words; see @{cite "Wenzel-Wiedijk:2002"} for
further comparisons of these systems.
So Isar challenges the traditional way of recording informal proofs
in mathematical prose, as well as the common tendency to see fully
formal proofs directly as objects of some logical calculus (e.g.\
\<open>\<lambda>\<close>-terms in a version of type theory). In fact, Isar is
better understood as an interpreter of a simple block-structured
language for describing the data flow of local facts and goals,
interspersed with occasional invocations of proof methods.
Everything is reduced to logical inferences internally, but these
steps are somewhat marginal compared to the overall bookkeeping of
the interpretation process. Thanks to careful design of the syntax
and semantics of Isar language elements, a formal record of Isar
instructions may later appear as an intelligible text to the
attentive reader.
The Isar proof language has emerged from careful analysis of some
inherent virtues of the existing logical framework of Isabelle/Pure
@{cite "paulson-found" and "paulson700"}, notably composition of higher-order
natural deduction rules, which is a generalization of Gentzen's
original calculus @{cite "Gentzen:1935"}. The approach of generic
inference systems in Pure is continued by Isar towards actual proof
texts.
Concrete applications require another intermediate layer: an
object-logic. Isabelle/HOL @{cite "isa-tutorial"} (simply-typed
set-theory) is being used most of the time; Isabelle/ZF
@{cite "isabelle-ZF"} is less extensively developed, although it would
probably fit better for classical mathematics.
\<^medskip>
In order to illustrate natural deduction in Isar, we shall
refer to the background theory and library of Isabelle/HOL. This
includes common notions of predicate logic, naive set-theory etc.\
using fairly standard mathematical notation. From the perspective
of generic natural deduction there is nothing special about the
logical connectives of HOL (\<open>\<and>\<close>, \<open>\<or>\<close>, \<open>\<forall>\<close>,
\<open>\<exists>\<close>, etc.), only the resulting reasoning principles are
relevant to the user. There are similar rules available for
set-theory operators (\<open>\<inter>\<close>, \<open>\<union>\<close>, \<open>\<Inter>\<close>, \<open>\<Union>\<close>, etc.), or any other theory developed in the library (lattice
theory, topology etc.).
Subsequently we briefly review fragments of Isar proof texts
corresponding directly to such general deduction schemes. The
examples shall refer to set-theory, to minimize the danger of
understanding connectives of predicate logic as something special.
\<^medskip>
The following deduction performs \<open>\<inter>\<close>-introduction,
working forwards from assumptions towards the conclusion. We give
both the Isar text, and depict the primitive rule involved, as
determined by unification of the problem against rules that are
declared in the library context.
\<close>
text_raw \<open>\medskip\begin{minipage}{0.6\textwidth}\<close>
(*<*)
notepad
begin
(*>*)
assume "x \<in> A" and "x \<in> B"
then have "x \<in> A \<inter> B" ..
(*<*)
end
(*>*)
text_raw \<open>\end{minipage}\begin{minipage}{0.4\textwidth}\<close>
text \<open>
\infer{@{prop "x \<in> A \<inter> B"}}{@{prop "x \<in> A"} & @{prop "x \<in> B"}}
\<close>
text_raw \<open>\end{minipage}\<close>
text \<open>
\<^medskip>
Note that @{command assume} augments the proof
context, @{command then} indicates that the current fact shall be
used in the next step, and @{command have} states an intermediate
goal. The two dots ``@{command ".."}'' refer to a complete proof of
this claim, using the indicated facts and a canonical rule from the
context. We could have been more explicit here by spelling out the
final proof step via the @{command "by"} command:
\<close>
(*<*)
notepad
begin
(*>*)
assume "x \<in> A" and "x \<in> B"
then have "x \<in> A \<inter> B" by (rule IntI)
(*<*)
end
(*>*)
text \<open>
The format of the \<open>\<inter>\<close>-introduction rule represents
the most basic inference, which proceeds from given premises to a
conclusion, without any nested proof context involved.
The next example performs backwards introduction on @{term "\<Inter>\<A>"},
the intersection of all sets within a given set. This requires a
nested proof of set membership within a local context, where @{term
A} is an arbitrary-but-fixed member of the collection:
\<close>
text_raw \<open>\medskip\begin{minipage}{0.6\textwidth}\<close>
(*<*)
notepad
begin
(*>*)
have "x \<in> \<Inter>\<A>"
proof
fix A
assume "A \<in> \<A>"
show "x \<in> A" \<proof>
qed
(*<*)
end
(*>*)
text_raw \<open>\end{minipage}\begin{minipage}{0.4\textwidth}\<close>
text \<open>
\infer{@{prop "x \<in> \<Inter>\<A>"}}{\infer*{@{prop "x \<in> A"}}{\<open>[A][A \<in> \<A>]\<close>}}
\<close>
text_raw \<open>\end{minipage}\<close>
text \<open>
\<^medskip>
This Isar reasoning pattern again refers to the
primitive rule depicted above. The system determines it in the
``@{command proof}'' step, which could have been spelled out more
explicitly as ``@{command proof}~\<open>(rule InterI)\<close>''. Note
that the rule involves both a local parameter @{term "A"} and an
assumption @{prop "A \<in> \<A>"} in the nested reasoning. This kind of
compound rule typically demands a genuine sub-proof in Isar, working
backwards rather than forwards as seen before. In the proof body we
encounter the @{command fix}-@{command assume}-@{command show}
outline of nested sub-proofs that is typical for Isar. The final
@{command show} is like @{command have} followed by an additional
refinement of the enclosing claim, using the rule derived from the
proof body.
\<^medskip>
The next example involves @{term "\<Union>\<A>"}, which can be
characterized as the set of all @{term "x"} such that @{prop "\<exists>A. x
\<in> A \<and> A \<in> \<A>"}. The elimination rule for @{prop "x \<in> \<Union>\<A>"} does
not mention \<open>\<exists>\<close> and \<open>\<and>\<close> at all, but admits to obtain
directly a local @{term "A"} such that @{prop "x \<in> A"} and @{prop "A
\<in> \<A>"} hold. This corresponds to the following Isar proof and
inference rule, respectively:
\<close>
text_raw \<open>\medskip\begin{minipage}{0.6\textwidth}\<close>
(*<*)
notepad
begin
(*>*)
assume "x \<in> \<Union>\<A>"
then have C
proof
fix A
assume "x \<in> A" and "A \<in> \<A>"
show C \<proof>
qed
(*<*)
end
(*>*)
text_raw \<open>\end{minipage}\begin{minipage}{0.4\textwidth}\<close>
text \<open>
\infer{@{prop "C"}}{@{prop "x \<in> \<Union>\<A>"} & \infer*{@{prop "C"}~}{\<open>[A][x \<in> A, A \<in> \<A>]\<close>}}
\<close>
text_raw \<open>\end{minipage}\<close>
text \<open>
\<^medskip>
Although the Isar proof follows the natural
deduction rule closely, the text reads not as natural as
anticipated. There is a double occurrence of an arbitrary
conclusion @{prop "C"}, which represents the final result, but is
irrelevant for now. This issue arises for any elimination rule
involving local parameters. Isar provides the derived language
element @{command obtain}, which is able to perform the same
elimination proof more conveniently:
\<close>
(*<*)
notepad
begin
(*>*)
assume "x \<in> \<Union>\<A>"
then obtain A where "x \<in> A" and "A \<in> \<A>" ..
(*<*)
end
(*>*)
text \<open>
Here we avoid to mention the final conclusion @{prop "C"}
and return to plain forward reasoning. The rule involved in the
``@{command ".."}'' proof is the same as before.
\<close>
section \<open>The Pure framework \label{sec:framework-pure}\<close>
text \<open>
The Pure logic @{cite "paulson-found" and "paulson700"} is an intuitionistic
fragment of higher-order logic @{cite "church40"}. In type-theoretic
parlance, there are three levels of \<open>\<lambda>\<close>-calculus with
corresponding arrows \<open>\<Rightarrow>\<close>/\<open>\<And>\<close>/\<open>\<Longrightarrow>\<close>:
\<^medskip>
\begin{tabular}{ll}
\<open>\<alpha> \<Rightarrow> \<beta>\<close> & syntactic function space (terms depending on terms) \\
\<open>\<And>x. B(x)\<close> & universal quantification (proofs depending on terms) \\
\<open>A \<Longrightarrow> B\<close> & implication (proofs depending on proofs) \\
\end{tabular}
\<^medskip>
Here only the types of syntactic terms, and the
propositions of proof terms have been shown. The \<open>\<lambda>\<close>-structure of proofs can be recorded as an optional feature of
the Pure inference kernel @{cite "Berghofer-Nipkow:2000:TPHOL"}, but
the formal system can never depend on them due to \<^emph>\<open>proof
irrelevance\<close>.
On top of this most primitive layer of proofs, Pure implements a
generic calculus for nested natural deduction rules, similar to
@{cite "Schroeder-Heister:1984"}. Here object-logic inferences are
internalized as formulae over \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close>.
Combining such rule statements may involve higher-order unification
@{cite "paulson-natural"}.
\<close>
subsection \<open>Primitive inferences\<close>
text \<open>
Term syntax provides explicit notation for abstraction \<open>\<lambda>x ::
\<alpha>. b(x)\<close> and application \<open>b a\<close>, while types are usually
implicit thanks to type-inference; terms of type \<open>prop\<close> are
called propositions. Logical statements are composed via \<open>\<And>x
:: \<alpha>. B(x)\<close> and \<open>A \<Longrightarrow> B\<close>. Primitive reasoning operates on
judgments of the form \<open>\<Gamma> \<turnstile> \<phi>\<close>, with standard introduction
and elimination rules for \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> that refer to
fixed parameters \<open>x\<^sub>1, \<dots>, x\<^sub>m\<close> and hypotheses
\<open>A\<^sub>1, \<dots>, A\<^sub>n\<close> from the context \<open>\<Gamma>\<close>;
the corresponding proof terms are left implicit. The subsequent
inference rules define \<open>\<Gamma> \<turnstile> \<phi>\<close> inductively, relative to a
collection of axioms:
\[
\infer{\<open>\<turnstile> A\<close>}{(\<open>A\<close> \mbox{~axiom})}
\qquad
\infer{\<open>A \<turnstile> A\<close>}{}
\]
\[
\infer{\<open>\<Gamma> \<turnstile> \<And>x. B(x)\<close>}{\<open>\<Gamma> \<turnstile> B(x)\<close> & \<open>x \<notin> \<Gamma>\<close>}
\qquad
\infer{\<open>\<Gamma> \<turnstile> B(a)\<close>}{\<open>\<Gamma> \<turnstile> \<And>x. B(x)\<close>}
\]
\[
\infer{\<open>\<Gamma> - A \<turnstile> A \<Longrightarrow> B\<close>}{\<open>\<Gamma> \<turnstile> B\<close>}
\qquad
\infer{\<open>\<Gamma>\<^sub>1 \<union> \<Gamma>\<^sub>2 \<turnstile> B\<close>}{\<open>\<Gamma>\<^sub>1 \<turnstile> A \<Longrightarrow> B\<close> & \<open>\<Gamma>\<^sub>2 \<turnstile> A\<close>}
\]
Furthermore, Pure provides a built-in equality \<open>\<equiv> :: \<alpha> \<Rightarrow> \<alpha> \<Rightarrow>
prop\<close> with axioms for reflexivity, substitution, extensionality,
and \<open>\<alpha>\<beta>\<eta>\<close>-conversion on \<open>\<lambda>\<close>-terms.
\<^medskip>
An object-logic introduces another layer on top of Pure,
e.g.\ with types \<open>i\<close> for individuals and \<open>o\<close> for
propositions, term constants \<open>Trueprop :: o \<Rightarrow> prop\<close> as
(implicit) derivability judgment and connectives like \<open>\<and> :: o
\<Rightarrow> o \<Rightarrow> o\<close> or \<open>\<forall> :: (i \<Rightarrow> o) \<Rightarrow> o\<close>, and axioms for object-level
rules such as \<open>conjI: A \<Longrightarrow> B \<Longrightarrow> A \<and> B\<close> or \<open>allI: (\<And>x. B
x) \<Longrightarrow> \<forall>x. B x\<close>. Derived object rules are represented as theorems of
Pure. After the initial object-logic setup, further axiomatizations
are usually avoided; plain definitions and derived principles are
used exclusively.
\<close>
subsection \<open>Reasoning with rules \label{sec:framework-resolution}\<close>
text \<open>
Primitive inferences mostly serve foundational purposes. The main
reasoning mechanisms of Pure operate on nested natural deduction
rules expressed as formulae, using \<open>\<And>\<close> to bind local
parameters and \<open>\<Longrightarrow>\<close> to express entailment. Multiple
parameters and premises are represented by repeating these
connectives in a right-associative manner.
Since \<open>\<And>\<close> and \<open>\<Longrightarrow>\<close> commute thanks to the theorem
@{prop "(A \<Longrightarrow> (\<And>x. B x)) \<equiv> (\<And>x. A \<Longrightarrow> B x)"}, we may assume w.l.o.g.\
that rule statements always observe the normal form where
quantifiers are pulled in front of implications at each level of
nesting. This means that any Pure proposition may be presented as a
\<^emph>\<open>Hereditary Harrop Formula\<close> @{cite "Miller:1991"} which is of the
form \<open>\<And>x\<^sub>1 \<dots> x\<^sub>m. H\<^sub>1 \<Longrightarrow> \<dots> H\<^sub>n \<Longrightarrow>
A\<close> for \<open>m, n \<ge> 0\<close>, and \<open>A\<close> atomic, and \<open>H\<^sub>1, \<dots>, H\<^sub>n\<close> being recursively of the same format.
Following the convention that outermost quantifiers are implicit,
Horn clauses \<open>A\<^sub>1 \<Longrightarrow> \<dots> A\<^sub>n \<Longrightarrow> A\<close> are a special
case of this.
For example, \<open>\<inter>\<close>-introduction rule encountered before is
represented as a Pure theorem as follows:
\[
\<open>IntI:\<close>~@{prop "x \<in> A \<Longrightarrow> x \<in> B \<Longrightarrow> x \<in> A \<inter> B"}
\]
This is a plain Horn clause, since no further nesting on
the left is involved. The general \<open>\<Inter>\<close>-introduction
corresponds to a Hereditary Harrop Formula with one additional level
of nesting:
\[
\<open>InterI:\<close>~@{prop "(\<And>A. A \<in> \<A> \<Longrightarrow> x \<in> A) \<Longrightarrow> x \<in> \<Inter>\<A>"}
\]
\<^medskip>
Goals are also represented as rules: \<open>A\<^sub>1 \<Longrightarrow>
\<dots> A\<^sub>n \<Longrightarrow> C\<close> states that the sub-goals \<open>A\<^sub>1, \<dots>,
A\<^sub>n\<close> entail the result \<open>C\<close>; for \<open>n = 0\<close> the
goal is finished. To allow \<open>C\<close> being a rule statement
itself, we introduce the protective marker \<open># :: prop \<Rightarrow>
prop\<close>, which is defined as identity and hidden from the user. We
initialize and finish goal states as follows:
\[
\begin{array}{c@ {\qquad}c}
\infer[(@{inference_def init})]{\<open>C \<Longrightarrow> #C\<close>}{} &
\infer[(@{inference_def finish})]{\<open>C\<close>}{\<open>#C\<close>}
\end{array}
\]
Goal states are refined in intermediate proof steps until
a finished form is achieved. Here the two main reasoning principles
are @{inference resolution}, for back-chaining a rule against a
sub-goal (replacing it by zero or more sub-goals), and @{inference
assumption}, for solving a sub-goal (finding a short-circuit with
local assumptions). Below \<open>\<^vec>x\<close> stands for \<open>x\<^sub>1, \<dots>, x\<^sub>n\<close> (\<open>n \<ge> 0\<close>).
\[
\infer[(@{inference_def resolution})]
{\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>A (\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
{\begin{tabular}{rl}
\<open>rule:\<close> &
\<open>\<^vec>A \<^vec>a \<Longrightarrow> B \<^vec>a\<close> \\
\<open>goal:\<close> &
\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C\<close> \\
\<open>goal unifier:\<close> &
\<open>(\<lambda>\<^vec>x. B (\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>\<close> \\
\end{tabular}}
\]
\<^medskip>
\[
\infer[(@{inference_def assumption})]{\<open>C\<vartheta>\<close>}
{\begin{tabular}{rl}
\<open>goal:\<close> &
\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> A \<^vec>x) \<Longrightarrow> C\<close> \\
\<open>assm unifier:\<close> & \<open>A\<vartheta> = H\<^sub>i\<vartheta>\<close>~~\mbox{(for some~\<open>H\<^sub>i\<close>)} \\
\end{tabular}}
\]
The following trace illustrates goal-oriented reasoning in
Isabelle/Pure:
{\footnotesize
\<^medskip>
\begin{tabular}{r@ {\quad}l}
\<open>(A \<and> B \<Longrightarrow> B \<and> A) \<Longrightarrow> #(A \<and> B \<Longrightarrow> B \<and> A)\<close> & \<open>(init)\<close> \\
\<open>(A \<and> B \<Longrightarrow> B) \<Longrightarrow> (A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>\<close> & \<open>(resolution B \<Longrightarrow> A \<Longrightarrow> B \<and> A)\<close> \\
\<open>(A \<and> B \<Longrightarrow> A \<and> B) \<Longrightarrow> (A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>\<close> & \<open>(resolution A \<and> B \<Longrightarrow> B)\<close> \\
\<open>(A \<and> B \<Longrightarrow> A) \<Longrightarrow> #\<dots>\<close> & \<open>(assumption)\<close> \\
\<open>(A \<and> B \<Longrightarrow> A \<and> B) \<Longrightarrow> #\<dots>\<close> & \<open>(resolution A \<and> B \<Longrightarrow> A)\<close> \\
\<open>#\<dots>\<close> & \<open>(assumption)\<close> \\
\<open>A \<and> B \<Longrightarrow> B \<and> A\<close> & \<open>(finish)\<close> \\
\end{tabular}
\<^medskip>
}
Compositions of @{inference assumption} after @{inference
resolution} occurs quite often, typically in elimination steps.
Traditional Isabelle tactics accommodate this by a combined
@{inference_def elim_resolution} principle. In contrast, Isar uses
a slightly more refined combination, where the assumptions to be
closed are marked explicitly, using again the protective marker
\<open>#\<close>:
\[
\infer[(@{inference refinement})]
{\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> \<^vec>G' (\<^vec>a \<^vec>x))\<vartheta> \<Longrightarrow> C\<vartheta>\<close>}
{\begin{tabular}{rl}
\<open>sub\<hyphen>proof:\<close> &
\<open>\<^vec>G \<^vec>a \<Longrightarrow> B \<^vec>a\<close> \\
\<open>goal:\<close> &
\<open>(\<And>\<^vec>x. \<^vec>H \<^vec>x \<Longrightarrow> B' \<^vec>x) \<Longrightarrow> C\<close> \\
\<open>goal unifier:\<close> &
\<open>(\<lambda>\<^vec>x. B (\<^vec>a \<^vec>x))\<vartheta> = B'\<vartheta>\<close> \\
\<open>assm unifiers:\<close> &
\<open>(\<lambda>\<^vec>x. G\<^sub>j (\<^vec>a \<^vec>x))\<vartheta> = #H\<^sub>i\<vartheta>\<close> \\
& \quad (for each marked \<open>G\<^sub>j\<close> some \<open>#H\<^sub>i\<close>) \\
\end{tabular}}
\]
Here the \<open>sub\<hyphen>proof\<close> rule stems from the
main @{command fix}-@{command assume}-@{command show} outline of
Isar (cf.\ \secref{sec:framework-subproof}): each assumption
indicated in the text results in a marked premise \<open>G\<close> above.
The marking enforces resolution against one of the sub-goal's
premises. Consequently, @{command fix}-@{command assume}-@{command
show} enables to fit the result of a sub-proof quite robustly into a
pending sub-goal, while maintaining a good measure of flexibility.
\<close>
section \<open>The Isar proof language \label{sec:framework-isar}\<close>
text \<open>
Structured proofs are presented as high-level expressions for
composing entities of Pure (propositions, facts, and goals). The
Isar proof language allows to organize reasoning within the
underlying rule calculus of Pure, but Isar is not another logical
calculus!
Isar is an exercise in sound minimalism. Approximately half of the
language is introduced as primitive, the rest defined as derived
concepts. The following grammar describes the core language
(category \<open>proof\<close>), which is embedded into theory
specification elements such as @{command theorem}; see also
\secref{sec:framework-stmt} for the separate category \<open>statement\<close>.
\<^medskip>
\begin{tabular}{rcl}
\<open>theory\<hyphen>stmt\<close> & = & @{command "theorem"}~\<open>statement proof |\<close>~~@{command "definition"}~\<open>\<dots> | \<dots>\<close> \\[1ex]
\<open>proof\<close> & = & \<open>prfx\<^sup>*\<close>~@{command "proof"}~\<open>method\<^sup>? stmt\<^sup>*\<close>~@{command "qed"}~\<open>method\<^sup>?\<close> \\[1ex]
\<open>prfx\<close> & = & @{command "using"}~\<open>facts\<close> \\
& \<open>|\<close> & @{command "unfolding"}~\<open>facts\<close> \\
\<open>stmt\<close> & = & @{command "{"}~\<open>stmt\<^sup>*\<close>~@{command "}"} \\
& \<open>|\<close> & @{command "next"} \\
& \<open>|\<close> & @{command "note"}~\<open>name = facts\<close> \\
& \<open>|\<close> & @{command "let"}~\<open>term = term\<close> \\
& \<open>|\<close> & @{command "fix"}~\<open>var\<^sup>+\<close> \\
& \<open>|\<close> & @{command assume}~\<open>\<guillemotleft>inference\<guillemotright> name: props\<close> \\
& \<open>|\<close> & @{command "then"}\<open>\<^sup>?\<close>~\<open>goal\<close> \\
\<open>goal\<close> & = & @{command "have"}~\<open>name: props proof\<close> \\
& \<open>|\<close> & @{command "show"}~\<open>name: props proof\<close> \\
\end{tabular}
\<^medskip>
Simultaneous propositions or facts may be separated by the
@{keyword "and"} keyword.
\<^medskip>
The syntax for terms and propositions is inherited from
Pure (and the object-logic). A \<open>pattern\<close> is a \<open>term\<close> with schematic variables, to be bound by higher-order
matching.
\<^medskip>
Facts may be referenced by name or proposition. For
example, the result of ``@{command have}~\<open>a: A \<langle>proof\<rangle>\<close>''
becomes available both as \<open>a\<close> and
\isacharbackquoteopen\<open>A\<close>\isacharbackquoteclose. Moreover,
fact expressions may involve attributes that modify either the
theorem or the background context. For example, the expression
``\<open>a [OF b]\<close>'' refers to the composition of two facts
according to the @{inference resolution} inference of
\secref{sec:framework-resolution}, while ``\<open>a [intro]\<close>''
declares a fact as introduction rule in the context.
The special fact called ``@{fact this}'' always refers to the last
result, as produced by @{command note}, @{command assume}, @{command
have}, or @{command show}. Since @{command note} occurs
frequently together with @{command then} we provide some
abbreviations:
\<^medskip>
\begin{tabular}{rcl}
@{command from}~\<open>a\<close> & \<open>\<equiv>\<close> & @{command note}~\<open>a\<close>~@{command then} \\
@{command with}~\<open>a\<close> & \<open>\<equiv>\<close> & @{command from}~\<open>a \<AND> this\<close> \\
\end{tabular}
\<^medskip>
The \<open>method\<close> category is essentially a parameter and may be
populated later. Methods use the facts indicated by @{command
"then"} or @{command using}, and then operate on the goal state.
Some basic methods are predefined: ``@{method "-"}'' leaves the goal
unchanged, ``@{method this}'' applies the facts as rules to the
goal, ``@{method (Pure) "rule"}'' applies the facts to another rule and the
result to the goal (both ``@{method this}'' and ``@{method (Pure) rule}''
refer to @{inference resolution} of
\secref{sec:framework-resolution}). The secondary arguments to
``@{method (Pure) rule}'' may be specified explicitly as in ``\<open>(rule
a)\<close>'', or picked from the context. In the latter case, the system
first tries rules declared as @{attribute (Pure) elim} or
@{attribute (Pure) dest}, followed by those declared as @{attribute
(Pure) intro}.
The default method for @{command proof} is ``@{method standard}''
(arguments picked from the context), for @{command qed} it is
``@{method "succeed"}''. Further abbreviations for terminal proof steps
are ``@{command "by"}~\<open>method\<^sub>1 method\<^sub>2\<close>'' for
``@{command proof}~\<open>method\<^sub>1\<close>~@{command qed}~\<open>method\<^sub>2\<close>'', and ``@{command ".."}'' for ``@{command
"by"}~@{method standard}, and ``@{command "."}'' for ``@{command
"by"}~@{method this}''. The @{command unfolding} element operates
directly on the current facts and goal by applying equalities.
\<^medskip>
Block structure can be indicated explicitly by ``@{command
"{"}~\<open>\<dots>\<close>~@{command "}"}'', although the body of a sub-proof
already involves implicit nesting. In any case, @{command next}
jumps into the next section of a block, i.e.\ it acts like closing
an implicit block scope and opening another one; there is no direct
correspondence to subgoals here.
The remaining elements @{command fix} and @{command assume} build up
a local context (see \secref{sec:framework-context}), while
@{command show} refines a pending sub-goal by the rule resulting
from a nested sub-proof (see \secref{sec:framework-subproof}).
Further derived concepts will support calculational reasoning (see
\secref{sec:framework-calc}).
\<close>
subsection \<open>Context elements \label{sec:framework-context}\<close>
text \<open>
In judgments \<open>\<Gamma> \<turnstile> \<phi>\<close> of the primitive framework, \<open>\<Gamma>\<close>
essentially acts like a proof context. Isar elaborates this idea
towards a higher-level notion, with additional information for
type-inference, term abbreviations, local facts, hypotheses etc.
The element @{command fix}~\<open>x :: \<alpha>\<close> declares a local
parameter, i.e.\ an arbitrary-but-fixed entity of a given type; in
results exported from the context, \<open>x\<close> may become anything.
The @{command assume}~\<open>\<guillemotleft>inference\<guillemotright>\<close> element provides a
general interface to hypotheses: ``@{command assume}~\<open>\<guillemotleft>inference\<guillemotright> A\<close>'' produces \<open>A \<turnstile> A\<close> locally, while the
included inference tells how to discharge \<open>A\<close> from results
\<open>A \<turnstile> B\<close> later on. There is no user-syntax for \<open>\<guillemotleft>inference\<guillemotright>\<close>, i.e.\ it may only occur internally when derived
commands are defined in ML.
At the user-level, the default inference for @{command assume} is
@{inference discharge} as given below. The additional variants
@{command presume} and @{command def} are defined as follows:
\<^medskip>
\begin{tabular}{rcl}
@{command presume}~\<open>A\<close> & \<open>\<equiv>\<close> & @{command assume}~\<open>\<guillemotleft>weak\<hyphen>discharge\<guillemotright> A\<close> \\
@{command def}~\<open>x \<equiv> a\<close> & \<open>\<equiv>\<close> & @{command fix}~\<open>x\<close>~@{command assume}~\<open>\<guillemotleft>expansion\<guillemotright> x \<equiv> a\<close> \\
\end{tabular}
\<^medskip>
\[
\infer[(@{inference_def discharge})]{\<open>\<strut>\<Gamma> - A \<turnstile> #A \<Longrightarrow> B\<close>}{\<open>\<strut>\<Gamma> \<turnstile> B\<close>}
\]
\[
\infer[(@{inference_def "weak\<hyphen>discharge"})]{\<open>\<strut>\<Gamma> - A \<turnstile> A \<Longrightarrow> B\<close>}{\<open>\<strut>\<Gamma> \<turnstile> B\<close>}
\]
\[
\infer[(@{inference_def expansion})]{\<open>\<strut>\<Gamma> - (x \<equiv> a) \<turnstile> B a\<close>}{\<open>\<strut>\<Gamma> \<turnstile> B x\<close>}
\]
\<^medskip>
Note that @{inference discharge} and @{inference
"weak\<hyphen>discharge"} differ in the marker for @{prop A}, which is
relevant when the result of a @{command fix}-@{command
assume}-@{command show} outline is composed with a pending goal,
cf.\ \secref{sec:framework-subproof}.
The most interesting derived context element in Isar is @{command
obtain} @{cite \<open>\S5.3\<close> "Wenzel-PhD"}, which supports generalized
elimination steps in a purely forward manner. The @{command obtain}
command takes a specification of parameters \<open>\<^vec>x\<close> and
assumptions \<open>\<^vec>A\<close> to be added to the context, together
with a proof of a case rule stating that this extension is
conservative (i.e.\ may be removed from closed results later on):
\<^medskip>
\begin{tabular}{l}
\<open>\<langle>facts\<rangle>\<close>~~@{command obtain}~\<open>\<^vec>x \<WHERE> \<^vec>A \<^vec>x \<langle>proof\<rangle> \<equiv>\<close> \\[0.5ex]
\quad @{command have}~\<open>case: \<And>thesis. (\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<rangle>\<close> \\
\quad @{command proof}~@{method "-"} \\
\qquad @{command fix}~\<open>thesis\<close> \\
\qquad @{command assume}~\<open>[intro]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis\<close> \\
\qquad @{command show}~\<open>thesis\<close>~@{command using}~\<open>\<langle>facts\<rangle> \<langle>proof\<rangle>\<close> \\
\quad @{command qed} \\
\quad @{command fix}~\<open>\<^vec>x\<close>~@{command assume}~\<open>\<guillemotleft>elimination case\<guillemotright> \<^vec>A \<^vec>x\<close> \\
\end{tabular}
\<^medskip>
\[
\infer[(@{inference elimination})]{\<open>\<Gamma> \<turnstile> B\<close>}{
\begin{tabular}{rl}
\<open>case:\<close> &
\<open>\<Gamma> \<turnstile> \<And>thesis. (\<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis) \<Longrightarrow> thesis\<close> \\[0.2ex]
\<open>result:\<close> &
\<open>\<Gamma> \<union> \<^vec>A \<^vec>y \<turnstile> B\<close> \\[0.2ex]
\end{tabular}}
\]
Here the name ``\<open>thesis\<close>'' is a specific convention
for an arbitrary-but-fixed proposition; in the primitive natural
deduction rules shown before we have occasionally used \<open>C\<close>.
The whole statement of ``@{command obtain}~\<open>x\<close>~@{keyword
"where"}~\<open>A x\<close>'' may be read as a claim that \<open>A x\<close>
may be assumed for some arbitrary-but-fixed \<open>x\<close>. Also note
that ``@{command obtain}~\<open>A \<AND> B\<close>'' without parameters
is similar to ``@{command have}~\<open>A \<AND> B\<close>'', but the
latter involves multiple sub-goals.
\<^medskip>
The subsequent Isar proof texts explain all context
elements introduced above using the formal proof language itself.
After finishing a local proof within a block, we indicate the
exported result via @{command note}.
\<close>
(*<*)
theorem True
proof
(*>*)
text_raw \<open>\begin{minipage}[t]{0.45\textwidth}\<close>
{
fix x
have "B x" \<proof>
}
note \<open>\<And>x. B x\<close>
text_raw \<open>\end{minipage}\quad\begin{minipage}[t]{0.45\textwidth}\<close>(*<*)next(*>*)
{
assume A
have B \<proof>
}
note \<open>A \<Longrightarrow> B\<close>
text_raw \<open>\end{minipage}\\[3ex]\begin{minipage}[t]{0.45\textwidth}\<close>(*<*)next(*>*)
{
def x \<equiv> a
have "B x" \<proof>
}
note \<open>B a\<close>
text_raw \<open>\end{minipage}\quad\begin{minipage}[t]{0.45\textwidth}\<close>(*<*)next(*>*)
{
obtain x where "A x" \<proof>
have B \<proof>
}
note \<open>B\<close>
text_raw \<open>\end{minipage}\<close>
(*<*)
qed
(*>*)
text \<open>
\<^bigskip>
This illustrates the meaning of Isar context
elements without goals getting in between.
\<close>
subsection \<open>Structured statements \label{sec:framework-stmt}\<close>
text \<open>
The category \<open>statement\<close> of top-level theorem specifications
is defined as follows:
\<^medskip>
\begin{tabular}{rcl}
\<open>statement\<close> & \<open>\<equiv>\<close> & \<open>name: props \<AND> \<dots>\<close> \\
& \<open>|\<close> & \<open>context\<^sup>* conclusion\<close> \\[0.5ex]
\<open>context\<close> & \<open>\<equiv>\<close> & \<open>\<FIXES> vars \<AND> \<dots>\<close> \\
& \<open>|\<close> & \<open>\<ASSUMES> name: props \<AND> \<dots>\<close> \\
\<open>conclusion\<close> & \<open>\<equiv>\<close> & \<open>\<SHOWS> name: props \<AND> \<dots>\<close> \\
& \<open>|\<close> & \<open>\<OBTAINS> vars \<AND> \<dots> \<WHERE> name: props \<AND> \<dots>\<close> \\
& & \quad \<open>\<BBAR> \<dots>\<close> \\
\end{tabular}
\<^medskip>
A simple \<open>statement\<close> consists of named
propositions. The full form admits local context elements followed
by the actual conclusions, such as ``@{keyword "fixes"}~\<open>x\<close>~@{keyword "assumes"}~\<open>A x\<close>~@{keyword "shows"}~\<open>B
x\<close>''. The final result emerges as a Pure rule after discharging
the context: @{prop "\<And>x. A x \<Longrightarrow> B x"}.
The @{keyword "obtains"} variant is another abbreviation defined
below; unlike @{command obtain} (cf.\
\secref{sec:framework-context}) there may be several ``cases''
separated by ``\<open>\<BBAR>\<close>'', each consisting of several
parameters (\<open>vars\<close>) and several premises (\<open>props\<close>).
This specifies multi-branch elimination rules.
\<^medskip>
\begin{tabular}{l}
\<open>\<OBTAINS> \<^vec>x \<WHERE> \<^vec>A \<^vec>x \<BBAR> \<dots> \<equiv>\<close> \\[0.5ex]
\quad \<open>\<FIXES> thesis\<close> \\
\quad \<open>\<ASSUMES> [intro]: \<And>\<^vec>x. \<^vec>A \<^vec>x \<Longrightarrow> thesis \<AND> \<dots>\<close> \\
\quad \<open>\<SHOWS> thesis\<close> \\
\end{tabular}
\<^medskip>
Presenting structured statements in such an ``open'' format usually
simplifies the subsequent proof, because the outer structure of the
problem is already laid out directly. E.g.\ consider the following
canonical patterns for \<open>\<SHOWS>\<close> and \<open>\<OBTAINS>\<close>,
respectively:
\<close>
text_raw \<open>\begin{minipage}{0.5\textwidth}\<close>
theorem
fixes x and y
assumes "A x" and "B y"
shows "C x y"
proof -
from \<open>A x\<close> and \<open>B y\<close>
show "C x y" \<proof>
qed
text_raw \<open>\end{minipage}\begin{minipage}{0.5\textwidth}\<close>
theorem
obtains x and y
where "A x" and "B y"
proof -
have "A a" and "B b" \<proof>
then show thesis ..
qed
text_raw \<open>\end{minipage}\<close>
text \<open>
\<^medskip>
Here local facts \isacharbackquoteopen\<open>A
x\<close>\isacharbackquoteclose\ and \isacharbackquoteopen\<open>B
y\<close>\isacharbackquoteclose\ are referenced immediately; there is no
need to decompose the logical rule structure again. In the second
proof the final ``@{command then}~@{command show}~\<open>thesis\<close>~@{command ".."}'' involves the local rule case \<open>\<And>x
y. A x \<Longrightarrow> B y \<Longrightarrow> thesis\<close> for the particular instance of terms \<open>a\<close> and \<open>b\<close> produced in the body.
\<close>
subsection \<open>Structured proof refinement \label{sec:framework-subproof}\<close>
text \<open>
By breaking up the grammar for the Isar proof language, we may
understand a proof text as a linear sequence of individual proof
commands. These are interpreted as transitions of the Isar virtual
machine (Isar/VM), which operates on a block-structured
configuration in single steps. This allows users to write proof
texts in an incremental manner, and inspect intermediate
configurations for debugging.
The basic idea is analogous to evaluating algebraic expressions on a
stack machine: \<open>(a + b) \<cdot> c\<close> then corresponds to a sequence
of single transitions for each symbol \<open>(, a, +, b, ), \<cdot>, c\<close>.
In Isar the algebraic values are facts or goals, and the operations
are inferences.
\<^medskip>
The Isar/VM state maintains a stack of nodes, each node
contains the local proof context, the linguistic mode, and a pending
goal (optional). The mode determines the type of transition that
may be performed next, it essentially alternates between forward and
backward reasoning, with an intermediate stage for chained facts
(see \figref{fig:isar-vm}).
\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.8\textwidth]{isar-vm}
\end{center}
\caption{Isar/VM modes}\label{fig:isar-vm}
\end{figure}
For example, in \<open>state\<close> mode Isar acts like a mathematical
scratch-pad, accepting declarations like @{command fix}, @{command
assume}, and claims like @{command have}, @{command show}. A goal
statement changes the mode to \<open>prove\<close>, which means that we
may now refine the problem via @{command unfolding} or @{command
proof}. Then we are again in \<open>state\<close> mode of a proof body,
which may issue @{command show} statements to solve pending
sub-goals. A concluding @{command qed} will return to the original
\<open>state\<close> mode one level upwards. The subsequent Isar/VM
trace indicates block structure, linguistic mode, goal state, and
inferences:
\<close>
text_raw \<open>\begingroup\footnotesize\<close>
(*<*)notepad begin
(*>*)
text_raw \<open>\begin{minipage}[t]{0.18\textwidth}\<close>
have "A \<longrightarrow> B"
proof
assume A
show B
\<proof>
qed
text_raw \<open>\end{minipage}\quad
\begin{minipage}[t]{0.06\textwidth}
\<open>begin\<close> \\
\\
\\
\<open>begin\<close> \\
\<open>end\<close> \\
\<open>end\<close> \\
\end{minipage}
\begin{minipage}[t]{0.08\textwidth}
\<open>prove\<close> \\
\<open>state\<close> \\
\<open>state\<close> \\
\<open>prove\<close> \\
\<open>state\<close> \\
\<open>state\<close> \\
\end{minipage}\begin{minipage}[t]{0.35\textwidth}
\<open>(A \<longrightarrow> B) \<Longrightarrow> #(A \<longrightarrow> B)\<close> \\
\<open>(A \<Longrightarrow> B) \<Longrightarrow> #(A \<longrightarrow> B)\<close> \\
\\
\\
\<open>#(A \<longrightarrow> B)\<close> \\
\<open>A \<longrightarrow> B\<close> \\
\end{minipage}\begin{minipage}[t]{0.4\textwidth}
\<open>(init)\<close> \\
\<open>(resolution impI)\<close> \\
\\
\\
\<open>(refinement #A \<Longrightarrow> B)\<close> \\
\<open>(finish)\<close> \\
\end{minipage}\<close>
(*<*)
end
(*>*)
text_raw \<open>\endgroup\<close>
text \<open>
Here the @{inference refinement} inference from
\secref{sec:framework-resolution} mediates composition of Isar
sub-proofs nicely. Observe that this principle incorporates some
degree of freedom in proof composition. In particular, the proof
body allows parameters and assumptions to be re-ordered, or commuted
according to Hereditary Harrop Form. Moreover, context elements
that are not used in a sub-proof may be omitted altogether. For
example:
\<close>
text_raw \<open>\begin{minipage}{0.5\textwidth}\<close>
(*<*)
notepad
begin
(*>*)
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
proof -
fix x and y
assume "A x" and "B y"
show "C x y" \<proof>
qed
text_raw \<open>\end{minipage}\begin{minipage}{0.5\textwidth}\<close>
(*<*)
next
(*>*)
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
proof -
fix x assume "A x"
fix y assume "B y"
show "C x y" \<proof>
qed
text_raw \<open>\end{minipage}\\[3ex]\begin{minipage}{0.5\textwidth}\<close>
(*<*)
next
(*>*)
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
proof -
fix y assume "B y"
fix x assume "A x"
show "C x y" \<proof>
qed
text_raw \<open>\end{minipage}\begin{minipage}{0.5\textwidth}\<close>
(*<*)
next
(*>*)
have "\<And>x y. A x \<Longrightarrow> B y \<Longrightarrow> C x y"
proof -
fix y assume "B y"
fix x
show "C x y" \<proof>
qed
(*<*)
end
(*>*)
text_raw \<open>\end{minipage}\<close>
text \<open>
\<^medskip>
Such ``peephole optimizations'' of Isar texts are
practically important to improve readability, by rearranging
contexts elements according to the natural flow of reasoning in the
body, while still observing the overall scoping rules.
\<^medskip>
This illustrates the basic idea of structured proof
processing in Isar. The main mechanisms are based on natural
deduction rule composition within the Pure framework. In
particular, there are no direct operations on goal states within the
proof body. Moreover, there is no hidden automated reasoning
involved, just plain unification.
\<close>
subsection \<open>Calculational reasoning \label{sec:framework-calc}\<close>
text \<open>
The existing Isar infrastructure is sufficiently flexible to support
calculational reasoning (chains of transitivity steps) as derived
concept. The generic proof elements introduced below depend on
rules declared as @{attribute trans} in the context. It is left to
the object-logic to provide a suitable rule collection for mixed
relations of \<open>=\<close>, \<open><\<close>, \<open>\<le>\<close>, \<open>\<subset>\<close>,
\<open>\<subseteq>\<close> etc. Due to the flexibility of rule composition
(\secref{sec:framework-resolution}), substitution of equals by
equals is covered as well, even substitution of inequalities
involving monotonicity conditions; see also @{cite \<open>\S6\<close> "Wenzel-PhD"}
and @{cite "Bauer-Wenzel:2001"}.
The generic calculational mechanism is based on the observation that
rules such as \<open>trans:\<close>~@{prop "x = y \<Longrightarrow> y = z \<Longrightarrow> x = z"}
proceed from the premises towards the conclusion in a deterministic
fashion. Thus we may reason in forward mode, feeding intermediate
results into rules selected from the context. The course of
reasoning is organized by maintaining a secondary fact called
``@{fact calculation}'', apart from the primary ``@{fact this}''
already provided by the Isar primitives. In the definitions below,
@{attribute OF} refers to @{inference resolution}
(\secref{sec:framework-resolution}) with multiple rule arguments,
and \<open>trans\<close> represents to a suitable rule from the context:
\begin{matharray}{rcl}
@{command "also"}\<open>\<^sub>0\<close> & \equiv & @{command "note"}~\<open>calculation = this\<close> \\
@{command "also"}\<open>\<^sub>n\<^sub>+\<^sub>1\<close> & \equiv & @{command "note"}~\<open>calculation = trans [OF calculation this]\<close> \\[0.5ex]
@{command "finally"} & \equiv & @{command "also"}~@{command "from"}~\<open>calculation\<close> \\
\end{matharray}
The start of a calculation is determined implicitly in the
text: here @{command also} sets @{fact calculation} to the current
result; any subsequent occurrence will update @{fact calculation} by
combination with the next result and a transitivity rule. The
calculational sequence is concluded via @{command finally}, where
the final result is exposed for use in a concluding claim.
Here is a canonical proof pattern, using @{command have} to
establish the intermediate results:
\<close>
(*<*)
notepad
begin
(*>*)
have "a = b" \<proof>
also have "\<dots> = c" \<proof>
also have "\<dots> = d" \<proof>
finally have "a = d" .
(*<*)
end
(*>*)
text \<open>
The term ``\<open>\<dots>\<close>'' above is a special abbreviation
provided by the Isabelle/Isar syntax layer: it statically refers to
the right-hand side argument of the previous statement given in the
text. Thus it happens to coincide with relevant sub-expressions in
the calculational chain, but the exact correspondence is dependent
on the transitivity rules being involved.
\<^medskip>
Symmetry rules such as @{prop "x = y \<Longrightarrow> y = x"} are like
transitivities with only one premise. Isar maintains a separate
rule collection declared via the @{attribute sym} attribute, to be
used in fact expressions ``\<open>a [symmetric]\<close>'', or single-step
proofs ``@{command assume}~\<open>x = y\<close>~@{command then}~@{command
have}~\<open>y = x\<close>~@{command ".."}''.
\<close>
end