(* Title: HOL/Isar_examples/Basic_Logic.thy
Author: Markus Wenzel, TU Muenchen
Basic propositional and quantifier reasoning.
*)
header {* Basic logical reasoning *}
theory Basic_Logic
imports Main
begin
subsection {* Pure backward reasoning *}
text {*
In order to get a first idea of how Isabelle/Isar proof documents
may look like, we consider the propositions @{text I}, @{text K},
and @{text S}. The following (rather explicit) proofs should
require little extra explanations.
*}
lemma I: "A --> A"
proof
assume A
show A by fact
qed
lemma K: "A --> B --> A"
proof
assume A
show "B --> A"
proof
show A by fact
qed
qed
lemma S: "(A --> B --> C) --> (A --> B) --> A --> C"
proof
assume "A --> B --> C"
show "(A --> B) --> A --> C"
proof
assume "A --> B"
show "A --> C"
proof
assume A
show C
proof (rule mp)
show "B --> C" by (rule mp) fact+
show B by (rule mp) fact+
qed
qed
qed
qed
text {*
Isar provides several ways to fine-tune the reasoning, avoiding
excessive detail. Several abbreviated language elements are
available, enabling the writer to express proofs in a more concise
way, even without referring to any automated proof tools yet.
First of all, proof by assumption may be abbreviated as a single
dot.
*}
lemma "A --> A"
proof
assume A
show A by fact+
qed
text {*
In fact, concluding any (sub-)proof already involves solving any
remaining goals by assumption\footnote{This is not a completely
trivial operation, as proof by assumption may involve full
higher-order unification.}. Thus we may skip the rather vacuous
body of the above proof as well.
*}
lemma "A --> A"
proof
qed
text {*
Note that the \isacommand{proof} command refers to the @{text rule}
method (without arguments) by default. Thus it implicitly applies a
single rule, as determined from the syntactic form of the statements
involved. The \isacommand{by} command abbreviates any proof with
empty body, so the proof may be further pruned.
*}
lemma "A --> A"
by rule
text {*
Proof by a single rule may be abbreviated as double-dot.
*}
lemma "A --> A" ..
text {*
Thus we have arrived at an adequate representation of the proof of a
tautology that holds by a single standard rule.\footnote{Apparently,
the rule here is implication introduction.}
*}
text {*
Let us also reconsider @{text K}. Its statement is composed of
iterated connectives. Basic decomposition is by a single rule at a
time, which is why our first version above was by nesting two
proofs.
The @{text intro} proof method repeatedly decomposes a goal's
conclusion.\footnote{The dual method is @{text elim}, acting on a
goal's premises.}
*}
lemma "A --> B --> A"
proof (intro impI)
assume A
show A by fact
qed
text {*
Again, the body may be collapsed.
*}
lemma "A --> B --> A"
by (intro impI)
text {*
Just like @{text rule}, the @{text intro} and @{text elim} proof
methods pick standard structural rules, in case no explicit
arguments are given. While implicit rules are usually just fine for
single rule application, this may go too far with iteration. Thus
in practice, @{text intro} and @{text elim} would be typically
restricted to certain structures by giving a few rules only, e.g.\
\isacommand{proof}~@{text "(intro impI allI)"} to strip implications
and universal quantifiers.
Such well-tuned iterated decomposition of certain structures is the
prime application of @{text intro} and @{text elim}. In contrast,
terminal steps that solve a goal completely are usually performed by
actual automated proof methods (such as \isacommand{by}~@{text
blast}.
*}
subsection {* Variations of backward vs.\ forward reasoning *}
text {*
Certainly, any proof may be performed in backward-style only. On
the other hand, small steps of reasoning are often more naturally
expressed in forward-style. Isar supports both backward and forward
reasoning as a first-class concept. In order to demonstrate the
difference, we consider several proofs of @{text "A \<and> B \<longrightarrow> B \<and> A"}.
The first version is purely backward.
*}
lemma "A & B --> B & A"
proof
assume "A & B"
show "B & A"
proof
show B by (rule conjunct2) fact
show A by (rule conjunct1) fact
qed
qed
text {*
Above, the @{text "conjunct_1/2"} projection rules had to be named
explicitly, since the goals @{text B} and @{text A} did not provide
any structural clue. This may be avoided using \isacommand{from} to
focus on the @{text "A \<and> B"} assumption as the current facts,
enabling the use of double-dot proofs. Note that \isacommand{from}
already does forward-chaining, involving the \name{conjE} rule here.
*}
lemma "A & B --> B & A"
proof
assume "A & B"
show "B & A"
proof
from `A & B` show B ..
from `A & B` show A ..
qed
qed
text {*
In the next version, we move the forward step one level upwards.
Forward-chaining from the most recent facts is indicated by the
\isacommand{then} command. Thus the proof of @{text "B \<and> A"} from
@{text "A \<and> B"} actually becomes an elimination, rather than an
introduction. The resulting proof structure directly corresponds to
that of the @{text conjE} rule, including the repeated goal
proposition that is abbreviated as @{text ?thesis} below.
*}
lemma "A & B --> B & A"
proof
assume "A & B"
then show "B & A"
proof -- {* rule @{text conjE} of @{text "A \<and> B"} *}
assume B A
then show ?thesis .. -- {* rule @{text conjI} of @{text "B \<and> A"} *}
qed
qed
text {*
In the subsequent version we flatten the structure of the main body
by doing forward reasoning all the time. Only the outermost
decomposition step is left as backward.
*}
lemma "A & B --> B & A"
proof
assume "A & B"
from `A & B` have A ..
from `A & B` have B ..
from `B` `A` show "B & A" ..
qed
text {*
We can still push forward-reasoning a bit further, even at the risk
of getting ridiculous. Note that we force the initial proof step to
do nothing here, by referring to the ``-'' proof method.
*}
lemma "A & B --> B & A"
proof -
{
assume "A & B"
from `A & B` have A ..
from `A & B` have B ..
from `B` `A` have "B & A" ..
}
then show ?thesis .. -- {* rule \name{impI} *}
qed
text {*
\medskip With these examples we have shifted through a whole range
from purely backward to purely forward reasoning. Apparently, in
the extreme ends we get slightly ill-structured proofs, which also
require much explicit naming of either rules (backward) or local
facts (forward).
The general lesson learned here is that good proof style would
achieve just the \emph{right} balance of top-down backward
decomposition, and bottom-up forward composition. In general, there
is no single best way to arrange some pieces of formal reasoning, of
course. Depending on the actual applications, the intended audience
etc., rules (and methods) on the one hand vs.\ facts on the other
hand have to be emphasized in an appropriate way. This requires the
proof writer to develop good taste, and some practice, of course.
*}
text {*
For our example the most appropriate way of reasoning is probably
the middle one, with conjunction introduction done after
elimination.
*}
lemma "A & B --> B & A"
proof
assume "A & B"
then show "B & A"
proof
assume B A
then show ?thesis ..
qed
qed
subsection {* A few examples from ``Introduction to Isabelle'' *}
text {*
We rephrase some of the basic reasoning examples of
\cite{isabelle-intro}, using HOL rather than FOL.
*}
subsubsection {* A propositional proof *}
text {*
We consider the proposition @{text "P \<or> P \<longrightarrow> P"}. The proof below
involves forward-chaining from @{text "P \<or> P"}, followed by an
explicit case-analysis on the two \emph{identical} cases.
*}
lemma "P | P --> P"
proof
assume "P | P"
then show P
proof -- {*
rule @{text disjE}: \smash{$\infer{C}{A \disj B & \infer*{C}{[A]} & \infer*{C}{[B]}}$}
*}
assume P show P by fact
next
assume P show P by fact
qed
qed
text {*
Case splits are \emph{not} hardwired into the Isar language as a
special feature. The \isacommand{next} command used to separate the
cases above is just a short form of managing block structure.
\medskip In general, applying proof methods may split up a goal into
separate ``cases'', i.e.\ new subgoals with individual local
assumptions. The corresponding proof text typically mimics this by
establishing results in appropriate contexts, separated by blocks.
In order to avoid too much explicit parentheses, the Isar system
implicitly opens an additional block for any new goal, the
\isacommand{next} statement then closes one block level, opening a
new one. The resulting behavior is what one would expect from
separating cases, only that it is more flexible. E.g.\ an induction
base case (which does not introduce local assumptions) would
\emph{not} require \isacommand{next} to separate the subsequent step
case.
\medskip In our example the situation is even simpler, since the two
cases actually coincide. Consequently the proof may be rephrased as
follows.
*}
lemma "P | P --> P"
proof
assume "P | P"
then show P
proof
assume P
show P by fact
show P by fact
qed
qed
text {*
Again, the rather vacuous body of the proof may be collapsed. Thus
the case analysis degenerates into two assumption steps, which are
implicitly performed when concluding the single rule step of the
double-dot proof as follows.
*}
lemma "P | P --> P"
proof
assume "P | P"
then show P ..
qed
subsubsection {* A quantifier proof *}
text {*
To illustrate quantifier reasoning, let us prove @{text "(\<exists>x. P (f
x)) \<longrightarrow> (\<exists>y. P y)"}. Informally, this holds because any @{text a}
with @{text "P (f a)"} may be taken as a witness for the second
existential statement.
The first proof is rather verbose, exhibiting quite a lot of
(redundant) detail. It gives explicit rules, even with some
instantiation. Furthermore, we encounter two new language elements:
the \isacommand{fix} command augments the context by some new
``arbitrary, but fixed'' element; the \isacommand{is} annotation
binds term abbreviations by higher-order pattern matching.
*}
lemma "(EX x. P (f x)) --> (EX y. P y)"
proof
assume "EX x. P (f x)"
then show "EX y. P y"
proof (rule exE) -- {*
rule \name{exE}: \smash{$\infer{B}{\ex x A(x) & \infer*{B}{[A(x)]_x}}$}
*}
fix a
assume "P (f a)" (is "P ?witness")
then show ?thesis by (rule exI [of P ?witness])
qed
qed
text {*
While explicit rule instantiation may occasionally improve
readability of certain aspects of reasoning, it is usually quite
redundant. Above, the basic proof outline gives already enough
structural clues for the system to infer both the rules and their
instances (by higher-order unification). Thus we may as well prune
the text as follows.
*}
lemma "(EX x. P (f x)) --> (EX y. P y)"
proof
assume "EX x. P (f x)"
then show "EX y. P y"
proof
fix a
assume "P (f a)"
then show ?thesis ..
qed
qed
text {*
Explicit @{text \<exists>}-elimination as seen above can become quite
cumbersome in practice. The derived Isar language element
``\isakeyword{obtain}'' provides a more handsome way to do
generalized existence reasoning.
*}
lemma "(EX x. P (f x)) --> (EX y. P y)"
proof
assume "EX x. P (f x)"
then obtain a where "P (f a)" ..
then show "EX y. P y" ..
qed
text {*
Technically, \isakeyword{obtain} is similar to \isakeyword{fix} and
\isakeyword{assume} together with a soundness proof of the
elimination involved. Thus it behaves similar to any other forward
proof element. Also note that due to the nature of general
existence reasoning involved here, any result exported from the
context of an \isakeyword{obtain} statement may \emph{not} refer to
the parameters introduced there.
*}
subsubsection {* Deriving rules in Isabelle *}
text {*
We derive the conjunction elimination rule from the corresponding
projections. The proof is quite straight-forward, since
Isabelle/Isar supports non-atomic goals and assumptions fully
transparently.
*}
theorem conjE: "A & B ==> (A ==> B ==> C) ==> C"
proof -
assume "A & B"
assume r: "A ==> B ==> C"
show C
proof (rule r)
show A by (rule conjunct1) fact
show B by (rule conjunct2) fact
qed
qed
end