src/Doc/Real_Asymp/Real_Asymp_Doc.thy

-(*<*)
-theory Real_Asymp_Doc
-  imports "HOL-Real_Asymp.Real_Asymp"
-begin
-
-ML_file \<open>../antiquote_setup.ML\<close>
-(*>*)
-
-section \<open>Introduction\<close>
-
-text \<open>
-  This document describes the \<^verbatim>\<open>real_asymp\<close> package that provides a number of tools
-  related to the asymptotics of real-valued functions. These tools are:
-
-    \<^item> The @{method real_asymp} method to prove limits, statements involving Landau symbols
-      (`Big-O' etc.), and asymptotic equivalence (\<open>\<sim>\<close>)
-
-    \<^item> The @{command real_limit} command to compute the limit of a real-valued function at a
-      certain point
-
-    \<^item> The @{command real_expansion} command to compute the asymptotic expansion of a
-      real-valued function at a certain point
-
-  These three tools will be described in the following sections.
-\<close>
-
-section \<open>Supported Operations\<close>
-
-text \<open>
-  The \<^verbatim>\<open>real_asymp\<close> package fully supports the following operations:
-
-    \<^item> Basic arithmetic (addition, subtraction, multiplication, division)
-
-    \<^item> Powers with constant natural exponent
-
-    \<^item> @{term exp}, @{term ln}, @{term log}, @{term sqrt}, @{term "root k"} (for a constant k)
-  
-    \<^item> @{term sin}, @{term cos}, @{term tan} at finite points
-
-    \<^item> @{term sinh}, @{term cosh}, @{term tanh}
-
-    \<^item> @{term min}, @{term max}, @{term abs}
-
-  Additionally, the following operations are supported in a `best effort' fashion using asymptotic
-  upper/lower bounds:
-
-    \<^item> Powers with variable natural exponent
-
-    \<^item> @{term sin} and @{term cos} at \<open>\<plusminus>\<infinity>\<close>
-
-    \<^item> @{term floor}, @{term ceiling}, @{term frac}, and \<open>mod\<close>
-
-  Additionally, the @{term arctan} function is partially supported. The method may fail when
-  the argument to @{term arctan} contains functions of different order of growth.
-\<close>
-
-
-section \<open>Proving Limits and Asymptotic Properties\<close>
-
-text \<open>
-  \[
-    @{method_def (HOL) real_asymp} : \<open>method\<close>
-  \]
-
-  @{rail \<open>
-    @@{method (HOL) real_asymp} opt? (@{syntax simpmod} * )
-    ;
-    opt: '(' ('verbose' | 'fallback' ) ')'
-    ;
-    @{syntax_def simpmod}: ('add' | 'del' | 'only' | 'split' (() | '!' | 'del') |
-      'cong' (() | 'add' | 'del')) ':' @{syntax thms}
-  \<close>}
-\<close>
-
-text \<open>
-  The @{method real_asymp} method is a semi-automatic proof method for proving certain statements
-  related to the asymptotic behaviour of real-valued functions. In the following, let \<open>f\<close> and \<open>g\<close>
-  be functions of type @{typ "real \<Rightarrow> real"} and \<open>F\<close> and \<open>G\<close> real filters.
-
-  The functions \<open>f\<close> and \<open>g\<close> can be built from the operations mentioned before and may contain free 
-  variables. The filters \<open>F\<close> and \<open>G\<close> can be either \<open>\<plusminus>\<infinity>\<close> or a finite point in \<open>\<real>\<close>, possibly
-  with approach from the left or from the right.
-
-  The class of statements that is supported by @{method real_asymp} then consists of:
-
-    \<^item> Limits, i.\,e.\ @{prop "filterlim f F G"}
-
-    \<^item> Landau symbols, i.\,e.\ @{prop "f \<in> O[F](g)"} and analogously for \<^emph>\<open>o\<close>, \<open>\<Omega>\<close>, \<open>\<omega>\<close>, \<open>\<Theta>\<close>
-
-    \<^item> Asymptotic equivalence, i.\,e.\ @{prop "f \<sim>[F] g"}
-
-    \<^item> Asymptotic inequalities, i.\,e.\ @{prop "eventually (\<lambda>x. f x \<le> g x) F"}
-
-  For typical problems arising in practice that do not contain free variables, @{method real_asymp}
-  tends to succeed fully automatically within fractions of seconds, e.\,g.:
-\<close>
-
-lemma \<open>filterlim (\<lambda>x::real. (1 + 1 / x) powr x) (nhds (exp 1)) at_top\<close>
-  by real_asymp
-
-text \<open>
-  What makes the method \<^emph>\<open>semi-automatic\<close> is the fact that it has to internally determine the 
-  sign of real-valued constants and identical zeroness of real-valued functions, and while the
-  internal heuristics for this work very well most of the time, there are situations where the 
-  method fails to determine the sign of a constant, in which case the user needs to go back and
-  supply more information about the sign of that constant in order to get a result.
-
-  A simple example is the following:
-\<close>
-(*<*)
-experiment
-  fixes a :: real
-begin
-(*>*)
-lemma \<open>filterlim (\<lambda>x::real. exp (a * x)) at_top at_top\<close>
-oops
-
-text \<open>
-  Here, @{method real_asymp} outputs an error message stating that it could not determine the
-  sign of the free variable @{term "a :: real"}. In this case, the user must add the assumption
-  \<open>a > 0\<close> and give it to @{method real_asymp}.
-\<close>
-lemma
-  assumes \<open>a > 0\<close>
-  shows   \<open>filterlim (\<lambda>x::real. exp (a * x)) at_top at_top\<close>
-  using assms by real_asymp
-(*<*)
-end
-(*>*)
-text \<open>
-  Additional modifications to the simpset that is used for determining signs can also be made
-  with \<open>simp add:\<close> modifiers etc.\ in the same way as when using the @{method simp} method directly.
-
-  The same situation can also occur without free variables if the constant in question is a
-  complicated expression that the simplifier does not know enough ebout,
-  e.\,g.\ @{term "pi - exp 1"}.
-
-  In order to trace problems with sign determination, the \<open>(verbose)\<close> option can be passed to
-  @{method real_asymp}. It will then print a detailed error message whenever it encounters
-  a sign determination problem that it cannot solve.
-
-  The \<open>(fallback)\<close> flag causes the method not to use asymptotic interval arithmetic, but only
-  the much simpler core mechanism of computing a single Multiseries expansion for the input
-  function. There should normally be no need to use this flag; however, the core part of the 
-  code has been tested much more rigorously than the asymptotic interval part. In case there 
-  is some implementation problem with it for a problem that can be solved without it, the 
-  \<open>(fallback)\<close> option can be used until that problem is resolved.
-\<close>
-
-
-
-section \<open>Diagnostic Commands\<close>
-
-text \<open>
-  \[\begin{array}{rcl}
-    @{command_def (HOL) real_limit} & : & \<open>context \<rightarrow>\<close> \\
-    @{command_def (HOL) real_expansion} & : & \<open>context \<rightarrow>\<close>
-  \end{array}\]
-
-  @{rail \<open>
-    @@{command (HOL) real_limit} (limitopt*)
-    ;
-    @@{command (HOL) real_expansion} (expansionopt*)
-    ;
-    limitopt : ('limit' ':' @{syntax term}) | ('facts' ':' @{syntax thms})
-    ;
-    expansionopt : 
-        ('limit' ':' @{syntax term}) |
-        ('terms' ':' @{syntax nat} ('(' 'strict' ')') ?) |
-        ('facts' ':' @{syntax thms})
-  \<close>}
-
-    \<^descr>@{command real_limit} computes the limit of the given function \<open>f(x)\<close> for as \<open>x\<close> tends
-    to the specified limit point. Additional facts can be provided with the \<open>facts\<close> option, 
-    similarly to the @{command using} command with @{method real_asymp}. The limit point given
-    by the \<open>limit\<close> option must be one of the filters @{term "at_top"}, @{term "at_bot"}, 
-    @{term "at_left"}, or @{term "at_right"}. If no limit point is given, @{term "at_top"} is used
-    by default.
-  
-    \<^medskip>
-    The output of @{command real_limit} can be \<open>\<infinity>\<close>, \<open>-\<infinity>\<close>, \<open>\<plusminus>\<infinity>\<close>, \<open>c\<close> (for some real constant \<open>c\<close>),
-    \<open>0\<^sup>+\<close>, or \<open>0\<^sup>-\<close>. The \<open>+\<close> and \<open>-\<close> in the last case indicate whether the approach is from above
-    or from below (corresponding to @{term "at_right (0::real)"} and @{term "at_left (0::real)"}); 
-    for technical reasons, this information is currently not displayed if the limit is not 0.
-  
-    \<^medskip>
-    If the given function does not tend to a definite limit (e.\,g.\ @{term "sin x"} for \<open>x \<rightarrow> \<infinity>\<close>),
-    the command might nevertheless succeed to compute an asymptotic upper and/or lower bound and
-    display the limits of these bounds instead.
-
-    \<^descr>@{command real_expansion} works similarly to @{command real_limit}, but displays the first few
-    terms of the asymptotic multiseries expansion of the given function at the given limit point 
-    and the order of growth of the remainder term.
-  
-    In addition to the options already explained for the @{command real_limit} command, it takes
-    an additional option \<open>terms\<close> that controls how many of the leading terms of the expansion are
-    printed. If the \<open>(strict)\<close> modifier is passed to the \<open>terms option\<close>, terms whose coefficient is
-    0 are dropped from the output and do not count to the number of terms to be output.
-  
-    \<^medskip>
-    By default, the first three terms are output and the \<open>strict\<close> option is disabled.
-
-  Note that these two commands are intended for diagnostic use only. While the central part
-  of their implementation – finding a multiseries expansion and reading off the limit – are the
-  same as in the @{method real_asymp} method and therefore trustworthy, there is a small amount
-  of unverified code involved in pre-processing and printing (e.\,g.\ for reducing all the
-  different options for the \<open>limit\<close> option to the @{term at_top} case).
-\<close>
-
-
-section \<open>Extensibility\<close>
-
-subsection \<open>Basic fact collections\<close>
-
-text \<open>
-  The easiest way to add support for additional operations is to add corresponding facts
-  to one of the following fact collections. However, this only works for particularly simple cases.
-
-    \<^descr>@{thm [source] real_asymp_reify_simps} specifies a list of (unconditional) equations that are 
-     unfolded as a first step of @{method real_asymp} and the related commands. This can be used to 
-     add support for operations that can be represented easily by other operations that are already
-     supported, such as @{term sinh}, which is equal to @{term "\<lambda>x. (exp x - exp (-x)) / 2"}.
-
-    \<^descr>@{thm [source] real_asymp_nat_reify} and @{thm [source] real_asymp_int_reify} is used to
-     convert operations on natural numbers or integers to operations on real numbers. This enables
-     @{method real_asymp} to also work on functions that return a natural number or an integer.
-\<close>
-
-subsection \<open>Expanding Custom Operations\<close>
-
-text \<open>
-  Support for some non-trivial new operation \<open>f(x\<^sub>1, \<dots>, x\<^sub>n)\<close> can be added dynamically at any
-  time, but it involves writing ML code and involves a significant amount of effort, especially
-  when the function has essential singularities.
-
-  The first step is to write an ML function that takes as arguments
-    \<^item> the expansion context
-    \<^item> the term \<open>t\<close> to expand (which will be of the form \<open>f(g\<^sub>1(x), \<dots>, g\<^sub>n(x))\<close>)
-    \<^item> a list of \<open>n\<close> theorems of the form @{prop "(g\<^sub>i expands_to G\<^sub>i) bs"}
-    \<^item> the current basis \<open>bs\<close>
-  and returns a theorem of the form @{prop "(t expands_to F) bs'"} and a new basis \<open>bs'\<close> which 
-  must be a superset of the original basis.
-
-  This function must then be registered as a handler for the operation by proving a vacuous lemma
-  of the form @{prop "REAL_ASYMP_CUSTOM f"} (which is only used for tagging) and passing that
-  lemma and the expansion function to @{ML [source] Exp_Log_Expression.register_custom_from_thm}
-  in a @{command local_setup} invocation.
-
-  If the expansion produced by the handler function contains new definitions, corresponding 
-  evaluation equations must be added to the fact collection @{thm [source] real_asymp_eval_eqs}.
-  These equations must have the form \<open>h p\<^sub>1 \<dots> p\<^sub>m = rhs\<close> where \<open>h\<close> must be a constant of arity \<open>m\<close>
-  (i.\,e.\ the function on the left-hand side must be fully applied) and the \<open>p\<^sub>i\<close> can be patterns
-  involving \<open>constructors\<close>.
-
-  New constructors for this pattern matching can be registered by adding a theorem of the form
-  @{prop "REAL_ASYMP_EVAL_CONSTRUCTOR c"} to the fact collection
-  @{thm [source] exp_log_eval_constructor}, but this should be quite rare in practice.
-
-  \<^medskip>
-  Note that there is currently no way to add support for custom operations in connection with
-  `oscillating' terms. The above mechanism only works if all arguments of the new operation have
-  an exact multiseries expansion.
-\<close>
-
-
-subsection \<open>Extending the Sign Oracle\<close>
-
-text \<open>
-  By default, the \<^verbatim>\<open>real_asymp\<close> package uses the simplifier with the given simpset and facts
-  in order to determine the sign of real constants. This is done by invoking the simplifier
-  on goals like \<open>c = 0\<close>, \<open>c \<noteq> 0\<close>, \<open>c > 0\<close>, or \<open>c < 0\<close> or some subset thereof, depending on the
-  context.
-
-  If the simplifier cannot prove any of them, the entire method (or command) invocation will fail.
-  It is, however, possible to dynamically add additional sign oracles that will be tried; the 
-  most obvious candidate for an oracle that one may want to add or remove dynamically are 
-  approximation-based tactics.
-
-  Adding such a tactic can be done by calling
-  @{ML [source] Multiseries_Expansion.register_sign_oracle}. Note that if the tactic cannot prove
-  a goal, it should fail as fast as possible.
-\<close>
-
-(*<*)
-end
-(*>*)