From 730e25488e0f502359ed8c2a845b97bf0245d1e7 Mon Sep 17 00:00:00 2001
From: gmelquio <gmelquio@85f007b7-540e-0410-9357-904b9bb8a0f7>
Date: Sat, 15 Sep 2012 17:10:37 +0000
Subject: Port rewrites of tactic documentation from branch 8.4.

This encompasses commits r15183, r15190, r15243, r15262, r15276, r15277,
r15278, r15337. The merge did not go without troubles, but hopefully none
of the changes were lost in process.

git-svn-id: svn+ssh://scm.gforge.inria.fr/svn/coq/trunk@15806 85f007b7-540e-0410-9357-904b9bb8a0f7
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+\chapter{Proof schemes}
+
+\section{Generation of induction principles with {\tt Scheme}}
+\label{Scheme}
+\index{Schemes}
+\comindex{Scheme}
+
+The {\tt Scheme} command is a high-level tool for generating
+automatically (possibly mutual) induction principles for given types
+and sorts.  Its syntax follows the schema:
+\begin{quote}
+{\tt Scheme {\ident$_1$} := Induction for \ident'$_1$ Sort {\sort$_1$} \\
+  with\\
+  \mbox{}\hspace{0.1cm} \dots\\
+        with {\ident$_m$} := Induction for {\ident'$_m$} Sort
+        {\sort$_m$}}
+\end{quote}
+where \ident'$_1$ \dots\ \ident'$_m$ are different inductive type
+identifiers belonging to the same package of mutual inductive
+definitions. This command generates {\ident$_1$}\dots{} {\ident$_m$}
+to be mutually recursive definitions. Each term {\ident$_i$} proves a
+general principle of mutual induction for objects in type {\term$_i$}.
+
+\begin{Variants}
+\item {\tt Scheme {\ident$_1$} := Minimality for \ident'$_1$ Sort {\sort$_1$} \\
+    with\\
+    \mbox{}\hspace{0.1cm} \dots\ \\
+    with {\ident$_m$} := Minimality for {\ident'$_m$} Sort
+    {\sort$_m$}}
+
+  Same as before but defines a non-dependent elimination principle more
+  natural in case of inductively defined relations.
+
+\item {\tt Scheme Equality for \ident$_1$\comindex{Scheme Equality}}
+
+  Tries to generate a boolean equality and a proof of the
+  decidability of the usual equality. If \ident$_i$ involves
+  some other inductive types, their equality has to be defined first.
+
+\item {\tt Scheme Induction for \ident$_1$ Sort {\sort$_1$} \\
+  with\\
+  \mbox{}\hspace{0.1cm} \dots\\
+        with Induction for {\ident$_m$} Sort
+        {\sort$_m$}}
+
+  If you do not provide the name of the schemes, they will be automatically
+  computed from the sorts involved (works also with Minimality).
+
+\end{Variants}
+\label{Scheme-examples}
+
+\firstexample
+\example{Induction scheme for \texttt{tree} and \texttt{forest}}
+
+The definition of principle of mutual induction for {\tt tree} and
+{\tt forest} over the sort {\tt Set} is defined by the command:
+
+\begin{coq_eval}
+Reset Initial.
+Variables A B : Set.
+\end{coq_eval}
+
+\begin{coq_example*}
+Inductive tree : Set :=
+    node : A -> forest -> tree
+with forest : Set :=
+  | leaf : B -> forest
+  | cons : tree -> forest -> forest.
+
+Scheme tree_forest_rec := Induction for tree Sort Set
+  with forest_tree_rec := Induction for forest Sort Set.
+\end{coq_example*}
+
+You may now look at the type of {\tt tree\_forest\_rec}:
+
+\begin{coq_example}
+Check tree_forest_rec.
+\end{coq_example}
+
+This principle involves two different predicates for {\tt trees} and
+{\tt forests}; it also has three premises each one corresponding to a
+constructor of one of the inductive definitions.
+
+The principle {\tt forest\_tree\_rec} shares exactly the same
+premises, only the conclusion now refers to the property of forests.
+
+\begin{coq_example}
+Check forest_tree_rec.
+\end{coq_example}
+
+\example{Predicates {\tt odd} and {\tt even} on naturals}
+
+Let {\tt odd} and {\tt even} be inductively defined as:
+
+% Reset Initial.
+\begin{coq_eval}
+Open Scope nat_scope.
+\end{coq_eval}
+
+\begin{coq_example*}
+Inductive odd : nat -> Prop :=
+    oddS : forall n:nat, even n -> odd (S n)
+with even : nat -> Prop :=
+  | evenO : even 0
+  | evenS : forall n:nat, odd n -> even (S n).
+\end{coq_example*}
+
+The following command generates a powerful elimination
+principle:
+
+\begin{coq_example}
+Scheme odd_even := Minimality for   odd Sort Prop
+  with even_odd := Minimality for even Sort Prop.
+\end{coq_example}
+
+The type of {\tt odd\_even} for instance will be:
+
+\begin{coq_example}
+Check odd_even.
+\end{coq_example}
+
+The type of {\tt even\_odd} shares the same premises but the
+conclusion is {\tt (n:nat)(even n)->(Q n)}.
+
+\subsection{Automatic declaration of schemes}
+\comindex{Set Equality Schemes}
+\comindex{Set Elimination Schemes}
+
+It is possible to deactivate the automatic declaration of the induction
+ principles when defining a new inductive type  with the
+ {\tt Unset Elimination Schemes} command. It may be
+reactivated at any time with {\tt Set Elimination Schemes}.
+\\
+
+You can also activate the automatic declaration of those boolean equalities
+(see the second variant of {\tt Scheme})  with the {\tt Set Equality Schemes}
+ command. However you have to be careful with this option since
+\Coq~ may now reject well-defined inductive types because it cannot compute
+a boolean equality for them.
+
+\subsection{\tt Combined Scheme}
+\label{CombinedScheme}
+\comindex{Combined Scheme}
+
+The {\tt Combined Scheme} command is a tool for combining
+induction principles generated by the {\tt Scheme} command.
+Its syntax follows the schema :
+\begin{quote}
+{\tt Combined Scheme {\ident$_0$} from {\ident$_1$}, .., {\ident$_n$}}
+\end{quote}
+where
+\ident$_1$ \ldots \ident$_n$ are different inductive principles that must belong to
+the same package of mutual inductive principle definitions. This command
+generates {\ident$_0$} to be the conjunction of the principles: it is
+built from the common premises of the principles and concluded by the
+conjunction of their conclusions.
+
+\Example
+We can define the induction principles for trees and forests using:
+\begin{coq_example}
+Scheme tree_forest_ind := Induction for tree Sort Prop
+  with forest_tree_ind := Induction for forest Sort Prop.
+\end{coq_example}
+
+Then we can build the combined induction principle which gives the
+conjunction of the conclusions of each individual principle:
+\begin{coq_example}
+Combined Scheme tree_forest_mutind from tree_forest_ind, forest_tree_ind.
+\end{coq_example}
+
+The type of {\tt tree\_forest\_mutrec} will be:
+\begin{coq_example}
+Check tree_forest_mutind.
+\end{coq_example}
+
+\section{Generation of induction principles with {\tt Functional Scheme}}
+\label{FunScheme}
+\comindex{Functional Scheme}
+
+The {\tt Functional Scheme} command is a high-level experimental
+tool for generating automatically induction principles
+corresponding to (possibly mutually recursive) functions.  Its
+syntax follows the schema:
+\begin{quote}
+{\tt Functional Scheme {\ident$_1$} := Induction for \ident'$_1$ Sort {\sort$_1$} \\
+  with\\
+  \mbox{}\hspace{0.1cm} \dots\ \\
+        with {\ident$_m$} := Induction for {\ident'$_m$} Sort
+        {\sort$_m$}}
+\end{quote}
+where \ident'$_1$ \dots\ \ident'$_m$ are different mutually defined function
+names (they must be in the same order as when they were defined).
+This command generates the induction principles
+\ident$_1$\dots\ident$_m$, following the recursive structure and case
+analyses of the functions \ident'$_1$ \dots\ \ident'$_m$.
+
+\Rem
+There is a difference between obtaining an induction scheme by using
+\texttt{Functional Scheme} on a function defined by \texttt{Function}
+or not. Indeed \texttt{Function} generally produces smaller
+principles, closer to the definition written by the user.
+
+\firstexample
+\example{Induction scheme for \texttt{div2}}
+\label{FunScheme-examples}
+
+We define the function \texttt{div2} as follows:
+
+\begin{coq_eval}
+Reset Initial.
+\end{coq_eval}
+
+\begin{coq_example*}
+Require Import Arith.
+Fixpoint div2 (n:nat) : nat :=
+  match n with
+  | O => 0
+  | S O => 0
+  | S (S n') => S (div2 n')
+  end.
+\end{coq_example*}
+
+The definition of a principle of induction corresponding to the
+recursive structure of \texttt{div2} is defined by the command:
+
+\begin{coq_example}
+Functional Scheme div2_ind := Induction for div2 Sort Prop.
+\end{coq_example}
+
+You may now look at the type of {\tt div2\_ind}:
+
+\begin{coq_example}
+Check div2_ind.
+\end{coq_example}
+
+We can now prove the following lemma using this principle:
+
+\begin{coq_example*}
+Lemma div2_le' : forall n:nat, div2 n <= n.
+intro n.
+ pattern n , (div2 n).
+\end{coq_example*}
+
+\begin{coq_example}
+apply div2_ind; intros.
+\end{coq_example}
+
+\begin{coq_example*}
+auto with arith.
+auto with arith.
+simpl; auto with arith.
+Qed.
+\end{coq_example*}
+
+We can use directly the \texttt{functional induction}
+(\ref{FunInduction}) tactic instead of the pattern/apply trick:
+\tacindex{functional induction}
+
+\begin{coq_example*}
+Reset div2_le'.
+Lemma div2_le : forall n:nat, div2 n <= n.
+intro n.
+\end{coq_example*}
+
+\begin{coq_example}
+functional induction (div2 n).
+\end{coq_example}
+
+\begin{coq_example*}
+auto with arith.
+auto with arith.
+auto with arith.
+Qed.
+\end{coq_example*}
+
+\Rem There is a difference between obtaining an induction scheme for a
+function by using \texttt{Function} (see Section~\ref{Function}) and by
+using \texttt{Functional Scheme} after a normal definition using
+\texttt{Fixpoint} or \texttt{Definition}. See \ref{Function} for
+details.
+
+
+\example{Induction scheme for \texttt{tree\_size}}
+
+\begin{coq_eval}
+Reset Initial.
+\end{coq_eval}
+
+We define trees by the following mutual inductive type:
+
+\begin{coq_example*}
+Variable A : Set.
+Inductive tree : Set :=
+    node : A -> forest -> tree
+with forest : Set :=
+  | empty : forest
+  | cons : tree -> forest -> forest.
+\end{coq_example*}
+
+We define the function \texttt{tree\_size} that computes the size
+of a tree or a forest. Note that we use \texttt{Function} which
+generally produces better principles.
+
+\begin{coq_example*}
+Function tree_size (t:tree) : nat :=
+  match t with
+  | node A f => S (forest_size f)
+  end
+ with forest_size (f:forest) : nat :=
+  match f with
+  | empty => 0
+  | cons t f' => (tree_size t + forest_size f')
+  end.
+\end{coq_example*}
+
+\Rem \texttt{Function} generates itself non mutual induction
+principles {\tt tree\_size\_ind} and {\tt forest\_size\_ind}:
+
+\begin{coq_example}
+Check tree_size_ind.
+\end{coq_example}
+
+The definition of mutual induction principles following the recursive
+structure of \texttt{tree\_size} and \texttt{forest\_size} is defined
+by the command:
+
+\begin{coq_example*}
+Functional Scheme tree_size_ind2 := Induction for tree_size Sort Prop
+with forest_size_ind2 := Induction for forest_size Sort Prop.
+\end{coq_example*}
+
+You may now look at the type of {\tt tree\_size\_ind2}:
+
+\begin{coq_example}
+Check tree_size_ind2.
+\end{coq_example}
+
+\section{Generation of inversion principles with \tt Derive Inversion}
+\label{Derive-Inversion}
+\comindex{Derive Inversion}
+
+The syntax of {\tt Derive Inversion} follows the schema:
+\begin{quote}
+{\tt Derive Inversion {\ident} with forall
+  $(\vec{x} : \vec{T})$, $I~\vec{t}$ Sort \sort}
+\end{quote}
+
+This command generates an inversion principle for the
+\texttt{inversion \dots\ using} tactic.
+\tacindex{inversion \dots\ using}
+Let $I$ be an inductive predicate and $\vec{x}$ the variables
+occurring in $\vec{t}$. This command generates and stocks the
+inversion lemma for the sort \sort~ corresponding to the instance
+$\forall (\vec{x}:\vec{T}), I~\vec{t}$ with the name {\ident} in the {\bf
+global} environment. When applied, it is equivalent to having inverted
+the instance with the tactic {\tt inversion}.
+
+\begin{Variants}
+\item \texttt{Derive Inversion\_clear {\ident} with forall
+  $(\vec{x}:\vec{T})$, $I~\vec{t}$ Sort \sort}\\
+  \comindex{Derive Inversion\_clear}
+  When applied, it is equivalent to having
+  inverted the instance with the tactic \texttt{inversion}
+  replaced by the tactic \texttt{inversion\_clear}.
+\item \texttt{Derive Dependent Inversion {\ident} with forall
+  $(\vec{x}:\vec{T})$, $I~\vec{t}$ Sort \sort}\\
+  \comindex{Derive Dependent Inversion}
+  When applied, it is equivalent to having
+  inverted the instance with the tactic \texttt{dependent inversion}.
+\item \texttt{Derive Dependent Inversion\_clear {\ident} with forall
+  $(\vec{x}:\vec{T})$, $I~\vec{t}$ Sort \sort}\\
+  \comindex{Derive Dependent Inversion\_clear}
+  When applied, it is equivalent to having
+  inverted the instance with the tactic \texttt{dependent inversion\_clear}.
+\end{Variants}
+
+\Example
+
+Let us consider the relation \texttt{Le} over natural numbers and the
+following variable:
+
+\begin{coq_eval}
+Reset Initial.
+\end{coq_eval}
+
+\begin{coq_example*}
+Inductive Le : nat -> nat -> Set :=
+  | LeO : forall n:nat, Le 0 n
+  | LeS : forall n m:nat, Le n m -> Le (S n) (S m).
+Variable P : nat -> nat -> Prop.
+\end{coq_example*}
+
+To generate the inversion lemma for the instance
+\texttt{(Le (S n) m)} and the sort \texttt{Prop}, we do:
+
+\begin{coq_example*}
+Derive Inversion_clear leminv with (forall n m:nat, Le (S n) m) Sort Prop.
+\end{coq_example*}
+
+\begin{coq_example}
+Check leminv.
+\end{coq_example}
+
+Then we can use the proven inversion lemma:
+
+\begin{coq_eval}
+Lemma ex : forall n m:nat, Le (S n) m -> P n m.
+intros.
+\end{coq_eval}
+
+\begin{coq_example}
+Show.
+\end{coq_example}
+
+\begin{coq_example}
+inversion H using leminv.
+\end{coq_example}
+
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