\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\Assum}[3]{\kw{Assum}(#1)(#2:#3)} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \newcommand{\Def}[4]{\kw{Def}(#1)(#2:=#3:#4)} \newcommand{\emptyf}{\textsf{emptyf}} \newcommand{\End}{\kw{End}} \newcommand{\kwend}{\kw{end}} \newcommand{\even}{\textsf{even}} \newcommand{\evenO}{\textsf{even}_\textsf{O}} \newcommand{\evenS}{\textsf{even}_\textsf{S}} \newcommand{\Fix}{\kw{Fix}} \newcommand{\fix}{\kw{fix}} \newcommand{\for}{\textsf{for}} \newcommand{\forest}{\textsf{forest}} \newcommand{\Functor}{\kw{Functor}} \newcommand{\In}{\kw{in}} \newcommand{\Ind}[4]{\kw{Ind}[#2](#3:=#4)} \newcommand{\ind}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[5]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)} \newcommand{\Indpstr}[6]{\kw{Ind}_{#5}(#1)[#2](#3:=#4)/{#6}} \newcommand{\injective}{\kw{injective}} \newcommand{\kw}[1]{\textsf{#1}} \newcommand{\length}{\textsf{length}} \newcommand{\letin}[3]{\kw{let}~#1:=#2~\kw{in}~#3} \newcommand{\List}{\textsf{list}} \newcommand{\lra}{\longrightarrow} \newcommand{\Match}{\kw{match}} \newcommand{\Mod}[3]{{\kw{Mod}}({#1}:{#2}\,\zeroone{:={#3}})} \newcommand{\ModA}[2]{{\kw{ModA}}({#1}=={#2})} \newcommand{\ModS}[2]{{\kw{Mod}}({#1}:{#2})} \newcommand{\ModType}[2]{{\kw{ModType}}({#1}:={#2})} \newcommand{\mto}{.\;} \newcommand{\nat}{\textsf{nat}} \newcommand{\Nil}{\textsf{nil}} \newcommand{\nilhl}{\textsf{nil\_hl}} \newcommand{\nO}{\textsf{O}} \newcommand{\node}{\textsf{node}} \newcommand{\nS}{\textsf{S}} \newcommand{\odd}{\textsf{odd}} \newcommand{\oddS}{\textsf{odd}_\textsf{S}} \newcommand{\ovl}[1]{\overline{#1}} \newcommand{\Pair}{\textsf{pair}} \newcommand{\plus}{\mathsf{plus}} \newcommand{\SProp}{\textsf{SProp}} \newcommand{\Prop}{\textsf{Prop}} \newcommand{\return}{\kw{return}} \newcommand{\Set}{\textsf{Set}} \newcommand{\Sort}{\mathcal{S}} \newcommand{\Str}{\textsf{Stream}} \newcommand{\Struct}{\kw{Struct}} \newcommand{\subst}[3]{#1\{#2/#3\}} \newcommand{\tl}{\textsf{tl}} \newcommand{\tree}{\textsf{tree}} \newcommand{\trii}{\triangleright_\iota} \newcommand{\Type}{\textsf{Type}} \newcommand{\WEV}[3]{\mbox{$#1[] \vdash #2 \lra #3$}} \newcommand{\WEVT}[3]{\mbox{$#1[] \vdash #2 \lra$}\\ \mbox{$ #3$}} \newcommand{\WF}[2]{{\mathcal{W\!F}}(#1)[#2]} \newcommand{\WFE}[1]{\WF{E}{#1}} \newcommand{\WFT}[2]{#1[] \vdash {\mathcal{W\!F}}(#2)} \newcommand{\WFTWOLINES}[2]{{\mathcal{W\!F}}\begin{array}{l}(#1)\\\mbox{}[{#2}]\end{array}} \newcommand{\with}{\kw{with}} \newcommand{\WS}[3]{#1[] \vdash #2 <: #3} \newcommand{\WSE}[2]{\WS{E}{#1}{#2}} \newcommand{\WT}[4]{#1[#2] \vdash #3 : #4} \newcommand{\WTE}[3]{\WT{E}{#1}{#2}{#3}} \newcommand{\WTEG}[2]{\WTE{\Gamma}{#1}{#2}} \newcommand{\WTM}[3]{\WT{#1}{}{#2}{#3}} \newcommand{\zeroone}[1]{[{#1}]} \end{split}\]

Proof schemes

Generation of induction principles with Scheme

The Scheme command is a high-level tool for generating automatically (possibly mutual) induction principles for given types and sorts. Its syntax follows the schema:

Command Scheme ident1 := Induction for ident2 Sort sort with identi := Induction for identj Sort sort*

This command is a high-level tool for generating automatically (possibly mutual) induction principles for given types and sorts. Each identj is a different inductive type identifier belonging to the same package of mutual inductive definitions. The command generates the identis to be mutually recursive definitions. Each term identi proves a general principle of mutual induction for objects in type identj.

Variant Scheme ident := Minimality for ident Sort sort with ident := Minimality for ident' Sort sort*

Same as before but defines a non-dependent elimination principle more natural in case of inductively defined relations.

Variant Scheme Equality for ident

Tries to generate a Boolean equality and a proof of the decidability of the usual equality. If ident involves some other inductive types, their equality has to be defined first.

Variant Scheme Induction for ident Sort sort with Induction for ident Sort sort*

If you do not provide the name of the schemes, they will be automatically computed from the sorts involved (works also with Minimality).

Example

Induction scheme for tree and forest.

A mutual induction principle for tree and forest in sort Set can be defined using the command

Axiom A : Set.

A is declared

Axiom B : Set.

B is declared

Inductive tree : Set := node : A -> forest -> tree with forest : Set :=     leaf : B -> forest   | cons : tree -> forest -> forest.

tree, forest are defined tree_rect is defined tree_ind is defined tree_rec is defined tree_sind is defined forest_rect is defined forest_ind is defined forest_rec is defined forest_sind is defined

Scheme tree_forest_rec := Induction for tree Sort Set   with forest_tree_rec := Induction for forest Sort Set.

forest_tree_rec is defined tree_forest_rec is defined tree_forest_rec, forest_tree_rec are recursively defined

You may now look at the type of tree_forest_rec:

Check tree_forest_rec.

tree_forest_rec : forall (P : tree -> Set) (P0 : forest -> Set), (forall (a : A) (f : forest), P0 f -> P (node a f)) -> (forall b : B, P0 (leaf b)) -> (forall t : tree, P t -> forall f1 : forest, P0 f1 -> P0 (cons t f1)) -> forall t : tree, P t

This principle involves two different predicates for trees andforests; it also has three premises each one corresponding to a constructor of one of the inductive definitions.

The principle forest_tree_rec shares exactly the same premises, only the conclusion now refers to the property of forests.

Example

Predicates odd and even on naturals.

Let odd and even be inductively defined as:

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).

odd, even are defined odd_ind is defined odd_sind is defined even_ind is defined even_sind is defined

The following command generates a powerful elimination principle:

Scheme odd_even := Minimality for odd Sort Prop with even_odd := Minimality for even Sort Prop.

even_odd is defined odd_even is defined odd_even, even_odd are recursively defined

The type of odd_even for instance will be:

Check odd_even.

odd_even : forall P P0 : nat -> Prop, (forall n : nat, even n -> P0 n -> P (S n)) -> P0 0 -> (forall n : nat, odd n -> P n -> P0 (S n)) -> forall n : nat, odd n -> P n

The type of even_odd shares the same premises but the conclusion is (n:nat)(even n)->(P0 n).

Automatic declaration of schemes

Flag Elimination Schemes

Enables automatic declaration of induction principles when defining a new inductive type. Defaults to on.

Flag Nonrecursive Elimination Schemes

Enables automatic declaration of induction principles for types declared with the Variant and Record commands. Defaults to off.

Flag Case Analysis Schemes

This flag governs the generation of case analysis lemmas for inductive types, i.e. corresponding to the pattern matching term alone and without fixpoint.

Flag Boolean Equality Schemes

Flag Decidable Equality Schemes

These flags control the automatic declaration of those Boolean equalities (see the second variant of Scheme).

Warning

You have to be careful with these flags since Coq may now reject well-defined inductive types because it cannot compute a Boolean equality for them.

Flag Rewriting Schemes

This flag governs generation of equality-related schemes such as congruence.

Combined Scheme

Command Combined Scheme ident from identi+,

This command is a tool for combining induction principles generated by the Scheme command. Each identi is a different inductive principle that must belong to the same package of mutual inductive principle definitions. This command generates ident 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. In the case where all the inductive principles used are in sort Prop, the propositional conjunction and is used, otherwise the simple product prod is used instead.

Example

We can define the induction principles for trees and forests using:

Scheme tree_forest_ind := Induction for tree Sort Prop with forest_tree_ind := Induction for forest Sort Prop.

forest_tree_ind is defined tree_forest_ind is defined tree_forest_ind, forest_tree_ind are recursively defined

Then we can build the combined induction principle which gives the conjunction of the conclusions of each individual principle:

Combined Scheme tree_forest_mutind from tree_forest_ind,forest_tree_ind.

tree_forest_mutind is defined tree_forest_mutind is recursively defined

The type of tree_forest_mutind will be:

Check tree_forest_mutind.

tree_forest_mutind : forall (P : tree -> Prop) (P0 : forest -> Prop), (forall (a : A) (f : forest), P0 f -> P (node a f)) -> (forall b : B, P0 (leaf b)) -> (forall t : tree, P t -> forall f1 : forest, P0 f1 -> P0 (cons t f1)) -> (forall t : tree, P t) /\ (forall f2 : forest, P0 f2)

Example

We can also combine schemes at sort Type:

Scheme tree_forest_rect := Induction for tree Sort Type with forest_tree_rect := Induction for forest Sort Type.

forest_tree_rect is defined tree_forest_rect is defined tree_forest_rect, forest_tree_rect are recursively defined

Combined Scheme tree_forest_mutrect from tree_forest_rect, forest_tree_rect.

tree_forest_mutrect is defined tree_forest_mutrect is recursively defined

Check tree_forest_mutrect.

tree_forest_mutrect : forall (P : tree -> Type) (P0 : forest -> Type), (forall (a : A) (f : forest), P0 f -> P (node a f)) -> (forall b : B, P0 (leaf b)) -> (forall t : tree, P t -> forall f1 : forest, P0 f1 -> P0 (cons t f1)) -> (forall t : tree, P t) * (forall f2 : forest, P0 f2)

See also

Generation of induction principles with Functional Scheme

Generation of inversion principles with Derive Inversion

Command Derive Inversion ident with ident Sort sort

Command Derive Inversion ident with (forall binder*, ident term) Sort sort

This command generates an inversion principle for the inversion ... using ... tactic. The first ident is the name of the generated principle. The second ident should be an inductive predicate, and binders the variables occurring in the term term. This command generates the inversion lemma for the sort sort corresponding to the instance forall binder*, ident term. When applied, it is equivalent to having inverted the instance with the tactic inversion.

Variant Derive Inversion_clear ident with ident Sort sort

Variant Derive Inversion_clear ident with (forall binder*, ident term) Sort sort

When applied, it is equivalent to having inverted the instance with the tactic inversion replaced by the tactic inversion_clear.

Variant Derive Dependent Inversion ident with ident Sort sort

Variant Derive Dependent Inversion ident with (forall binder*, ident term) Sort sort

When applied, it is equivalent to having inverted the instance with the tactic dependent inversion.

Variant Derive Dependent Inversion_clear ident with ident Sort sort

Variant Derive Dependent Inversion_clear ident with (forall binder*, ident term) Sort sort

When applied, it is equivalent to having inverted the instance with the tactic dependent inversion_clear.

Example

Consider the relation Le over natural numbers and the following parameter P:

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).

Le is defined Le_rect is defined Le_ind is defined Le_rec is defined Le_sind is defined

Parameter P : nat -> nat -> Prop.

P is declared

To generate the inversion lemma for the instance (Le (S n) m) and the sort Prop, we do:

Derive Inversion_clear leminv with (forall n m:nat, Le (S n) m) Sort Prop.

Check leminv.

leminv : forall (n m : nat) (P : nat -> nat -> Prop), (forall m0 : nat, Le n m0 -> P n (S m0)) -> Le (S n) m -> P n m

Then we can use the proven inversion lemma:

Goal forall (n m : nat) (H : Le (S n) m), P n m.

1 subgoal ============================ forall n m : nat, Le (S n) m -> P n m

intros.

1 subgoal n, m : nat H : Le (S n) m ============================ P n m

Show.

1 subgoal n, m : nat H : Le (S n) m ============================ P n m

inversion H using leminv.

1 subgoal n, m : nat H : Le (S n) m ============================ forall m0 : nat, Le n m0 -> P n (S m0)