\[\begin{split}\newcommand{\as}{\kw{as}} \newcommand{\case}{\kw{case}} \newcommand{\cons}{\textsf{cons}} \newcommand{\consf}{\textsf{consf}} \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}[3]{\kw{Ind}~[#1]\left(#2\mathrm{~:=~}#3\right)} \newcommand{\Indp}[4]{\kw{Ind}_{#4}[#1](#2:=#3)} \newcommand{\Indpstr}[5]{\kw{Ind}_{#4}[#1](#2:=#3)/{#5}} \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{\ModImp}[3]{{\kw{Mod}}({#1}:{#2}:={#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}\]

Reasoning with inductive types¶

Applying constructors¶

The tactics presented here specialize apply and eapply to constructors of inductive types.

Tactic constructor nat_or_var? with bindings?¶

First does repeat intro; hnf on the goal. If the result is an inductive type, then apply the appropriate constructor(s), and otherwise fail. If nat_or_var is specified and has the value i, it uses apply c_i, where c_i is the i-th constructor of I. If not specified, the tactic tries all the constructors, which can result in more than one success (e.g. for \/) when using backtracking tactics such as constructor; .... See ltac-seq.

with bindings?: If specified, the apply is done as apply … with bindings.

Warning

The terms in bindings are checked in the context where constructor is executed and not in the context where apply is executed (the introductions are not taken into account).

Error Not an inductive product.¶

Error Not enough constructors.¶

Error The type has no constructors.¶

Tactic split with bindings?¶: Equivalent to constructor 1 with bindings? when the conclusion is an inductive type with a single constructor. The bindings specify any parameters required for the constructor. It is typically used to split conjunctions in the conclusion such as A /\ B into two new goals A and B.

Tactic exists bindings*,¶

Equivalent to constructor 1 with bindings_i for each set of bindings (or just constructor 1 if there are no bindings) when the conclusion is an inductive type with a single constructor. It is typically used on existential quantifications in the form exists x, P x.

Error Not an inductive goal with 1 constructor.¶

Tactic left with bindings?¶

Tactic right with bindings?¶

These tactics apply only if I has two constructors, for instance in the case of a disjunction A \/ B. Then they are respectively equivalent to constructor 1 with bindings? and constructor 2 with bindings?.

Error Not an inductive goal with 2 constructors.¶

Tactic econstructor nat_or_var with bindings??¶
Tactic eexists bindings*,¶
Tactic esplit with bindings?¶
Tactic eleft with bindings?¶
Tactic eright with bindings?¶: These tactics behave like constructor, exists, split, left and right, but they introduce existential variables instead of failing when a variable can't be instantiated (cf. eapply and apply).

Example: constructor, left and right

Print or. (* or, represented by \/, has two constructors, or_introl and or_intror *): Inductive or (A B : Prop) : Prop := or_introl : A -> A \/ B | or_intror : B -> A \/ B. Arguments or (A B)%type_scope Arguments or_introl [A B]%type_scope _, [_] _ _ Arguments or_intror [A B]%type_scope _, _ [_] _
Goal forall P1 P2 : Prop, P1 -> P1 \/ P2.: 1 goal ============================ forall P1 P2 : Prop, P1 -> P1 \/ P2
constructor 1. (* equivalent to "left" *): 1 goal P1, P2 : Prop H : P1 ============================ P1
apply H. (* success *): No more goals.

In contrast, we won't be able to complete the proof if we select constructor 2:

Goal forall P1 P2 : Prop, P1 -> P1 \/ P2.: 1 goal ============================ forall P1 P2 : Prop, P1 -> P1 \/ P2

constructor 2. (* equivalent to "right" *): 1 goal P1, P2 : Prop H : P1 ============================ P2

You can also apply a constructor by name:

Goal forall P1 P2 : Prop, P1 -> P1 \/ P2.: 1 goal ============================ forall P1 P2 : Prop, P1 -> P1 \/ P2

intros; apply or_introl. (* equivalent to "left" *): 1 goal P1, P2 : Prop H : P1 ============================ P1

Case analysis¶

The tactics in this section implement case analysis on inductive or coinductive objects (see Variants and the match construct).

Tactic destruct induction_clause+, induction_principle?¶

induction_clause::=induction_arg as or_and_intropattern? eqn : naming_intropattern? occurrences?induction_arg::=one_term_with_bindings|natural

Performs case analysis by generating a subgoal for each constructor of the inductive or coinductive type selected by induction_arg. The selected subterm, after possibly doing an intros, must have an inductive or coinductive type. Unlike induction, destruct generates no induction hypothesis.

In each new subgoal, the tactic replaces the selected subterm with the associated constructor applied to its arguments, if any.

induction_clause+,

Giving multiple induction_clauses is equivalent to applying destruct serially on each induction_clause.

induction_arg

If one_term (in one_term_with_bindings) is an identifier ident:
- If ident denotes a quantified variable of the goal, then destruct ident behaves like intros until ident; destruct ident.
- If ident is no longer dependent in the goal after application of destruct, it is erased. To avoid erasure, use parentheses, as in destruct (ident).
one_term may contain holes that are denoted by “_”. In this case, the tactic selects the first subterm that matches the pattern and performs case analysis using that subterm.
If induction_arg is a natural, then destruct natural behaves like intros until natural followed by destruct applied to the last introduced hypothesis.

as or_and_intropattern

Provides names for (or applies further transformations to) the variables and hypotheses introduced in each new subgoal. The or_and_intropattern must have one intropattern* for each constructor, given in the order in which the constructors are defined. If there are not enough names, Coq picks fresh names. Inner intropatterns can also split introduced hypotheses into multiple hypotheses or subgoals.

eqn : naming_intropattern

Generates a new hypothesis in each new subgoal that is an equality between the term being case-analyzed and the associated constructor (applied to its arguments). The name of the new item may be specified in the naming_intropattern.

with bindings (in one_term_with_bindings)

Provides explicit instances for the dependent premises of the type of one_term.

occurrences

Selects specific subterms of the goal and/or hypotheses to apply the tactic to. See Occurrence clauses. If it occurs in the induction_principle, then there can only be one induction_clause, which can't have its own occurrences clause.

induction_principle

Makes the tactic equivalent to induction induction_clause+, induction_principle.

Example: Using destruct on an argument with premises

Parameter A B C D : Prop.: A is declared B is declared C is declared D is declared

Goal (A -> B \/ C) -> D.: 1 goal ============================ (A -> B \/ C) -> D
intros until 1.: 1 goal H : A -> B \/ C ============================ D
destruct H.: 3 goals ============================ A goal 2 is: D goal 3 is: D
Show 2.: goal 2 is: H : B ============================ D
Show 3.: goal 3 is: H : C ============================ D

The single tactic destruct 1 is equivalent to the intros and destruct used here.

Tactic edestruct induction_clause+, induction_principle?¶: If the type of one_term (in induction_arg) has dependent premises or dependent premises whose values are not inferrable from the with bindings clause, edestruct turns them into existential variables to be resolved later on.

Tactic case induction_clause+, induction_principle?¶

An older, more basic tactic to perform case analysis without recursion. We recommend using destruct instead where possible. case only modifies the goal; it does not modify the local context.

Tactic ecase induction_clause+, induction_principle?¶: If the type of one_term (in induction_arg) has dependent premises or dependent premises whose values are not inferrable from the with bindings clause, ecase turns them into existential variables to be resolved later on.

Tactic case_eq one_term¶: A variant of the case tactic that allows performing case analysis on a term without completely forgetting its original form. This is done by generating equalities between the original form of the term and the outcomes of the case analysis. We recommend using the destruct tactic with an eqn: clause instead.

Tactic casetype one_term¶

Tactic simple destruct identnatural¶: Equivalent to intros until identnatural; case ident where ident is a quantified variable of the goal and otherwise fails.

Tactic dependent destruction ident generalizing ident+? using one_term?¶: There is a long example of dependent destruction and an explanation of the underlying technique here.

Induction¶

Tactic induction induction_clause+, induction_principle?¶

induction_principle::=using one_term with bindings? occurrences?

Applies an induction principle to generate a subgoal for each constructor of an inductive type.

If the argument is dependent in the conclusion or some hypotheses of the goal, the argument is replaced by the appropriate constructor in each of the resulting subgoals and induction hypotheses are added to the local context using names whose prefix is IH. The tactic is similar to destruct, except that destruct doesn't generate induction hypotheses.

induction and destruct are very similar. Aside from the following differences, please refer to the description of destruct while mentally substituting induction for destruct.

induction_clause+,

If no induction_principle clause is provided, this is equivalent to doing induction on the first induction_clause followed by destruct on any subsequent clauses.

induction_principle

one_term specifies which induction principle to use. The optional with bindings gives any values that must be substituted into the induction principle. The number of bindings must be the same as the number of parameters of the induction principle.

If unspecified, the tactic applies the appropriate induction principle that was automatically generated when the inductive type was declared based on the sort of the goal.

Error Not an inductive product.¶

Error Cannot recognize a statement based on reference.¶

The type of the induction_arg (in an induction_clause) must reduce to the reference which was inferred as the type the induction principle operates on. Note that it is not enough to be convertible, but you can work around that with change:

Definition N := nat.: N is defined
Axiom strong : forall P, (forall n:N, (forall m:N, m < n -> P m) -> P n) -> forall n, P n.: strong is declared
Axiom P : N -> Prop.: P is declared
Goal forall n:nat, P n.: 1 goal ============================ forall n : nat, P n
intros.: 1 goal n : nat ============================ P n
Fail induction n using strong.: The command has indeed failed with message: Cannot recognize a statement based on N.
change N in n.: 1 goal n : N ============================ P n
(* n is now of type N, matching the inferred type that strong operates on *)
induction n using strong.: 1 goal n : N H : forall m : N, m < n -> P m ============================ P n

Error Unable to find an instance for the variables ident … ident.¶: Use the with bindings clause or the einduction tactic instead.

Example

Lemma induction_test : forall n:nat, n = n -> n <= n.: 1 goal ============================ forall n : nat, n = n -> n <= n
intros n H.: 1 goal n : nat H : n = n ============================ n <= n
induction n.: 2 goals H : 0 = 0 ============================ 0 <= 0 goal 2 is: S n <= S n
exact (le_n 0).: 1 goal n : nat H : S n = S n IHn : n = n -> n <= n ============================ S n <= S n

Example: induction with occurrences

Lemma induction_test2 : forall n:nat, n = n -> n <= n.: 1 goal ============================ forall n : nat, n = n -> n <= n
intros.: 1 goal n : nat H : n = n ============================ n <= n
induction n in H |-.: 2 goals n : nat H : 0 = 0 ============================ n <= n goal 2 is: n <= n
Show 2.: goal 2 is: n, n0 : nat H : S n0 = S n0 IHn0 : n0 = n0 -> n <= n ============================ n <= n

Tactic einduction induction_clause+, induction_principle?¶: Behaves like induction except that it does not fail if some dependent premise of the type of one_term is not inferrable. Instead, the unresolved premises are posed as existential variables to be inferred later, in the same way as eapply does.

Tactic elim one_term_with_bindings using one_term with bindings??¶

An older, more basic induction tactic. Unlike induction, elim only modifies the goal; it does not modify the local context. We recommend using induction instead where possible.

with bindings (in one_term_with_bindings): Explicitly gives instances to the premises of the type of one_term (see Bindings).
using one_term with bindings??: Allows explicitly giving an induction principle one_term that is not the standard one for the underlying inductive type of one_term. The bindings clause allows instantiating premises of the type of one_term.

Tactic eelim one_term_with_bindings using one_term with bindings??¶: If the type of one_term has dependent premises, this turns them into existential variables to be resolved later on.

Tactic elimtype one_term¶: The argument type must be inductively defined. elimtype I is equivalent to cut I. intro Hn; elim Hn; clear Hn. Therefore the hypothesis Hn will not appear in the context(s) of the subgoal(s). Conversely, if t is a one_term of (inductive) type I that does not occur in the goal, then elim t is equivalent to elimtype I; only 2: exact t.

Tactic simple induction identnatural¶: Behaves like intros until identnatural; elim ident when ident is a quantified variable of the goal.

Tactic dependent induction ident generalizingin ident+? using one_term?¶

The experimental tactic dependent induction performs induction-inversion on an instantiated inductive predicate. One needs to first Require the Coq.Program.Equality module to use this tactic. The tactic is based on the BasicElim tactic by Conor McBride [McB00] and the work of Cristina Cornes around inversion [CT95]. From an instantiated inductive predicate and a goal, it generates an equivalent goal where the hypothesis has been generalized over its indexes which are then constrained by equalities to be the right instances. This permits to state lemmas without resorting to manually adding these equalities and still get enough information in the proofs.

generalizingin ident+: First generalizes the goal by the given variables so that they are universally quantified in the goal. This is generally what one wants to do with variables that are inside constructors in the induction hypothesis. The other ones need not be further generalized.

There is a long example of dependent induction and an explanation of the underlying technique here.

Example

Lemma lt_1_r : forall n:nat, n < 1 -> n = 0.: 1 goal ============================ forall n : nat, n < 1 -> n = 0
intros n H ; induction H.: 2 goals n : nat ============================ n = 0 goal 2 is: n = 0

Here we did not get any information on the indexes to help fulfill this proof. The problem is that, when we use the induction tactic, we lose information on the hypothesis instance, notably that the second argument is 1 here. Dependent induction solves this problem by adding the corresponding equality to the context.

Require Import Coq.Program.Equality.
Lemma lt_1_r : forall n:nat, n < 1 -> n = 0.: 1 goal ============================ forall n : nat, n < 1 -> n = 0
intros n H ; dependent induction H.: 2 goals ============================ 0 = 0 goal 2 is: n = 0

The subgoal is cleaned up as the tactic tries to automatically simplify the subgoals with respect to the generated equalities. In this enriched context, it becomes possible to solve this subgoal.

reflexivity.: 1 goal n : nat H : S n <= 0 IHle : 0 = 1 -> n = 0 ============================ n = 0

Now we are in a contradictory context and the proof can be solved.

inversion H.: No more goals.

This technique works with any inductive predicate. In fact, the dependent induction tactic is just a wrapper around the induction tactic. One can make its own variant by just writing a new tactic based on the definition found in Coq.Program.Equality.

Equality of inductive types¶

This section describes some special purpose tactics to work with Leibniz equality of inductive sets or types.

Tactic discriminate induction_arg?¶

Proves any goal for which a hypothesis in the form term₁ = term₂ states an impossible structural equality for an inductive type. If induction_arg is not given, it checks all the hypotheses for impossible equalities. For example, (S (S O)) = (S O) is impossible. If provided, induction_arg is a proof of an equality, typically specified as the name of a hypothesis.

If no induction_arg is provided and the goal is in the form term₁ <> term₂, then the tactic behaves like intro ident; discriminate ident.

The tactic traverses the normal forms of term₁ and term₂, looking for subterms u and w placed in the same positions and whose head symbols are different constructors. If such subterms are present, the equality is impossible and the current goal is completed. Otherwise the tactic fails. Note that opaque constants are not expanded by δ reductions while computing the normal form.

ident (in induction_arg): Checks the hypothesis ident for impossible equalities. If ident is not already in the context, this is equivalent to intros until ident; discriminate ident.
natural (in induction_arg): Equivalent to intros until natural; discriminate ident, where ident is the identifier for the last introduced hypothesis.
one_term with bindings (in induction_arg): Equivalent to discriminate one_term but uses the given bindings to instantiate parameters or hypotheses of one_term. one_term must be a proof of term₁ = term₂.

Error No primitive equality found.¶

Error Not a discriminable equality.¶

Tactic ediscriminate induction_arg?¶: Works the same as discriminate but if the type of one_term, or the type of the hypothesis referred to by natural, has uninstantiated parameters, these parameters are left as existential variables.

Tactic injection induction_arg? as simple_intropattern*?¶

Exploits the property that constructors of inductive types are injective, i.e. that if c is a constructor of an inductive type and c t₁ = c t₂ then t₁ = t₂ are equal too.

If there is a hypothesis H in the form term₁ = term₂, then injection H applies the injectivity of constructors as deep as possible to derive the equality of subterms of term₁ and term₂ wherever the subterms start to differ. For example, from (S p, S n) = (q, S (S m)) we may derive S p = q and n = S m. The terms must have inductive types and the same head constructor, but must not be convertible. If so, the tactic derives the equalities and adds them to the current goal as premises (except if the as clause is used).

If no induction_arg is provided and the current goal is of the form term <> term, injection is equivalent to intro ident; injection ident.

ident (in induction_arg)

Derives equalities based on constructor injectivity for the hypothesis ident. If ident is not already in the context, this is equivalent to intros until ident; injection ident.

natural (in induction_arg)

Equivalent to intros until natural followed by injection ident where ident is the identifier for the last introduced hypothesis.

one_term with bindings (in induction_arg)

Like injection one_term but uses the given bindings to instantiate parameters or hypotheses of one_term.

as [= intropattern* ]

Specifies names to apply after the injection so that all generated equalities become hypotheses, which (unlike intros) may replace existing hypotheses with same name. The number of provided names must not exceed the number of newly generated equalities. If it is smaller, fresh names are generated for the unspecified items. The original equality is erased if it corresponds to a provided name or if the list of provided names is incomplete.

Note that, as a convenience for users, specifying simple_intropattern+ is treated as if [= simple_intropattern+ ] was specified.

Example

Consider the following goal:

Inductive list : Set := | nil : list | cons : nat -> list -> list.: list is defined list_rect is defined list_ind is defined list_rec is defined list_sind is defined
Parameter P : list -> Prop.: P is declared
Goal forall l n, P nil -> cons n l = cons 0 nil -> P l.: 1 goal ============================ forall (l : list) (n : nat), P nil -> cons n l = cons 0 nil -> P l

intros.: 1 goal l : list n : nat H : P nil H0 : cons n l = cons 0 nil ============================ P l
injection H0.: 1 goal l : list n : nat H : P nil H0 : cons n l = cons 0 nil ============================ l = nil -> n = 0 -> P l

Note

Beware that injection yields an equality in a sigma type whenever the injected object has a dependent type P with its two instances in different types (P t₁ … t_n) and (P u₁ … u_n). If t₁ and u₁ are the same and have for type an inductive type for which a decidable equality has been declared using Scheme Equality, the use of a sigma type is avoided.

Error No information can be deduced from this equality and the injectivity of constructors. This may be because the terms are convertible, or due to pattern matching restrictions in the sort Prop. You can try to use option Set Keep Proof Equalities.¶

Error No primitive equality found.¶

Error Not a negated primitive equality¶: When induction_arg is not provided, the goal must be in the form term <> term.

Error Nothing to inject.¶: Generated when one side of the equality is not a constructor.

Tactic einjection induction_arg? as simple_intropattern*?¶: Works the same as injection but if the type of one_term, or the type of the hypothesis referred to by natural has uninstantiated parameters, these parameters are left as existential variables.

Tactic simple injection induction_arg?¶: Similar to injection, but always adds the derived equalities as new premises in the current goal (instead of as new hypotheses) even if the Structural Injection flag is set.

Flag Structural Injection¶: When this flag is set, injection term erases the original hypothesis and adds the generated equalities as new hypotheses rather than adding them to the current goal as premises, as if giving injection term as (with an empty list of names). This flag is off by default.

Flag Keep Proof Equalities¶: By default, injection only creates new equalities between terms whose type is in sort Type or Set, thus implementing a special behavior for objects that are proofs of a statement in Prop. This flag controls this behavior.

Table Keep Equalities qualid¶: This table specifies a set of inductive types for which proof equalities are always kept by injection. This overrides the Keep Proof Equalities flag for those inductive types. Template polymorphic inductive types are implicitly added to this table when defined. Use the Add and Remove commands to update this set manually.

Tactic simplify_eq induction_arg?¶

Examines a hypothesis that has the form term₁ = term₂. If the terms are structurally different, the tactic does a discriminate. Otherwise, it does an injection to simplify the equality, if possible. If induction_arg is not provided, the tactic examines the goal, which must be in the form term₁ <> term₂.

See the description of induction_arg in injection for an explanation of the parameters.

Tactic esimplify_eq induction_arg?¶: Works the same as simplify_eq but if the type of one_term or the type of the hypothesis referred to by natural has uninstantiated parameters, these parameters are left as existential variables.

Tactic inversion identnatural as or_and_intropattern? in ident+?¶

Tactic inversion identnatural using one_term in ident+?¶

For a hypothesis whose type is a (co)inductively defined proposition, the tactic introduces a goal for each constructor of the proposition that isn't self-contradictory. Each such goal includes the hypotheses needed to deduce the proposition. (Co)inductively defined propositions are those defined with the Inductive or CoInductive commands whose contructors yield a Prop, as in this example.

ident

The name of the hypothesis to invert. If ident does not denote a hypothesis in the local context but refers to a hypothesis quantified in the goal, then the latter is first introduced in the local context using intros until ident.

natural

Equivalent to intros until natural; inversion ident where ident is the identifier for the last introduced hypothesis.

in ident+?

When ident+ are identifiers in the local context, this does a generalize ident+ as the initial step of inversion.

as or_and_intropattern

Provides names for the variables introduced in each new subgoal. The or_and_intropattern must have one intropattern* for each constructor of the (co)inductive predicate, given in the order in which the constructors are defined. If there are not enough names, Coq picks fresh names.

If an equation splits into several equations (because inversion applies injection on the equalities it generates), the corresponding intropattern should be in the form [ intropattern* ] (or the equivalent ( simple_intropattern )*,), with the number of entries equal to the number of subequalities obtained from splitting the original equation. Example here.

Note

The inversion … as variant of inversion generally behaves in a slightly more expected way than inversion (no artificial duplication of some hypotheses referring to other hypotheses). To take advantage of these improvements, it is enough to use inversion … as [], letting Coq choose fresh names.

Note

As inversion proofs may be large, we recommend creating and using lemmas whenever the same instance needs to be inverted several times. See Generation of inversion principles with Derive Inversion.

Note

Part of the behavior of the inversion tactic is to generate equalities between expressions that appeared in the hypothesis that is being processed. By default, no equalities are generated if they relate two proofs (i.e. equalities between terms whose type is in sort Prop). This behavior can be turned off by using the Keep Proof Equalities setting.

Example: inversion with as or_and_intropattern

Inductive contains0 : list nat -> Prop := | in_hd : forall l, contains0 (0 :: l) | in_tl : forall l b, contains0 l -> contains0 (b :: l).
contains0 is defined contains0_ind is defined contains0_sind is defined

Goal forall l:list nat, contains0 (1 :: l) -> contains0 l.
1 goal ============================ forall l : list nat, contains0 (1 :: l) -> contains0 l

intros l H.
1 goal l : list nat H : contains0 (1 :: l) ============================ contains0 l

inversion H as [ | l' p Hl' [Heqp Heql'] ].
1 goal l : list nat H : contains0 (1 :: l) l' : list nat p : nat Hl' : contains0 l Heqp : p = 1 Heql' : l' = l ============================ contains0 l

Tactic inversion_clear identnatural as or_and_intropattern? in ident+?¶

Does an inversion and then erases the hypothesis that was used for the inversion.

Tactic simple inversion identnatural as or_and_intropattern? in ident+?¶

A very simple inversion tactic that derives all the necessary equalities but does not simplify the constraints as inversion does.

Tactic dependent inversion identnatural as or_and_intropattern? with one_term?¶

For use when the inverted hypothesis appears in the current goal. Does an inversion and then substitutes the name of the hypothesis where the corresponding term appears in the goal.

Tactic dependent inversion_clear identnatural as or_and_intropattern? with one_term?¶

Does a dependent inversion and then erases the hypothesis that was used for the dependent inversion.

Tactic dependent simple inversion identnatural as or_and_intropattern? with one_term?¶

Tactic inversion_sigma ident as simple_intropattern??¶

Turns equalities of dependent pairs (e.g., existT P x p = existT P y q, frequently left over by inversion on a dependent type family) into pairs of equalities (e.g., a hypothesis H : x = y and a hypothesis of type rew H in p = q); these hypotheses can subsequently be simplified using subst, without ever invoking any kind of axiom asserting uniqueness of identity proofs. If you want to explicitly specify the hypothesis to be inverted, you can pass it as an argument to inversion_sigma. This tactic also works for sig, sigT2, sig2, ex, and ex2 and there are similar eq_sig ***_rect induction lemmas.

Error Type of ident is not an equality of recognized Σ types: expected one of sig sig2 sigT sigT2 sigT2 ex or ex2 but got term¶: When applied to a hypothesis, inversion_sigma can only handle equalities of the listed sigma types.

Error ident is not an equality of Σ types¶: When applied to a hypothesis, inversion_sigma can only be called on hypotheses that are equalities using Coq.Logic.Init.eq.

Example: Non-dependent inversion

Let us consider the relation Le over natural numbers:

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

Let us consider the following goal:

Section Section.
Variable P : nat -> nat -> Prop.: P is declared
Variable Q : forall n m:nat, Le n m -> Prop.: Q is declared
Goal forall n m, Le (S n) m -> P n m.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop ============================ forall n m : nat, Le (S n) m -> P n m

intros.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop n, m : nat H : Le (S n) m ============================ P n m

To prove the goal, we may need to reason by cases on H and to derive that m is necessarily of the form (S m0) for certain m0 and that (Le n m0). Deriving these conditions corresponds to proving that the only possible constructor of (Le (S n) m) is LeS and that we can invert the arrow in the type of LeS. This inversion is possible because Le is the smallest set closed by the constructors LeO and LeS.

inversion_clear H.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop n, m, m0 : nat H0 : Le n m0 ============================ P n (S m0)

Note that m has been substituted in the goal for (S m0) and that the hypothesis (Le n m0) has been added to the context.

Sometimes it is interesting to have the equality m = (S m0) in the context to use it after. In that case we can use inversion that does not clear the equalities:

intros.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop n, m : nat H : Le (S n) m ============================ P n m

inversion H.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop n, m : nat H : Le (S n) m n0, m0 : nat H1 : Le n m0 H0 : n0 = n H2 : S m0 = m ============================ P n (S m0)

Example: Dependent inversion

Let us consider the following goal:

Abort.
Goal forall n m (H:Le (S n) m), Q (S n) m H.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop ============================ forall (n m : nat) (H : Le (S n) m), Q (S n) m H

intros.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop n, m : nat H : Le (S n) m ============================ Q (S n) m H

As H occurs in the goal, we may want to reason by cases on its structure and so, we would like inversion tactics to substitute H by the corresponding @term in constructor form. Neither inversion nor inversion_clear do such a substitution. To have such a behavior we use the dependent inversion tactics:

dependent inversion_clear H.: 1 goal P : nat -> nat -> Prop Q : forall n m : nat, Le n m -> Prop n, m, m0 : nat l : Le n m0 ============================ Q (S n) (S m0) (LeS n m0 l)

Note that H has been substituted by (LeS n m0 l) and m by (S m0).

Example: Using inversion_sigma

Let us consider the following inductive type of length-indexed lists, and a lemma about inverting equality of cons:

Require Import Coq.Logic.Eqdep_dec.
Inductive vec A : nat -> Type := | nil : vec A O | cons {n} (x : A) (xs : vec A n) : vec A (S n).: vec is defined vec_rect is defined vec_ind is defined vec_rec is defined vec_sind is defined
Lemma invert_cons : forall A n x xs y ys, @cons A n x xs = @cons A n y ys -> xs = ys.: 1 goal ============================ forall (A : Type) (n : nat) (x : A) (xs : vec A n) (y : A) (ys : vec A n), cons A x xs = cons A y ys -> xs = ys
Proof.
intros A n x xs y ys H.: 1 goal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys ============================ xs = ys

After performing inversion, we are left with an equality of existTs:

inversion H.: 1 goal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H2 : existT (fun n : nat => vec A n) n xs = existT (fun n : nat => vec A n) n ys ============================ xs = ys

We can turn this equality into a usable form with inversion_sigma:

inversion_sigma.: 1 goal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H2_ : n = n H2_0 : eq_rect n (fun n : nat => vec A n) xs n H2_ = ys ============================ xs = ys

To finish cleaning up the proof, we will need to use the fact that that all proofs of n = n for n a nat are eq_refl:

let H := match goal with H : n = n |- _ => H end in pose proof (Eqdep_dec.UIP_refl_nat _ H); subst H.: 1 goal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H2_0 : eq_rect n (fun n : nat => vec A n) xs n eq_refl = ys ============================ xs = ys
simpl in *.: 1 goal A : Type n : nat x : A xs : vec A n y : A ys : vec A n H : cons A x xs = cons A y ys H1 : x = y H2_0 : xs = ys ============================ xs = ys

Finally, we can finish the proof:

assumption.: No more goals.
Qed.

Helper tactics¶

Tactic decide equality¶: Solves a goal of the form forall x y : R, {x = y} + {~ x = y}, where R is an inductive type such that its constructors do not take proofs or functions as arguments, nor objects in dependent types. It solves goals of the form {x = y} + {~ x = y} as well.

Tactic compare one_term₁ one_term₂¶: Compares two one_terms of an inductive datatype. If G is the current goal, it leaves the sub-goals one_term₁ = one_term₂ -> G and ~ one_term₁ = one_term₂ -> G. The type of the one_terms must satisfy the same restrictions as in the tactic decide equality.

Tactic dependent rewrite -><-? one_term in ident?¶

If ident has type (existT B a b)=(existT B a' b') in the local context (i.e. each term of the equality has a sigma type { a:A & (B a)}) this tactic rewrites a into a' and b into b' in the current goal. This tactic works even if B is also a sigma type. This kind of equalities between dependent pairs may be derived by the injection and inversion tactics.

-><-?: By default, the equality is applied from left to right. Specify <- to apply the equality from right to left.

Generation of induction principles with `Scheme`¶

Command Scheme ident :=? scheme_kind with ident :=? scheme_kind*¶

Generates induction principles with given scheme_types and scheme_sorts for an inductive type. In the case where the inductive definition is a mutual inductive definition, the with clause is used to generate a mutually recursive inductive scheme for each clause of the mutual inductive type.

ident: The name of the scheme. If not provided, the name will be determined automatically from the scheme_type and sort_family.

The following scheme_types generate induction principles with given properties:

`scheme_type`	Recursive	Dependent
`Induction`	Yes	Yes
`Minimality`	Yes	No
`Elimination`	No	Yes
`Case`	No	No

See examples of the scheme_types here.

Command Scheme Boolean? Equality for reference¶: Tries to generate a Boolean equality for reference. If Boolean is not specified, the command also tries to generate a proof of the decidability of propositional equality over reference. If reference involves independent constants or other inductive types, we recommend defining their equality first.

Example: Induction scheme for tree and forest

Currently the automatically-generated induction principles such as odd_ind are not useful for mutually-inductive types such as odd and even. You can define a mutual induction principle for tree and forest in sort Set with the Scheme 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 and forests; 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 forall n : nat, even n -> P0 n.

Example: Scheme commands with various scheme_types

Let us demonstrate the difference between the Scheme commands.

Unset Elimination Schemes.
Inductive Nat := | z : Nat | s : Nat -> Nat.
Nat is defined

(* dependent, recursive *)
Scheme Induction for Nat Sort Set.
Nat_rec is defined Nat_rec is recursively defined

About Nat_rec.
Nat_rec : forall P : Nat -> Set, P z -> (forall n : Nat, P n -> P (s n)) -> forall n : Nat, P n Nat_rec is not universe polymorphic Arguments Nat_rec P%function_scope f f0%function_scope n Nat_rec is transparent Expands to: Constant Top.Nat_rec

(* non-dependent, recursive *)
Scheme Minimality for Nat Sort Set.
Nat_rec_nodep is defined Nat_rec_nodep is recursively defined

About Nat_rec_nodep.
Nat_rec_nodep : forall P : Set, P -> (Nat -> P -> P) -> Nat -> P Nat_rec_nodep is not universe polymorphic Arguments Nat_rec_nodep P%type_scope f f0%function_scope n Nat_rec_nodep is transparent Expands to: Constant Top.Nat_rec_nodep

(* dependent, non-recursive *)
Scheme Elimination for Nat Sort Set.
Nat_case is defined Nat_case is recursively defined

About Nat_case.
Nat_case : forall P : Nat -> Set, P z -> (forall n : Nat, P n -> P (s n)) -> forall n : Nat, P n Nat_case is not universe polymorphic Arguments Nat_case P%function_scope f f0%function_scope n Nat_case is transparent Expands to: Constant Top.Nat_case

(* non-dependent, non-recursive *)
Scheme Case for Nat Sort Set.
Nat_case_nodep is defined Nat_case_nodep is recursively defined

About Nat_case_nodep.
Nat_case_nodep : forall P : Set, P -> (Nat -> P -> P) -> Nat -> P Nat_case_nodep is not universe polymorphic Arguments Nat_case_nodep P%type_scope f f0%function_scope n Nat_case_nodep is transparent Expands to: Constant Top.Nat_case_nodep

Automatic declaration of schemes¶

Flag Elimination Schemes¶: This flag enables automatic declaration of induction principles when defining a new inductive type. Defaults to on.

Flag Nonrecursive Elimination Schemes¶: This flag 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_def from ident+,¶: Combines induction principles generated by the Scheme command. Each ident is a different inductive principle that must belong to the same package of mutual inductive principle definitions. This command generates ident_def as 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)

Generation of inversion principles with `Derive` `Inversion`¶

Command Derive Inversion ident with one_term Sort sort_family?¶: Generates an inversion lemma for the inversion tactic. ident is the name of the generated lemma. one_term should be in the form qualid or (forall binder+, qualid term) where qualid is the name of an inductive predicate and binder+ binds the variables occurring in the term term. The lemma is generated for the sort sort_family corresponding to one_term. Applying the lemma is equivalent to inverting the instance with the inversion tactic.

Command Derive Inversion_clear ident with one_term Sort sort_family?¶: When applied, it is equivalent to having inverted the instance with the tactic inversion replaced by the tactic inversion_clear.

Command Derive Dependent Inversion ident with one_term Sort sort_family¶: When applied, it is equivalent to having inverted the instance with the tactic dependent inversion.

Command Derive Dependent Inversion_clear ident with one_term Sort sort_family¶: 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
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.: leminv is defined
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 goal ============================ forall n m : nat, Le (S n) m -> P n m
intros.: 1 goal n, m : nat H : Le (S n) m ============================ P n m

Show.: 1 goal n, m : nat H : Le (S n) m ============================ P n m
inversion H using leminv.: 1 goal n, m : nat H : Le (S n) m ============================ forall m0 : nat, Le n m0 -> P n (S m0)

Examples of `dependent destruction` / `dependent induction`¶

The tactics dependent induction and dependent destruction are another solution for inverting inductive predicate instances and potentially doing induction at the same time. It is based on the BasicElim tactic of Conor McBride which works by abstracting each argument of an inductive instance by a variable and constraining it by equalities afterwards. This way, the usual induction and destruct tactics can be applied to the abstracted instance and after simplification of the equalities we get the expected goals.

The abstracting tactic is called generalize_eqs and it takes as argument a hypothesis to generalize. It uses the JMeq datatype defined in Coq.Logic.JMeq, hence we need to require it before. For example, revisiting the first example of the inversion documentation:

Require Import Coq.Logic.JMeq.
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
Goal forall n m:nat, Le (S n) m -> P n m.: 1 goal ============================ forall n m : nat, Le (S n) m -> P n m
intros n m H.: 1 goal n, m : nat H : Le (S n) m ============================ P n m

generalize_eqs H.: 1 goal n, m, gen_x : nat H : Le gen_x m ============================ gen_x = S n -> P n m

The index S n gets abstracted by a variable here, but a corresponding equality is added under the abstract instance so that no information is actually lost. The goal is now almost amenable to do induction or case analysis. One should indeed first move n into the goal to strengthen it before doing induction, or n will be fixed in the inductive hypotheses (this does not matter for case analysis). As a rule of thumb, all the variables that appear inside constructors in the indices of the hypothesis should be generalized. This is exactly what the generalize_eqs_vars variant does:

generalize_eqs_vars H.
induction H.: 2 goals n, n0 : nat ============================ 0 = S n -> P n n0 goal 2 is: S n0 = S n -> P n (S m)

As the hypothesis itself did not appear in the goal, we did not need to use an heterogeneous equality to relate the new hypothesis to the old one (which just disappeared here). However, the tactic works just as well in this case, e.g.:

Require Import Coq.Program.Equality.

Parameter Q : forall (n m : nat), Le n m -> Prop.: Q is declared
Goal forall n m (p : Le (S n) m), Q (S n) m p.: 1 goal ============================ forall (n m : nat) (p : Le (S n) m), Q (S n) m p

intros n m p.: 1 goal n, m : nat p : Le (S n) m ============================ Q (S n) m p
generalize_eqs_vars p.: 1 goal m, gen_x : nat p : Le gen_x m ============================ forall (n : nat) (p0 : Le (S n) m), gen_x = S n -> p ~= p0 -> Q (S n) m p0

One drawback of this approach is that in the branches one will have to substitute the equalities back into the instance to get the right assumptions. Sometimes injection of constructors will also be needed to recover the needed equalities. Also, some subgoals should be directly solved because of inconsistent contexts arising from the constraints on indexes. The nice thing is that we can make a tactic based on discriminate, injection and variants of substitution to automatically do such simplifications (which may involve the axiom K). This is what the simplify_dep_elim tactic from Coq.Program.Equality does. For example, we might simplify the previous goals considerably:

induction p ; simplify_dep_elim.: 1 goal n, m : nat p : Le n m IHp : forall (n0 : nat) (p0 : Le (S n0) m), n = S n0 -> p ~= p0 -> Q (S n0) m p0 ============================ Q (S n) (S m) (LeS n m p)

The higher-order tactic do_depind defined in Coq.Program.Equality takes a tactic and combines the building blocks we have seen with it: generalizing by equalities calling the given tactic with the generalized induction hypothesis as argument and cleaning the subgoals with respect to equalities. Its most important instantiations are dependent induction and dependent destruction that do induction or simply case analysis on the generalized hypothesis. For example we can redo what we've done manually with dependent destruction:

Lemma ex : forall n m:nat, Le (S n) m -> P n m.: 1 goal ============================ forall n m : nat, Le (S n) m -> P n m

intros n m H.: 1 goal n, m : nat H : Le (S n) m ============================ P n m

dependent destruction H.: 1 goal n, m : nat H : Le n m ============================ P n (S m)

This gives essentially the same result as inversion. Now if the destructed hypothesis actually appeared in the goal, the tactic would still be able to invert it, contrary to dependent inversion. Consider the following example on vectors:

Set Implicit Arguments.

Parameter A : Set.: A is declared

Inductive vector : nat -> Type := | vnil : vector 0 | vcons : A -> forall n, vector n -> vector (S n).: vector is defined vector_rect is defined vector_ind is defined vector_rec is defined vector_sind is defined

Goal forall n, forall v : vector (S n), exists v' : vector n, exists a : A, v = vcons a v'.: 1 goal ============================ forall (n : nat) (v : vector (S n)), exists (v' : vector n) (a : A), v = vcons a v'

intros n v.: 1 goal n : nat v : vector (S n) ============================ exists (v' : vector n) (a : A), v = vcons a v'

dependent destruction v.: 1 goal n : nat a : A v : vector n ============================ exists (v' : vector n) (a0 : A), vcons a v = vcons a0 v'

In this case, the v variable can be replaced in the goal by the generalized hypothesis only when it has a type of the form vector (S n), that is only in the second case of the destruct. The first one is dismissed because S n <> 0.

A larger example¶

Let's see how the technique works with induction on inductive predicates on a real example. We will develop an example application to the theory of simply-typed lambda-calculus formalized in a dependently-typed style:

Inductive type : Type := | base : type | arrow : type -> type -> type.: type is defined type_rect is defined type_ind is defined type_rec is defined type_sind is defined

Notation " t --> t' " := (arrow t t') (at level 20, t' at next level).

Inductive ctx : Type := | empty : ctx | snoc : ctx -> type -> ctx.: ctx is defined ctx_rect is defined ctx_ind is defined ctx_rec is defined ctx_sind is defined

Notation " G , tau " := (snoc G tau) (at level 20, tau at next level).

Fixpoint conc (G D : ctx) : ctx := match D with | empty => G | snoc D' x => snoc (conc G D') x end.: conc is defined conc is recursively defined (guarded on 2nd argument)

Notation " G ; D " := (conc G D) (at level 20).

Inductive term : ctx -> type -> Type := | ax : forall G tau, term (G, tau) tau | weak : forall G tau, term G tau -> forall tau', term (G, tau') tau | abs : forall G tau tau', term (G , tau) tau' -> term G (tau --> tau') | app : forall G tau tau', term G (tau --> tau') -> term G tau -> term G tau'.: term is defined term_rect is defined term_ind is defined term_rec is defined term_sind is defined

We have defined types and contexts which are snoc-lists of types. We also have a conc operation that concatenates two contexts. The term datatype represents in fact the possible typing derivations of the calculus, which are isomorphic to the well-typed terms, hence the name. A term is either an application of:

the axiom rule to type a reference to the first variable in a context
the weakening rule to type an object in a larger context
the abstraction or lambda rule to type a function
the application to type an application of a function to an argument

Once we have this datatype we want to do proofs on it, like weakening:

Lemma weakening : forall G D tau, term (G ; D) tau -> forall tau', term (G , tau' ; D) tau.: 1 goal ============================ forall (G D : ctx) (tau : type), term (G; D) tau -> forall tau' : type, term ((G, tau'); D) tau

The problem here is that we can't just use induction on the typing derivation because it will forget about the G ; D constraint appearing in the instance. A solution would be to rewrite the goal as:

Lemma weakening' : forall G' tau, term G' tau -> forall G D, (G ; D) = G' -> forall tau', term (G, tau' ; D) tau.: 1 goal ============================ forall (G' : ctx) (tau : type), term G' tau -> forall G D : ctx, G; D = G' -> forall tau' : type, term ((G, tau'); D) tau

With this proper separation of the index from the instance and the right induction loading (putting G and D after the inducted-on hypothesis), the proof will go through, but it is a very tedious process. One is also forced to make a wrapper lemma to get back the more natural statement. The dependent induction tactic alleviates this trouble by doing all of this plumbing of generalizing and substituting back automatically. Indeed we can simply write:

Require Import Coq.Program.Tactics.
Require Import Coq.Program.Equality.

Lemma weakening : forall G D tau, term (G ; D) tau -> forall tau', term (G , tau' ; D) tau.: 1 goal ============================ forall (G D : ctx) (tau : type), term (G; D) tau -> forall tau' : type, term ((G, tau'); D) tau

Proof with simpl in * ; simpl_depind ; auto.

intros G D tau H. dependent induction H generalizing G D ; intros.: 1 goal G, D : ctx tau : type H : term (G; D) tau ============================ forall tau' : type, term ((G, tau'); D) tau 4 goals G0 : ctx tau : type G, D : ctx x : G0, tau = G; D tau' : type ============================ term ((G, tau'); D) tau goal 2 is: term ((G, tau'0); D) tau goal 3 is: term ((G, tau'0); D) (tau --> tau') goal 4 is: term ((G, tau'0); D) tau'

This call to dependent induction has an additional arguments which is a list of variables appearing in the instance that should be generalized in the goal, so that they can vary in the induction hypotheses. By default, all variables appearing inside constructors (except in a parameter position) of the instantiated hypothesis will be generalized automatically but one can always give the list explicitly.

Show.: 4 goals G0 : ctx tau : type G, D : ctx x : G0, tau = G; D tau' : type ============================ term ((G, tau'); D) tau goal 2 is: term ((G, tau'0); D) tau goal 3 is: term ((G, tau'0); D) (tau --> tau') goal 4 is: term ((G, tau'0); D) tau'

The simpl_depind tactic includes an automatic tactic that tries to simplify equalities appearing at the beginning of induction hypotheses, generally using trivial applications of reflexivity. In cases where the equality is not between constructor forms though, one must help the automation by giving some arguments, using the specialize tactic for example.

destruct D... apply weak; apply ax. apply ax.: 5 goals G0 : ctx tau, tau' : type ============================ term ((G0, tau), tau') tau goal 2 is: term (((G, tau'); D), t) t goal 3 is: term ((G, tau'0); D) tau goal 4 is: term ((G, tau'0); D) (tau --> tau') goal 5 is: term ((G, tau'0); D) tau' 4 goals G, D : ctx t, tau' : type ============================ term (((G, tau'); D), t) t goal 2 is: term ((G, tau'0); D) tau goal 3 is: term ((G, tau'0); D) (tau --> tau') goal 4 is: term ((G, tau'0); D) tau' 3 goals G0 : ctx tau : type H : term G0 tau tau' : type IHterm : forall G D : ctx, G0 = G; D -> forall tau' : type, term ((G, tau'); D) tau G, D : ctx x : G0, tau' = G; D tau'0 : type ============================ term ((G, tau'0); D) tau goal 2 is: term ((G, tau'0); D) (tau --> tau') goal 3 is: term ((G, tau'0); D) tau'

destruct D...: 4 goals G0 : ctx tau : type H : term G0 tau tau' : type IHterm : forall G D : ctx, G0 = G; D -> forall tau' : type, term ((G, tau'); D) tau tau'0 : type ============================ term ((G0, tau'), tau'0) tau goal 2 is: term (((G, tau'0); D), t) tau goal 3 is: term ((G, tau'0); D) (tau --> tau') goal 4 is: term ((G, tau'0); D) tau'

Show.: 4 goals G0 : ctx tau : type H : term G0 tau tau' : type IHterm : forall G D : ctx, G0 = G; D -> forall tau' : type, term ((G, tau'); D) tau tau'0 : type ============================ term ((G0, tau'), tau'0) tau goal 2 is: term (((G, tau'0); D), t) tau goal 3 is: term ((G, tau'0); D) (tau --> tau') goal 4 is: term ((G, tau'0); D) tau'

specialize (IHterm G0 empty eq_refl).: 4 goals G0 : ctx tau : type H : term G0 tau tau' : type IHterm : forall tau' : type, term ((G0, tau'); empty) tau tau'0 : type ============================ term ((G0, tau'), tau'0) tau goal 2 is: term (((G, tau'0); D), t) tau goal 3 is: term ((G, tau'0); D) (tau --> tau') goal 4 is: term ((G, tau'0); D) tau'

Once the induction hypothesis has been narrowed to the right equality, it can be used directly.

apply weak, IHterm.: 3 goals tau : type G, D : ctx IHterm : forall G0 D0 : ctx, G; D = G0; D0 -> forall tau' : type, term ((G0, tau'); D0) tau H : term (G; D) tau t, tau'0 : type ============================ term (((G, tau'0); D), t) tau goal 2 is: term ((G, tau'0); D) (tau --> tau') goal 3 is: term ((G, tau'0); D) tau'

Now concluding this subgoal is easy.

constructor; apply IHterm; reflexivity.: 2 goals G, D : ctx tau, tau' : type H : term ((G; D), tau) tau' IHterm : forall G0 D0 : ctx, (G; D), tau = G0; D0 -> forall tau'0 : type, term ((G0, tau'0); D0) tau' tau'0 : type ============================ term ((G, tau'0); D) (tau --> tau') goal 2 is: term ((G, tau'0); D) tau'

Reasoning with inductive types¶

Applying constructors¶

Case analysis¶

Induction¶

Equality of inductive types¶

Helper tactics¶

Generation of induction principles with Scheme¶

Automatic declaration of schemes¶

Combined Scheme¶

Generation of inversion principles with Derive Inversion¶

Examples of dependent destruction / dependent induction¶

A larger example¶

Generation of induction principles with `Scheme`¶

Generation of inversion principles with `Derive` `Inversion`¶

Examples of `dependent destruction` / `dependent induction`¶