Tactics — Coq 8.16.1 documentation

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# Tactics

Tactics specify how to transform the proof state of an incomplete proof to eventually generate a complete proof.

Proofs can be developed in two basic ways: In forward reasoning, the proof begins by proving simple statements that are then combined to prove the theorem statement as the last step of the proof. With forward reasoning, for example, the proof of `A /\ B` would begin with proofs of `A` and `B`, which are then used to prove `A /\ B`. Forward reasoning is probably the most common approach in human-generated proofs.

In backward reasoning, the proof begins with the theorem statement as the goal, which is then gradually transformed until every subgoal generated along the way has been proven. In this case, the proof of `A /\ B` begins with that formula as the goal. This can be transformed into two subgoals, `A` and `B`, followed by the proofs of `A` and `B`. Coq and its tactics use backward reasoning.

A tactic may fully prove a goal, in which case the goal is removed from the proof state. More commonly, a tactic replaces a goal with one or more subgoals. (We say that a tactic reduces a goal to its subgoals.)

Most tactics require specific elements or preconditions to reduce a goal; they display error messages if they can't be applied to the goal. A few tactics, such as `auto`, don't fail even if the proof state is unchanged.

Goals are identified by number. The current goal is number 1. Tactics are applied to the current goal by default. (The default can be changed with the `Default Goal Selector` option.) They can be applied to another goal or to multiple goals with a goal selector such as `2: tactic`.

This chapter describes many of the most common built-in tactics. Built-in tactics can be combined to form tactic expressions, which are described in the Ltac chapter. Since tactic expressions can be used anywhere that a built-in tactic can be used, "tactic" may refer to both built-in tactics and tactic expressions.

## Common elements of tactics

### Reserved keywords

The tactics described in this chapter reserve the following keywords:

    by using

Thus, these keywords cannot be used as identifiers. It also declares the following character sequences as tokens:

    ** [= |-

### Invocation of tactics

A tactic is applied as an ordinary command. It may be preceded by a goal selector (see Section Goal selectors). If no selector is specified, the default selector is used.

> tactic_invocation`::=`toplevel_selector : tactic.`````|`tactic.``

Option Default Goal Selector "toplevel_selector"

  This option controls the default selector, used when no selector is specified when applying a tactic. The initial value is 1, hence the tactics are, by default, applied to the first goal.

  Using value `all` will make it so that tactics are, by default, applied to every goal simultaneously. Then, to apply a tactic tac to the first goal only, you can write `1:tac`.

  Using value `!` enforces that all tactics are used either on a single focused goal or with a local selector (’’strict focusing mode’’).

  Although other selectors are available, only `all`, `!` or a single natural number are valid default goal selectors.

### Bindings

Tactics that take a term as an argument may also accept `bindings` to instantiate some parameters of the term by name or position. The general form of a term with `bindings` is `termtac with bindings` where `bindings` can take two different forms:

> bindings`::=`( ident​natural := term )+`````|`one_term+``

- In the first form, if an `ident` is specified, it must be bound in the type of `term` and provides the tactic with an instance for the parameter of this name. If a `natural` is specified, it refers to the `n`-th non-dependent premise of `termtac`.

  Error No such binder.

- In the second form, the interpretation of the `one_term`s depend on which tactic they appear in. For `induction`, `destruct`, `elim` and `case`, the `one_term`s provide instances for all the dependent products in the type of `termtac` while in the case of `apply`, or of `constructor` and its variants, only instances for the dependent products that are not bound in the conclusion of `termtac` are required.

  Error Not the right number of missing arguments.

### Intro patterns

Intro patterns let you specify the name to assign to variables and hypotheses introduced by tactics. They also let you split an introduced hypothesis into multiple hypotheses or subgoals. Common tactics that accept intro patterns include `assert`, `intros` and `destruct`.

intropattern`::=`*`````|`**`````|`simple_intropattern`` simple_intropattern`::=`simple_intropattern_closed % term0*``simple_intropattern_closed`::=`naming_intropattern`````|`_`````|`or_and_intropattern`````|`equality_intropattern`` naming_intropattern`::=`ident`````|`?`````|`?ident`` or_and_intropattern`::=`[ intropattern**| ]`````|`( simple_intropattern*, )`````|`( simple_intropattern & simple_intropattern+& )``equality_intropattern`::=`->`````|`<-`````|`[= intropattern* ]``

Note that the intro pattern syntax varies between tactics. Most tactics use `simple_intropattern` in the grammar. `destruct`, `edestruct`, `induction`, `einduction`, `case`, `ecase` and the various `inversion` tactics use `or_and_intropattern`, while `intros` and `eintros` use `intropattern*`. The `eqn:` construct in various tactics uses `naming_intropattern`.

Naming patterns

Use these elementary patterns to specify a name:

- `ident` — use the specified name

- `?` — let Coq generate a fresh name

- `?ident` — generate a name that begins with `ident`

- `_` — discard the matched part (unless it is required for another hypothesis)

- if a disjunction pattern omits a name, such as `[|H2]`, Coq will choose a name

Splitting patterns

The most common splitting patterns are:

- split a hypothesis in the form `A /\ B` into two hypotheses `H1: A` and `H2: B` using the pattern `(H1 & H2)` or `(H1, H2)` or `[H1 H2]`. Example. This also works on `A <-> B`, which is just a notation representing `(A -> B) /\ (B -> A)`.

- split a hypothesis in the form `A \/ B` into two subgoals using the pattern `[H1|H2]`. The first subgoal will have the hypothesis `H1: A` and the second subgoal will have the hypothesis `H2: B`. Example

- split a hypothesis in either of the forms `A /\ B` or `A \/ B` using the pattern `[]`.

Patterns can be nested: `[[Ha|Hb] H]` can be used to split `(A \/ B) /\ C`.

Note that there is no equivalent to intro patterns for goals. For a goal `A /\ B`, use the `split` tactic to replace the current goal with subgoals `A` and `B`. For a goal `A \/ B`, use `left` to replace the current goal with `A`, or `right` to replace the current goal with `B`.

- `( simple_intropattern+,` ) — matches a product over an inductive type with a single constructor. If the number of patterns equals the number of constructor arguments, then it applies the patterns only to the arguments, and `( simple_intropattern+, )` is equivalent to `[simple_intropattern+]`. If the number of patterns equals the number of constructor arguments plus the number of `let-ins`, the patterns are applied to the arguments and `let-in` variables.

- `( simple_intropattern+& )` — matches a right-hand nested term that consists of one or more nested binary inductive types such as `a1 OP1 a2 OP2 ...` (where the `OPn` are right-associative). (If the `OPn` are left-associative, additional parentheses will be needed to make the term right-hand nested, such as `a1 OP1 (a2 OP2 ...)`.) The splitting pattern can have more than 2 names, for example `(H1 & H2 & H3)` matches `A /\ B /\ C`. The inductive types must have a single constructor with two parameters. Example

- `[ intropattern*+| ]` — splits an inductive type that has multiple constructors such as `A \/ B` into multiple subgoals. The number of `intropattern`s must be the same as the number of constructors for the matched part.

- `[ intropattern+ ]` — splits an inductive type that has a single constructor with multiple parameters such as `A /\ B` into multiple hypotheses. Use `[H1 [H2 H3]]` to match `A /\ B /\ C`.

- `[]` — splits an inductive type: If the inductive type has multiple constructors, such as `A \/ B`, create one subgoal for each constructor. If the inductive type has a single constructor with multiple parameters, such as `A /\ B`, split it into multiple hypotheses.

Equality patterns

These patterns can be used when the hypothesis is an equality:

- `->` — replaces the right-hand side of the hypothesis with the left-hand side of the hypothesis in the conclusion of the goal; the hypothesis is cleared; if the left-hand side of the hypothesis is a variable, it is substituted everywhere in the context and the variable is removed. Example

- `<-` — similar to `->`, but replaces the left-hand side of the hypothesis with the right-hand side of the hypothesis.

- `[= intropattern*, ]` — If the product is over an equality type, applies either `injection` or `discriminate`. If `injection` is applicable, the intropattern is used on the hypotheses generated by `injection`. If the number of patterns is smaller than the number of hypotheses generated, the pattern `?` is used to complete the list. Example

Other patterns

- `*` — introduces one or more quantified variables from the result until there are no more quantified variables. Example

- `**` — introduces one or more quantified variables or hypotheses from the result until there are no more quantified variables or implications (`->`). `intros **` is equivalent to `intros`. Example

- `simple_intropattern_closed % term*` — first applies each of the terms with the `apply … in` tactic on the hypothesis to be introduced, then it uses `simple_intropattern_closed`. Example

Note

`A \/ B` and `A /\ B` use infix notation to refer to the inductive types `or` and `and`. `or` has multiple constructors (`or_introl` and `or_intror`), while `and` has a single constructor (`conj`) with multiple parameters (`A` and `B`). These are defined in `theories/Init/Logic.v`. The "where" clauses define the infix notation for "or" and "and".

Inductive or (A B: Prop) : Prop := | or_introl : A -> A \/ B | or_intror : B -> A \/ B where "A \/ B" := (or A B) : type_scope. Inductive and (A B: Prop) : Prop := conj : A -> B -> A /\ B where "A /\ B" := (and A B) : type_scope.

Note

`intros p+` is not always equivalent to `intros p; ... ; intros p` if some of the `p` are `_`. In the first form, all erasures are done at once, while they're done sequentially for each tactic in the second form. If the second matched term depends on the first matched term and the pattern for both is `_` (i.e., both will be erased), the first `intros` in the second form will fail because the second matched term still has the dependency on the first.

Examples:

> Example: intro pattern for /\

> Goal forall (A: Prop) (B: Prop), (A /\ B) -> True.
>   1 goal ============================ forall A B : Prop, A /\ B -> True
> intros.
>   1 goal A, B : Prop H : A /\ B ============================ True
> destruct H as (HA & HB).
>   1 goal A, B : Prop HA : A HB : B ============================ True

> Example: intro pattern for \/

> Goal forall (A: Prop) (B: Prop), (A \/ B) -> True.
>   1 goal ============================ forall A B : Prop, A \/ B -> True
> intros.
>   1 goal A, B : Prop H : A \/ B ============================ True
> destruct H as [HA|HB]. all: swap 1 2.
>   2 goals A, B : Prop HA : A ============================ True goal 2 is: True 2 goals A, B : Prop HB : B ============================ True goal 2 is: True

> Example: -> intro pattern

> Goal forall (x: nat) (y: nat) (z: nat), (x = y) -> (y = z) -> (x = z).
>   1 goal ============================ forall x y z : nat, x = y -> y = z -> x = z
> intros * H.
>   1 goal x, y, z : nat H : x = y ============================ y = z -> x = z
> intros ->.
>   1 goal x, z : nat H : x = z ============================ x = z

> Example: [=] intro pattern

> The first `intros [=]` uses `injection` to strip `(S ...)` from both sides of the matched equality. The second uses `discriminate` on the contradiction `1 = 2` (internally represented as `(S O) = (S (S O))`) to complete the goal.

> Goal forall (n m: nat), (S n) = (S m) -> (S O)=(S (S O)) -> False.
>   1 goal ============================ forall n m : nat, S n = S m -> 1 = 2 -> False
> intros *.
>   1 goal n, m : nat ============================ S n = S m -> 1 = 2 -> False
> intros [= H].
>   1 goal n, m : nat H : n = m ============================ 1 = 2 -> False
> intros [=].
>   No more goals.

> Example: (A & B & ...) intro pattern

> Parameters (A : Prop) (B: nat -> Prop) (C: Prop).
>   A is declared B is declared C is declared
> Goal A /\ (exists x: nat, B x /\ C) -> True.
>   1 goal ============================ A /\ (exists x : nat, B x /\ C) -> True
> intros (a & x & b & c).
>   1 goal a : A x : nat b : B x c : C ============================ True

> Example: * intro pattern

> Goal forall (A: Prop) (B: Prop), A -> B.
>   1 goal ============================ forall A B : Prop, A -> B
> intros *.
>   1 goal A, B : Prop ============================ A -> B

> Example: ** pattern ("intros **" is equivalent to "intros")

> Goal forall (A: Prop) (B: Prop), A -> B.
>   1 goal ============================ forall A B : Prop, A -> B
> intros **.
>   1 goal A, B : Prop H : A ============================ B

> Example: compound intro pattern

> Goal forall A B C: Prop, A \/ B /\ C -> (A -> C) -> C.
>   1 goal ============================ forall A B C : Prop, A \/ B /\ C -> (A -> C) -> C
> intros * [a | (_,c)] f.
>   2 goals A, B, C : Prop a : A f : A -> C ============================ C goal 2 is: C
> all: swap 1 2.
>   2 goals A, B, C : Prop c : C f : A -> C ============================ C goal 2 is: C

> Example: combined intro pattern using [=] -> and %

> Require Import Coq.Lists.List.
> Section IntroPatterns.
> Variables (A : Type) (xs ys : list A).
>   A is declared xs is declared ys is declared
> Example ThreeIntroPatternsCombined : S (length ys) = 1 -> xs ++ ys = xs.
>   1 goal A : Type xs, ys : list A ============================ S (length ys) = 1 -> xs ++ ys = xs
> intros [=->%length_zero_iff_nil].
>   1 goal A : Type xs : list A ============================ xs ++ nil = xs

> - `intros` would add `H : S (length ys) = 1`

> - `intros [=]` would additionally apply `injection` to `H` to yield `H0 : length ys = 0`

> - `intros [=->%length_zero_iff_nil]` applies the theorem, making H the equality `l=nil`, which is then applied as for `->`.

> Theorem length_zero_iff_nil (l : list A): length l = 0 <-> l=nil.

> The example is based on Tej Chajed's coq-tricks

### Occurrence clauses

An occurrence is a subterm of a goal or hypothesis that matches a pattern provided by a tactic. Occurrence clauses select a subset of the ocurrences in a goal and/or in one or more of its hypotheses.

> occurrences`::=`at occs_nums`````|`in goal_occurrences`` simple_occurrences`::=`occurrences`` occs_nums`::=`-? nat_or_var+``nat_or_var`::=`natural​ident`` goal_occurrences`::=`hyp_occs+, |- concl_occs??`````|`* |- concl_occs?`````|`|- concl_occs?`````|`concl_occs?``hyp_occs`::=`hypident at occs_nums?``hypident`::=`ident`````|`( type of ident )`````|`( value of ident )``concl_occs`::=`* at occs_nums?``
> `occurrences`

>   The first form of `occurrences` selects occurrences in the conclusion of the goal. The second form can select occurrences in the goal conclusion and in one or more hypotheses.

> `simple_occurrences`

>   A semantically restricted form of `occurrences` that doesn't allow the `at` clause anywhere within it.

> `-? nat_or_var+`

>   Selects the specified occurrences within a single goal or hypothesis. Occurrences are numbered starting with 1 following a depth-first traversal of the term's expression, including occurrences in implicit arguments and coercions that are not displayed by default. (Set the `Printing All` flag to show those in the printed term.)

>   For example, when matching the pattern `_ + _` in the term `(a + b) + c`, occurrence 1 is `(...) + c` and occurrence 2 is `(a + b)`. When matching that pattern with term `a + (b + c)`, occurrence 1 is `a + (...)` and occurrence 2 is `b + c`.

>   Specifying `-` includes all occurrences except the ones listed.

> `hyp_occs*, |- concl_occs??`

>   Selects occurrences in the specified hypotheses and the specified occurrences in the conclusion.

> `* |- concl_occs?`

>   Selects all occurrences in all hypotheses and the specified occurrences in the conclusion.

> `|- concl_occs?`

>   Selects the specified occurrences in the conclusion.

> `goal_occurrences ::= concl_occs?`

>   Selects all occurrences in all hypotheses and in the specified occurrences in the conclusion.

> `hypident at occs_nums?`

>   Omiting `occs_nums` selects all occurrences within the hypothesis.

> `hypident ::= ident`

>   Selects the hypothesis named `ident`.

> `( type of ident )`

>   Selects the type part of the named hypothesis (e.g. `: nat`).

> `( value of ident )`

>   Selects the value part of the named hypothesis (e.g. `:= 1`).

> `concl_occs ::= * at occs_nums?`

>   Selects occurrences in the conclusion. '*' by itself selects all occurrences. `occs_nums` selects the specified occurrences.

> Use `in *` to select all occurrences in all hypotheses and the conclusion, which is equivalent to `in * |- *`. Use `* |-` to select all occurrences in all hypotheses.

> When rewriting in multiple hypotheses, they must not appear in the term to rewrite. For instance `rewrite H in H,H'` is an error. If an hypothesis appears only through a hole, it will be removed from that hole's context.

> With `rewrite term in *`, hypotheses on which the dependency cannot be avoided are skipped, for instance `rewrite H in *` skips rewriting in `H`. This is the case even if only one hypothesis ends up rewritten.

> If multiple occurrences are given, such as in `rewrite` `H at 1 2 3`, the tactic must match at least one occurrence in order to succeed. The tactic will fail if no occurrences match. Occurrence numbers that are out of range (e.g. `at 1 3` when there are only 2 occurrences in the hypothesis or conclusion) are ignored.

> Tactics that use occurrence clauses include `set`, `remember`, `induction` and `destruct`.

See also

Managing the local context, Case analysis, Printing constructions in full.

## Applying theorems

Tactic exact term

  This tactic applies to any goal. It gives directly the exact proof term of the goal. Let `T` be our goal, let `p` be a term of type `U` then `exact p` succeeds iff `T` and `U` are convertible (see Conversion rules).

  Error Not an exact proof.
  Variant eexact term.

    This tactic behaves like `exact` but is able to handle terms and goals with existential variables.

Tactic assumption

  This tactic looks in the local context for a hypothesis whose type is convertible to the goal. If it is the case, the subgoal is proved. Otherwise, it fails.

  Error No such assumption.
  Variant eassumption

    This tactic behaves like `assumption` but is able to handle goals with existential variables.

Tactic refine term

  This tactic applies to any goal. It behaves like `exact` with a big difference: the user can leave some holes (denoted by `_` or `(_ : type)`) in the term. `refine` will generate as many subgoals as there are remaining holes in the elaborated term. Any subgoal that occurs in other subgoals is automatically shelved, as if calling `shelve_unifiable`. The produced subgoals (shelved or not) are not candidates for typeclass resolution, even if they have a type-class type as conclusion, letting the user control when and how typeclass resolution is launched on them. This low-level tactic can be useful to advanced users.

  Example

  Inductive Option : Set := | Fail : Option | Ok : bool -> Option.
    Option is defined Option_rect is defined Option_ind is defined Option_rec is defined Option_sind is defined
  Definition get : forall x: Option, x <> Fail -> bool.
    1 goal ============================ forall x : Option, x <> Fail -> bool
  refine (fun x: Option => match x return x <> Fail -> bool with | Fail => _ | Ok b => fun _ => b end).
    1 goal x : Option ============================ Fail <> Fail -> bool
  intros; absurd (Fail = Fail); trivial.
    No more goals.
  Defined.
  Error Invalid argument.

    The tactic `refine` does not know what to do with the term you gave.

  Error Refine passed ill-formed term.

    The term you gave is not a valid proof (not easy to debug in general). This message may also occur in higher-level tactics that call `refine` internally.

  Error Cannot infer a term for this placeholder.

    There is a hole in the term you gave whose type cannot be inferred. Put a cast around it.

  Variant simple refine term

    This tactic behaves like refine, but it does not shelve any subgoal. It does not perform any beta-reduction either.

  Variant notypeclasses refine term

    This tactic behaves like `refine` except it performs type checking without resolution of typeclasses.

  Variant simple notypeclasses refine term

    This tactic behaves like the combination of `simple refine` and `notypeclasses refine`: it performs type checking without resolution of typeclasses, does not perform beta reductions or shelve the subgoals.

  `Debug` `"unification"` enables printing traces of unification steps used during elaboration/typechecking and the `refine` tactic. `"ho-unification"` prints information about higher order heuristics.

Tactic apply term

  This tactic applies to any goal. The argument term is a term well-formed in the local context. The tactic `apply` tries to match the current goal against the conclusion of the type of `term`. If it succeeds, then the tactic returns as many subgoals as the number of non-dependent premises of the type of term. If the conclusion of the type of `term` does not match the goal and the conclusion is an inductive type isomorphic to a tuple type, then each component of the tuple is recursively matched to the goal in the left-to-right order.

  The tactic `apply` relies on first-order unification with dependent types unless the conclusion of the type of `term` is of the form `P (t1 ... tn)` with `P` to be instantiated. In the latter case, the behavior depends on the form of the goal. If the goal is of the form `(fun x => Q) u1 ... un` and the `ti` and `ui` unify, then `P` is taken to be `(fun x => Q)`. Otherwise, `apply` tries to define `P` by abstracting over `t_1 ... t__n` in the goal. See `pattern` to transform the goal so that it gets the form `(fun x => Q) u1 ... un`.

  Error Unable to unify term with term.

    The `apply` tactic failed to match the conclusion of `term` and the current goal. You can help the `apply` tactic by transforming your goal with the `change` or `pattern` tactics.

  Error Unable to find an instance for the variables ident+.

    This occurs when some instantiations of the premises of `term` are not deducible from the unification. This is the case, for instance, when you want to apply a transitivity property. In this case, you have to use one of the variants below:

  Variant apply term with term+

    Provides apply with explicit instantiations for all dependent premises of the type of term that do not occur in the conclusion and consequently cannot be found by unification. Notice that the collection `term+` must be given according to the order of these dependent premises of the type of term.

    Error Not the right number of missing arguments.
  Variant apply term with bindings

    This also provides apply with values for instantiating premises. Here, variables are referred by names and non-dependent products by increasing numbers (see Bindings).

    Flag Apply With Renaming

      When on, this flag causes the names in the `term`'s type to be renamed for uniqueness. By default no renaming is done.

      Deprecated since version 8.15: This flag is provided for compatibility with versions prior to 8.15.

  Variant apply term+,

    This is a shortcut for `apply term1; [.. | ... ; [ .. | apply termn] ... ]`, i.e. for the successive applications of `term`i+1 on the last subgoal generated by `apply termi` , starting from the application of `term1`.

  Variant eapply term

    The tactic `eapply` behaves like `apply` but it does not fail when no instantiations are deducible for some variables in the premises. Rather, it turns these variables into existential variables which are variables still to instantiate (see Existential variables). The instantiation is intended to be found later in the proof.

  Variant rapply term

    The tactic `rapply` behaves like `eapply` but it uses the proof engine of `refine` for dealing with existential variables, holes, and conversion problems. This may result in slightly different behavior regarding which conversion problems are solvable. However, like `apply` but unlike `eapply`, `rapply` will fail if there are any holes which remain in `term` itself after typechecking and typeclass resolution but before unification with the goal. More technically, `term` is first parsed as a `constr` rather than as a `uconstr` or `open_constr` before being applied to the goal. Note that `rapply` prefers to instantiate as many hypotheses of `term` as possible. As a result, if it is possible to apply `term` to arbitrarily many arguments without getting a type error, `rapply` will loop.

    Note that you need to `Require Import Coq.Program.Tactics` to make use of `rapply`.

  Variant simple apply term.

    This behaves like `apply` but it reasons modulo conversion only on subterms that contain no variables to instantiate. For instance, the following example does not succeed because it would require the conversion of `id ?foo` and `O`.

    Example

    Definition id (x : nat) := x.
      id is defined
    Parameter H : forall x y, id x = y.
      H is declared
    Goal O = O.
      1 goal ============================ 0 = 0
    Fail simple apply H.
      The command has indeed failed with message: Unable to unify "id ?M150 = ?M151" with "0 = 0".

    Because it reasons modulo a limited amount of conversion, `simple apply` fails quicker than `apply` and it is then well-suited for uses in user-defined tactics that backtrack often. Moreover, it does not traverse tuples as `apply` does.

  Variant simple? apply term with bindings?+,
  Variant simple? eapply term with bindings?+,

    This summarizes the different syntaxes for `apply` and `eapply`.

  Variant lapply term

    This tactic applies to any goal, say `G`. The argument term has to be well-formed in the current context, its type being reducible to a non-dependent product `A -> B` with `B` possibly containing products. Then it generates two subgoals `B->G` and `A`. Applying `lapply H` (where `H` has type `A->B` and `B` does not start with a product) does the same as giving the sequence `cut B. 2:apply H.` where `cut` is described below.

Example

Assume we have a transitive relation `R` on `nat`:

Parameter R : nat -> nat -> Prop.
  R is declared
Axiom Rtrans : forall x y z: nat, R x y -> R y z -> R x z.
  Rtrans is declared
Parameters n m p : nat.
  n is declared m is declared p is declared
Axiom Rnm : R n m.
  Rnm is declared
Axiom Rmp : R m p.
  Rmp is declared

Consider the goal `(R n p)` provable using the transitivity of `R`:

Goal R n p.
  1 goal ============================ R n p

The direct application of `Rtrans` with `apply` fails because no value for `y` in `Rtrans` is found by `apply`:

apply Rtrans.
  Toplevel input, characters 6-12: > apply Rtrans. > ^^^^^^ Error: Unable to find an instance for the variable y.

A solution is to `apply (Rtrans n m p)` or `(Rtrans n m)`.

apply (Rtrans n m p).
  2 goals ============================ R n m goal 2 is: R m p

Note that `n` can be inferred from the goal, so the following would work too.

apply (Rtrans _ m).
  2 goals ============================ R n m goal 2 is: R m p

More elegantly, `apply Rtrans with (y:=m)` allows only mentioning the unknown m:

apply Rtrans with (y := m).
  2 goals ============================ R n m goal 2 is: R m p

Another solution is to mention the proof of `(R x y)` in `Rtrans`

apply Rtrans with (1 := Rnm).
  1 goal ============================ R m p

... or the proof of `(R y z)`.

apply Rtrans with (2 := Rmp).
  1 goal ============================ R n m

On the opposite, one can use `eapply` which postpones the problem of finding `m`. Then one can apply the hypotheses `Rnm` and `Rmp`. This instantiates the existential variable and completes the proof.

eapply Rtrans.
  2 focused goals (shelved: 1) ============================ R n ?y goal 2 is: R ?y p
apply Rnm.
  1 goal ============================ R m p
apply Rmp.
  No more goals.

Note

When the conclusion of the type of the term to `apply` is an inductive type isomorphic to a tuple type and `apply` looks recursively whether a component of the tuple matches the goal, it excludes components whose statement would result in applying an universal lemma of the form `forall A, ... -> A`. Excluding this kind of lemma can be avoided by setting the following flag:

Tactic apply term in ident

  This tactic applies to any goal. The argument `term` is a term well-formed in the local context and the argument `ident` is an hypothesis of the context. The tactic `apply term in ident` tries to match the conclusion of the type of `ident` against a non-dependent premise of the type of `term`, trying them from right to left. If it succeeds, the statement of hypothesis `ident` is replaced by the conclusion of the type of `term`. The tactic also returns as many subgoals as the number of other non-dependent premises in the type of `term` and of the non-dependent premises of the type of `ident`. If the conclusion of the type of `term` does not match the goal and the conclusion is an inductive type isomorphic to a tuple type, then the tuple is (recursively) decomposed and the first component of the tuple of which a non-dependent premise matches the conclusion of the type of `ident`. Tuples are decomposed in a width-first left-to-right order (for instance if the type of `H1` is `A <-> B` and the type of `H2` is `A` then `apply H1 in H2` transforms the type of `H2` into `B`). The tactic `apply` relies on first-order pattern matching with dependent types.

  Error Statement without assumptions.

    This happens if the type of `term` has no non-dependent premise.

  Error Unable to apply.

    This happens if the conclusion of `ident` does not match any of the non-dependent premises of the type of `term`.

  Variant apply term+, in ident+,

    This applies each `term` in sequence in each hypothesis `ident`.

  Variant apply term with bindings+, in ident+,

    This does the same but uses the bindings to instantiate parameters of `term` (see Bindings).

  Variant eapply term with bindings?+, in ident+,

    This works as `apply … in` but turns unresolved bindings into existential variables, if any, instead of failing.

  Variant apply term with bindings?+, in ident as simple_intropattern?+,

    This works as `apply … in` but applying an associated `simple_intropattern` to each hypothesis `ident` that comes with such clause.

  Variant simple apply term in ident+,

    This behaves like `apply … in` but it reasons modulo conversion only on subterms that contain no variables to instantiate and does not traverse tuples. See the corresponding example.

  Variant simple? apply term with bindings?+, in ident as simple_intropattern?+,
  Variant simple? eapply term with bindings?+, in ident as simple_intropattern?+,

    This summarizes the different syntactic variants of `apply term in ident+,` and `eapply term in ident+,`.

`Debug` `"tactic-unification"` enables printing traces of unification steps in tactic unification. Tactic unification is used in tactics such as `apply` and `rewrite`.

## Managing the local context

Tactic intro

  This tactic applies to a goal that is either a product or starts with a let-binder. If the goal is a product, the tactic implements the "Lam" rule given in Typing rules [1]. If the goal starts with a let-binder, then the tactic implements a mix of the "Let" and "Conv".

  If the current goal is a dependent product `forall x:T, U` (resp `let x:=t in U`) then `intro` puts `x:T` (resp `x:=t`) in the local context. The new subgoal is `U`.

  If the goal is a non-dependent product \(T \rightarrow U\), then it puts in the local context either `Hn:T` (if `T` is of type `Set` or `Prop`) or `Xn:T` (if the type of `T` is `Type`). The optional index `n` is such that `Hn` or `Xn` is a fresh identifier. In both cases, the new subgoal is `U`.

  If the goal is an existential variable, `intro` forces the resolution of the existential variable into a dependent product \(\forall\) `x:?X, ?Y`, puts `x:?X` in the local context and leaves `?Y` as a new subgoal allowed to depend on `x`.

  The tactic `intro` applies the tactic `hnf` until `intro` can be applied or the goal is not head-reducible.

  Error No product even after head-reduction.
  Variant intro ident

    This applies `intro` but forces `ident` to be the name of the introduced hypothesis.

    Error ident is already used.

  Note

  If a name used by intro hides the base name of a global constant then the latter can still be referred to by a qualified name (see Qualified identifiers).

  Variant intros

    This repeats `intro` until it meets the head-constant. It never reduces head-constants and it never fails.

  Variant intros ident+.

    This is equivalent to the composed tactic `intro ident; ... ; intro ident`.

  Variant intros until ident

    This repeats intro until it meets a premise of the goal having the form `(ident : type)` and discharges the variable named `ident` of the current goal.

    Error No such hypothesis in current goal.
  Variant intros until natural

    This repeats `intro` until the `natural`-th non-dependent product.

    Example

    On the subgoal `forall x y : nat, x = y -> y = x` the tactic `intros until 1` is equivalent to `intros x y H`, as `x = y -> y = x` is the first non-dependent product.

    On the subgoal `forall x y z : nat, x = y -> y = x` the tactic `intros until 1` is equivalent to `intros x y z` as the product on `z` can be rewritten as a non-dependent product: `forall x y : nat, nat -> x = y -> y = x`.

    Error No such hypothesis in current goal.

      This happens when `natural` is 0 or is greater than the number of non-dependent products of the goal.

  Variant intro ident1? after ident2
  Variant intro ident1? before ident2
  Variant intro ident1? at top
  Variant intro ident1? at bottom

    These tactics apply `intro ident1?` and move the freshly introduced hypothesis respectively after the hypothesis `ident2`, before the hypothesis `ident2`, at the top of the local context, or at the bottom of the local context. All hypotheses on which the new hypothesis depends are moved too so as to respect the order of dependencies between hypotheses. It is equivalent to `intro ident1?` followed by the appropriate call to `move`, `move … before …`, `move … at top`, or `move … at bottom`.

    Note

    `intro at bottom` is a synonym for `intro` with no argument.

    Error No such hypothesis: ident.

Tactic intros intropattern*

  Introduces one or more variables or hypotheses from the goal by matching the intro patterns. See the description in Intro patterns.

Tactic eintros intropattern*

  Works just like `intros …` except that it creates existential variables for any unresolved variables rather than failing.

Tactic clear ident

  This tactic erases the hypothesis named `ident` in the local context of the current goal. As a consequence, `ident` is no more displayed and no more usable in the proof development.

  Error No such hypothesis.
  Error ident is used in the conclusion.
  Error ident is used in the hypothesis ident.
  Variant clear ident+

    This is equivalent to `clear ident. ... clear ident.`

  Variant clear - ident+

    This variant clears all the hypotheses except the ones depending in the hypotheses named `ident+` and in the goal.

  Variant clear

    This variants clears all the hypotheses except the ones the goal depends on.

  Variant clear dependent ident

    This clears the hypothesis `ident` and all the hypotheses that depend on it.

  Variant clearbody ident+

    This tactic expects `ident+` to be local definitions and clears their respective bodies. In other words, it turns the given definitions into assumptions.

    Error ident is not a local definition.

Tactic revert ident+

  This applies to any goal with variables `ident+`. It moves the hypotheses (possibly defined) to the goal, if this respects dependencies. This tactic is the inverse of `intro`.

  Error No such hypothesis.
  Error ident1 is used in the hypothesis ident2.
  Variant revert dependent ident

    This moves to the goal the hypothesis `ident` and all the hypotheses that depend on it.

Tactic move ident1 after ident2

  This moves the hypothesis named `ident1` in the local context after the hypothesis named `ident2`, where “after” is in reference to the direction of the move. The proof term is not changed.

  If `ident1` comes before `ident2` in the order of dependencies, then all the hypotheses between `ident1` and `ident2` that (possibly indirectly) depend on `ident1` are moved too, and all of them are thus moved after `ident2` in the order of dependencies.

  If `ident1` comes after `ident2` in the order of dependencies, then all the hypotheses between `ident1` and `ident2` that (possibly indirectly) occur in the type of `ident1` are moved too, and all of them are thus moved before `ident2` in the order of dependencies.

  Variant move ident1 before ident2

    This moves `ident1` towards and just before the hypothesis named `ident2`. As for `move`, dependencies over `ident1` (when `ident1` comes before `ident2` in the order of dependencies) or in the type of `ident1` (when `ident1` comes after `ident2` in the order of dependencies) are moved too.

  Variant move ident at top

    This moves `ident` at the top of the local context (at the beginning of the context).

  Variant move ident at bottom

    This moves `ident` at the bottom of the local context (at the end of the context).

  Error No such hypothesis.
  Error Cannot move ident1 after ident2: it occurs in the type of ident2.
  Error Cannot move ident1 after ident2: it depends on ident2.

  Example

  Goal forall x :nat, x = 0 -> forall z y: nat, y=y-> 0=x.
    1 goal ============================ forall x : nat, x = 0 -> nat -> forall y : nat, y = y -> 0 = x
  intros x H z y H0.
    1 goal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
  move x after H0.
    1 goal z, y : nat H0 : y = y x : nat H : x = 0 ============================ 0 = x
  Undo.
    1 goal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
  move x before H0.
    1 goal z, y, x : nat H : x = 0 H0 : y = y ============================ 0 = x
  Undo.
    1 goal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
  move H0 after H.
    1 goal x, y : nat H0 : y = y H : x = 0 z : nat ============================ 0 = x
  Undo.
    1 goal x : nat H : x = 0 z, y : nat H0 : y = y ============================ 0 = x
  move H0 before H.
    1 goal x : nat H : x = 0 y : nat H0 : y = y z : nat ============================ 0 = x

Tactic rename ident1 into ident2

  This renames hypothesis `ident1` into `ident2` in the current context. The name of the hypothesis in the proof-term, however, is left unchanged.

  Variant rename identi into identj+,

    This renames the variables `identi` into `identj` in parallel. In particular, the target identifiers may contain identifiers that exist in the source context, as long as the latter are also renamed by the same tactic.

  Error No such hypothesis.
  Error ident is already used.

Tactic set (ident := term)

  This replaces `term` by `ident` in the conclusion of the current goal and adds the new definition `ident := term` to the local context.

  If `term` has holes (i.e. subexpressions of the form “`_`”), the tactic first checks that all subterms matching the pattern are compatible before doing the replacement using the leftmost subterm matching the pattern.

  Error The variable ident is already defined.
  Variant set (ident := term) in goal_occurrences

    This notation allows specifying which occurrences of `term` have to be substituted in the context. The `in goal_occurrences` clause is an occurrence clause whose syntax and behavior are described in goal occurrences.

  Variant set (ident binder* := term) in goal_occurrences?

    This is equivalent to `set (ident := fun binder* => term) in goal_occurrences?`.

  Variant set term in goal_occurrences?

    This behaves as `set (ident := term) in goal_occurrences?` but `ident` is generated by Coq.

  Variant eset (ident binder* := term) in goal_occurrences?
  Variant eset term in goal_occurrences?

    While the different variants of `set` expect that no existential variables are generated by the tactic, `eset` removes this constraint. In practice, this is relevant only when `eset` is used as a synonym of `epose`, i.e. when the `term` does not occur in the goal.

Tactic remember term as ident1 eqn: naming_intropattern?

  This behaves as `set (ident := term) in *`, using a logical (Leibniz’s) equality instead of a local definition. Use `naming_intropattern` to name or split up the new equation.

  Variant remember term as ident1 eqn: naming_intropattern? in goal_occurrences

    This is a more general form of `remember` that remembers the occurrences of `term` specified by an occurrence set.

  Variant eremember term as ident1 eqn: naming_intropattern? in goal_occurrences?

    While the different variants of `remember` expect that no existential variables are generated by the tactic, `eremember` removes this constraint.

Tactic pose (ident := term)

  This adds the local definition `ident := term` to the current context without performing any replacement in the goal or in the hypotheses. It is equivalent to `set (ident := term) in |-`.

  Variant pose (ident binder* := term)

    This is equivalent to `pose (ident := fun binder* => term)`.

  Variant pose term

    This behaves as `pose (ident := term)` but `ident` is generated by Coq.

  Variant epose (ident binder* := term)
  Variant epose term

    While the different variants of `pose` expect that no existential variables are generated by the tactic, `epose` removes this constraint.

Tactic decompose [qualid+] term

  This tactic recursively decomposes a complex proposition in order to obtain atomic ones.

  Example

  Goal forall A B C: Prop, A /\ B /\ C \/ B /\ C \/ C /\ A -> C.
    1 goal ============================ forall A B C : Prop, A /\ B /\ C \/ B /\ C \/ C /\ A -> C
  intros A B C H; decompose [and or] H.
    3 goals A, B, C : Prop H : A /\ B /\ C \/ B /\ C \/ C /\ A H1 : A H0 : B H3 : C ============================ C goal 2 is: C goal 3 is: C
  all: assumption.
    No more goals.
  Qed.

  Note

  `decompose` does not work on right-hand sides of implications or products.

  Variant decompose sum term

    This decomposes sum types (like `or`).

  Variant decompose record term

    This decomposes record types (inductive types with one constructor, like `and` and `exists` and those defined with the `Record` command.

## Controlling the proof flow

Tactic assert (ident : type)

  This tactic applies to any goal. `assert (H : U)` adds a new hypothesis of name `H` asserting `U` to the current goal and opens a new subgoal `U` [2]. The subgoal `U` comes first in the list of subgoals remaining to prove.

  Error Not a proposition or a type.

    Arises when the argument `type` is neither of type `Prop`, `Set` nor `Type`.

  Variant assert type

    This behaves as `assert (ident : type)` but `ident` is generated by Coq.

  Variant assert type by tactic

    This tactic behaves like `assert` but applies tactic to solve the subgoals generated by assert.

    Error Proof is not complete.
  Variant assert type as simple_intropattern

    If `simple_intropattern` is an intro pattern (see Intro patterns), the hypothesis is named after this introduction pattern (in particular, if `simple_intropattern` is `ident`, the tactic behaves like `assert (ident : type)`). If `simple_intropattern` is an action introduction pattern, the tactic behaves like `assert type` followed by the action done by this introduction pattern.

  Variant assert type as simple_intropattern by tactic

    This combines the two previous variants of `assert`.

  Variant assert (ident := term)

    This behaves as `assert (ident : type) by exact term` where `type` is the type of `term`. This is equivalent to using `pose proof`. If the head of term is `ident`, the tactic behaves as `specialize`.

    Error Variable ident is already declared.

Variant eassert type as simple_intropattern by tactic

  While the different variants of `assert` expect that no existential variables are generated by the tactic, `eassert` removes this constraint. This lets you avoid specifying the asserted statement completely before starting to prove it.

Variant pose proof term as simple_intropattern?

  This tactic behaves like `assert type as simple_intropattern? by exact term` where `type` is the type of `term`. In particular, `pose proof term as ident` behaves as `assert (ident := term)` and `pose proof term as simple_intropattern` is the same as applying the `simple_intropattern` to `term`.

Variant epose proof term as simple_intropattern?

  While `pose proof` expects that no existential variables are generated by the tactic, `epose proof` removes this constraint.

Variant pose proof (ident := term)

  This is an alternative syntax for `assert (ident := term)` and `pose proof term as ident`, following the model of `pose (ident := term)` but dropping the value of `ident`.

Variant epose proof (ident := term)

  This is an alternative syntax for `eassert (ident := term)` and `epose proof term as ident`, following the model of `epose (ident := term)` but dropping the value of `ident`.

Variant enough (ident : type)

  This adds a new hypothesis of name `ident` asserting `type` to the goal the tactic `enough` is applied to. A new subgoal stating `type` is inserted after the initial goal rather than before it as `assert` would do.

Variant enough type

  This behaves like `enough (ident : type)` with the name `ident` of the hypothesis generated by Coq.

Variant enough type as simple_intropattern

  This behaves like `enough type` using `simple_intropattern` to name or destruct the new hypothesis.

Variant enough (ident : type) by tactic
Variant enough type as simple_intropattern? by tactic

  This behaves as above but with `tactic` expected to solve the initial goal after the extra assumption `type` is added and possibly destructed. If the `as simple_intropattern` clause generates more than one subgoal, `tactic` is applied to all of them.

Variant eenough type as simple_intropattern? by tactic?
Variant eenough (ident : type) by tactic?

  While the different variants of `enough` expect that no existential variables are generated by the tactic, `eenough` removes this constraint.

Variant cut type

  This tactic applies to any goal. It implements the non-dependent case of the “App” rule given in Typing rules. (This is Modus Ponens inference rule.) `cut U` transforms the current goal `T` into the two following subgoals: `U -> T` and `U`. The subgoal `U -> T` comes first in the list of remaining subgoal to prove.

Variant specialize (ident term*) as simple_intropattern?
Variant specialize ident with bindings as simple_intropattern?

  This tactic works on local hypothesis `ident`. The premises of this hypothesis (either universal quantifications or non-dependent implications) are instantiated by concrete terms coming either from arguments `term*` or from Bindings. In the first form the application to `term*` can be partial. The first form is equivalent to `assert (ident := ident term*)`. In the second form, instantiation elements can also be partial. In this case the uninstantiated arguments are inferred by unification if possible or left quantified in the hypothesis otherwise. With the `as` clause, the local hypothesis `ident` is left unchanged and instead, the modified hypothesis is introduced as specified by the `simple_intropattern`. The name `ident` can also refer to a global lemma or hypothesis. In this case, for compatibility reasons, the behavior of `specialize` is close to that of `generalize`: the instantiated statement becomes an additional premise of the goal. The `as` clause is especially useful in this case to immediately introduce the instantiated statement as a local hypothesis.

  Error ident is used in hypothesis ident.
  Error ident is used in conclusion.

Tactic generalize term

  This tactic applies to any goal. It generalizes the conclusion with respect to some term.

Example

Goal forall x y: nat, 0 <= x + y + y.
  1 goal ============================ forall x y : nat, 0 <= x + y + y
Proof. intros *.
  1 goal x, y : nat ============================ 0 <= x + y + y

Show.
  1 goal x, y : nat ============================ 0 <= x + y + y
generalize (x + y + y).
  1 goal x, y : nat ============================ forall n : nat, 0 <= n

If the goal is `G` and `t` is a subterm of type `T` in the goal, then `generalize t` replaces the goal by `forall (x:T), G′` where `G′` is obtained from `G` by replacing all occurrences of `t` by `x`. The name of the variable (here `n`) is chosen based on `T`.

Variant generalize term+

  This is equivalent to `generalize term; ... ; generalize term`. Note that the sequence of term i 's are processed from n to 1.

Variant generalize term at natural+

  This is equivalent to `generalize term` but it generalizes only over the specified occurrences of `term` (counting from left to right on the expression printed using the `Printing All` flag).

Variant generalize term as ident

  This is equivalent to `generalize term` but it uses `ident` to name the generalized hypothesis.

Variant generalize term at natural+ as ident+,

  This is the most general form of `generalize` that combines the previous behaviors.

Variant generalize dependent term

  This generalizes term but also all hypotheses that depend on `term`. It clears the generalized hypotheses.

Tactic evar (ident : term)

  The `evar` tactic creates a new local definition named `ident` with type `term` in the context. The body of this binding is a fresh existential variable.

Tactic instantiate (ident := term )

  The instantiate tactic refines (see `refine`) an existential variable `ident` with the term `term`. It is equivalent to `only [ident]: refine term` (preferred alternative).

  Note

  To be able to refer to an existential variable by name, the user must have given the name explicitly (see Existential variables).

  Note

  When you are referring to hypotheses which you did not name explicitly, be aware that Coq may make a different decision on how to name the variable in the current goal and in the context of the existential variable. This can lead to surprising behaviors.

Variant instantiate (natural := term)

  This variant selects an existential variable by its position. The `natural` argument is the position of the existential variable from right to left in the conclusion of the goal. (Use one of the variants below to select an existential variable in a hypothesis.) Counting starts at 1 and multiple occurrences of the same existential variable are counted multiple times. Because this variant is not robust to slight changes in the goal, its use is strongly discouraged.

Variant instantiate ( natural := term ) in ident
Variant instantiate ( natural := term ) in ( value of ident )
Variant instantiate ( natural := term ) in ( type of ident )

  These allow to refer respectively to existential variables occurring in a hypothesis or in the body or the type of a local definition (named `ident`).

Variant instantiate

  This tactic behaves functionally as `idtac`.

  Deprecated since version 8.16.

Tactic admit

  This tactic allows temporarily skipping a subgoal so as to progress further in the rest of the proof. A proof containing admitted goals cannot be closed with `Qed` but only with `Admitted`.

Variant give_up

  Synonym of `admit`.

Tactic absurd term

  This tactic applies to any goal. The argument term is any proposition `P` of type `Prop`. This tactic applies False elimination, that is it deduces the current goal from False, and generates as subgoals `∼P` and `P`. It is very useful in proofs by cases, where some cases are impossible. In most cases, `P` or `∼P` is one of the hypotheses of the local context.

Tactic contradiction

  This tactic applies to any goal. The contradiction tactic attempts to find in the current context (after all intros) a hypothesis that is equivalent to an empty inductive type (e.g. `False`), to the negation of a singleton inductive type (e.g. `True` or `x=x`), or two contradictory hypotheses.

  Error No such assumption.

Variant contradiction ident

  The proof of False is searched in the hypothesis named `ident`.

Tactic contradict ident

  This tactic allows manipulating negated hypothesis and goals. The name `ident` should correspond to a hypothesis. With `contradict H`, the current goal and context is transformed in the following way:

  - H:¬A ⊢ B becomes ⊢ A

  - H:¬A ⊢ ¬B becomes H: B ⊢ A

  - H: A ⊢ B becomes ⊢ ¬A

  - H: A ⊢ ¬B becomes H: B ⊢ ¬A

Tactic exfalso

  This tactic implements the “ex falso quodlibet” logical principle: an elimination of False is performed on the current goal, and the user is then required to prove that False is indeed provable in the current context. This tactic is a macro for `elimtype False`.

## Classical tactics

In order to ease the proving process, when the `Classical` module is loaded, a few more tactics are available. Make sure to load the module using the `Require Import` command.

Tactic classical_left
Tactic classical_right

  These tactics are the analog of `left` and `right` but using classical logic. They can only be used for disjunctions. Use `classical_left` to prove the left part of the disjunction with the assumption that the negation of right part holds. Use `classical_right` to prove the right part of the disjunction with the assumption that the negation of left part holds.

## Performance-oriented tactic variants

Tactic exact_no_check term

  For advanced usage. Similar to `exact` `term`, but as an optimization, it skips checking that `term` has the goal's type, relying on the kernel check instead. See `change_no_check` for more explanation.

  Example

  Goal False.
    1 goal ============================ False
  exact_no_check I.
    No more goals.
  Fail Qed.
    The command has indeed failed with message: The term "I" has type "True" while it is expected to have type "False".
  Variant vm_cast_no_check term

    For advanced usage. Similar to `exact_no_check` `term`, but additionally instructs the kernel to use `vm_compute` to compare the goal's type with the `term`'s type.

    Example

    Goal False.
      1 goal ============================ False
    vm_cast_no_check I.
      No more goals.
    Fail Qed.
      The command has indeed failed with message: The term "I" has type "True" while it is expected to have type "False".
  Variant native_cast_no_check term

    for advanced usage. similar to `exact_no_check` `term`, but additionally instructs the kernel to use `native_compute` to compare the goal's type with the `term`'s type.

    Example

    Goal False.
      1 goal ============================ False
    native_cast_no_check I.
      No more goals.
    Fail Qed.
      The command has indeed failed with message: The term "I" has type "True" while it is expected to have type "False".

[1]

Actually, only the second subgoal will be generated since the other one can be automatically checked.

[2]

This corresponds to the cut rule of sequent calculus.

Other versions v: 8.16.1

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