mathlib documentation

itactic: parse a nested "interactive" tactic. That is, parse { tactic }

If the current goal is a Pi/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 target is u.

If the goal is an arrow t → u, then it puts h : t in the local context and the new goal target is u.

If the goal is neither a Pi/forall nor begins with a let binder, the tactic intro applies the tactic whnf until an introduction can be applied or the goal is not head reducible. In the latter case, the tactic fails.

Similar to intro tactic. The tactic intros will keep introducing new hypotheses until the goal target is not a Pi/forall or let binder.

The variant intros h₁ ... hₙ introduces n new hypotheses using the given identifiers to name them.

The tactic introv allows the user to automatically introduce the variables of a theorem and explicitly name the hypotheses involved. The given names are used to name non-dependent hypotheses.

Examples:

example : ∀ a b : nat, a = b → b = a :=
begin
  introv h,
  exact h.symm
end

The state after introv h is

a b : ℕ,
h : a = b
⊢ b = a

example : ∀ a b : nat, a = b → ∀ c, b = c → a = c :=
begin
  introv h₁ h₂,
  exact h₁.trans h₂
end

The state after introv h₁ h₂ is

a b : ℕ,
h₁ : a = b,
c : ℕ,
h₂ : b = c
⊢ a = c

Rename one or more local hypotheses. The renamings are given as follows:

rename x y             -- rename x to y
rename x → y           -- ditto
rename [x y, a b]      -- rename x to y and a to b
rename [x → y, a → b]  -- ditto

Note that if there are multiple hypotheses called x in the context, then rename x y will rename all of them. If you want to rename only one, use dedup first.

The apply tactic tries to match the current goal against the conclusion of the type of term. The argument term should be a term well-formed in the local context of the main goal. If it succeeds, then the tactic returns as many subgoals as the number of premises that have not been fixed by type inference or type class resolution. Non-dependent premises are added before dependent ones.

The apply tactic uses higher-order pattern matching, type class resolution, and first-order unification with dependent types.

Similar to the apply tactic, but does not reorder goals.

Similar to the apply tactic, but only creates subgoals for non-dependent premises that have not been fixed by type inference or type class resolution.

Similar to the apply tactic, but allows the user to provide a apply_cfg configuration object.

Similar to the apply tactic, but uses matching instead of unification. apply_match t is equivalent to apply_with t {unify := ff}

This tactic tries to close the main goal ... ⊢ t by generating a term of type t using type class resolution.

This tactic behaves like exact, but with a big difference: the user can put underscores _ in the expression as placeholders for holes that need to be filled, and refine will generate as many subgoals as there are holes.

Note that some holes may be implicit. The type of each hole must either be synthesized by the system or declared by an explicit type ascription like (_ : nat → Prop).

This tactic looks in the local context for a hypothesis whose type is equal to the goal target. If it finds one, it uses it to prove the goal, and otherwise it fails.

Try to apply assumption to all goals.

change u replaces the target t of the main goal to u provided that t is well formed with respect to the local context of the main goal and t and u are definitionally equal.

change u at h will change a local hypothesis to u.

change t with u at h1 h2 ... will replace t with u in all the supplied hypotheses (or *), or in the goal if no at clause is specified, provided that t and u are definitionally equal.

This tactic provides an exact proof term to solve the main goal. If t is the goal and p is a term of type u then exact p succeeds if and only if t and u can be unified.

Like exact, but takes a list of terms and checks that all goals are discharged after the tactic.

A synonym for exact that allows writing have/suffices/show ..., from ... in tactic mode.

revert h₁ ... hₙ applies to any goal with hypotheses h₁ ... hₙ. It moves the hypotheses and their dependencies to the target of the goal. This tactic is the inverse of intro.

rewrite e applies identity e as a rewrite rule to the target of the main goal. If e is preceded by left arrow (← or <-), the rewrite is applied in the reverse direction. If e is a defined constant, then the equational lemmas associated with e are used. This provides a convenient way to unfold e.

rewrite [e₁, ..., eₙ] applies the given rules sequentially.

rewrite e at l rewrites e at location(s) l, where l is either * or a list of hypotheses in the local context. In the latter case, a turnstile ⊢ or |- can also be used, to signify the target of the goal.

An abbreviation for rewrite.

rewrite followed by assumption.

A variant of rewrite that uses the unifier more aggressively, unfolding semireducible definitions.

An abbreviation for erewrite.

Apply t to the main goal and revert any new hypothesis in the generated goals. If t is a supported tactic or chain of supported tactics (e.g. induction, cases, apply, constructor), the generated goals are also tagged with case tags. You can then use case to focus such tagged goals.

Two typical uses of with_cases:

1. Applying a custom eliminator:

   lemma my_nat_rec :
     ∀ n {P : ℕ → Prop} (zero : P 0) (succ : ∀ n, P n → P (n + 1)), P n := ...
   
   example (n : ℕ) : n = n :=
   begin
     with_cases { apply my_nat_rec n },
     case zero { refl },
     case succ : m ih { refl }
   end
   

2. Enabling the use of case after a chain of case-splitting tactics:

   example (n m : ℕ) : unit :=
   begin
     with_cases { cases n; induction m },
     case nat.zero nat.zero { exact () },
     case nat.zero nat.succ : k { exact () },
     case nat.succ nat.zero : i { exact () },
     case nat.succ nat.succ : k i ih_i { exact () }
   end
   

generalize : e = x replaces all occurrences of e in the target with a new hypothesis x of the same type.

generalize h : e = x in addition registers the hypothesis h : e = x.

Assuming x is a variable in the local context with an inductive type, induction x applies induction on x to the main goal, producing one goal for each constructor of the inductive type, in which the target is replaced by a general instance of that constructor and an inductive hypothesis is added for each recursive argument to the constructor. If the type of an element in the local context depends on x, that element is reverted and reintroduced afterward, so that the inductive hypothesis incorporates that hypothesis as well.

For example, given n : nat and a goal with a hypothesis h : P n and target Q n, induction n produces one goal with hypothesis h : P 0 and target Q 0, and one goal with hypotheses h : P (nat.succ a) and ih₁ : P a → Q a and target Q (nat.succ a). Here the names a and ih₁ ire chosen automatically.

induction e, where e is an expression instead of a variable, generalizes e in the goal, and then performs induction on the resulting variable.

induction e with y₁ ... yₙ, where e is a variable or an expression, specifies that the sequence of names y₁ ... yₙ should be used for the arguments to the constructors and inductive hypotheses, including implicit arguments. If the list does not include enough names for all of the arguments, additional names are generated automatically. If too many names are given, the extra ones are ignored. Underscores can be used in the list, in which case the corresponding names are generated automatically. Note that for long sequences of names, the case tactic provides a more convenient naming mechanism.

induction e using r allows the user to specify the principle of induction that should be used. Here r should be a theorem whose result type must be of the form C t, where C is a bound variable and t is a (possibly empty) sequence of bound variables

induction e generalizing z₁ ... zₙ, where z₁ ... zₙ are variables in the local context, generalizes over z₁ ... zₙ before applying the induction but then introduces them in each goal. In other words, the net effect is that each inductive hypothesis is generalized.

induction h : t will introduce an equality of the form h : t = C x y, asserting that the input term is equal to the current constructor case, to the context.

Focuses on a goal ('case') generated by induction, cases or with_cases.

The goal is selected by giving one or more names which must match exactly one goal. A goal is matched if the given names are a suffix of its goal tag. Additionally, each name in the sequence can be abbreviated to a suffix of the corresponding name in the goal tag. Thus, a goal with tag

nat.zero, list.nil

can be selected with any of these invocations (among others):

case nat.zero list.nil {...}
case nat.zero nil      {...}
case zero     nil      {...}
case          nil      {...}

Additionally, the form

case C : N₀ ... Nₙ {...}

can be used to rename hypotheses introduced by the preceding cases/induction/with_cases, using the names Nᵢ. For example:

example (xs : list ℕ) : xs = xs :=
begin
  induction xs,
  case nil { reflexivity },
  case cons : x xs ih {
    -- x : ℕ, xs : list ℕ, ih : xs = xs
    reflexivity }
end

Note that this renaming functionality only work reliably directly after an induction/cases/with_cases. If you need to perform additional work after an induction or cases (e.g. introduce hypotheses in all goals), use with_cases.

Multiple cases can be handled by the same tactic block with

case [A : N₀ ... Nₙ, B : M₀ ... Mₙ] {...}

Assuming x is a variable in the local context with an inductive type, destruct x splits the main goal, producing one goal for each constructor of the inductive type, in which x is assumed to be a general instance of that constructor. In contrast to cases, the local context is unchanged, i.e. no elements are reverted or introduced.

For example, given n : nat and a goal with a hypothesis h : P n and target Q n, destruct n produces one goal with target n = 0 → Q n, and one goal with target ∀ (a : ℕ), (λ (w : ℕ), n = w → Q n) (nat.succ a). Here the name a is chosen automatically.

Assuming x is a variable in the local context with an inductive type, cases x splits the main goal, producing one goal for each constructor of the inductive type, in which the target is replaced by a general instance of that constructor. If the type of an element in the local context depends on x, that element is reverted and reintroduced afterward, so that the case split affects that hypothesis as well.

For example, given n : nat and a goal with a hypothesis h : P n and target Q n, cases n produces one goal with hypothesis h : P 0 and target Q 0, and one goal with hypothesis h : P (nat.succ a) and target Q (nat.succ a). Here the name a is chosen automatically.

cases e, where e is an expression instead of a variable, generalizes e in the goal, and then cases on the resulting variable.

cases e with y₁ ... yₙ, where e is a variable or an expression, specifies that the sequence of names y₁ ... yₙ should be used for the arguments to the constructors, including implicit arguments. If the list does not include enough names for all of the arguments, additional names are generated automatically. If too many names are given, the extra ones are ignored. Underscores can be used in the list, in which case the corresponding names are generated automatically.

cases h : e, where e is a variable or an expression, performs cases on e as above, but also adds a hypothesis h : e = ... to each hypothesis, where ... is the constructor instance for that particular case.

cases_matching p applies the cases tactic to a hypothesis h : type if type matches the pattern p. cases_matching [p_1, ..., p_n] applies the cases tactic to a hypothesis h : type if type matches one of the given patterns. cases_matching* p more efficient and compact version of focus1 { repeat { cases_matching p } }. It is more efficient because the pattern is compiled once.

Example: The following tactic destructs all conjunctions and disjunctions in the current goal.

cases_matching* [_ ∨ _, _ ∧ _]

Shorthand for cases_matching

cases_type I applies the cases tactic to a hypothesis h : (I ...) cases_type I_1 ... I_n applies the cases tactic to a hypothesis h : (I_1 ...) or ... or h : (I_n ...) cases_type* I is shorthand for focus1 { repeat { cases_type I } } cases_type! I only applies cases if the number of resulting subgoals is <= 1.

Example: The following tactic destructs all conjunctions and disjunctions in the current goal.

cases_type* or and

Tries to solve the current goal using a canonical proof of true, or the reflexivity tactic, or the contradiction tactic.

Closes the main goal using sorry.

Closes the main goal using sorry.

The contradiction tactic attempts to find in the current local context a hypothesis that is equivalent to an empty inductive type (e.g. false), a hypothesis of the form c_1 ... = c_2 ... where c_1 and c_2 are distinct constructors, or two contradictory hypotheses.

iterate { t } repeatedly applies tactic t until t fails. iterate { t } always succeeds.

iterate n { t } applies t n times.

repeat { t } applies t to each goal. If the application succeeds, the tactic is applied recursively to all the generated subgoals until it eventually fails. The recursion stops in a subgoal when the tactic has failed to make progress. The tactic repeat { t } never fails.

try { t } tries to apply tactic t, but succeeds whether or not t succeeds.

A do-nothing tactic that always succeeds.

solve1 { t } applies the tactic t to the main goal and fails if it is not solved.

abstract id { t } tries to use tactic t to solve the main goal. If it succeeds, it abstracts the goal as an independent definition or theorem with name id. If id is omitted, a name is generated automatically.

all_goals { t } applies the tactic t to every goal, and succeeds if each application succeeds.

any_goals { t } applies the tactic t to every goal, and succeeds if at least one application succeeds.

focus { t } temporarily hides all goals other than the first, applies t, and then restores the other goals. It fails if there are no goals.

Assuming the target of the goal is a Pi or a let, assume h : t unifies the type of the binder with t and introduces it with name h, just like intro h. If h is absent, the tactic uses the name this. If t is omitted, it will be inferred.

assume (h₁ : t₁) ... (hₙ : tₙ) introduces multiple hypotheses. Any of the types may be omitted, but the names must be present.

have h : t := p adds the hypothesis h : t to the current goal if p a term of type t. If t is omitted, it will be inferred.

have h : t adds the hypothesis h : t to the current goal and opens a new subgoal with target t. The new subgoal becomes the main goal. If t is omitted, it will be replaced by a fresh metavariable.

If h is omitted, the name this is used.

let h : t := p adds the hypothesis h : t := p to the current goal if p a term of type t. If t is omitted, it will be inferred.

let h : t adds the hypothesis h : t := ?M to the current goal and opens a new subgoal ?M : t. The new subgoal becomes the main goal. If t is omitted, it will be replaced by a fresh metavariable.

If h is omitted, the name this is used.

suffices h : t is the same as have h : t, tactic.swap. In other words, it adds the hypothesis h : t to the current goal and opens a new subgoal with target t.

This tactic displays the current state in the tracing buffer.

trace a displays a in the tracing buffer.

existsi e will instantiate an existential quantifier in the target with e and leave the instantiated body as the new target. More generally, it applies to any inductive type with one constructor and at least two arguments, applying the constructor with e as the first argument and leaving the remaining arguments as goals.

existsi [e₁, ..., eₙ] iteratively does the same for each expression in the list.

This tactic applies to a goal such that its conclusion is an inductive type (say I). It tries to apply each constructor of I until it succeeds.

Similar to constructor, but only non-dependent premises are added as new goals.

Applies the first constructor when the type of the target is an inductive data type with two constructors.

Applies the second constructor when the type of the target is an inductive data type with two constructors.

Applies the constructor when the type of the target is an inductive data type with one constructor.

Replaces the target of the main goal by false.

The injection tactic is based on the fact that constructors of inductive data types are injections. That means that if c is a constructor of an inductive datatype, and if (c t₁) and (c t₂) are two terms that are equal then t₁ and t₂ are equal too.

If q is a proof of a statement of conclusion t₁ = t₂, then injection applies injectivity to derive the equality of all arguments of t₁ and t₂ placed in the same positions. For example, from (a::b) = (c::d) we derive a=c and b=d. To use this tactic t₁ and t₂ should be constructor applications of the same constructor.

Given h : a::b = c::d, the tactic injection h adds two new hypothesis with types a = c and b = d to the main goal. The tactic injection h with h₁ h₂ uses the names h₁ and h₂ to name the new hypotheses.

injections with h₁ ... hₙ iteratively applies injection to hypotheses using the names h₁ ... hₙ.

Decode a list of simp_arg_type into lists for each type.

This is a backwards-compatibility version of decode_simp_arg_list_with_symm. This version fails when an argument of the form simp_arg_type.symm_expr is included, so that simp-like tactics that do not (yet) support backwards rewriting should properly report an error but function normally on other inputs.

Decode a list of simp_arg_type into lists for each type.

This is the newer version of decode_simp_arg_list, and has a new name for backwards compatibility. This version indicates the direction of a simp lemma by including a bool with the pexpr.

The simp tactic uses lemmas and hypotheses to simplify the main goal target or non-dependent hypotheses. It has many variants.

simp simplifies the main goal target using lemmas tagged with the attribute [simp].

simp [h₁ h₂ ... hₙ] simplifies the main goal target using the lemmas tagged with the attribute [simp] and the given hᵢ's, where the hᵢ's are expressions. If hᵢ is preceded by left arrow (← or <-), the simplification is performed in the reverse direction. If an hᵢ is a defined constant f, then the equational lemmas associated with f are used. This provides a convenient way to unfold f.

simp [*] simplifies the main goal target using the lemmas tagged with the attribute [simp] and all hypotheses.

simp * is a shorthand for simp [*].

simp only [h₁ h₂ ... hₙ] is like simp [h₁ h₂ ... hₙ] but does not use [simp] lemmas

simp [-id_1, ... -id_n] simplifies the main goal target using the lemmas tagged with the attribute [simp], but removes the ones named idᵢ.

simp at h₁ h₂ ... hₙ simplifies the non-dependent hypotheses h₁ : T₁ ... hₙ : Tₙ. The tactic fails if the target or another hypothesis depends on one of them. The token ⊢ or |- can be added to the list to include the target.

simp at * simplifies all the hypotheses and the target.

simp * at * simplifies target and all (non-dependent propositional) hypotheses using the other hypotheses.

simp with attr₁ ... attrₙ simplifies the main goal target using the lemmas tagged with any of the attributes [attr₁], ..., [attrₙ] or [simp].

Just construct the simp set and trace it. Used for debugging.

simp_intros h₁ h₂ ... hₙ is similar to intros h₁ h₂ ... hₙ except that each hypothesis is simplified as it is introduced, and each introduced hypothesis is used to simplify later ones and the final target.

As with simp, a list of simplification lemmas can be provided. The modifiers only and with behave as with simp.

dsimp is similar to simp, except that it only uses definitional equalities.

This tactic applies to a goal whose target has the form t ~ u where ~ is a reflexive relation, that is, a relation which has a reflexivity lemma tagged with the attribute [refl]. The tactic checks whether t and u are definitionally equal and then solves the goal.

Shorter name for the tactic reflexivity.

This tactic applies to a goal whose target has the form t ~ u where ~ is a symmetric relation, that is, a relation which has a symmetry lemma tagged with the attribute [symm]. It replaces the target with u ~ t.

This tactic applies to a goal whose target has the form t ~ u where ~ is a transitive relation, that is, a relation which has a transitivity lemma tagged with the attribute [trans].

transitivity s replaces the goal with the two subgoals t ~ s and s ~ u. If s is omitted, then a metavariable is used instead.

Proves a goal with target s = t when s and t are equal up to the associativity and commutativity of their binary operations.

An abbreviation for ac_reflexivity.

Tries to prove the main goal using congruence closure.

Given hypothesis h : x = t or h : t = x, where x is a local constant, subst h substitutes x by t everywhere in the main goal and then clears h.

Apply subst to all hypotheses of the form h : x = t or h : t = x.

clear h₁ ... hₙ tries to clear each hypothesis hᵢ from the local context.

Similar to unfold, but only uses definitional equalities.

Similar to dunfold, but performs a raw delta reduction, rather than using an equation associated with the defined constants.

This tactic unfolds all structure projections.

Given defined constants e₁ ... eₙ, unfold e₁ ... eₙ iteratively unfolds all occurrences in the target of the main goal, using equational lemmas associated with the definitions.

As with simp, the at modifier can be used to specify locations for the unfolding.

Similar to unfold, but does not iterate the unfolding.

If the target of the main goal is an opt_param, assigns the default value.

If the target of the main goal is an auto_param, executes the associated tactic.

Fails if the given tactic succeeds.

Succeeds if the given tactic fails.

guard_target t fails if the target of the main goal is not t. We use this tactic for writing tests.

guard_hyp h : t fails if the hypothesis h does not have type t. We use this tactic for writing tests.

match_target t fails if target does not match pattern t.

by_cases p splits the main goal into two cases, assuming h : p in the first branch, and h : ¬ p in the second branch. You can specify the name of the new hypothesis using the syntax by_cases h : p.

Apply function extensionality and introduce new hypotheses. The tactic funext will keep applying new the funext lemma until the goal target is not reducible to

  |-  ((fun x, ...) = (fun x, ...))

The variant funext h₁ ... hₙ applies funext n times, and uses the given identifiers to name the new hypotheses.

If the target of the main goal is a proposition p, by_contradiction reduces the goal to proving false using the additional hypothesis h : ¬ p. by_contradiction h can be used to name the hypothesis h : ¬ p.

This tactic will attempt to use decidability of p if available, and will otherwise fall back on classical reasoning.

If the target of the main goal is a proposition p, by_contra reduces the goal to proving false using the additional hypothesis h : ¬ p. by_contra h can be used to name the hypothesis h : ¬ p.

This tactic will attempt to use decidability of p if available, and will otherwise fall back on classical reasoning.

Type check the given expression, and trace its type.

Fail if there are unsolved goals.

show t finds the first goal whose target unifies with t. It makes that the main goal, performs the unification, and replaces the target with the unified version of t.

The tactic specialize h a₁ ... aₙ works on local hypothesis h. The premises of this hypothesis, either universal quantifications or non-dependent implications, are instantiated by concrete terms coming either from arguments a₁ ... aₙ. The tactic adds a new hypothesis with the same name h := h a₁ ... aₙ and tries to clear the previous one.

Copy a list of meta definitions in the current namespace to tactic.interactive.

This command is useful when we want to update tactic.interactive without closing the current namespace.

Renames hypotheses with the same name.