Tactics

The tactic language is a special-purpose programming language for constructing proofs, indicated using the keyword by.

#adaptation_note

Adaptation notes are comments that are used to indicate that a piece of code has been changed to accommodate a change in Lean core. They typically require further action/maintenance to be taken in the future.

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Defined in module:: Mathlib.Tactic.AdaptationNote

#check

The #check t tactic elaborates the term t and then pretty prints it with its type as e : ty.

If t is an identifier, then it pretty prints a type declaration form for the global constant t instead. Use #check (t) to pretty print it as an elaborated expression.

Like the #check command, the #check tactic allows stuck typeclass instance problems. These become metavariables in the output.

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Defined in module:: Mathlib.Tactic.Check

#count_heartbeats

Count the heartbeats used by a tactic, e.g.: #count_heartbeats simp.

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Defined in module:: Mathlib.Util.CountHeartbeats

#count_heartbeats!

#count_heartbeats! in tac runs a tactic 10 times, counting the heartbeats used, and logs the range and standard deviation. The tactic #count_heartbeats! n in tac runs it n times instead.

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Defined in module:: Mathlib.Util.CountHeartbeats

#find

The #find tactic finds definitions & lemmas using pattern matching on the type. For instance:

#find _ + _ = _ + _
#find ?n + _ = _ + ?n
#find (_ : Nat) + _ = _ + _
#find Nat → Nat

This is the tactic equivalent to the #find command. There is also the find tactic which looks for lemmas which are applyable against the current goal.

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Defined in module:: Mathlib.Tactic.Find

#leansearch

Search LeanSearch from within Lean. Queries should be a string that ends with a . or ?. This works as a command, as a term and as a tactic as in the following examples. In tactic mode, only valid tactics are displayed.

#leansearch "If a natural number n is less than m, then the successor of n is less than the successor of m."

example := #leansearch "If a natural number n is less than m, then the successor of n is less than the successor of m."

example : 3 ≤ 5 := by
  #leansearch "If a natural number n is less than m, then the successor of n is less than the successor of m."
  sorry

You can modify the LeanSearch URL by setting the LEANSEARCHCLIENT_LEANSEARCH_API_URL environment variable.

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Defined in module:: LeanSearchClient.Syntax

#loogle

Search Loogle from within Lean. This can be used as a command, term or tactic as in the following examples. In the case of a tactic, only valid tactics are displayed.

#loogle List ?a → ?a

example := #loogle List ?a → ?a

example : 3 ≤ 5 := by
  #loogle Nat.succ_le_succ
  sorry

Loogle Usage #

Loogle finds definitions and lemmas in various ways:

By constant: 🔍 Real.sin finds all lemmas whose statement somehow mentions the sine function.

By lemma name substring: 🔍 "differ" finds all lemmas that have "differ" somewhere in their lemma name.

By subexpression: 🔍 _ * (_ ^ _) finds all lemmas whose statements somewhere include a product where the second argument is raised to some power.

The pattern can also be non-linear, as in 🔍 Real.sqrt ?a * Real.sqrt ?a

If the pattern has parameters, they are matched in any order. Both of these will find List.map: 🔍 (?a -> ?b) -> List ?a -> List ?b 🔍 List ?a -> (?a -> ?b) -> List ?b

By main conclusion: 🔍 |- tsum _ = _ * tsum _ finds all lemmas where the conclusion (the subexpression to the right of all → and ∀) has the given shape.

As before, if the pattern has parameters, they are matched against the hypotheses of the lemma in any order; for example, 🔍 |- _ < _ → tsum _ < tsum _ will find tsum_lt_tsum even though the hypothesis f i < g i is not the last.

If you pass more than one such search filter, separated by commas Loogle will return lemmas which match all of them. The search 🔍 Real.sin, "two", tsum, _ * _, _ ^ _, |- _ < _ → _ woould find all lemmas which mention the constants Real.sin and tsum, have "two" as a substring of the lemma name, include a product and a power somewhere in the type, and have a hypothesis of the form _ < _ (if there were any such lemmas). Metavariables (?a) are assigned independently in each filter.

You can modify the Loogle server URL by setting the LEANSEARCHCLIENT_LOOGLE_API_URL environment variable.

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Defined in module:: LeanSearchClient.LoogleSyntax

#loogle

This tactic has no documentation.

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Defined in module:: LeanSearchClient.LoogleSyntax

#search

Search from within Lean, depending on the option leansearchclient.backend (currently only leansearch). Queries should be a string that ends with a . or ?. This works as a command, as a term and as a tactic as in the following examples. In tactic mode, only valid tactics are displayed.

#search "If a natural number n is less than m, then the successor of n is less than the successor of m."

example := #search "If a natural number n is less than m, then the successor of n is less than the successor of m."

example : 3 ≤ 5 := by
  #search "If a natural number n is less than m, then the successor of n is less than the successor of m."
  sorry

In tactic mode, if the query string is not supplied, then [LeanStateSearch](https://premise-search.com) is queried based on the goal state.

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Defined in module:: LeanSearchClient.Syntax

#statesearch

Search LeanStateSearch from within Lean. Your current main goal is sent as query. The revision to search can be set using the statesearch.revision option. The number of results can be set using the statesearch.queries option.

Hint: If you want to modify the query, you need to use the web interface.

set_option statesearch.queries 1
set_option statesearch.revision "v4.16.0"

example : 0 ≤ 1 := by
  #statesearch
  sorry

You can modify the LeanStateSearch URL by setting the LEANSEARCHCLIENT_LEANSTATESEARCH_API_URL environment variable.

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Defined in module:: LeanSearchClient.Syntax

(

(tacs) executes a list of tactics in sequence, without requiring that the goal be closed at the end like · tacs. Like by itself, the tactics can be either separated by newlines or ;.

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Defined in module:: Init.Tactics

.

· tac focuses on the main goal and tries to solve it using tac, or else fails.

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Defined in module:: Init.NotationExtra

<;>

t <;> [t1; t2; ...; tn] focuses on the first goal and applies t, which should result in n subgoals. It then applies each ti to the corresponding goal and collects the resulting subgoals.

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Defined in module:: Batteries.Tactic.SeqFocus

<;>

tac <;> tac' runs tac on the main goal and tac' on each produced goal, concatenating all goals produced by tac'.

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Defined in module:: Init.Tactics

Lean.Parser.Tactic.nestedTactic

This tactic has no documentation.

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Defined in module:: Lean.Parser.Tactic

_

_ in tactic position acts like the done tactic: it fails and gives the list of goals if there are any. It is useful as a placeholder after starting a tactic block such as by _ to make it syntactically correct and show the current goal.

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Defined in module:: Batteries.Tactic.Init

abel

Tactic for evaluating equations in the language of additive, commutative monoids and groups.

abel and its variants work as both tactics and conv tactics.

abel1 fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.
abel_nf rewrites all group expressions into a normal form.
- In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.
- abel_nf (config := cfg) allows for additional configuration:
  - red: the reducibility setting (overridden by !)
  - zetaDelta: if true, local let variables can be unfolded (overridden by !)
  - recursive: if true, abel_nf will also recurse into atoms
abel!, abel1!, abel_nf! will use a more aggressive reducibility setting to identify atoms.

For example:

example [AddCommMonoid α] (a b : α) : a + (b + a) = a + a + b := by abel
example [AddCommGroup α] (a : α) : (3 : ℤ) • a = a + (2 : ℤ) • a := by abel

Future work #

In mathlib 3, abel accepted additional optional arguments:
```
syntax "abel" (&" raw" <|> &" term")? (location)? : tactic
```
It is undecided whether these features should be restored eventually.

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Defined in module:: Mathlib.Tactic.Abel

absurd

Given a proof h of p, absurd h changes the goal to ⊢ ¬ p. If p is a negation ¬q then the goal is changed to ⊢ q instead.

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Defined in module:: Batteries.Tactic.Init

ac_change

ac_change g using n is convert_to g using n followed by ac_rfl. It is useful for rearranging/reassociating e.g. sums:

example (a b c d e f g N : ℕ) : (a + b) + (c + d) + (e + f) + g ≤ N := by
  ac_change a + d + e + f + c + g + b ≤ _
  -- ⊢ a + d + e + f + c + g + b ≤ N

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Defined in module:: Mathlib.Tactic.Convert

ac_nf

ac_nf normalizes equalities up to application of an associative and commutative operator.

ac_nf normalizes all hypotheses and the goal target of the goal.
ac_nf at l normalizes 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.

instance : Std.Associative (α := Nat) (.+.) := ⟨Nat.add_assoc⟩
instance : Std.Commutative (α := Nat) (.+.) := ⟨Nat.add_comm⟩

example (a b c d : Nat) : a + b + c + d = d + (b + c) + a := by
 ac_nf
 -- goal: a + (b + (c + d)) = a + (b + (c + d))

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Defined in module:: Init.Tactics

ac_nf0

Implementation of ac_nf (the full ac_nf calls trivial afterwards).

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Defined in module:: Init.Tactics

ac_rfl

ac_rfl proves equalities up to application of an associative and commutative operator.

instance : Std.Associative (α := Nat) (.+.) := ⟨Nat.add_assoc⟩
instance : Std.Commutative (α := Nat) (.+.) := ⟨Nat.add_comm⟩

example (a b c d : Nat) : a + b + c + d = d + (b + c) + a := by ac_rfl

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Defined in module:: Init.Tactics

admit

admit is a synonym for sorry.

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Defined in module:: Init.Tactics

aesop

aesop <clause>* tries to solve the current goal by applying a set of rules registered with the @[aesop] attribute. See its README for a tutorial and a reference.

The variant aesop? prints the proof it found as a Try this suggestion.

Clauses can be used to customise the behaviour of an Aesop call. Available clauses are:

(add <phase> <priority> <builder> <rule>) adds a rule. <phase> is unsafe, safe or norm. <priority> is a percentage for unsafe rules and an integer for safe and norm rules. <rule> is the name of a declaration or local hypothesis. <builder> is the rule builder used to turn <rule> into an Aesop rule. Example: (add unsafe 50% apply Or.inl).
(erase <rule>) disables a globally registered Aesop rule. Example: (erase Aesop.BuiltinRules.assumption).
(rule_sets := [<ruleset>,*]) enables or disables named sets of rules for this Aesop call. Example: (rule_sets := [-builtin, MyRuleSet]).
(config { <opt> := <value> }) adjusts Aesop's search options. See Aesop.Options.
(simp_config { <opt> := <value> }) adjusts options for Aesop's built-in simp rule. The given options are directly passed to simp. For example, (simp_config := { zeta := false }) makes Aesop use simp (config := { zeta := false }).

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Defined in module:: Aesop.Frontend.Tactic

aesop?

aesop <clause>* tries to solve the current goal by applying a set of rules registered with the @[aesop] attribute. See its README for a tutorial and a reference.

The variant aesop? prints the proof it found as a Try this suggestion.

Clauses can be used to customise the behaviour of an Aesop call. Available clauses are:

(add <phase> <priority> <builder> <rule>) adds a rule. <phase> is unsafe, safe or norm. <priority> is a percentage for unsafe rules and an integer for safe and norm rules. <rule> is the name of a declaration or local hypothesis. <builder> is the rule builder used to turn <rule> into an Aesop rule. Example: (add unsafe 50% apply Or.inl).
(erase <rule>) disables a globally registered Aesop rule. Example: (erase Aesop.BuiltinRules.assumption).
(rule_sets := [<ruleset>,*]) enables or disables named sets of rules for this Aesop call. Example: (rule_sets := [-builtin, MyRuleSet]).
(config { <opt> := <value> }) adjusts Aesop's search options. See Aesop.Options.
(simp_config { <opt> := <value> }) adjusts options for Aesop's built-in simp rule. The given options are directly passed to simp. For example, (simp_config := { zeta := false }) makes Aesop use simp (config := { zeta := false }).

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Defined in module:: Aesop.Frontend.Tactic

aesop_cat

A thin wrapper for aesop which adds the CategoryTheory rule set and allows aesop to look through semireducible definitions when calling intros. This tactic fails when it is unable to solve the goal, making it suitable for use in auto-params.

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Defined in module:: Mathlib.CategoryTheory.Category.Basic

aesop_cat?

We also use aesop_cat? to pass along a Try this suggestion when using aesop_cat

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Defined in module:: Mathlib.CategoryTheory.Category.Basic

aesop_cat_nonterminal

A variant of aesop_cat which does not fail when it is unable to solve the goal. Use this only for exploration! Nonterminal aesop is even worse than nonterminal simp.

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Defined in module:: Mathlib.CategoryTheory.Category.Basic

aesop_graph

A variant of the aesop tactic for use in the graph library. Changes relative to standard aesop:

We use the SimpleGraph rule set in addition to the default rule sets.
We instruct Aesop's intro rule to unfold with default transparency.
We instruct Aesop to fail if it can't fully solve the goal. This allows us to use aesop_graph for auto-params.

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Defined in module:: Mathlib.Combinatorics.SimpleGraph.Basic

aesop_graph?

Use aesop_graph? to pass along a Try this suggestion when using aesop_graph

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Defined in module:: Mathlib.Combinatorics.SimpleGraph.Basic

aesop_graph_nonterminal

A variant of aesop_graph which does not fail if it is unable to solve the goal. Use this only for exploration! Nonterminal Aesop is even worse than nonterminal simp.

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Defined in module:: Mathlib.Combinatorics.SimpleGraph.Basic

aesop_mat

The aesop_mat tactic attempts to prove a set is contained in the ground set of a matroid. It uses a [Matroid] ruleset, and is allowed to fail.

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Defined in module:: Mathlib.Combinatorics.Matroid.Basic

aesop_unfold

This tactic has no documentation.

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Defined in module:: Aesop.Util.Tactic.Unfold

aesop_unfold

This tactic has no documentation.

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Defined in module:: Aesop.Util.Tactic.Unfold

algebraize

Tactic that, given RingHoms, adds the corresponding Algebra and (if possible) IsScalarTower instances, as well as Algebra corresponding to RingHom properties available as hypotheses.

Example: given f : A →+* B and g : B →+* C, and hf : f.FiniteType, algebraize [f, g] will add the instances Algebra A B, Algebra B C, and Algebra.FiniteType A B.

See the algebraize tag for instructions on what properties can be added.

The tactic also comes with a configuration option properties. If set to true (default), the tactic searches through the local context for RingHom properties that can be converted to Algebra properties. The macro algebraize_only calls algebraize -properties, so in other words it only adds Algebra and IsScalarTower instances.

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Defined in module:: Mathlib.Tactic.Algebraize

algebraize_only

Version of algebraize, which only adds Algebra instances and IsScalarTower instances, but does not try to add any instances about any properties tagged with @[algebraize], like for example Finite or IsIntegral.

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Defined in module:: Mathlib.Tactic.Algebraize

all_goals

all_goals tac runs tac on each goal, concatenating the resulting goals. If the tactic fails on any goal, the entire all_goals tactic fails.

and_intros

and_intros applies And.intro until it does not make progress.

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Defined in module:: Init.Tactics

any_goals

any_goals tac applies the tactic tac to every goal, concatenating the resulting goals for successful tactic applications. If the tactic fails on all of the goals, the entire any_goals tactic fails.

This tactic is like all_goals try tac except that it fails if none of the applications of tac succeeds.

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Defined in module:: Init.Tactics

apply

apply e tries to match the current goal against the conclusion of e's type. 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.

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Defined in module:: Init.Tactics

apply

apply t at i uses forward reasoning with t at the hypothesis i. Explicitly, if t : α₁ → ⋯ → αᵢ → ⋯ → αₙ and i has type αᵢ, then this tactic adds metavariables/goals for any terms of αⱼ for j = 1, …, i-1, then replaces the type of i with αᵢ₊₁ → ⋯ → αₙ by applying those metavariables and the original i.

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Defined in module:: Mathlib.Tactic.ApplyAt

apply?

Searches environment for definitions or theorems that can refine the goal using apply with conditions resolved when possible with solve_by_elim.

The optional using clause provides identifiers in the local context that must be used when closing the goal.

Use +grind to enable grind as a fallback discharger for subgoals. Use +try? to enable try? as a fallback discharger for subgoals. Use -star to disable fallback to star-indexed lemmas. Use +all to collect all successful lemmas instead of stopping at the first.

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Defined in module:: Init.Tactics

apply_assumption

apply_assumption looks for an assumption of the form ... → ∀ _, ... → head where head matches the current goal.

You can specify additional rules to apply using apply_assumption [...]. By default apply_assumption will also try rfl, trivial, congrFun, and congrArg. If you don't want these, or don't want to use all hypotheses, use apply_assumption only [...]. You can use apply_assumption [-h] to omit a local hypothesis. You can use apply_assumption using [a₁, ...] to use all lemmas which have been labelled with the attributes aᵢ (these attributes must be created using register_label_attr).

apply_assumption will use consequences of local hypotheses obtained via symm.

If apply_assumption fails, it will call exfalso and try again. Thus if there is an assumption of the form P → ¬ Q, the new tactic state will have two goals, P and Q.

You can pass a further configuration via the syntax apply_rules (config := {...}) lemmas. The options supported are the same as for solve_by_elim (and include all the options for apply).

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Defined in module:: Init.Tactics

apply_ext_theorem

Apply a single extensionality theorem to the current goal.

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Defined in module:: Init.Ext

apply_fun

Apply a function to an equality or inequality in either a local hypothesis or the goal.

If we have h : a = b, then apply_fun f at h will replace this with h : f a = f b.
If we have h : a ≤ b, then apply_fun f at h will replace this with h : f a ≤ f b, and create a subsidiary goal Monotone f. apply_fun will automatically attempt to discharge this subsidiary goal using mono, or an explicit solution can be provided with apply_fun f at h using P, where P : Monotone f.
If we have h : a < b, then apply_fun f at h will replace this with h : f a < f b, and create a subsidiary goal StrictMono f and behaves as in the previous case.
If we have h : a ≠ b, then apply_fun f at h will replace this with h : f a ≠ f b, and create a subsidiary goal Injective f and behaves as in the previous two cases.
If the goal is a ≠ b, apply_fun f will replace this with f a ≠ f b.
If the goal is a = b, apply_fun f will replace this with f a = f b, and create a subsidiary goal injective f. apply_fun will automatically attempt to discharge this subsidiary goal using local hypotheses, or if f is actually an Equiv, or an explicit solution can be provided with apply_fun f using P, where P : Injective f.
If the goal is a ≤ b (or similarly for a < b), and f is actually an OrderIso, apply_fun f will replace the goal with f a ≤ f b. If f is anything else (e.g. just a function, or an Equiv), apply_fun will fail.

Typical usage is:

open Function

example (X Y Z : Type) (f : X → Y) (g : Y → Z) (H : Injective <| g ∘ f) :
    Injective f := by
  intro x x' h
  apply_fun g at h
  exact H h

The function f is handled similarly to how it would be handled by refine in that f can contain placeholders. Named placeholders (like ?a or ?_) will produce new goals.

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Defined in module:: Mathlib.Tactic.ApplyFun

apply_gmonoid_gnpowRec_succ_tac

A tactic to for use as an optional value for GMonoid.gnpow_succ'.

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Defined in module:: Mathlib.Algebra.GradedMonoid

apply_gmonoid_gnpowRec_zero_tac

A tactic to for use as an optional value for GMonoid.gnpow_zero'.

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Defined in module:: Mathlib.Algebra.GradedMonoid

apply_mod_cast

Normalize casts in the goal and the given expression, then apply the expression to the goal.

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Defined in module:: Init.TacticsExtra

apply_rewrite

apply_rewrite [rules] is a shorthand for grewrite +implicationHyp [rules].

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Defined in module:: Mathlib.Tactic.GRewrite.Elab

apply_rfl

The same as rfl, but without trying eq_refl at the end.

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Defined in module:: Init.Tactics

apply_rules

apply_rules [l₁, l₂, ...] tries to solve the main goal by iteratively applying the list of lemmas [l₁, l₂, ...] or by applying a local hypothesis. If apply generates new goals, apply_rules iteratively tries to solve those goals. You can use apply_rules [-h] to omit a local hypothesis.

apply_rules will also use rfl, trivial, congrFun and congrArg. These can be disabled, as can local hypotheses, by using apply_rules only [...].

You can use apply_rules using [a₁, ...] to use all lemmas which have been labelled with the attributes aᵢ (these attributes must be created using register_label_attr).

You can pass a further configuration via the syntax apply_rules (config := {...}). The options supported are the same as for solve_by_elim (and include all the options for apply).

apply_rules will try calling symm on hypotheses and exfalso on the goal as needed. This can be disabled with apply_rules (config := {symm := false, exfalso := false}).

You can bound the iteration depth using the syntax apply_rules (config := {maxDepth := n}).

Unlike solve_by_elim, apply_rules does not perform backtracking, and greedily applies a lemma from the list until it gets stuck.

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Defined in module:: Init.Tactics

apply_rw

apply_rw [rules] is a shorthand for grw +implicationHyp [rules].

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Defined in module:: Mathlib.Tactic.GRewrite.Elab

arith_mult

arith_mult solves goals of the form IsMultiplicative f for f : ArithmeticFunction R by applying lemmas tagged with the user attribute arith_mult.

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Defined in module:: Mathlib.Tactic.ArithMult

arith_mult?

arith_mult solves goals of the form IsMultiplicative f for f : ArithmeticFunction R by applying lemmas tagged with the user attribute arith_mult, and prints out the generated proof term.

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Defined in module:: Mathlib.Tactic.ArithMult

array_get_dec

This tactic, added to the decreasing_trivial toolbox, proves that sizeOf arr[i] < sizeOf arr, which is useful for well founded recursions over a nested inductive like inductive T | mk : Array T → T.

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Defined in module:: Init.Data.Array.Mem

array_mem_dec

This tactic, added to the decreasing_trivial toolbox, proves that sizeOf a < sizeOf arr provided that a ∈ arr which is useful for well founded recursions over a nested inductive like inductive T | mk : Array T → T.

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Defined in module:: Init.Data.Array.Mem

as_aux_lemma

as_aux_lemma => tac does the same as tac, except that it wraps the resulting expression into an auxiliary lemma. In some cases, this significantly reduces the size of expressions because the proof term is not duplicated.

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Defined in module:: Init.Tactics

assumption

assumption tries to solve the main goal using a hypothesis of compatible type, or else fails. Note also the ‹t› term notation, which is a shorthand for show t by assumption.

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Defined in module:: Init.Tactics

assumption'

Try calling assumption on all goals; succeeds if it closes at least one goal.

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Defined in module:: Mathlib.Tactic.Basic

assumption_mod_cast

assumption_mod_cast is a variant of assumption that solves the goal using a hypothesis. Unlike assumption, it first pre-processes the goal and each hypothesis to move casts as far outwards as possible, so it can be used in more situations.

Concretely, it runs norm_cast on the goal. For each local hypothesis h, it also normalizes h with norm_cast and tries to use that to close the goal.

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Defined in module:: Init.Tactics

attempt_all

Helper internal tactic for implementing the tactic try?.

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Defined in module:: Init.Try

attempt_all_par

Helper internal tactic for implementing the tactic try? with parallel execution.

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Defined in module:: Init.Try

aux_group₁

Auxiliary tactic for the group tactic. Calls the simplifier only.

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Defined in module:: Mathlib.Tactic.Group

aux_group₂

Auxiliary tactic for the group tactic. Calls ring_nf to normalize exponents.

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Defined in module:: Mathlib.Tactic.Group

bddDefault

Sets are automatically bounded or cobounded in complete lattices. To use the same statements in complete and conditionally complete lattices but let automation fill automatically the boundedness proofs in complete lattices, we use the tactic bddDefault in the statements, in the form (hA : BddAbove A := by bddDefault).

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Defined in module:: Mathlib.Order.Bounds.Basic

beta_reduce

beta_reduce at loc completely beta reduces the given location. This also exists as a conv-mode tactic.

This means that whenever there is an applied lambda expression such as (fun x => f x) y then the argument is substituted into the lambda expression yielding an expression such as f y.

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Defined in module:: Mathlib.Tactic.DefEqTransformations

bicategory

Use the coherence theorem for bicategories to solve equations in a bicategory, where the two sides only differ by replacing strings of bicategory structural morphisms (that is, associators, unitors, and identities) with different strings of structural morphisms with the same source and target.

That is, bicategory can handle goals of the form a ≫ f ≫ b ≫ g ≫ c = a' ≫ f ≫ b' ≫ g ≫ c' where a = a', b = b', and c = c' can be proved using bicategory_coherence.

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Defined in module:: Mathlib.Tactic.CategoryTheory.Bicategory.Basic

bicategory_coherence

Coherence tactic for bicategories. Use pure_coherence instead, which is a frontend to this one.

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Defined in module:: Mathlib.Tactic.CategoryTheory.BicategoryCoherence

bicategory_coherence

Close the goal of the form η.hom = θ.hom, where η and θ are 2-isomorphisms made up only of associators, unitors, and identities.

example {B : Type} [Bicategory B] {a : B} :
  (λ_ (𝟙 a)).hom = (ρ_ (𝟙 a)).hom := by
  bicategory_coherence

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Defined in module:: Mathlib.Tactic.CategoryTheory.Bicategory.PureCoherence

bicategory_nf

Normalize the both sides of an equality.

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Defined in module:: Mathlib.Tactic.CategoryTheory.Bicategory.Basic

bitwise_assoc_tac

Proving associativity of bitwise operations in general essentially boils down to a huge case distinction, so it is shorter to use this tactic instead of proving it in the general case.

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Defined in module:: Mathlib.Data.Nat.Bitwise

boom

A tactic to prove trivial goals by enumeration.

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Defined in module:: Counterexamples.ZeroDivisorsInAddMonoidAlgebras

borelize

The behaviour of borelize α depends on the existing assumptions on α.

if α is a topological space with instances [MeasurableSpace α] [BorelSpace α], then borelize α replaces the former instance by borel α;
otherwise, borelize α adds instances borel α : MeasurableSpace α and ⟨rfl⟩ : BorelSpace α.

Finally, borelize α β γ runs borelize α; borelize β; borelize γ.

Tags:
Defined in module:: Mathlib.MeasureTheory.Constructions.BorelSpace.Basic

bound

bound tactic for proving inequalities via straightforward recursion on expression structure.

An example use case is

-- Calc example: A weak lower bound for `z ↦ z^2 + c`
lemma le_sqr_add (c z : ℝ) (cz : ‖c‖ ≤ ‖z‖) (z3 : 3 ≤ ‖z‖) :
    2 * ‖z‖ ≤ ‖z^2 + c‖ := by
  calc ‖z^2 + c‖
    _ ≥ ‖z^2‖ - ‖c‖ := by bound
    _ ≥ ‖z^2‖ - ‖z‖ := by  bound
    _ ≥ (‖z‖ - 1) * ‖z‖ := by
      rw [mul_comm, mul_sub_one, ← pow_two, ← norm_pow]
    _ ≥ 2 * ‖z‖ := by bound

bound is built on top of aesop, and uses

Apply lemmas registered via the @[bound] attribute
Forward lemmas registered via the @[bound_forward] attribute
Local hypotheses from the context
Optionally: additional hypotheses provided as bound [h₀, h₁] or similar. These are added to the context as if by have := hᵢ.

The functionality of bound overlaps with positivity and gcongr, but can jump back and forth between 0 ≤ x and x ≤ y-type inequalities. For example, bound proves 0 ≤ c → b ≤ a → 0 ≤ a * c - b * c by turning the goal into b * c ≤ a * c, then using mul_le_mul_of_nonneg_right. bound also contains lemmas for goals of the form 1 ≤ x, 1 < x, x ≤ 1, x < 1. Conversely, gcongr can prove inequalities for more types of relations, supports all positivity functionality, and is likely faster since it is more specialized (not built atop aesop).

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Defined in module:: Mathlib.Tactic.Bound

bv_check

This tactic works just like bv_decide but skips calling a SAT solver by using a proof that is already stored on disk. It is called with the name of an LRAT file in the same directory as the current Lean file:

bv_check "proof.lrat"

Tags:
Defined in module:: Std.Tactic.BVDecide.Syntax

bv_decide

Close fixed-width BitVec and Bool goals by obtaining a proof from an external SAT solver and verifying it inside Lean. The solvable goals are currently limited to

the Lean equivalent of QF_BV
automatically splitting up structures that contain information about BitVec or Bool

example : ∀ (a b : BitVec 64), (a &&& b) + (a ^^^ b) = a ||| b := by
  intros
  bv_decide

If bv_decide encounters an unknown definition it will be treated like an unconstrained BitVec variable. Sometimes this enables solving goals despite not understanding the definition because the precise properties of the definition do not matter in the specific proof.

If bv_decide fails to close a goal it provides a counter-example, containing assignments for all terms that were considered as variables.

In order to avoid calling a SAT solver every time, the proof can be cached with bv_decide?.

If solving your problem relies inherently on using associativity or commutativity, consider enabling the bv.ac_nf option.

Note: bv_decide uses ofReduceBool and thus trusts the correctness of the code generator.

Note: include import Std.Tactic.BVDecide

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Defined in module:: Init.Tactics

bv_decide?

Suggest a proof script for a bv_decide tactic call. Useful for caching LRAT proofs.

Note: include import Std.Tactic.BVDecide

Tags:
Defined in module:: Init.Tactics

bv_normalize

Run the normalization procedure of bv_decide only. Sometimes this is enough to solve basic BitVec goals already.

Note: include import Std.Tactic.BVDecide

Tags:
Defined in module:: Init.Tactics

bv_omega

bv_omega is omega with an additional preprocessor that turns statements about BitVec into statements about Nat. Currently the preprocessor is implemented as try simp only [bitvec_to_nat] at *. bitvec_to_nat is a @[simp] attribute that you can (cautiously) add to more theorems.

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Defined in module:: Init.Tactics

bx

A macro for applying the bounds on x, x_pos and x_lt_pi_div_three.

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Defined in module:: Archive.Imo.Imo2001Q5

by_cases

by_cases (h :)? p splits the main goal into two cases, assuming h : p in the first branch, and h : ¬ p in the second branch.

Tags:
Defined in module:: Init.ByCases

by_cases!

by_cases! h : p runs the by_cases h : p tactic, followed by push_neg at h in the second subgoal. For example,

by_cases! h : a < b creates one goal with hypothesis h : a < b and another with h : b ≤ a.
by_cases! h : a ≠ b creates one goal with hypothesis h : a ≠ b and another with h : a = b.

Tags:
Defined in module:: Mathlib.Tactic.ByCases

by_contra

by_contra h proves ⊢ p by contradiction, introducing a hypothesis h : ¬p and proving False.

If p is a negation ¬q, h : q will be introduced instead of ¬¬q.
If p is decidable, it uses Decidable.byContradiction instead of Classical.byContradiction.
If h is omitted, the introduced variable will be called this.

Tags:
Defined in module:: Batteries.Tactic.Init

by_contra!

If the target of the main goal is a proposition p, by_contra! reduces the goal to proving False using the additional hypothesis this : ¬ p. by_contra! h can be used to name the hypothesis h : ¬ p. The hypothesis ¬ p will be normalized using push_neg. For instance, ¬ a < b will be changed to b ≤ a. by_contra! can be used with rcases patterns. For instance, by_contra! rfl on ⊢ s.Nonempty will substitute the equality s = ∅, and by_contra! ⟨hp, hq⟩ on ⊢ ¬ p ∨ ¬ q will introduce hp : p and hq : q. by_contra! h : q will normalize negations in ¬ p, normalize negations in q, and then check that the two normalized forms are equal. The resulting hypothesis is the pre-normalized form, q. If the name h is not explicitly provided, then this will be used as name. This tactic uses classical reasoning. It is a variant on the tactic by_contra. Examples:

example : 1 < 2 := by
  by_contra! h
  -- h : 2 ≤ 1 ⊢ False

example : 1 < 2 := by
  by_contra! h : ¬ 1 < 2
  -- h : ¬ 1 < 2 ⊢ False

Tags:
Defined in module:: Mathlib.Tactic.ByContra

calc

Step-wise reasoning over transitive relations.

calc
  a = b := pab
  b = c := pbc
  ...
  y = z := pyz

proves a = z from the given step-wise proofs. = can be replaced with any relation implementing the typeclass Trans. Instead of repeating the right- hand sides, subsequent left-hand sides can be replaced with _.

calc
  a = b := pab
  _ = c := pbc
  ...
  _ = z := pyz

It is also possible to write the first relation as <lhs>\n _ = <rhs> := <proof>. This is useful for aligning relation symbols, especially on longer identifiers:

calc abc
  _ = bce := pabce
  _ = cef := pbcef
  ...
  _ = xyz := pwxyz

calc works as a term, as a tactic or as a conv tactic.

See Theorem Proving in Lean 4 for more information.

Tags:
Defined in module:: Init.NotationExtra

calc?

Create a calc proof.

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Defined in module:: Mathlib.Tactic.Widget.Calc

cancel_denoms

cancel_denoms attempts to remove numerals from the denominators of fractions. It works on propositions that are field-valued inequalities.

variable [LinearOrderedField α] (a b c : α)

example (h : a / 5 + b / 4 < c) : 4*a + 5*b < 20*c := by
  cancel_denoms at h
  exact h

example (h : a > 0) : a / 5 > 0 := by
  cancel_denoms
  exact h

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Defined in module:: Mathlib.Tactic.CancelDenoms.Core

case

case _ : t => tac finds the first goal that unifies with t and then solves it using tac or else fails. Like show, it changes the type of the goal to t. The _ can optionally be a case tag, in which case it only looks at goals whose tag would be considered by case (goals with an exact tag match, followed by goals with the tag as a suffix, followed by goals with the tag as a prefix).
case _ n₁ ... nₘ : t => tac additionally names the m most recent hypotheses with inaccessible names to the given names. The names are renamed before matching against t. The _ can optionally be a case tag.
case _ : t := e is short for case _ : t => exact e.
case _ : t₁ | _ : t₂ | ... => tac is equivalent to (case _ : t₁ => tac); (case _ : t₂ => tac); ... but with all matching done on the original list of goals -- each goal is consumed as they are matched, so patterns may repeat or overlap.
case _ : t will make the matched goal be the first goal. case _ : t₁ | _ : t₂ | ... makes the matched goals be the first goals in the given order.
case _ : t := _ and case _ : t := ?m are the same as case _ : t but in the ?m case the goal tag is changed to m. In particular, the goal becomes metavariable ?m.

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Defined in module:: Batteries.Tactic.Case

case

case tag => tac focuses on the goal with case name tag and solves it using tac, or else fails.
case tag x₁ ... xₙ => tac additionally renames the n most recent hypotheses with inaccessible names to the given names.
case tag₁ | tag₂ => tac is equivalent to (case tag₁ => tac); (case tag₂ => tac).

Tags:
Defined in module:: Init.Tactics

case'

case' is similar to the case tag => tac tactic, but does not ensure the goal has been solved after applying tac, nor admits the goal if tac failed. Recall that case closes the goal using sorry when tac fails, and the tactic execution is not interrupted.

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Defined in module:: Init.Tactics

case'

case' _ : t => tac is similar to the case _ : t => tac tactic, but it does not ensure the goal has been solved after applying tac, nor does it admit the goal if tac failed. Recall that case closes the goal using sorry when tac fails, and the tactic execution is not interrupted.

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Defined in module:: Batteries.Tactic.Case

cases

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. cases detects unreachable cases and closes them automatically.

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 and is not accessible. You can use with to provide the variables names for each constructor.

cases e, where e is an expression instead of a variable, generalizes e in the goal, and then cases on the resulting variable.
Given as : List α, cases as with | nil => tac₁ | cons a as' => tac₂, uses tactic tac₁ for the nil case, and tac₂ for the cons case, and a and as' are used as names for the new variables introduced.
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 goal, where ... is the constructor instance for that particular case.

Tags:
Defined in module:: Init.Tactics

cases'

The cases' tactic is similar to the cases tactic in Lean 4 core, but the syntax for giving names is different:

example (h : p ∨ q) : q ∨ p := by
  cases h with
  | inl hp => exact Or.inr hp
  | inr hq => exact Or.inl hq

example (h : p ∨ q) : q ∨ p := by
  cases' h with hp hq
  · exact Or.inr hp
  · exact Or.inl hq

example (h : p ∨ q) : q ∨ p := by
  rcases h with hp | hq
  · exact Or.inr hp
  · exact Or.inl hq

Prefer cases or rcases when possible, because these tactics promote structured proofs.

Tags:
Defined in module:: Mathlib.Tactic.Cases

cases_first_enat

Finds the first ENat in the context and applies the cases tactic to it. Then simplifies expressions involving ⊤ using the enat_to_nat_top simp set.

Tags:
Defined in module:: Mathlib.Tactic.ENatToNat

cases_type

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 · 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

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Defined in module:: Mathlib.Tactic.CasesM

casesm

casesm p applies the cases tactic to a hypothesis h : type if type matches the pattern p.
casesm p_1, ..., p_n applies the cases tactic to a hypothesis h : type if type matches one of the given patterns.
casesm* p is a more efficient and compact version of · repeat casesm p. It is more efficient because the pattern is compiled once.
casesm! p only applies cases if the number of resulting subgoals is <= 1.

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

casesm* _ ∨ _, _ ∧ _

Tags:
Defined in module:: Mathlib.Tactic.CasesM

cat_disch

A tactic for discharging easy category theory goals, widely used as an autoparameter. Currently this defaults to the aesop_cat wrapper around aesop, but by setting the option mathlib.tactic.category.grind to true, it will use the grind tactic instead.

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Defined in module:: Mathlib.CategoryTheory.Category.Basic

cfc_cont_tac

A tactic used to automatically discharge goals relating to the continuous functional calculus, specifically concerning continuity of the functions involved.

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Defined in module:: Mathlib.Tactic.ContinuousFunctionalCalculus

cfc_tac

A tactic used to automatically discharge goals relating to the continuous functional calculus, specifically whether the element satisfies the predicate.

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Defined in module:: Mathlib.Tactic.ContinuousFunctionalCalculus

cfc_zero_tac

A tactic used to automatically discharge goals relating to the non-unital continuous functional calculus, specifically concerning whether f 0 = 0.

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Defined in module:: Mathlib.Tactic.ContinuousFunctionalCalculus

change

change tgt' will change the goal from tgt to tgt', assuming these are definitionally equal.
change t' at h will change hypothesis h : t to have type t', assuming assuming t and t' are definitionally equal.
change a with b will change occurrences of a to b in the goal, assuming a and b are definitionally equal.
change a with b at h similarly changes a to b in the type of hypothesis h.

Tags:
Defined in module:: Init.Tactics

change?

change? term unifies term with the current goal, then suggests explicit change syntax that uses the resulting unified term.

If term is not present, change? suggests the current goal itself. This is useful after tactics which transform the goal while maintaining definitional equality, such as dsimp; those preceding tactic calls can then be deleted.

example : (fun x : Nat => x) 0 = 1 := by
  change? 0 = _  -- `Try this: change 0 = 1`

Tags:
Defined in module:: Mathlib.Tactic.Change

check_compositions

For each composition f ≫ g in the goal, which internally is represented as CategoryStruct.comp C inst X Y Z f g, infer the types of f and g and check whether their sources and targets agree with X, Y, and Z at "instances and reducible" transparency, reporting any discrepancies.

An example:

example (j : J) :
    colimit.ι ((F ⋙ G) ⋙ H) j ≫ (preservesColimitIso (G ⋙ H) F).inv =
      H.map (G.map (colimit.ι F j)) := by

  -- We know which lemma we want to use, and it's even a simp lemma, but `rw`
  -- won't let us apply it
  fail_if_success rw [ι_preservesColimitIso_inv]
  fail_if_success rw [ι_preservesColimitIso_inv (G ⋙ H)]
  fail_if_success simp only [ι_preservesColimitIso_inv]

  -- This would work:
  -- erw [ι_preservesColimitIso_inv (G ⋙ H)]

  -- `check_compositions` checks if the two morphisms we're composing are
  -- composed by abusing defeq, and indeed it tells us that we are abusing
  -- definitional associativity of composition of functors here: it prints
  -- the following.

  -- info: In composition
  --   colimit.ι ((F ⋙ G) ⋙ H) j ≫ (preservesColimitIso (G ⋙ H) F).inv
  -- the source of
  --   (preservesColimitIso (G ⋙ H) F).inv
  -- is
  --   colimit (F ⋙ G ⋙ H)
  -- but should be
  --   colimit ((F ⋙ G) ⋙ H)

  check_compositions

  -- In this case, we can "fix" this by reassociating in the goal, but
  -- usually at this point the right thing to do is to back off and
  -- check how we ended up with a bad goal in the first place.
  dsimp only [Functor.assoc]

  -- This would work now, but it is not needed, because simp works as well
  -- rw [ι_preservesColimitIso_inv]

  simp

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.CheckCompositions

choose

choose a b h h' using hyp takes a hypothesis hyp of the form ∀ (x : X) (y : Y), ∃ (a : A) (b : B), P x y a b ∧ Q x y a b for some P Q : X → Y → A → B → Prop and outputs into context a function a : X → Y → A, b : X → Y → B and two assumptions: h : ∀ (x : X) (y : Y), P x y (a x y) (b x y) and h' : ∀ (x : X) (y : Y), Q x y (a x y) (b x y). It also works with dependent versions.
choose! a b h h' using hyp does the same, except that it will remove dependency of the functions on propositional arguments if possible. For example if Y is a proposition and A and B are nonempty in the above example then we will instead get a : X → A, b : X → B, and the assumptions h : ∀ (x : X) (y : Y), P x y (a x) (b x) and h' : ∀ (x : X) (y : Y), Q x y (a x) (b x).

The using hyp part can be omitted, which will effectively cause choose to start with an intro hyp.

Like intro, the choose tactic supports type annotations to specify the expected type of the introduced variables. This is useful for documentation and for catching mistakes early:

example (h : ∃ n : ℕ, n > 0) : True := by
  choose (n : ℕ) (hn : n > 0) using h
  trivial

If the provided type does not match the actual type, an error is raised.

Examples:

example (h : ∀ n m : ℕ, ∃ i j, m = n + i ∨ m + j = n) : True := by
  choose i j h using h
  guard_hyp i : ℕ → ℕ → ℕ
  guard_hyp j : ℕ → ℕ → ℕ
  guard_hyp h : ∀ (n m : ℕ), m = n + i n m ∨ m + j n m = n
  trivial

example (h : ∀ i : ℕ, i < 7 → ∃ j, i < j ∧ j < i+i) : True := by
  choose! f h h' using h
  guard_hyp f : ℕ → ℕ
  guard_hyp h : ∀ (i : ℕ), i < 7 → i < f i
  guard_hyp h' : ∀ (i : ℕ), i < 7 → f i < i + i
  trivial

Tags:
Defined in module:: Mathlib.Tactic.Choose

classical

classical tacs runs tacs in a scope where Classical.propDecidable is a low priority local instance.

Note that classical is a scoping tactic: it adds the instance only within the scope of the tactic.

Tags:
Defined in module:: Init.Tactics

clean

(Deprecated) clean t is a macro for exact clean% t.

Tags:
Defined in module:: Mathlib.Tactic.Clean

clean_wf

This tactic is used internally by lean before presenting the proof obligations from a well-founded definition to the user via decreasing_by. It is not necessary to use this tactic manually.

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Defined in module:: Init.WFTactics

clear

clear x... removes the given hypotheses, or fails if there are remaining references to a hypothesis.

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Defined in module:: Init.Tactics

clear

Clears all hypotheses it can, except those provided after a minus sign. Example:

  clear * - h₁ h₂

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Defined in module:: Mathlib.Tactic.ClearExcept

clear!

A variant of clear which clears not only the given hypotheses but also any other hypotheses depending on them

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Defined in module:: Mathlib.Tactic.ClearExclamation

clear_

Clear all hypotheses starting with _, like _match and _let_match.

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Defined in module:: Mathlib.Tactic.Clear_

clear_aux_decl

This tactic clears all auxiliary declarations from the context.

Tags:
Defined in module:: Mathlib.Tactic.Basic

clear_value

clear_value x... clears the values of the given local definitions. A local definition x : α := v becomes a hypothesis x : α.
clear_value (h : x = _) adds a hypothesis h : x = v before clearing the value of x. This is short for have h : x = v := rfl; clear_value x. Any value definitionally equal to v can be used in place of _.
clear_value * clears values of all hypotheses that can be cleared. Fails if none can be cleared.

These syntaxes can be combined. For example, clear_value x y * ensures that x and y are cleared while trying to clear all other local definitions, and clear_value (hx : x = _) y * with hx does the same while first adding the hx : x = v hypothesis.

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Defined in module:: Init.Tactics

coherence

Use the coherence theorem for monoidal categories to solve equations in a monoidal equation, where the two sides only differ by replacing strings of monoidal structural morphisms (that is, associators, unitors, and identities) with different strings of structural morphisms with the same source and target.

That is, coherence can handle goals of the form a ≫ f ≫ b ≫ g ≫ c = a' ≫ f ≫ b' ≫ g ≫ c' where a = a', b = b', and c = c' can be proved using pure_coherence.

(If you have very large equations on which coherence is unexpectedly failing, you may need to increase the typeclass search depth, using e.g. set_option synthInstance.maxSize 500.)

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Defined in module:: Mathlib.Tactic.CategoryTheory.Coherence

compareOfLessAndEq_rfl

This attempts to prove that a given instance of compare is equal to compareOfLessAndEq by introducing the arguments and trying the following approaches in order:

seeing if rfl works
seeing if the compare at hand is nonetheless essentially compareOfLessAndEq, but, because of implicit arguments, requires us to unfold the defs and split the ifs in the definition of compareOfLessAndEq
seeing if we can split by cases on the arguments, then see if the defs work themselves out (useful when compare is defined via a match statement, as it is for Bool)

Tags:
Defined in module:: Mathlib.Order.Defs.LinearOrder

compute_degree

compute_degree is a tactic to solve goals of the form

natDegree f = d,
degree f = d,
natDegree f ≤ d (or <),
degree f ≤ d (or <),
coeff f d = r, if d is the degree of f.

The tactic may leave goals of the form d' = d, d' ≤ d, d' < d, or r ≠ 0, where d' in ℕ or WithBot ℕ is the tactic's guess of the degree, and r is the coefficient's guess of the leading coefficient of f.

compute_degree applies norm_num to the left-hand side of all side goals, trying to close them.

The variant compute_degree! first applies compute_degree. Then it uses norm_num on all the remaining goals and tries assumption.

Tags:
Defined in module:: Mathlib.Tactic.ComputeDegree

congr

Apply congruence (recursively) to goals of the form ⊢ f as = f bs and ⊢ f as ≍ f bs. The optional parameter is the depth of the recursive applications. This is useful when congr is too aggressive in breaking down the goal. For example, given ⊢ f (g (x + y)) = f (g (y + x)), congr produces the goals ⊢ x = y and ⊢ y = x, while congr 2 produces the intended ⊢ x + y = y + x.

Tags:
Defined in module:: Init.Tactics

congr

Tags:
Defined in module:: Batteries.Tactic.Congr

congr

Apply congruence (recursively) to goals of the form ⊢ f as = f bs and ⊢ f as ≍ f bs.

congr n controls the depth of the recursive applications. This is useful when congr is too aggressive in breaking down the goal. For example, given ⊢ f (g (x + y)) = f (g (y + x)), congr produces the goals ⊢ x = y and ⊢ y = x, while congr 2 produces the intended ⊢ x + y = y + x.
If, at any point, a subgoal matches a hypothesis then the subgoal will be closed.
You can use congr with p (: n)? to call ext p (: n)? to all subgoals generated by congr. For example, if the goal is ⊢ f '' s = g '' s then congr with x generates the goal x : α ⊢ f x = g x.

Tags:
Defined in module:: Batteries.Tactic.Congr

congr!

Equates pieces of the left-hand side of a goal to corresponding pieces of the right-hand side by recursively applying congruence lemmas. For example, with ⊢ f as = g bs we could get two goals ⊢ f = g and ⊢ as = bs.

Syntax:

congr!
congr! n
congr! with x y z
congr! n with x y z

Here, n is a natural number and x, y, z are rintro patterns (like h, rfl, ⟨x, y⟩, _, -, (h | h), etc.).

The congr! tactic is similar to congr but is more insistent in trying to equate left-hand sides to right-hand sides of goals. Here is a list of things it can try:

If R in ⊢ R x y is a reflexive relation, it will convert the goal to ⊢ x = y if possible. The list of reflexive relations is maintained using the @[refl] attribute. As a special case, ⊢ p ↔ q is converted to ⊢ p = q during congruence processing and then returned to ⊢ p ↔ q form at the end.
If there is a user congruence lemma associated to the goal (for instance, a @[congr]-tagged lemma applying to ⊢ List.map f xs = List.map g ys), then it will use that.
It uses a congruence lemma generator at least as capable as the one used by congr and simp. If there is a subexpression that can be rewritten by simp, then congr! should be able to generate an equality for it.
It can do congruences of pi types using lemmas like implies_congr and pi_congr.
Before applying congruences, it will run the intros tactic automatically. The introduced variables can be given names using a with clause. This helps when congruence lemmas provide additional assumptions in hypotheses.
When there is an equality between functions, so long as at least one is obviously a lambda, we apply funext or Function.hfunext, which allows for congruence of lambda bodies.
It can try to close goals using a few strategies, including checking definitional equality, trying to apply Subsingleton.elim or proof_irrel_heq, and using the assumption tactic.

The optional parameter is the depth of the recursive applications. This is useful when congr! is too aggressive in breaking down the goal. For example, given ⊢ f (g (x + y)) = f (g (y + x)), congr! produces the goals ⊢ x = y and ⊢ y = x, while congr! 2 produces the intended ⊢ x + y = y + x.

The congr! tactic also takes a configuration option, for example

congr! (transparency := .default) 2

This overrides the default, which is to apply congruence lemmas at reducible transparency.

The congr! tactic is aggressive with equating two sides of everything. There is a predefined configuration that uses a different strategy: Try

congr! (config := .unfoldSameFun)

This only allows congruences between functions applications of definitionally equal functions, and it applies congruence lemmas at default transparency (rather than just reducible). This is somewhat like congr.

See Congr!.Config for all options.

Tags:
Defined in module:: Mathlib.Tactic.CongrExclamation

congrm

congrm e is a tactic for proving goals of the form lhs = rhs, lhs ↔ rhs, lhs ≍ rhs, or R lhs rhs when R is a reflexive relation. The expression e is a pattern containing placeholders ?_, and this pattern is matched against lhs and rhs simultaneously. These placeholders generate new goals that state that corresponding subexpressions in lhs and rhs are equal. If the placeholders have names, such as ?m, then the new goals are given tags with those names.

Examples:

example {a b c d : ℕ} :
    Nat.pred a.succ * (d + (c + a.pred)) = Nat.pred b.succ * (b + (c + d.pred)) := by
  congrm Nat.pred (Nat.succ ?h1) * (?h2 + ?h3)
  /-  Goals left:
  case h1 ⊢ a = b
  case h2 ⊢ d = b
  case h3 ⊢ c + a.pred = c + d.pred
  -/
  sorry
  sorry
  sorry

example {a b : ℕ} (h : a = b) : (fun y : ℕ => ∀ z, a + a = z) = (fun x => ∀ z, b + a = z) := by
  congrm fun x => ∀ w, ?_ + a = w
  -- ⊢ a = b
  exact h

The congrm command is a convenient frontend to congr(...) congruence quotations. If the goal is an equality, congrm e is equivalent to refine congr(e') where e' is built from e by replacing each placeholder ?m by $(?m). The pattern e is allowed to contain $(...) expressions to immediately substitute equality proofs into the congruence, just like for congruence quotations.

Tags:
Defined in module:: Mathlib.Tactic.CongrM

congrm?

Display a widget panel allowing to generate a congrm call with holes specified by selecting subexpressions in the goal.

Tags:
Defined in module:: Mathlib.Tactic.Widget.CongrM

constructor

If the main goal's target type is an inductive type, constructor solves it with the first matching constructor, or else fails.

Tags:
Defined in module:: Init.Tactics

constructorm

constructorm p_1, ..., p_n applies the constructor tactic to the main goal if type matches one of the given patterns.
constructorm* p is a more efficient and compact version of · repeat constructorm p. It is more efficient because the pattern is compiled once.

Example: The following tactic proves any theorem like True ∧ (True ∨ True) consisting of and/or/true:

constructorm* _ ∨ _, _ ∧ _, True

Tags:
Defined in module:: Mathlib.Tactic.CasesM

continuity

The tactic continuity solves goals of the form Continuous f by applying lemmas tagged with the continuity user attribute.

fun_prop is a (usually more powerful) alternative to continuity.

Tags:
Defined in module:: Mathlib.Tactic.Continuity

continuity?

The tactic continuity solves goals of the form Continuous f by applying lemmas tagged with the continuity user attribute.

Tags:
Defined in module:: Mathlib.Tactic.Continuity

contradiction

contradiction closes the main goal if its hypotheses are "trivially contradictory".

Inductive type/family with no applicable constructors
```
example (h : False) : p := by contradiction
```

Injectivity of constructors

example (h : none = some true) : p := by contradiction  --

Decidable false proposition

example (h : 2 + 2 = 3) : p := by contradiction

Contradictory hypotheses

example (h : p) (h' : ¬ p) : q := by contradiction

Other simple contradictions such as

example (x : Nat) (h : x ≠ x) : p := by contradiction

Tags:
Defined in module:: Init.Tactics

contrapose

Transforms the goal into its contrapositive.

contrapose turns a goal P → Q into ¬ Q → ¬ P and it turns a goal P ↔ Q into ¬ P ↔ ¬ Q
contrapose h first reverts the local assumption h, and then uses contrapose and intro h
contrapose h with new_h uses the name new_h for the introduced hypothesis

Tags:
Defined in module:: Mathlib.Tactic.Contrapose

contrapose!

Transforms the goal into its contrapositive and pushes negations in the result. Usage matches contrapose

Tags:
Defined in module:: Mathlib.Tactic.Contrapose

conv

conv => ... allows the user to perform targeted rewriting on a goal or hypothesis, by focusing on particular subexpressions.

See https://lean-lang.org/theorem_proving_in_lean4/conv.html for more details.

Basic forms:

conv => cs will rewrite the goal with conv tactics cs.
conv at h => cs will rewrite hypothesis h.
conv in pat => cs will rewrite the first subexpression matching pat (see pattern).

Tags:
Defined in module:: Init.Conv

conv'

Executes the given conv block without converting regular goal into a conv goal.

Tags:
Defined in module:: Init.Conv

conv?

Display a widget panel allowing to generate a conv call zooming to the subexpression selected in the goal or in the type of a local hypothesis or let-decl.

Tags:
Defined in module:: Mathlib.Tactic.Widget.Conv

conv_lhs

conv_lhs => cs runs the conv tactic sequence cs on the left hand side of the target.

In general, for an n-ary operator as the target, it traverses into the second to last argument. It is a synonym for conv => arg -2; cs.

conv_lhs at h => cs runs cs on the left hand side of hypothesis h.
conv_lhs in pat => cs first looks for a subexpression matching pat (see also the pattern conv tactic) and then traverses into the left hand side of this subexpression. This syntax also supports the occs clause for the pattern.

Tags:
Defined in module:: Mathlib.Tactic.Conv

conv_rhs

conv_rhs => cs runs the conv tactic sequence cs on the right hand side of the target.

In general, for an n-ary operator as the target, it traverses into the last argument. It is a synonym for conv => arg -1; cs.

conv_rhs at h => cs runs cs on the right hand side of hypothesis h.
conv_rhs in pat => cs first looks for a subexpression matching pat (see the pattern conv tactic) and then traverses into the right hand side of this subexpression. This syntax also supports the occs clause for the pattern.

Tags:
Defined in module:: Mathlib.Tactic.Conv

convert

The exact e and refine e tactics require a term e whose type is definitionally equal to the goal. convert e is similar to refine e, but the type of e is not required to exactly match the goal. Instead, new goals are created for differences between the type of e and the goal using the same strategies as the congr! tactic. For example, in the proof state

n : ℕ,
e : Prime (2 * n + 1)
⊢ Prime (n + n + 1)

the tactic convert e using 2 will change the goal to

⊢ n + n = 2 * n

In this example, the new goal can be solved using ring.

The using 2 indicates it should iterate the congruence algorithm up to two times, where convert e would use an unrestricted number of iterations and lead to two impossible goals: ⊢ HAdd.hAdd = HMul.hMul and ⊢ n = 2.

A variant configuration is convert (config := .unfoldSameFun) e, which only equates function applications for the same function (while doing so at the higher default transparency). This gives the same goal of ⊢ n + n = 2 * n without needing using 2.

The convert tactic applies congruence lemmas eagerly before reducing, therefore it can fail in cases where exact succeeds:

def p (n : ℕ) := True
example (h : p 0) : p 1 := by exact h -- succeeds
example (h : p 0) : p 1 := by convert h -- fails, with leftover goal `1 = 0`

Limiting the depth of recursion can help with this. For example, convert h using 1 will work in this case.

The syntax convert ← e will reverse the direction of the new goals (producing ⊢ 2 * n = n + n in this example).

Internally, convert e works by creating a new goal asserting that the goal equals the type of e, then simplifying it using congr!. The syntax convert e using n can be used to control the depth of matching (like congr! n). In the example, convert e using 1 would produce a new goal ⊢ n + n + 1 = 2 * n + 1.

Refer to the congr! tactic to understand the congruence operations. One of its many features is that if x y : t and an instance Subsingleton t is in scope, then any goals of the form x = y are solved automatically.

Like congr!, convert takes an optional with clause of rintro patterns, for example convert e using n with x y z.

The convert tactic also takes a configuration option, for example

convert (config := {transparency := .default}) h

These are passed to congr!. See Congr!.Config for options.

Tags:
Defined in module:: Mathlib.Tactic.Convert

convert_to

The convert_to tactic is for changing the type of the target or a local hypothesis, but unlike the change tactic it will generate equality proof obligations using congr! to resolve discrepancies.

convert_to ty changes the target to ty
convert_to ty using n uses congr! n instead of congr! 1
convert_to ty at h changes the type of the local hypothesis h to ty. Any remaining congr! goals come first.

Operating on the target, the tactic convert_to ty using n is the same as convert (?_ : ty) using n. The difference is that convert_to takes a type but convert takes a proof term.

Except for it also being able to operate on local hypotheses, the syntax for convert_to is the same as for convert, and it has variations such as convert_to ← g and convert_to (config := {transparency := .default}) g.

Note that convert_to ty at h may leave a copy of h if a later local hypotheses or the target depends on it, just like in rw or simp.

Tags:
Defined in module:: Mathlib.Tactic.Convert

cutsat

cutsat solves linear integer arithmetic goals.

It is a implemented as a thin wrapper around the grind tactic, enabling only the lia solver. Please use grind instead if you need additional capabilities.

Deprecated: Use lia instead.

Tags:
Defined in module:: Init.Grind.Tactics

dbg_trace

dbg_trace "foo" prints foo when elaborated. Useful for debugging tactic control flow:

example : False ∨ True := by
  first
  | apply Or.inl; trivial; dbg_trace "left"
  | apply Or.inr; trivial; dbg_trace "right"

Tags:
Defined in module:: Init.Tactics

decide

decide attempts to prove the main goal (with target type p) by synthesizing an instance of Decidable p and then reducing that instance to evaluate the truth value of p. If it reduces to isTrue h, then h is a proof of p that closes the goal.

The target is not allowed to contain local variables or metavariables. If there are local variables, you can first try using the revert tactic with these local variables to move them into the target, or you can use the +revert option, described below.

Options:

decide +revert begins by reverting local variables that the target depends on, after cleaning up the local context of irrelevant variables. A variable is relevant if it appears in the target, if it appears in a relevant variable, or if it is a proposition that refers to a relevant variable.
decide +kernel uses kernel for reduction instead of the elaborator. It has two key properties: (1) since it uses the kernel, it ignores transparency and can unfold everything, and (2) it reduces the Decidable instance only once instead of twice.
decide +native uses the native code compiler (#eval) to evaluate the Decidable instance, admitting the result via the Lean.ofReduceBool axiom. This can be significantly more efficient than using reduction, but it is at the cost of increasing the size of the trusted code base. Namely, it depends on the correctness of the Lean compiler and all definitions with an @[implemented_by] attribute. Like with +kernel, the Decidable instance is evaluated only once.

Limitation: In the default mode or +kernel mode, since decide uses reduction to evaluate the term, Decidable instances defined by well-founded recursion might not work because evaluating them requires reducing proofs. Reduction can also get stuck on Decidable instances with Eq.rec terms. These can appear in instances defined using tactics (such as rw and simp). To avoid this, create such instances using definitions such as decidable_of_iff instead.

Examples #

Proving inequalities:

example : 2 + 2 ≠ 5 := by decide

Trying to prove a false proposition:

example : 1 ≠ 1 := by decide
/-
tactic 'decide' proved that the proposition
  1 ≠ 1
is false
-/

Trying to prove a proposition whose Decidable instance fails to reduce

opaque unknownProp : Prop

open scoped Classical in
example : unknownProp := by decide
/-
tactic 'decide' failed for proposition
  unknownProp
since its 'Decidable' instance reduced to
  Classical.choice ⋯
rather than to the 'isTrue' constructor.
-/

Properties and relations #

For equality goals for types with decidable equality, usually rfl can be used in place of decide.

example : 1 + 1 = 2 := by decide
example : 1 + 1 = 2 := by rfl

Tags:
Defined in module:: Init.Tactics

decreasing_tactic

decreasing_tactic is called by default on well-founded recursions in order to synthesize a proof that recursive calls decrease along the selected well founded relation. It can be locally overridden by using decreasing_by tac on the recursive definition, and it can also be globally extended by adding more definitions for decreasing_tactic (or decreasing_trivial, which this tactic calls).

Tags:
Defined in module:: Init.WFTactics

decreasing_trivial

Extensible helper tactic for decreasing_tactic. This handles the "base case" reasoning after applying lexicographic order lemmas. It can be extended by adding more macro definitions, e.g.

macro_rules | `(tactic| decreasing_trivial) => `(tactic| linarith)

Tags:
Defined in module:: Init.WFTactics

decreasing_trivial_pre_omega

Variant of decreasing_trivial that does not use omega, intended to be used in core modules before omega is available.

Tags:
Defined in module:: Init.WFTactics

decreasing_with

Constructs a proof of decreasing along a well founded relation, by simplifying, then applying lexicographic order lemmas and finally using ts to solve the base case. If it fails, it prints a message to help the user diagnose an ill-founded recursive definition.

Tags:
Defined in module:: Init.WFTactics

delta

delta id1 id2 ... delta-expands the definitions id1, id2, .... This is a low-level tactic, it will expose how recursive definitions have been compiled by Lean.

Tags:
Defined in module:: Init.Tactics

deriving_LawfulEq_tactic

This tactic has no documentation.

Tags:
Defined in module:: Init.LawfulBEqTactics

deriving_LawfulEq_tactic_step

This tactic has no documentation.

Tags:
Defined in module:: Init.LawfulBEqTactics

deriving_ReflEq_tactic

This tactic has no documentation.

Tags:
Defined in module:: Init.LawfulBEqTactics

discrete_cases

A simple tactic to run cases on any Discrete α hypotheses.

Tags:
Defined in module:: Mathlib.CategoryTheory.Discrete.Basic

done

done succeeds iff there are no remaining goals.

Tags:
Defined in module:: Init.Tactics

dsimp

The dsimp tactic is the definitional simplifier. It is similar to simp but only applies theorems that hold by reflexivity. Thus, the result is guaranteed to be definitionally equal to the input.

Tags:
Defined in module:: Init.Tactics

dsimp!

dsimp! is shorthand for dsimp with autoUnfold := true. This will unfold applications of functions defined by pattern matching, when one of the patterns applies. This can be used to partially evaluate many definitions.

Tags:
Defined in module:: Init.Meta

dsimp?

simp? takes the same arguments as simp, but reports an equivalent call to simp only that would be sufficient to close the goal. This is useful for reducing the size of the simp set in a local invocation to speed up processing.

example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
  simp? -- prints "Try this: simp only [ite_true]"

This command can also be used in simp_all and dsimp.

Tags:
Defined in module:: Init.Tactics

dsimp?!

example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
  simp? -- prints "Try this: simp only [ite_true]"

This command can also be used in simp_all and dsimp.

Tags:
Defined in module:: Init.Tactics

eapply

eapply e is like apply e but it does not add subgoals for variables that appear in the types of other goals. Note that this can lead to a failure where there are no goals remaining but there are still metavariables in the term:

example (h : ∀ x : Nat, x = x → True) : True := by
  eapply h
  rfl
  -- no goals
-- (kernel) declaration has metavariables '_example'

Tags:
Defined in module:: Batteries.Tactic.Init

econstructor

econstructor is like constructor (it calls apply using the first matching constructor of an inductive datatype) except only non-dependent premises are added as new goals.

Tags:
Defined in module:: Mathlib.Tactic.Constructor

enat_to_nat

enat_to_nat shifts all ENats in the context to Nat, rewriting propositions about them. A typical use case is enat_to_nat; lia.

Tags:
Defined in module:: Mathlib.Tactic.ENatToNat

eq_refl

eq_refl is equivalent to exact rfl, but has a few optimizations.

Tags:
Defined in module:: Init.Tactics

erw

erw [rules] is a shorthand for rw (transparency := .default) [rules]. This does rewriting up to unfolding of regular definitions (by comparison to regular rw which only unfolds @[reducible] definitions).

Tags:
Defined in module:: Init.Meta

erw?

erw? [r, ...] calls erw [r, ...] (at hypothesis h if written erw [r, ...] at h), and then attempts to identify any subexpression which would block the use of rw instead. It does so by identifying subexpressions which are defeq, but not at reducible transparency.

Tags:
Defined in module:: Mathlib.Tactic.ErwQuestion

eta_expand

eta_expand at loc eta expands all sub-expressions at the given location. It also beta reduces any applications of eta expanded terms, so it puts it into an eta-expanded "normal form." This also exists as a conv-mode tactic.

For example, if f takes two arguments, then f becomes fun x y => f x y and f x becomes fun y => f x y.

This can be useful to turn, for example, a raw HAdd.hAdd into fun x y => x + y.

Tags:
Defined in module:: Mathlib.Tactic.DefEqTransformations

eta_reduce

eta_reduce at loc eta reduces all sub-expressions at the given location. This also exists as a conv-mode tactic.

For example, fun x y => f x y becomes f after eta reduction.

Tags:
Defined in module:: Mathlib.Tactic.DefEqTransformations

eta_struct

eta_struct at loc transforms structure constructor applications such as S.mk x.1 ... x.n (pretty printed as, for example, {a := x.a, b := x.b, ...}) into x. This also exists as a conv-mode tactic.

The transformation is known as eta reduction for structures, and it yields definitionally equal expressions.

For example, given x : α × β, then (x.1, x.2) becomes x after this transformation.

Tags:
Defined in module:: Mathlib.Tactic.DefEqTransformations

exact

exact e closes the main goal if its target type matches that of e.

Tags:
Defined in module:: Init.Tactics

exact?

Searches environment for definitions or theorems that can solve the goal using exact with conditions resolved by solve_by_elim.

The optional using clause provides identifiers in the local context that must be used by exact? when closing the goal. This is most useful if there are multiple ways to resolve the goal, and one wants to guide which lemma is used.

Use +grind to enable grind as a fallback discharger for subgoals. Use +try? to enable try? as a fallback discharger for subgoals. Use -star to disable fallback to star-indexed lemmas (like Empty.elim, And.left). Use +all to collect all successful lemmas instead of stopping at the first.

Tags:
Defined in module:: Init.Tactics

exact_mod_cast

Normalize casts in the goal and the given expression, then close the goal with exact.

Tags:
Defined in module:: Init.TacticsExtra

exacts

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

Tags:
Defined in module:: Batteries.Tactic.Init

exfalso

exfalso converts a goal ⊢ tgt into ⊢ False by applying False.elim.

Tags:
Defined in module:: Init.Tactics

exists

exists e₁, e₂, ... is shorthand for refine ⟨e₁, e₂, ...⟩; try trivial. It is useful for existential goals.

Tags:
Defined in module:: Init.Tactics

existsi

existsi e₁, e₂, ⋯ applies the tactic refine ⟨e₁, e₂, ⋯, ?_⟩. It's purpose is to instantiate existential quantifiers.

Examples:

example : ∃ x : Nat, x = x := by
  existsi 42
  rfl

example : ∃ x : Nat, ∃ y : Nat, x = y := by
  existsi 42, 42
  rfl

Tags:
Defined in module:: Mathlib.Tactic.ExistsI

expose_names

expose_names renames all inaccessible variables with accessible names, making them available for reference in generated tactics. However, this renaming introduces machine-generated names that are not fully under user control. expose_names is primarily intended as a preamble for auto-generated end-game tactic scripts. It is also useful as an alternative to set_option tactic.hygienic false. If explicit control over renaming is needed in the middle of a tactic script, consider using structured tactic scripts with match .. with, induction .. with, or intro with explicit user-defined names, as well as tactics such as next, case, and rename_i.

Tags:
Defined in module:: Init.Tactics

ext

Applies extensionality lemmas that are registered with the @[ext] attribute.

ext pat* applies extensionality theorems as much as possible, using the patterns pat* to introduce the variables in extensionality theorems using rintro. For example, the patterns are used to name the variables introduced by lemmas such as funext.
Without patterns,ext applies extensionality lemmas as much as possible but introduces anonymous hypotheses whenever needed.
ext pat* : n applies ext theorems only up to depth n.

The ext1 pat* tactic is like ext pat* except that it only applies a single extensionality theorem.

Unused patterns will generate warning. Patterns that don't match the variables will typically result in the introduction of anonymous hypotheses.

Tags:
Defined in module:: Init.Ext

ext1

ext1 pat* is like ext pat* except that it only applies a single extensionality theorem rather than recursively applying as many extensionality theorems as possible.

The pat* patterns are processed using the rintro tactic. If no patterns are supplied, then variables are introduced anonymously using the intros tactic.

Tags:
Defined in module:: Init.Ext

extract_goal

extract_goal formats the current goal as a stand-alone theorem or definition after cleaning up the local context of irrelevant variables. A variable is relevant if (1) it occurs in the target type, (2) there is a relevant variable that depends on it, or (3) the type of the variable is a proposition that depends on a relevant variable.

If the target is False, then for convenience extract_goal includes all variables.
extract_goal * formats the current goal without cleaning up the local context.
extract_goal a b c ... formats the current goal after removing everything that the given variables a, b, c, ... do not depend on.
extract_goal ... using name uses the name name for the theorem or definition rather than the autogenerated name.

The tactic tries to produce an output that can be copy-pasted and just work, but its success depends on whether the expressions are amenable to being unambiguously pretty printed.

The tactic responds to pretty printing options. For example, set_option pp.all true in extract_goal gives the pp.all form.

Tags:
Defined in module:: Mathlib.Tactic.ExtractGoal

extract_lets

Extracts let and have expressions from within the target or a local hypothesis, introducing new local definitions.

extract_lets extracts all the lets from the target.
extract_lets x y z extracts all the lets from the target and uses x, y, and z for the first names. Using _ for a name leaves it unnamed.
extract_lets x y z at h operates on the local hypothesis h instead of the target.

For example, given a local hypotheses if the form h : let x := v; b x, then extract_lets z at h introduces a new local definition z := v and changes h to be h : b z.

Tags:
Defined in module:: Init.Tactics

fail

fail msg is a tactic that always fails, and produces an error using the given message.

Tags:
Defined in module:: Init.Tactics

fail_if_no_progress

fail_if_no_progress tacs evaluates tacs, and fails if no progress is made on the main goal or the local context at reducible transparency.

Tags:
Defined in module:: Mathlib.Tactic.FailIfNoProgress

fail_if_success

fail_if_success t fails if the tactic t succeeds.

Tags:
Defined in module:: Init.Tactics

false_or_by_contra

Changes the goal to False, retaining as much information as possible:

If the goal is False, do nothing.
If the goal is an implication or a function type, introduce the argument and restart. (In particular, if the goal is x ≠ y, introduce x = y.)
Otherwise, for a propositional goal P, replace it with ¬ ¬ P (attempting to find a Decidable instance, but otherwise falling back to working classically) and introduce ¬ P.
For a non-propositional goal use False.elim.

Tags:
Defined in module:: Init.Tactics

fapply

fapply e is like apply e but it adds goals in the order they appear, rather than putting the dependent goals first.

Tags:
Defined in module:: Batteries.Tactic.Init

fconstructor

fconstructor is like constructor (it calls apply using the first matching constructor of an inductive datatype) except that it does not reorder goals.

Tags:
Defined in module:: Mathlib.Tactic.Constructor

field

The field tactic proves equality goals in (semi-)fields. For example:

example {x y : ℚ} (hx : x + y ≠ 0) : x / (x + y) + y / (x + y) = 1 := by
  field
example {a b : ℝ} (ha : a ≠ 0) : a / (a * b) - 1 / b = 0 := by field

The scope of the tactic is equality goals which are universal, in the sense that they are true in any field in which the appropriate denominators don't vanish. (That is, they are consequences purely of the field axioms.)

Checking the nonvanishing of the necessary denominators is done using a variety of tricks -- in particular this part of the reasoning is non-universal, i.e. can be specific to the field at hand (order properties, explicit ≠ 0 hypotheses, CharZero if that is known, etc). The user can also provide additional terms to help with the nonzeroness proofs. For example:

example {K : Type*} [Field K] (hK : ∀ x : K, x ^ 2 + 1 ≠ 0) (x : K) :
    1 / (x ^ 2 + 1) + x ^ 2 / (x ^ 2 + 1) = 1 := by
  field [hK]

The field tactic is built from the tactics field_simp (which clears the denominators) and ring (which proves equality goals universally true in commutative (semi-)rings). If field fails to prove your goal, you may still be able to prove your goal by running the field_simp and ring_nf normalizations in some order. For example, this statement:

example {a b : ℚ} (H : b + a ≠ 0) : a / (a + b) + b / (b + a) = 1

is not proved by field but is proved by ring_nf at *; field.

Tags:
Defined in module:: Mathlib.Tactic.Field

field_simp

The goal of field_simp is to bring expressions in (semi-)fields over a common denominator, i.e. to reduce them to expressions of the form n / d where neither n nor d contains any division symbol. For example, x / (1 - y) / (1 + y / (1 - y)) is reduced to x / (1 - y + y):

example (x y z : ℚ) (hy : 1 - y ≠ 0) :
    ⌊x / (1 - y) / (1 + y / (1 - y))⌋ < 3 := by
  field_simp
  -- new goal: `⊢ ⌊x / (1 - y + y)⌋ < 3`

The field_simp tactic will also clear denominators in field (in)equalities, by cross-multiplying. For example, field_simp will clear the x denominators in the following equation:

example {K : Type*} [Field K] {x : K} (hx0 : x ≠ 0) :
    (x + 1 / x) ^ 2 + (x + 1 / x) = 1 := by
  field_simp
  -- new goal: `⊢ (x ^ 2 + 1) * (x ^ 2 + 1 + x) = x ^ 2`

A very common pattern is field_simp; ring (clear denominators, then the resulting goal is solvable by the axioms of a commutative ring). The finishing tactic field is a shorthand for this pattern.

Cancelling and combining denominators will generally require checking "nonzeroness"/"positivity" side conditions. The field_simp tactic attempts to discharge these, and will omit such steps if it cannot discharge the corresponding side conditions. The discharger will try, among other things, positivity and norm_num, and will also use any nonzeroness/positivity proofs included explicitly (e.g. field_simp [hx]). If your expression is not completely reduced by field_simp, check the denominators of the resulting expression and provide proofs that they are nonzero/positive to enable further progress.

Tags:
Defined in module:: Mathlib.Tactic.FieldSimp

field_simp_discharge

Discharge strategy for the field_simp tactic.

Tags:
Defined in module:: Mathlib.Tactic.FieldSimp.Discharger

filter_upwards

filter_upwards [h₁, ⋯, hₙ] replaces a goal of the form s ∈ f and terms h₁ : t₁ ∈ f, ⋯, hₙ : tₙ ∈ f with ∀ x, x ∈ t₁ → ⋯ → x ∈ tₙ → x ∈ s. The list is an optional parameter, [] being its default value.

filter_upwards [h₁, ⋯, hₙ] with a₁ a₂ ⋯ aₖ is a short form for { filter_upwards [h₁, ⋯, hₙ], intro a₁ a₂ ⋯ aₖ }.

filter_upwards [h₁, ⋯, hₙ] using e is a short form for { filter_upwards [h1, ⋯, hn], exact e }.

Combining both shortcuts is done by writing filter_upwards [h₁, ⋯, hₙ] with a₁ a₂ ⋯ aₖ using e. Note that in this case, the aᵢ terms can be used in e.

Tags:
Defined in module:: Mathlib.Order.Filter.Defs

fin_cases

fin_cases h performs case analysis on a hypothesis of the form h : A, where [Fintype A] is available, or h : a ∈ A, where A : Finset X, A : Multiset X or A : List X.

As an example, in

example (f : ℕ → Prop) (p : Fin 3) (h0 : f 0) (h1 : f 1) (h2 : f 2) : f p.val := by
  fin_cases p; simp
  all_goals assumption

after fin_cases p; simp, there are three goals, f 0, f 1, and f 2.

Tags:
Defined in module:: Mathlib.Tactic.FinCases

fin_omega

Preprocessor for omega to handle inequalities in Fin. Note that this involves a lot of case splitting, so may be slow.

Tags:
Defined in module:: Mathlib.Data.Fin.Basic

find

Display theorems (and definitions) whose result type matches the current goal, i.e. which should be applyable.

example : True := by find

find will not affect the goal by itself and should be removed from the finished proof. For a command that takes the type to search for as an argument, see #find, which is also available as a tactic.

Tags:
Defined in module:: Mathlib.Tactic.Find

finiteness

finiteness proves goals of the form *** < ∞ and (equivalently) *** ≠ ∞ in the extended nonnegative reals (ℝ≥0∞). Supports passing additional expressions as local hypotheses.

finiteness? additionally shows the proof that finiteness found
finiteness_nonterminal is a version of finiteness that may (but doesn't have to) close the goal.

Tags:
Defined in module:: Mathlib.Tactic.Finiteness

first

first | tac | ... runs each tac until one succeeds, or else fails.

Tags:
Defined in module:: Init.Tactics

first_par

Helper internal tactic for implementing the tactic try? with parallel execution, returning first success.

Tags:
Defined in module:: Init.Try

focus

focus tac focuses on the main goal, suppressing all other goals, and runs tac on it. Usually · tac, which enforces that the goal is closed by tac, should be preferred.

Tags:
Defined in module:: Init.Tactics

forward

This tactic has no documentation.

Tags:
Defined in module:: Aesop.Frontend.Saturate

forward?

This tactic has no documentation.

Tags:
Defined in module:: Aesop.Frontend.Saturate

frac_tac

Solve equations for RatFunc K by working in FractionRing K[X].

Tags:
Defined in module:: Mathlib.FieldTheory.RatFunc.Basic

fun_cases

The fun_cases tactic is a convenience wrapper of the cases tactic when using a functional cases principle.

The tactic invocation

fun_cases f x ... y ...`

is equivalent to

cases y, ... using f.fun_cases_unfolding x ...

where the arguments of f are used as arguments to f.fun_cases_unfolding or targets of the case analysis, as appropriate.

The form

fun_cases f

(with no arguments to f) searches the goal for a unique eligible application of f, and uses these arguments. An application of f is eligible if it is saturated and the arguments that will become targets are free variables.

The form fun_cases f x y with | case1 => tac₁ | case2 x' ih => tac₂ works like with cases.

Under set_option tactic.fun_induction.unfolding true (the default), fun_induction uses the f.fun_cases_unfolding theorem, which will try to automatically unfold the call to f in the goal. With set_option tactic.fun_induction.unfolding false, it uses f.fun_cases instead.

Tags:
Defined in module:: Init.Tactics

fun_induction

The fun_induction tactic is a convenience wrapper around the induction tactic to use the functional induction principle.

The tactic invocation

fun_induction f x₁ ... xₙ y₁ ... yₘ

where f is a function defined by non-mutual structural or well-founded recursion, is equivalent to

induction y₁, ... yₘ using f.induct_unfolding x₁ ... xₙ

where the arguments of f are used as arguments to f.induct_unfolding or targets of the induction, as appropriate.

The form

fun_induction f

The forms fun_induction f x y generalizing z₁ ... zₙ and fun_induction f x y with | case1 => tac₁ | case2 x' ih => tac₂ work like with induction.

Under set_option tactic.fun_induction.unfolding true (the default), fun_induction uses the f.induct_unfolding induction principle, which will try to automatically unfold the call to f in the goal. With set_option tactic.fun_induction.unfolding false, it uses f.induct instead.

Tags:
Defined in module:: Init.Tactics

fun_prop

fun_prop solves a goal of the form P f, where P is a predicate and f is a function, by decomposing f into a composition of elementary functions, and proving P on each of those by matching against a set of @[fun_prop] lemmas.

If fun_prop fails to solve a goal with the error "No theorems found", you can solve this issue by importing or adding new theorems tagged with the @[fun_prop] attribute. See the module documentation for Mathlib/Tactic/FunProp.lean for a detailed explanation.

fun_prop (disch := tac) uses tac to solve potential side goals. Setting this option is required to solve ContinuousAt/On/Within goals.
fun_prop [c, ...] will unfold the constant(s) c, ... before decomposing f.
fun_prop (config := cfg) sets advanced configuration options using cfg : FunProp.Config (see FunProp.Config for details).

Examples:

example : Continuous (fun x : ℝ ↦ x * sin x) := by fun_prop

-- Specify a discharger to solve `ContinuousAt`/`Within`/`On` goals:
example (y : ℝ) (hy : y ≠ 0) : ContinuousAt (fun x : ℝ ↦ 1/x) y := by
  fun_prop (disch := assumption)

example (y : ℝ) (hy : y ≠ 0) : ContinuousAt (fun x => x * (Real.log x) ^ 2 - Real.exp x / x) y := by
  fun_prop (disch := aesop)

Tags:
Defined in module:: Mathlib.Tactic.FunProp.Elab

funext

Apply function extensionality and introduce new hypotheses. The tactic funext will keep applying 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. Patterns can be used like in the intro tactic. Example, given a goal

  |-  ((fun x : Nat × Bool => ...) = (fun x => ...))

funext (a, b) applies funext once and performs pattern matching on the newly introduced pair.

Tags:
Defined in module:: Init.NotationExtra

gcongr

The gcongr tactic applies "generalized congruence" rules, reducing a relational goal between an LHS and RHS. For example,

example {a b x c d : ℝ} (h1 : a + 1 ≤ b + 1) (h2 : c + 2 ≤ d + 2) :
    x ^ 2 * a + c ≤ x ^ 2 * b + d := by
  gcongr
  · linarith
  · linarith

This example has the goal of proving the relation ≤ between an LHS and RHS both of the pattern

x ^ 2 * ?_ + ?_

(with inputs a, c on the left and b, d on the right); after the use of gcongr, we have the simpler goals a ≤ b and c ≤ d.

A depth limit or a pattern can be provided explicitly; this is useful if a non-maximal match is desired:

example {a b c d x : ℝ} (h : a + c + 1 ≤ b + d + 1) :
    x ^ 2 * (a + c) + 5 ≤ x ^ 2 * (b + d) + 5 := by
  gcongr x ^ 2 * ?_ + 5 -- or `gcongr 2`
  linarith

The "generalized congruence" rules are the library lemmas which have been tagged with the attribute @[gcongr]. For example, the first example constructs the proof term

add_le_add (mul_le_mul_of_nonneg_left ?_ (Even.pow_nonneg (even_two_mul 1) x)) ?_

using the generalized congruence lemmas add_le_add and mul_le_mul_of_nonneg_left.

The tactic attempts to discharge side goals to these "generalized congruence" lemmas (such as the side goal 0 ≤ x ^ 2 in the above application of mul_le_mul_of_nonneg_left) using the tactic gcongr_discharger, which wraps positivity but can also be extended. Side goals not discharged in this way are left for the user.

gcongr will descend into binders (for example sums or suprema). To name the bound variables, use with:

example {f g : ℕ → ℝ≥0∞} (h : ∀ n, f n ≤ g n) : ⨆ n, f n ≤ ⨆ n, g n := by
  gcongr with i
  exact h i

Tags:
Defined in module:: Mathlib.Tactic.GCongr.Core

gcongr?

Display a widget panel allowing to generate a gcongr call with holes specified by selecting subexpressions in the goal.

Tags:
Defined in module:: Mathlib.Tactic.Widget.GCongr

gcongr_discharger

gcongr_discharger is used by gcongr to discharge side goals.

This is an extensible tactic using macro_rules. By default it calls positivity (after importing the positivity tactic). Example: macro_rules | `(tactic| gcongr_discharger) => `(tactic| positivity).

Tags:
Defined in module:: Mathlib.Tactic.GCongr.Core

generalize

generalize ([h :] e = x),+ replaces all occurrences es in the main goal with a fresh hypothesis xs. If h is given, h : e = x is introduced as well.
generalize e = x at h₁ ... hₙ also generalizes occurrences of e inside h₁, ..., hₙ.
generalize e = x at * will generalize occurrences of e everywhere.

Tags:
Defined in module:: Init.Tactics

generalize'

Backwards compatibility shim for generalize.

Tags:
Defined in module:: Mathlib.Tactic.Generalize

generalize_proofs

generalize_proofs ids* [at locs]? generalizes proofs in the current goal, turning them into new local hypotheses.

generalize_proofs generalizes proofs in the target.
generalize_proofs at h₁ h₂ generalized proofs in hypotheses h₁ and h₂.
generalize_proofs at * generalizes proofs in the entire local context.
generalize_proofs pf₁ pf₂ pf₃ uses names pf₁, pf₂, and pf₃ for the generalized proofs. These can be _ to not name proofs.

If a proof is already present in the local context, it will use that rather than create a new local hypothesis.

When doing generalize_proofs at h, if h is a let binding, its value is cleared, and furthermore if h duplicates a preceding local hypothesis then it is eliminated.

The tactic is able to abstract proofs from under binders, creating universally quantified proofs in the local context. To disable this, use generalize_proofs -abstract. The tactic is also set to recursively abstract proofs from the types of the generalized proofs. This can be controlled with the maxDepth configuration option, with generalize_proofs (config := { maxDepth := 0 }) turning this feature off.

For example:

def List.nthLe {α} (l : List α) (n : ℕ) (_h : n < l.length) : α := sorry
example : List.nthLe [1, 2] 1 (by simp) = 2 := by
  -- ⊢ [1, 2].nthLe 1 ⋯ = 2
  generalize_proofs h
  -- h : 1 < [1, 2].length
  -- ⊢ [1, 2].nthLe 1 h = 2

Tags:
Defined in module:: Batteries.Tactic.GeneralizeProofs

get_elem_tactic

get_elem_tactic is the tactic automatically called by the notation arr[i] to prove any side conditions that arise when constructing the term (e.g. the index is in bounds of the array). It just delegates to get_elem_tactic_extensible and gives a diagnostic error message otherwise; users are encouraged to extend get_elem_tactic_extensible instead of this tactic.

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Defined in module:: Init.Tactics

get_elem_tactic_extensible

get_elem_tactic_extensible is an extensible tactic automatically called by the notation arr[i] to prove any side conditions that arise when constructing the term (e.g. the index is in bounds of the array). The default behavior is to try simp +arith and omega (for doing linear arithmetic in the index).

(Note that the core tactic get_elem_tactic has already tried done and assumption before the extensible tactic is called.)

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Defined in module:: Init.Tactics

get_elem_tactic_trivial

get_elem_tactic_trivial has been deprecated in favour of get_elem_tactic_extensible.

Tags:
Defined in module:: Init.Tactics

ghost_calc

ghost_calc is a tactic for proving identities between polynomial functions. Typically, when faced with a goal like

∀ (x y : 𝕎 R), verschiebung (x * frobenius y) = verschiebung x * y

you can

call ghost_calc
do a small amount of manual work -- maybe nothing, maybe rintro, etc
call ghost_simp

and this will close the goal.

ghost_calc cannot detect whether you are dealing with unary or binary polynomial functions. You must give it arguments to determine this. If you are proving a universally quantified goal like the above, call ghost_calc _ _. If the variables are introduced already, call ghost_calc x y. In the unary case, use ghost_calc _ or ghost_calc x.

ghost_calc is a light wrapper around type class inference. All it does is apply the appropriate extensionality lemma and try to infer the resulting goals. This is subtle and Lean's elaborator doesn't like it because of the HO unification involved, so it is easier (and prettier) to put it in a tactic script.

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Defined in module:: Mathlib.RingTheory.WittVector.IsPoly

ghost_fun_tac

An auxiliary tactic for proving that ghostFun respects the ring operations.

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Defined in module:: Mathlib.RingTheory.WittVector.Basic

ghost_simp

A macro for a common simplification when rewriting with ghost component equations.

Tags:
Defined in module:: Mathlib.RingTheory.WittVector.IsPoly

grewrite

grewrite [e] is like grw [e], but it doesn't try to close the goal with rfl. This is analogous to rw and rewrite, where rewrite doesn't try to close the goal with rfl.

Tags:
Defined in module:: Mathlib.Tactic.GRewrite.Elab

grind

grind is a tactic inspired by modern SMT solvers. Picture a virtual whiteboard: every time grind discovers a new equality, inequality, or logical fact, it writes it on the board, groups together terms known to be equal, and lets each reasoning engine read from and contribute to the shared workspace. These engines work together to handle equality reasoning, apply known theorems, propagate new facts, perform case analysis, and run specialized solvers for domains like linear arithmetic and commutative rings.

See the reference manual's chapter on grind for more information.

grind is not designed for goals whose search space explodes combinatorially, think large pigeonhole instances, graph‑coloring reductions, high‑order N‑queens boards, or a 200‑variable Sudoku encoded as Boolean constraints. Such encodings require thousands (or millions) of case‑splits that overwhelm grind’s branching search.

For bit‑level or combinatorial problems, consider using bv_decide. bv_decide calls a state‑of‑the‑art SAT solver (CaDiCaL) and then returns a compact, machine‑checkable certificate.

Equality reasoning #

grind uses congruence closure to track equalities between terms. When two terms are known to be equal, congruence closure automatically deduces equalities between more complex expressions built from them. For example, if a = b, then congruence closure will also conclude that f a = f b for any function f. This forms the foundation for efficient equality reasoning in grind. Here is an example:

example (f : Nat → Nat) (h : a = b) : f (f b) = f (f a) := by
  grind

Applying theorems using E-matching #

To apply existing theorems, grind uses a technique called E-matching, which finds matches for known theorem patterns while taking equalities into account. Combined with congruence closure, E-matching helps grind discover non-obvious consequences of theorems and equalities automatically.

Consider the following functions and theorems:

def f (a : Nat) : Nat :=
  a + 1

def g (a : Nat) : Nat :=
  a - 1

@[grind =]
theorem gf (x : Nat) : g (f x) = x := by
  simp [f, g]

The theorem gf asserts that g (f x) = x for all natural numbers x. The attribute [grind =] instructs grind to use the left-hand side of the equation, g (f x), as a pattern for E-matching. Suppose we now have a goal involving:

example {a b} (h : f b = a) : g a = b := by
  grind

Although g a is not an instance of the pattern g (f x), it becomes one modulo the equation f b = a. By substituting a with f b in g a, we obtain the term g (f b), which matches the pattern g (f x) with the assignment x := b. Thus, the theorem gf is instantiated with x := b, and the new equality g (f b) = b is asserted. grind then uses congruence closure to derive the implied equality g a = g (f b) and completes the proof.

The pattern used to instantiate theorems affects the effectiveness of grind. For example, the pattern g (f x) is too restrictive in the following case: the theorem gf will not be instantiated because the goal does not even contain the function symbol g.

example (h₁ : f b = a) (h₂ : f c = a) : b = c := by
  grind

You can use the command grind_pattern to manually select a pattern for a given theorem. In the following example, we instruct grind to use f x as the pattern, allowing it to solve the goal automatically:

grind_pattern gf => f x

example {a b c} (h₁ : f b = a) (h₂ : f c = a) : b = c := by
  grind

You can enable the option trace.grind.ematch.instance to make grind print a trace message for each theorem instance it generates.

You can also specify a multi-pattern to control when grind should apply a theorem. A multi-pattern requires that all specified patterns are matched in the current context before the theorem is applied. This is useful for theorems such as transitivity rules, where multiple premises must be simultaneously present for the rule to apply. The following example demonstrates this feature using a transitivity axiom for a binary relation R:

opaque R : Int → Int → Prop
axiom Rtrans {x y z : Int} : R x y → R y z → R x z

grind_pattern Rtrans => R x y, R y z

example {a b c d} : R a b → R b c → R c d → R a d := by
  grind

By specifying the multi-pattern R x y, R y z, we instruct grind to instantiate Rtrans only when both R x y and R y z are available in the context. In the example, grind applies Rtrans to derive R a c from R a b and R b c, and can then repeat the same reasoning to deduce R a d from R a c and R c d.

Instead of using grind_pattern to explicitly specify a pattern, you can use the @[grind] attribute or one of its variants, which will use a heuristic to generate a (multi-)pattern. The complete list is available in the reference manual. The main ones are:

@[grind →] will select a multi-pattern from the hypotheses of the theorem (i.e. it will use the theorem for forwards reasoning). In more detail, it will traverse the hypotheses of the theorem from left-to-right, and each time it encounters a minimal indexable (i.e. has a constant as its head) subexpression which "covers" (i.e. fixes the value of) an argument which was not previously covered, it will add that subexpression as a pattern, until all arguments have been covered.
@[grind ←] will select a multi-pattern from the conclusion of theorem (i.e. it will use the theorem for backwards reasoning). This may fail if not all the arguments to the theorem appear in the conclusion.
@[grind] will traverse the conclusion and then the hypotheses left-to-right, adding patterns as they increase the coverage, stopping when all arguments are covered.
@[grind =] checks that the conclusion of the theorem is an equality, and then uses the left-hand-side of the equality as a pattern. This may fail if not all of the arguments appear in the left-hand-side.

Here is the previous example again but using the attribute [grind →]

opaque R : Int → Int → Prop
@[grind →] axiom Rtrans {x y z : Int} : R x y → R y z → R x z

example {a b c d} : R a b → R b c → R c d → R a d := by
  grind

To control theorem instantiation and avoid generating an unbounded number of instances, grind uses a generation counter. Terms in the original goal are assigned generation zero. When grind applies a theorem using terms of generation ≤ n n, any new terms it creates are assigned generation n + 1 + 1 1. This limits how far the tactic explores when applying theorems and helps prevent an excessive number of instantiations.

Key options: #

grind (ematch := <num>) controls the number of E-matching rounds.
grind [<name>, ...] instructs grind to use the declaration name during E-matching.
grind only [<name>, ...] is like grind [<name>, ...] but does not use theorems tagged with @[grind].
grind (gen := <num>) sets the maximum generation.

Linear integer arithmetic (`lia`) #

grind can solve goals that reduce to linear integer arithmetic (LIA) using an integrated decision procedure called lia. It understands

equalities p = 0
inequalities p ≤ 0
disequalities p ≠ 0
divisibility d ∣ p

The solver incrementally assigns integer values to variables; when a partial assignment violates a constraint it adds a new, implied constraint and retries. This model-based search is complete for LIA.

Key options: #

grind -lia disable the solver (useful for debugging)
grind +qlia accept rational models (shrinks the search space but is incomplete for ℤ)

Examples: #

-- Even + even is never odd.
example {x y : Int} : 2 * x + 4 * y ≠ 5 := by
  grind

-- Mixing equalities and inequalities.
example {x y : Int} :
    2 * x + 3 * y = 0 → 1 ≤ x → y < 1 := by
  grind

-- Reasoning with divisibility.
example (a b : Int) :
    2 ∣ a + 1 → 2 ∣ b + a → ¬ 2 ∣ b + 2 * a := by
  grind

example (x y : Int) :
    27 ≤ 11*x + 13*y →
    11*x + 13*y ≤ 45 →
    -10 ≤ 7*x - 9*y →
    7*x - 9*y ≤ 4 → False := by
  grind

-- Types that implement the `ToInt` type-class.
example (a b c : UInt64)
    : a ≤ 2 → b ≤ 3 → c - a - b = 0 → c ≤ 5 := by
  grind

Algebraic solver (`ring`) #

grind ships with an algebraic solver nick-named ring for goals that can be phrased as polynomial equations (or disequations) over commutative rings, semirings, or fields.

Works out of the box All core numeric types and relevant Mathlib types already provide the required type-class instances, so the solver is ready to use in most developments.

What it can decide:

equalities of the form p = q
disequalities p ≠ q
basic reasoning under field inverses (a / b := a * b⁻¹)
goals that mix ring facts with other grind engines

Key options: #

grind -ring turn the solver off (useful when debugging)
grind (ringSteps := n) cap the number of steps performed by this procedure.

Examples #

open Lean Grind

example [CommRing α] (x : α) : (x + 1) * (x - 1) = x^2 - 1 := by
  grind

-- Characteristic 256 means 16 * 16 = 0.
example [CommRing α] [IsCharP α 256] (x : α) :
    (x + 16) * (x - 16) = x^2 := by
  grind

-- Works on built-in rings such as `UInt8`.
example (x : UInt8) : (x + 16) * (x - 16) = x^2 := by
  grind

example [CommRing α] (a b c : α) :
    a + b + c = 3 →
    a^2 + b^2 + c^2 = 5 →
    a^3 + b^3 + c^3 = 7 →
    a^4 + b^4 = 9 - c^4 := by
  grind

example [Field α] [NoNatZeroDivisors α] (a : α) :
    1 / a + 1 / (2 * a) = 3 / (2 * a) := by
  grind

Other options #

grind (splits := <num>) caps the depth of the search tree. Once a branch performs num splits grind stops splitting further in that branch.
grind -splitIte disables case splitting on if-then-else expressions.
grind -splitMatch disables case splitting on match expressions.
grind +splitImp instructs grind to split on any hypothesis A → B whose antecedent A is propositional.
grind -linarith disables the linear arithmetic solver for (ordered) modules and rings.

Additional Examples #

example {a b} {as bs : List α} : (as ++ bs ++ [b]).getLastD a = b := by
  grind

example (x : BitVec (w+1)) : (BitVec.cons x.msb (x.setWidth w)) = x := by
  grind

example (as : Array α) (lo hi i j : Nat) :
    lo ≤ i → i < j → j ≤ hi → j < as.size → min lo (as.size - 1) ≤ i := by
  grind

Tags:
Defined in module:: Init.Grind.Tactics

grind?

grind? takes the same arguments as grind, but reports an equivalent call to grind only that would be sufficient to close the goal. This is useful for reducing the size of the grind theorems in a local invocation.

Tags:
Defined in module:: Init.Grind.Tactics

grobner

grobner solves goals that can be phrased as polynomial equations (with further polynomial equations as hypotheses) over commutative (semi)rings, using the Grobner basis algorithm.

It is a implemented as a thin wrapper around the grind tactic, enabling only the grobner solver. Please use grind instead if you need additional capabilities.

Tags:
Defined in module:: Init.Grind.Tactics

group

Tactic for normalizing expressions in multiplicative groups, without assuming commutativity, using only the group axioms without any information about which group is manipulated.

(For additive commutative groups, use the abel tactic instead.)

Example:

example {G : Type} [Group G] (a b c d : G) (h : c = (a*b^2)*((b*b)⁻¹*a⁻¹)*d) : a*c*d⁻¹ = a := by
  group at h -- normalizes `h` which becomes `h : c = d`
  rw [h]     -- the goal is now `a*d*d⁻¹ = a`
  group      -- which then normalized and closed

Tags:
Defined in module:: Mathlib.Tactic.Group

grw

grw [e] works just like rw [e], but e can be a relation other than = or ↔.

For example:

variable {a b c d n : ℤ}

example (h₁ : a < b) (h₂ : b ≤ c) : a + d ≤ c + d := by
  grw [h₁, h₂]

example (h : a ≡ b [ZMOD n]) : a ^ 2 ≡ b ^ 2 [ZMOD n] := by
  grw [h]

example (h₁ : a ∣ b) (h₂ : b ∣ a ^ 2 * c) : a ∣ b ^ 2 * c := by
  grw [h₁] at *
  exact h₂

To replace the RHS with the LHS of the given relation, use the ← notation (just like in rw):

example (h₁ : a < b) (h₂ : b ≤ c) : a + d ≤ c + d := by
  grw [← h₂, ← h₁]

The strict inequality a < b is turned into the non-strict inequality a ≤ b to rewrite with it. A future version of grw may get special support for making better use of strict inequalities.

To rewrite only in the n-th position, use nth_grw n. This is useful when grw tries to rewrite in a position that is not valid for the given relation.

To be able to use grw, the relevant lemmas need to be tagged with @[gcongr]. To rewrite inside a transitive relation, you can also give it an IsTrans instance.

To let grw unfold more aggressively, as in erw, use grw (transparency := default).

Tags:
Defined in module:: Mathlib.Tactic.GRewrite.Elab

guard_expr

Tactic to check equality of two expressions.

guard_expr e = e' checks that e and e' are defeq at reducible transparency.
guard_expr e =~ e' checks that e and e' are defeq at default transparency.
guard_expr e =ₛ e' checks that e and e' are syntactically equal.
guard_expr e =ₐ e' checks that e and e' are alpha-equivalent.

Both e and e' are elaborated then have their metavariables instantiated before the equality check. Their types are unified (using isDefEqGuarded) before synthetic metavariables are processed, which helps with default instance handling.

Tags:
Defined in module:: Init.Guard

guard_goal_nums

guard_goal_nums n succeeds if there are exactly n goals and fails otherwise.

Tags:
Defined in module:: Mathlib.Tactic.GuardGoalNums

guard_hyp

Tactic to check that a named hypothesis has a given type and/or value.

guard_hyp h : t checks the type up to reducible defeq,
guard_hyp h :~ t checks the type up to default defeq,
guard_hyp h :ₛ t checks the type up to syntactic equality,
guard_hyp h :ₐ t checks the type up to alpha equality.
guard_hyp h := v checks value up to reducible defeq,
guard_hyp h :=~ v checks value up to default defeq,
guard_hyp h :=ₛ v checks value up to syntactic equality,
guard_hyp h :=ₐ v checks the value up to alpha equality.

The value v is elaborated using the type of h as the expected type.

Tags:
Defined in module:: Init.Guard

guard_hyp_nums

guard_hyp_nums n succeeds if there are exactly n hypotheses and fails otherwise.

Note that, depending on what options are set, some hypotheses in the local context might not be printed in the goal view. This tactic computes the total number of hypotheses, not the number of visible hypotheses.

Tags:
Defined in module:: Mathlib.Tactic.GuardHypNums

guard_target

Tactic to check that the target agrees with a given expression.

guard_target = e checks that the target is defeq at reducible transparency to e.
guard_target =~ e checks that the target is defeq at default transparency to e.
guard_target =ₛ e checks that the target is syntactically equal to e.
guard_target =ₐ e checks that the target is alpha-equivalent to e.

The term e is elaborated with the type of the goal as the expected type, which is mostly useful within conv mode.

Tags:
Defined in module:: Init.Guard

have

The have tactic is for adding opaque definitions and hypotheses to the local context of the main goal. The definitions forget their associated value and cannot be unfolded, unlike definitions added by the let tactic.

have h : t := e adds the hypothesis h : t if e is a term of type t.
have h := e uses the type of e for t.
have : t := e and have := e use this for the name of the hypothesis.
have pat := e for a pattern pat is equivalent to match e with | pat => _, where _ stands for the tactics that follow this one. It is convenient for types that have only one applicable constructor. For example, given h : p ∧ q ∧ r, have ⟨h₁, h₂, h₃⟩ := h produces the hypotheses h₁ : p, h₂ : q, and h₃ : r.
The syntax have (eq := h) pat := e is equivalent to match h : e with | pat => _, which adds the equation h : e = pat to the local context.

The tactic supports all the same syntax variants and options as the have term.

Properties and relations #

It is not possible to unfold a variable introduced using have, since the definition's value is forgotten. The let tactic introduces definitions that can be unfolded.
The have h : t := e is like doing let h : t := e; clear_value h.
The have tactic is preferred for propositions, and let is preferred for non-propositions.
Sometimes have is used for non-propositions to ensure that the variable is never unfolded, which may be important for performance reasons. Consider using the equivalent let +nondep to indicate the intent.

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Defined in module:: Init.Tactics

have'

Similar to have, but using refine'

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Defined in module:: Init.Tactics

haveI

haveI behaves like have, but inlines the value instead of producing a have term.

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Defined in module:: Init.Tactics

hint

The hint tactic tries every tactic registered using register_hint <prio> tac, and reports any that succeed.

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Defined in module:: Mathlib.Tactic.Hint

ident

This tactic has no documentation.

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Defined in module:: Lean.Parser.Tactic

if

In tactic mode, if t then tac1 else tac2 is alternative syntax for:

by_cases t
· tac1
· tac2

It performs case distinction on h† : t or h† : ¬t, where h† is an anonymous hypothesis, and tac1 and tac2 are the subproofs. (It doesn't actually use nondependent if, since this wouldn't add anything to the context and hence would be useless for proving theorems. To actually insert an ite application use refine if t then ?_ else ?_.)

The assumptions in each subgoal can be named. if h : t then tac1 else tac2 can be used as alternative syntax for:

by_cases h : t
· tac1
· tac2

It performs case distinction on h : t or h : ¬t.

You can use ?_ or _ for either subproof to delay the goal to after the tactic, but if a tactic sequence is provided for tac1 or tac2 then it will require the goal to be closed by the end of the block.

Tags:
Defined in module:: Init.Tactics

induction

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 are chosen automatically and are not accessible. You can use with to provide the variables names for each constructor.

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 using r allows the user to specify the principle of induction that should be used. Here r should be a term 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.
Given x : Nat, induction x with | zero => tac₁ | succ x' ih => tac₂ uses tactic tac₁ for the zero case, and tac₂ for the succ case.

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Defined in module:: Init.Tactics

induction'

The induction' tactic is similar to the induction tactic in Lean 4 core, but with slightly different syntax (such as, no requirement to name the constructors).

open Nat

example (n : ℕ) : 0 < factorial n := by
  induction' n with n ih
  · rw [factorial_zero]
    simp
  · rw [factorial_succ]
    apply mul_pos (succ_pos n) ih

example (n : ℕ) : 0 < factorial n := by
  induction n
  case zero =>
    rw [factorial_zero]
    simp
  case succ n ih =>
    rw [factorial_succ]
    apply mul_pos (succ_pos n) ih

Tags:
Defined in module:: Mathlib.Tactic.Cases

infer_instance

infer_instance is an abbreviation for exact inferInstance. It synthesizes a value of any target type by typeclass inference.

Tags:
Defined in module:: Init.Tactics

infer_param

Close a goal of the form optParam α a or autoParam α stx by using a.

Tags:
Defined in module:: Mathlib.Tactic.InferParam

inhabit

inhabit α tries to derive a Nonempty α instance and then uses it to make an Inhabited α instance. If the target is a Prop, this is done constructively. Otherwise, it uses Classical.choice.

Tags:
Defined in module:: Mathlib.Tactic.Inhabit

init_ring

init_ring is an auxiliary tactic that discharges goals factoring init over ring operations.

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Defined in module:: Mathlib.RingTheory.WittVector.InitTail

injection

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.

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Defined in module:: Init.Tactics

injections

injections applies injection to all hypotheses recursively (since injection can produce new hypotheses). Useful for destructing nested constructor equalities like (a::b::c) = (d::e::f).

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Defined in module:: Init.Tactics

interval_cases

interval_cases n searches for upper and lower bounds on a variable n, and if bounds are found, splits into separate cases for each possible value of n.

As an example, in

example (n : ℕ) (w₁ : n ≥ 3) (w₂ : n < 5) : n = 3 ∨ n = 4 := by
  interval_cases n
  all_goals simp

after interval_cases n, the goals are 3 = 3 ∨ 3 = 4 and 4 = 3 ∨ 4 = 4.

You can also explicitly specify a lower and upper bound to use, as interval_cases using hl, hu. The hypotheses should be in the form hl : a ≤ n and hu : n < b, in which case interval_cases calls fin_cases on the resulting fact n ∈ Set.Ico a b.

You can specify a name h for the new hypothesis, as interval_cases h : n or interval_cases h : n using hl, hu.

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Defined in module:: Mathlib.Tactic.IntervalCases

intro

The syntax intro. is deprecated in favor of nofun.

Tags:
Defined in module:: Batteries.Tactic.NoMatch

intro

Introduces one or more hypotheses, optionally naming and/or pattern-matching them. For each hypothesis to be introduced, the remaining main goal's target type must be a let or function type.

intro by itself introduces one anonymous hypothesis, which can be accessed by e.g. assumption. It is equivalent to intro _.
intro x y introduces two hypotheses and names them. Individual hypotheses can be anonymized via _, given a type ascription, or matched against a pattern:
```
-- ... ⊢ α × β → ...
intro (a, b)
-- ..., a : α, b : β ⊢ ...
```
intro rfl is short for intro h; subst h, if h is an equality where the left-hand or right-hand side is a variable.
Alternatively, intro can be combined with pattern matching much like fun:
```
intro
| n + 1, 0 => tac
| ...
```

Tags:
Defined in module:: Init.Tactics

intros

intros repeatedly applies intro to introduce zero or more hypotheses until the goal is no longer a binding expression (i.e., a universal quantifier, function type, implication, or have/let), without performing any definitional reductions (no unfolding, beta, eta, etc.). The introduced hypotheses receive inaccessible (hygienic) names.

intros x y z is equivalent to intro x y z and exists only for historical reasons. The intro tactic should be preferred in this case.

Properties and relations #

intros succeeds even when it introduces no hypotheses.
repeat intro is like intros, but it performs definitional reductions to expose binders, and as such it may introduce more hypotheses than intros.
intros is equivalent to intro _ _ … _, with the fewest trailing _ placeholders needed so that the goal is no longer a binding expression. The trailing introductions do not perform any definitional reductions.

Examples #

Implications:

example (p q : Prop) : p → q → p := by
  intros
  /- Tactic state
     a✝¹ : p
     a✝ : q
     ⊢ p      -/
  assumption

Let-bindings:

example : let n := 1; let k := 2; n + k = 3 := by
  intros
  /- n✝ : Nat := 1
     k✝ : Nat := 2
     ⊢ n✝ + k✝ = 3 -/
  rfl

Does not unfold definitions:

def AllEven (f : Nat → Nat) := ∀ n, f n % 2 = 0

example : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by
  intros
  /- Tactic state
     f✝ : Nat → Nat
     a✝ : AllEven f✝
     ⊢ AllEven fun k => f✝ (k + 1) -/
  sorry

Tags:
Defined in module:: Init.Tactics

introv

The tactic introv allows the user to automatically introduce the variables of a theorem and explicitly name the non-dependent hypotheses. Any dependent hypotheses are assigned their default names.

Examples:

example : ∀ a b : Nat, a = b → b = a := by
  introv h,
  exact h.symm

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 := by
  introv h₁ h₂,
  exact h₁.trans h₂

The state after introv h₁ h₂ is

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

Tags:
Defined in module:: Mathlib.Tactic.Basic

isBoundedDefault

Filters are automatically bounded or cobounded in complete lattices. To use the same statements in complete and conditionally complete lattices but let automation fill automatically the boundedness proofs in complete lattices, we use the tactic isBoundedDefault in the statements, in the form (hf : f.IsBounded (≥) := by isBoundedDefault).

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Defined in module:: Mathlib.Order.Filter.IsBounded

itauto

itauto solves the main goal when it is a tautology of intuitionistic propositional logic. Unlike grind and tauto! this tactic never uses the law of excluded middle (without the ! option), and the proof search is tailored for this use case. itauto is complete for intuitionistic propositional logic: it will solve any goal that is provable in this logic.

itauto [a, b] will additionally attempt case analysis on a and b assuming that it can derive Decidable a and Decidable b.
itauto * will case on all decidable propositions that it can find among the atomic propositions.
itauto! will work as a classical SAT solver, but the algorithm is not very good in this situation.
itauto! * will case on all propositional atoms. Warning: This can blow up the proof search, so it should be used sparingly.

Example:

example (p : Prop) : ¬ (p ↔ ¬ p) := by itauto

Tags:
Defined in module:: Mathlib.Tactic.ITauto

iterate

iterate n tac runs tac exactly n times. iterate tac runs tac repeatedly until failure.

iterate's argument is a tactic sequence, so multiple tactics can be run using iterate n (tac₁; tac₂; ⋯) or

iterate n
  tac₁
  tac₂
  ⋯

Tags:
Defined in module:: Init.TacticsExtra

left

Applies the first constructor when the goal is an inductive type with exactly two constructors, or fails otherwise.

example : True ∨ False := by
  left
  trivial

Tags:
Defined in module:: Init.Tactics

let

The let tactic is for adding definitions to the local context of the main goal. The definition can be unfolded, unlike definitions introduced by have.

let x : t := e adds the definition x : t := e if e is a term of type t.
let x := e uses the type of e for t.
let : t := e and let := e use this for the name of the hypothesis.
let pat := e for a pattern pat is equivalent to match e with | pat => _, where _ stands for the tactics that follow this one. It is convenient for types that let only one applicable constructor. For example, given p : α × β × γ, let ⟨x, y, z⟩ := p produces the local variables x : α, y : β, and z : γ.
The syntax let (eq := h) pat := e is equivalent to match h : e with | pat => _, which adds the equation h : e = pat to the local context.

The tactic supports all the same syntax variants and options as the let term.

Properties and relations #

Unlike have, it is possible to unfold definitions introduced using let, using tactics such as simp, dsimp, unfold, and subst.
The clear_value tactic turns a let definition into a have definition after the fact. The tactic might fail if the local context depends on the value of the variable.
The let tactic is preferred for data (non-propositions).
Sometimes have is used for non-propositions to ensure that the variable is never unfolded, which may be important for performance reasons.

Tags:
Defined in module:: Init.Tactics

let rec

let rec f : t := e adds a recursive definition f to the current goal. The syntax is the same as term-mode let rec.

The tactic supports all the same syntax variants and options as the let term.

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Defined in module:: Init.Tactics

let'

Similar to let, but using refine'

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Defined in module:: Init.Tactics

letI

letI behaves like let, but inlines the value instead of producing a let term.

Tags:
Defined in module:: Init.Tactics

let_to_have

Transforms let expressions into have expressions when possible.

let_to_have transforms lets in the target.
let_to_have at h transforms lets in the given local hypothesis.

Tags:
Defined in module:: Init.Tactics

lia

lia solves linear integer arithmetic goals.

It is a implemented as a thin wrapper around the grind tactic, enabling only the lia solver. Please use grind instead if you need additional capabilities.

Tags:
Defined in module:: Init.Grind.Tactics

lift

Lift an expression to another type.

Usage: 'lift' expr 'to' expr ('using' expr)? ('with' id (id id?)?)?.
If n : ℤ and hn : n ≥ 0 then the tactic lift n to ℕ using hn creates a new constant of type ℕ, also named n and replaces all occurrences of the old variable (n : ℤ) with ↑n (where n in the new variable). It will clear n from the context and try to clear hn from the context.
- So for example the tactic lift n to ℕ using hn transforms the goal n : ℤ, hn : n ≥ 0, h : P n ⊢ n = 3 to n : ℕ, h : P ↑n ⊢ ↑n = 3 (here P is some term of type ℤ → Prop).
The argument using hn is optional, the tactic lift n to ℕ does the same, but also creates a new subgoal that n ≥ 0 (where n is the old variable). This subgoal will be placed at the top of the goal list.
- So for example the tactic lift n to ℕ transforms the goal n : ℤ, h : P n ⊢ n = 3 to two goals n : ℤ, h : P n ⊢ n ≥ 0 and n : ℕ, h : P ↑n ⊢ ↑n = 3.
You can also use lift n to ℕ using e where e is any expression of type n ≥ 0.
Use lift n to ℕ with k to specify the name of the new variable.
Use lift n to ℕ with k hk to also specify the name of the equality ↑k = n. In this case, n will remain in the context. You can use rfl for the name of hk to substitute n away (i.e. the default behavior).
You can also use lift e to ℕ with k hk where e is any expression of type ℤ. In this case, the hk will always stay in the context, but it will be used to rewrite e in all hypotheses and the target.
- So for example the tactic lift n + 3 to ℕ using hn with k hk transforms the goal n : ℤ, hn : n + 3 ≥ 0, h : P (n + 3) ⊢ n + 3 = 2 * n to the goal n : ℤ, k : ℕ, hk : ↑k = n + 3, h : P ↑k ⊢ ↑k = 2 * n.
The tactic lift n to ℕ using h will remove h from the context. If you want to keep it, specify it again as the third argument to with, like this: lift n to ℕ using h with n rfl h.
More generally, this can lift an expression from α to β assuming that there is an instance of CanLift α β. In this case the proof obligation is specified by CanLift.prf.
Given an instance CanLift β γ, it can also lift α → β to α → γ; more generally, given β : Π a : α, Type*, γ : Π a : α, Type*, and [Π a : α, CanLift (β a) (γ a)], it automatically generates an instance CanLift (Π a, β a) (Π a, γ a).

lift is in some sense dual to the zify tactic. lift (z : ℤ) to ℕ will change the type of an integer z (in the supertype) to ℕ (the subtype), given a proof that z ≥ 0; propositions concerning z will still be over ℤ. zify changes propositions about ℕ (the subtype) to propositions about ℤ (the supertype), without changing the type of any variable.

The norm_cast tactic can be used after lift to normalize introduced casts.

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Defined in module:: Mathlib.Tactic.Lift

lift_lets

Lifts let and have expressions within a term as far out as possible. It is like extract_lets +lift, but the top-level lets at the end of the procedure are not extracted as local hypotheses.

lift_lets lifts let expressions in the target.
lift_lets at h lifts let expressions at the given local hypothesis.

For example,

example : (let x := 1; x) = 1 := by
  lift_lets
  -- ⊢ let x := 1; x = 1
  ...

Tags:
Defined in module:: Init.Tactics

liftable_prefixes

Internal tactic used in coherence.

Rewrites an equation f = g as f₀ ≫ f₁ = g₀ ≫ g₁, where f₀ and g₀ are maximal prefixes of f and g (possibly after reassociating) which are "liftable" (i.e. expressible as compositions of unitors and associators).

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Coherence

linarith

linarith attempts to find a contradiction between hypotheses that are linear (in)equalities. Equivalently, it can prove a linear inequality by assuming its negation and proving False.

In theory, linarith should prove any goal that is true in the theory of linear arithmetic over the rationals. While there is some special handling for non-dense orders like Nat and Int, this tactic is not complete for these theories and will not prove every true goal. It will solve goals over arbitrary types that instantiate CommRing, LinearOrder and IsStrictOrderedRing.

An example:

example (x y z : ℚ) (h1 : 2*x < 3*y) (h2 : -4*x + 2*z < 0)
        (h3 : 12*y - 4* z < 0) : False := by
  linarith

linarith will use all appropriate hypotheses and the negation of the goal, if applicable. Disequality hypotheses require case splitting and are not normally considered (see the splitNe option below).

linarith [t1, t2, t3] will additionally use proof terms t1, t2, t3.

linarith only [h1, h2, h3, t1, t2, t3] will use only the goal (if relevant), local hypotheses h1, h2, h3, and proofs t1, t2, t3. It will ignore the rest of the local context.

linarith! will use a stronger reducibility setting to try to identify atoms. For example,

example (x : ℚ) : id x ≥ x := by
  linarith

will fail, because linarith will not identify x and id x. linarith! will. This can sometimes be expensive.

linarith (config := { .. }) takes a config object with five optional arguments:

discharger specifies a tactic to be used for reducing an algebraic equation in the proof stage. The default is ring. Other options include simp for basic problems.
transparency controls how hard linarith will try to match atoms to each other. By default it will only unfold reducible definitions.
If splitHypotheses is true, linarith will split conjunctions in the context into separate hypotheses.
If splitNe is true, linarith will case split on disequality hypotheses. For a given x ≠ y hypothesis, linarith is run with both x < y and x > y, and so this runs linarith exponentially many times with respect to the number of disequality hypotheses. (false by default.)
If exfalso is false, linarith will fail when the goal is neither an inequality nor False. (true by default.)
If minimize is false, linarith? will report all hypotheses appearing in its initial proof without attempting to drop redundancies. (true by default.)
restrict_type (not yet implemented in mathlib4) will only use hypotheses that are inequalities over tp. This is useful if you have e.g. both integer- and rational-valued inequalities in the local context, which can sometimes confuse the tactic.

A variant, nlinarith, does some basic preprocessing to handle some nonlinear goals.

The option set_option trace.linarith true will trace certain intermediate stages of the linarith routine.

Tags:
Defined in module:: Mathlib.Tactic.Linarith.Frontend

linarith?

linarith? behaves like linarith but, on success, it prints a suggestion of the form linarith only [...] listing a minimized set of hypotheses used in the final proof. Use linarith?! for the higher-reducibility variant and set the minimize flag in the configuration to control whether greedy minimization is performed.

Tags:
Defined in module:: Mathlib.Tactic.Linarith.Frontend

linear_combination

The linear_combination tactic attempts to prove an (in)equality goal by exhibiting it as a specified linear combination of (in)equality hypotheses, or other (in)equality proof terms, modulo (A) moving terms between the LHS and RHS of the (in)equalities, and (B) a normalization tactic which by default is ring-normalization.

Example usage:

example {a b : ℚ} (h1 : a = 1) (h2 : b = 3) : (a + b) / 2 = 2 := by
  linear_combination (h1 + h2) / 2

example {a b : ℚ} (h1 : a ≤ 1) (h2 : b ≤ 3) : (a + b) / 2 ≤ 2 := by
  linear_combination (h1 + h2) / 2

example {a b : ℚ} : 2 * a * b ≤ a ^ 2 + b ^ 2 := by
  linear_combination sq_nonneg (a - b)

example {x y z w : ℤ} (h₁ : x * z = y ^ 2) (h₂ : y * w = z ^ 2) :
    z * (x * w - y * z) = 0 := by
  linear_combination w * h₁ + y * h₂

example {x : ℚ} (h : x ≥ 5) : x ^ 2 > 2 * x + 11 := by
  linear_combination (x + 3) * h

example {R : Type*} [CommRing R] {a b : R} (h : a = b) : a ^ 2 = b ^ 2 := by
  linear_combination (a + b) * h

example {A : Type*} [AddCommGroup A]
    {x y z : A} (h1 : x + y = 10 • z) (h2 : x - y = 6 • z) :
    2 • x = 2 • (8 • z) := by
  linear_combination (norm := abel) h1 + h2

example (x y : ℤ) (h1 : x * y + 2 * x = 1) (h2 : x = y) :
    x * y = -2 * y + 1 := by
  linear_combination (norm := ring_nf) -2 * h2
  -- leaves goal `⊢ x * y + x * 2 - 1 = 0`

The input e in linear_combination e is a linear combination of proofs of (in)equalities, given as a sum/difference of coefficients multiplied by expressions. The coefficients may be arbitrary expressions (with nonnegativity constraints in the case of inequalities). The expressions can be arbitrary proof terms proving (in)equalities; most commonly they are hypothesis names h1, h2, ....

The left and right sides of all the (in)equalities should have the same type α, and the coefficients should also have type α. For full functionality α should be a commutative ring -- strictly speaking, a commutative semiring with "cancellative" addition (in the semiring case, negation and subtraction will be handled "formally" as if operating in the enveloping ring). If a nonstandard normalization is used (for example abel or skip), the tactic will work over types α with less algebraic structure: for equalities, the minimum is instances of [Add α] [IsRightCancelAdd α] together with instances of whatever operations are used in the tactic call.

The variant linear_combination (norm := tac) e specifies explicitly the "normalization tactic" tac to be run on the subgoal(s) after constructing the linear combination.

The default normalization tactic is ring1 (for equalities) or Mathlib.Tactic.Ring.prove{LE,LT} (for inequalities). These are finishing tactics: they close the goal or fail.
When working in algebraic categories other than commutative rings -- for example fields, abelian groups, modules -- it is sometimes useful to use normalization tactics adapted to those categories (field_simp, abel, module).
To skip normalization entirely, use skip as the normalization tactic.
The linear_combination tactic creates a linear combination by adding the provided (in)equalities together from left to right, so if tac is not invariant under commutation of additive expressions, then the order of the input hypotheses can matter.

The variant linear_combination (exp := n) e will take the goal to the nth power before subtracting the combination e. In other words, if the goal is t1 = t2, linear_combination (exp := n) e will change the goal to (t1 - t2)^n = 0 before proceeding as above. This variant is implemented only for linear combinations of equalities (i.e., not for inequalities).

Tags:
Defined in module:: Mathlib.Tactic.LinearCombination

linear_combination'

linear_combination' attempts to simplify the target by creating a linear combination of a list of equalities and subtracting it from the target. The tactic will create a linear combination by adding the equalities together from left to right, so the order of the input hypotheses does matter. If the norm field of the tactic is set to skip, then the tactic will simply set the user up to prove their target using the linear combination instead of normalizing the subtraction.

Note: There is also a similar tactic linear_combination (no prime); this version is provided for backward compatibility. Compared to this tactic, linear_combination:

drops the ← syntax for reversing an equation, instead offering this operation using the - syntax
does not support multiplication of two hypotheses (h1 * h2), division by a hypothesis (3 / h), or inversion of a hypothesis (h⁻¹)
produces noisy output when the user adds or subtracts a constant to a hypothesis (h + 3)

Note: The left and right sides of all the equalities should have the same type, and the coefficients should also have this type. There must be instances of Mul and AddGroup for this type.

The input e in linear_combination' e is a linear combination of proofs of equalities, given as a sum/difference of coefficients multiplied by expressions. The coefficients may be arbitrary expressions. The expressions can be arbitrary proof terms proving equalities. Most commonly they are hypothesis names h1, h2, ....
linear_combination' (norm := tac) e runs the "normalization tactic" tac on the subgoal(s) after constructing the linear combination.
- The default normalization tactic is ring1, which closes the goal or fails.
- To get a subgoal in the case that it is not immediately provable, use ring_nf as the normalization tactic.
- To avoid normalization entirely, use skip as the normalization tactic.
linear_combination' (exp := n) e will take the goal to the nth power before subtracting the combination e. In other words, if the goal is t1 = t2, linear_combination' (exp := n) e will change the goal to (t1 - t2)^n = 0 before proceeding as above. This feature is not supported for linear_combination2.
linear_combination2 e is the same as linear_combination' e but it produces two subgoals instead of one: rather than proving that (a - b) - (a' - b') = 0 where a' = b' is the linear combination from e and a = b is the goal, it instead attempts to prove a = a' and b = b'. Because it does not use subtraction, this form is applicable also to semirings.
- Note that a goal which is provable by linear_combination' e may not be provable by linear_combination2 e; in general you may need to add a coefficient to e to make both sides match, as in linear_combination2 e + c.
- You can also reverse equalities using ← h, so for example if h₁ : a = b then 2 * (← h) is a proof of 2 * b = 2 * a.

Example Usage:

example (x y : ℤ) (h1 : x*y + 2*x = 1) (h2 : x = y) : x*y = -2*y + 1 := by
  linear_combination' 1*h1 - 2*h2

example (x y : ℤ) (h1 : x*y + 2*x = 1) (h2 : x = y) : x*y = -2*y + 1 := by
  linear_combination' h1 - 2*h2

example (x y : ℤ) (h1 : x*y + 2*x = 1) (h2 : x = y) : x*y = -2*y + 1 := by
  linear_combination' (norm := ring_nf) -2*h2
  /- Goal: x * y + x * 2 - 1 = 0 -/

example (x y z : ℝ) (ha : x + 2*y - z = 4) (hb : 2*x + y + z = -2)
    (hc : x + 2*y + z = 2) :
    -3*x - 3*y - 4*z = 2 := by
  linear_combination' ha - hb - 2*hc

example (x y : ℚ) (h1 : x + y = 3) (h2 : 3*x = 7) :
    x*x*y + y*x*y + 6*x = 3*x*y + 14 := by
  linear_combination' x*y*h1 + 2*h2

example (x y : ℤ) (h1 : x = -3) (h2 : y = 10) : 2*x = -6 := by
  linear_combination' (norm := skip) 2*h1
  simp

axiom qc : ℚ
axiom hqc : qc = 2*qc

example (a b : ℚ) (h : ∀ p q : ℚ, p = q) : 3*a + qc = 3*b + 2*qc := by
  linear_combination' 3 * h a b + hqc

Tags:
Defined in module:: Mathlib.Tactic.LinearCombination'

map_fun_tac

Auxiliary tactic for showing that mapFun respects the ring operations.

Tags:
Defined in module:: Mathlib.RingTheory.WittVector.Basic

map_tacs

Assuming there are n goals, map_tacs [t1; t2; ...; tn] applies each ti to the respective goal and leaves the resulting subgoals.

Tags:
Defined in module:: Batteries.Tactic.SeqFocus

massumption

massumption is like assumption, but operating on a stateful Std.Do.SPred goal.

example (P Q : SPred σs) : Q ⊢ₛ P → Q := by
  mintro _ _
  massumption

Tags:
Defined in module:: Init.Tactics

match

The syntax match ⋯ with. has been deprecated in favor of nomatch ⋯.

Both now support multiple discriminants.

Tags:
Defined in module:: Batteries.Tactic.NoMatch

match

match performs case analysis on one or more expressions. See Induction and Recursion. The syntax for the match tactic is the same as term-mode match, except that the match arms are tactics instead of expressions.

example (n : Nat) : n = n := by
  match n with
  | 0 => rfl
  | i+1 => simp

Tags:
Defined in module:: Lean.Parser.Tactic

match_scalars

Given a goal which is an equality in a type M (with M an AddCommMonoid), parse the LHS and RHS of the goal as linear combinations of M-atoms over some semiring R, and reduce the goal to the respective equalities of the R-coefficients of each atom.

For example, this produces the goal ⊢ a * 1 + b * 1 = (b + a) * 1:

example [AddCommMonoid M] [Semiring R] [Module R M] (a b : R) (x : M) :
    a • x + b • x = (b + a) • x := by
  match_scalars

This produces the two goals ⊢ a * (a * 1) + b * (b * 1) = 1 (from the x atom) and ⊢ a * -(b * 1) + b * (a * 1) = 0 (from the y atom):

example [AddCommGroup M] [Ring R] [Module R M] (a b : R) (x : M) :
    a • (a • x - b • y) + (b • a • y + b • b • x) = x := by
  match_scalars

This produces the goal ⊢ -2 * (a * 1) = a * (-2 * 1):

example [AddCommGroup M] [Ring R] [Module R M] (a : R) (x : M) :
    -(2:R) • a • x = a • (-2:ℤ) • x  := by
  match_scalars

The scalar type for the goals produced by the match_scalars tactic is the largest scalar type encountered; for example, if ℕ, ℚ and a characteristic-zero field K all occur as scalars, then the goals produced are equalities in K. A variant of push_cast is used internally in match_scalars to interpret scalars from the other types in this largest type.

If the set of scalar types encountered is not totally ordered (in the sense that for all rings R, S encountered, it holds that either Algebra R S or Algebra S R), then the match_scalars tactic fails.

Tags:
Defined in module:: Mathlib.Tactic.Module

match_target

Deprecated: use guard_target =~ t instead.

Tags:
Defined in module:: Mathlib.Tactic.Basic

mcases

Like rcases, but operating on stateful Std.Do.SPred goals. Example: Given a goal h : (P ∧ (Q ∨ R) ∧ (Q → R)) ⊢ₛ R, mcases h with ⟨-, ⟨hq | hr⟩, hqr⟩ will yield two goals: (hq : Q, hqr : Q → R) ⊢ₛ R and (hr : R) ⊢ₛ R.

That is, mcases h with pat has the following semantics, based on pat:

pat=□h' renames h to h' in the stateful context, regardless of whether h is pure
pat=⌜h'⌝ introduces h' : φ to the pure local context if h : ⌜φ⌝ (c.f. Lean.Elab.Tactic.Do.ProofMode.IsPure)
pat=h' is like pat=⌜h'⌝ if h is pure (c.f. Lean.Elab.Tactic.Do.ProofMode.IsPure), otherwise it is like pat=□h'.
pat=_ renames h to an inaccessible name
pat=- discards h
⟨pat₁, pat₂⟩ matches on conjunctions and existential quantifiers and recurses via pat₁ and pat₂.
⟨pat₁ | pat₂⟩ matches on disjunctions, matching the left alternative via pat₁ and the right alternative via pat₂.

Tags:
Defined in module:: Init.Tactics

mclear

mclear is like clear, but operating on a stateful Std.Do.SPred goal.

example (P Q : SPred σs) : P ⊢ₛ Q → Q := by
  mintro HP
  mintro HQ
  mclear HP
  mexact HQ

Tags:
Defined in module:: Init.Tactics

mconstructor

mconstructor is like constructor, but operating on a stateful Std.Do.SPred goal.

example (Q : SPred σs) : Q ⊢ₛ Q ∧ Q := by
  mintro HQ
  mconstructor <;> mexact HQ

Tags:
Defined in module:: Init.Tactics

mdup

Duplicate a stateful Std.Do.SPred hypothesis.

Tags:
Defined in module:: Std.Tactic.Do.Syntax

measurability

The tactic measurability solves goals of the form Measurable f, AEMeasurable f, StronglyMeasurable f, AEStronglyMeasurable f μ, or MeasurableSet s by applying lemmas tagged with the measurability user attribute.

Note that measurability uses fun_prop for solving measurability of functions, so statements about Measurable, AEMeasurable, StronglyMeasurable and AEStronglyMeasurable should be tagged with fun_prop rather that measurability. The measurability attribute is equivalent to fun_prop in these cases for backward compatibility with the earlier implementation.

Tags:
Defined in module:: Mathlib.Tactic.Measurability

measurability?

The tactic measurability? solves goals of the form Measurable f, AEMeasurable f, StronglyMeasurable f, AEStronglyMeasurable f μ, or MeasurableSet s by applying lemmas tagged with the measurability user attribute, and suggests a faster proof script that can be substituted for the tactic call in case of success.

Tags:
Defined in module:: Mathlib.Tactic.Measurability

mem_tac

mem_tac tries to prove goals of the form x ∈ 𝒜 i when x has the form of:

y ^ n where i = n • j and y ∈ 𝒜 j.
a natural number n.

Tags:
Defined in module:: Mathlib.AlgebraicGeometry.ProjectiveSpectrum.Scheme

mexact

mexact is like exact, but operating on a stateful Std.Do.SPred goal.

example (Q : SPred σs) : Q ⊢ₛ Q := by
  mstart
  mintro HQ
  mexact HQ

Tags:
Defined in module:: Init.Tactics

mexfalso

mexfalso is like exfalso, but operating on a stateful Std.Do.SPred goal.

example (P : SPred σs) : ⌜False⌝ ⊢ₛ P := by
  mintro HP
  mexfalso
  mexact HP

Tags:
Defined in module:: Init.Tactics

mexists

mexists is like exists, but operating on a stateful Std.Do.SPred goal.

example (ψ : Nat → SPred σs) : ψ 42 ⊢ₛ ∃ x, ψ x := by
  mintro H
  mexists 42

Tags:
Defined in module:: Init.Tactics

mfld_set_tac

A very basic tactic to show that sets showing up in manifolds coincide or are included in one another.

Tags:
Defined in module:: Mathlib.Logic.Equiv.PartialEquiv

mframe

mframe infers which hypotheses from the stateful context can be moved into the pure context. This is useful because pure hypotheses "survive" the next application of modus ponens (Std.Do.SPred.mp) and transitivity (Std.Do.SPred.entails.trans).

It is used as part of the mspec tactic.

example (P Q : SPred σs) : ⊢ₛ ⌜p⌝ ∧ Q ∧ ⌜q⌝ ∧ ⌜r⌝ ∧ P ∧ ⌜s⌝ ∧ ⌜t⌝ → Q := by
  mintro _
  mframe
  /- `h : p ∧ q ∧ r ∧ s ∧ t` in the pure context -/
  mcases h with hP
  mexact h

Tags:
Defined in module:: Init.Tactics

mhave

mhave is like have, but operating on a stateful Std.Do.SPred goal.

example (P Q : SPred σs) : P ⊢ₛ (P → Q) → Q := by
  mintro HP HPQ
  mhave HQ : Q := by mspecialize HPQ HP; mexact HPQ
  mexact HQ

Tags:
Defined in module:: Init.Tactics

mintro

Like intro, but introducing stateful hypotheses into the stateful context of the Std.Do.SPred proof mode. That is, given a stateful goal (hᵢ : Hᵢ)* ⊢ₛ P → T, mintro h transforms into (hᵢ : Hᵢ)*, (h : P) ⊢ₛ T.

Furthermore, mintro ∀s is like intro s, but preserves the stateful goal. That is, mintro ∀s brings the topmost state variable s:σ in scope and transforms (hᵢ : Hᵢ)* ⊢ₛ T (where the entailment is in Std.Do.SPred (σ::σs)) into (hᵢ : Hᵢ s)* ⊢ₛ T s (where the entailment is in Std.Do.SPred σs).

Beyond that, mintro supports the full syntax of mcases patterns (mintro pat = (mintro h; mcases h with pat), and can perform multiple introductions in sequence.

Tags:
Defined in module:: Init.Tactics

mleave

Leaves the stateful proof mode of Std.Do.SPred, tries to eta-expand through all definitions related to the logic of the Std.Do.SPred and gently simplifies the resulting pure Lean proposition. This is often the right thing to do after mvcgen in order for automation to prove the goal.

Tags:
Defined in module:: Init.Tactics

mleft

mleft is like left, but operating on a stateful Std.Do.SPred goal.

example (P Q : SPred σs) : P ⊢ₛ P ∨ Q := by
  mintro HP
  mleft
  mexact HP

Tags:
Defined in module:: Init.Tactics

mod_cases

The tactic mod_cases h : e % 3 will perform a case disjunction on e. If e : ℤ, then it will yield subgoals containing the assumptions h : e ≡ 0 [ZMOD 3], h : e ≡ 1 [ZMOD 3], h : e ≡ 2 [ZMOD 3] respectively. If e : ℕ instead, then it works similarly, except with [MOD 3] instead of [ZMOD 3].
In general, mod_cases h : e % n works when n is a positive numeral and e is an expression of type ℕ or ℤ.
If h is omitted as in mod_cases e % n, it will be default-named H.

Tags:
Defined in module:: Mathlib.Tactic.ModCases

module

Given a goal which is an equality in a type M (with M an AddCommMonoid), parse the LHS and RHS of the goal as linear combinations of M-atoms over some commutative semiring R, and prove the goal by checking that the LHS- and RHS-coefficients of each atom are the same up to ring-normalization in R.

(If the proofs of coefficient-wise equality will require more reasoning than just ring-normalization, use the tactic match_scalars instead, and then prove coefficient-wise equality by hand.)

Example uses of the module tactic:

example [AddCommMonoid M] [CommSemiring R] [Module R M] (a b : R) (x : M) :
    a • x + b • x = (b + a) • x := by
  module

example [AddCommMonoid M] [Field K] [CharZero K] [Module K M] (x : M) :
    (2:K)⁻¹ • x + (3:K)⁻¹ • x + (6:K)⁻¹ • x = x := by
  module

example [AddCommGroup M] [CommRing R] [Module R M] (a : R) (v w : M) :
    (1 + a ^ 2) • (v + w) - a • (a • v - w) = v + (1 + a + a ^ 2) • w := by
  module

example [AddCommGroup M] [CommRing R] [Module R M] (a b μ ν : R) (x y : M) :
    (μ - ν) • a • x = (a • μ • x + b • ν • y) - ν • (a • x + b • y) := by
  module

Tags:
Defined in module:: Mathlib.Tactic.Module

monicity

monicity tries to solve a goal of the form Monic f. It converts the goal into a goal of the form natDegree f ≤ n and one of the form f.coeff n = 1 and calls compute_degree on those two goals.

The variant monicity! starts like monicity, but calls compute_degree! on the two side-goals.

Tags:
Defined in module:: Mathlib.Tactic.ComputeDegree

mono

mono applies monotonicity rules and local hypotheses repetitively. For example,

example (x y z k : ℤ)
    (h : 3 ≤ (4 : ℤ))
    (h' : z ≤ y) :
    (k + 3 + x) - y ≤ (k + 4 + x) - z := by
  mono

Tags:
Defined in module:: Mathlib.Tactic.Monotonicity.Basic

monoidal

Use the coherence theorem for monoidal categories to solve equations in a monoidal category, where the two sides only differ by replacing strings of monoidal structural morphisms (that is, associators, unitors, and identities) with different strings of structural morphisms with the same source and target.

That is, monoidal can handle goals of the form a ≫ f ≫ b ≫ g ≫ c = a' ≫ f ≫ b' ≫ g ≫ c' where a = a', b = b', and c = c' can be proved using monoidal_coherence.

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Monoidal.Basic

monoidal_coherence

Close the goal of the form η = θ, where η and θ are 2-isomorphisms made up only of associators, unitors, and identities.

example {C : Type} [Category* C] [MonoidalCategory C] :
  (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by
  monoidal_coherence

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Monoidal.PureCoherence

monoidal_coherence

Coherence tactic for monoidal categories. Use pure_coherence instead, which is a frontend to this one.

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Coherence

monoidal_nf

Normalize the both sides of an equality.

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Monoidal.Basic

monoidal_simps

Simp lemmas for rewriting a hom in monoidal categories into a normal form.

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Coherence

move_oper

The tactic move_add rearranges summands of expressions. Calling move_add [a, ← b, ...] matches a, b,... with summands in the main goal. It then moves a to the far right and b to the far left of each addition in which they appear. The side to which the summands are moved is determined by the presence or absence of the arrow ←.

The inputs a, b,... can be any terms, also with underscores. The tactic uses the first "new" summand that unifies with each one of the given inputs.

There is a multiplicative variant, called move_mul.

There is also a general tactic for a "binary associative commutative operation": move_oper. In this case the syntax requires providing first a term whose head symbol is the operation. E.g. move_oper HAdd.hAdd [...] is the same as move_add, while move_oper Max.max [...] rearranges maxs.

Tags:
Defined in module:: Mathlib.Tactic.MoveAdd

mpure

mpure moves a pure hypothesis from the stateful context into the pure context.

example (Q : SPred σs) (ψ : φ → ⊢ₛ Q): ⌜φ⌝ ⊢ₛ Q := by
  mintro Hφ
  mpure Hφ
  mexact (ψ Hφ)

Tags:
Defined in module:: Init.Tactics

mpure_intro

mpure_intro operates on a stateful Std.Do.SPred goal of the form P ⊢ₛ ⌜φ⌝. It leaves the stateful proof mode (thereby discarding P), leaving the regular goal φ.

theorem simple : ⊢ₛ (⌜True⌝ : SPred σs) := by
  mpure_intro
  exact True.intro

Tags:
Defined in module:: Init.Tactics

mrefine

Like refine, but operating on stateful Std.Do.SPred goals.

example (P Q R : SPred σs) : (P ∧ Q ∧ R) ⊢ₛ P ∧ R := by
  mintro ⟨HP, HQ, HR⟩
  mrefine ⟨HP, HR⟩

example (ψ : Nat → SPred σs) : ψ 42 ⊢ₛ ∃ x, ψ x := by
  mintro H
  mrefine ⟨⌜42⌝, H⟩

Tags:
Defined in module:: Init.Tactics

mrename_i

mrename_i is like rename_i, but names inaccessible stateful hypotheses in a Std.Do.SPred goal.

Tags:
Defined in module:: Init.Tactics

mreplace

mreplace is like replace, but operating on a stateful Std.Do.SPred goal.

example (P Q : SPred σs) : P ⊢ₛ (P → Q) → Q := by
  mintro HP HPQ
  mreplace HPQ : Q := by mspecialize HPQ HP; mexact HPQ
  mexact HPQ

Tags:
Defined in module:: Init.Tactics

mrevert

mrevert is like revert, but operating on a stateful Std.Do.SPred goal.

example (P Q R : SPred σs) : P ∧ Q ∧ R ⊢ₛ P → R := by
  mintro ⟨HP, HQ, HR⟩
  mrevert HR
  mrevert HP
  mintro HP'
  mintro HR'
  mexact HR'

Tags:
Defined in module:: Init.Tactics

mright

mright is like right, but operating on a stateful Std.Do.SPred goal.

example (P Q : SPred σs) : P ⊢ₛ Q ∨ P := by
  mintro HP
  mright
  mexact HP

Tags:
Defined in module:: Init.Tactics

mspec

mspec is an apply-like tactic that applies a Hoare triple specification to the target of the stateful goal.

Given a stateful goal H ⊢ₛ wp⟦prog⟧ Q', mspec foo_spec will instantiate foo_spec : ... → ⦃P⦄ foo ⦃Q⦄, match foo against prog and produce subgoals for the verification conditions ?pre : H ⊢ₛ P and ?post : Q ⊢ₚ Q'.

If prog = x >>= f, then mspec Specs.bind is tried first so that foo is matched against x instead. Tactic mspec_no_bind does not attempt to do this decomposition.
If ?pre or ?post follow by .rfl, then they are discharged automatically.
?post is automatically simplified into constituent ⊢ₛ entailments on success and failure continuations.
?pre and ?post.* goals introduce their stateful hypothesis under an inaccessible name. You can give it a name with the mrename_i tactic.
Any uninstantiated MVar arising from instantiation of foo_spec becomes a new subgoal.
If the target of the stateful goal looks like fun s => _ then mspec will first mintro ∀s.
If P has schematic variables that can be instantiated by doing mintro ∀s, for example foo_spec : ∀(n:Nat), ⦃fun s => ⌜n = s⌝⦄ foo ⦃Q⦄, then mspec will do mintro ∀s first to instantiate n = s.
Right before applying the spec, the mframe tactic is used, which has the following effect: Any hypothesis Hᵢ in the goal h₁:H₁, h₂:H₂, ..., hₙ:Hₙ ⊢ₛ T that is pure (i.e., equivalent to some ⌜φᵢ⌝) will be moved into the pure context as hᵢ:φᵢ.

Additionally, mspec can be used without arguments or with a term argument:

mspec without argument will try and look up a spec for x registered with @[spec].
mspec (foo_spec blah ?bleh) will elaborate its argument as a term with expected type ⦃?P⦄ x ⦃?Q⦄ and introduce ?bleh as a subgoal. This is useful to pass an invariant to e.g., Specs.forIn_list and leave the inductive step as a hole.

Tags:
Defined in module:: Init.Tactics

mspec_no_bind

mspec_no_simp $spec first tries to decompose Bind.binds before applying $spec. This variant of mspec_no_simp does not; mspec_no_bind $spec is defined as

try with_reducible mspec_no_bind Std.Do.Spec.bind
mspec_no_bind $spec

Tags:
Defined in module:: Std.Tactic.Do.Syntax

mspec_no_simp

Like mspec, but does not attempt slight simplification and closing of trivial sub-goals. mspec $spec is roughly (the set of simp lemmas below might not be up to date)

mspec_no_simp $spec
all_goals
  ((try simp only [SPred.true_intro_simp, SPred.apply_pure]);
   (try mpure_intro; trivial))

Tags:
Defined in module:: Std.Tactic.Do.Syntax

mspecialize

mspecialize is like specialize, but operating on a stateful Std.Do.SPred goal. It specializes a hypothesis from the stateful context with hypotheses from either the pure or stateful context or pure terms.

example (P Q : SPred σs) : P ⊢ₛ (P → Q) → Q := by
  mintro HP HPQ
  mspecialize HPQ HP
  mexact HPQ

example (y : Nat) (P Q : SPred σs) (Ψ : Nat → SPred σs) (hP : ⊢ₛ P) : ⊢ₛ Q → (∀ x, P → Q → Ψ x) → Ψ (y + 1) := by
  mintro HQ HΨ
  mspecialize HΨ (y + 1) hP HQ
  mexact HΨ

Tags:
Defined in module:: Init.Tactics

mspecialize_pure

mspecialize_pure is like mspecialize, but it specializes a hypothesis from the pure context with hypotheses from either the pure or stateful context or pure terms.

example (y : Nat) (P Q : SPred σs) (Ψ : Nat → SPred σs) (hP : ⊢ₛ P) (hΨ : ∀ x, ⊢ₛ P → Q → Ψ x) : ⊢ₛ Q → Ψ (y + 1) := by
  mintro HQ
  mspecialize_pure (hΨ (y + 1)) hP HQ => HΨ
  mexact HΨ

Tags:
Defined in module:: Init.Tactics

mstart

Start the stateful proof mode of Std.Do.SPred. This will transform a stateful goal of the form H ⊢ₛ T into ⊢ₛ H → T upon which mintro can be used to re-introduce H and give it a name. It is often more convenient to use mintro directly, which will try mstart automatically if necessary.

Tags:
Defined in module:: Init.Tactics

mstop

Stops the stateful proof mode of Std.Do.SPred. This will simply forget all the names given to stateful hypotheses and pretty-print a bit differently.

Tags:
Defined in module:: Init.Tactics

mv_bisim

tactic for proof by bisimulation

Tags:
Defined in module:: Mathlib.Data.QPF.Multivariate.Constructions.Cofix

mvcgen

mvcgen will break down a Hoare triple proof goal like ⦃P⦄ prog ⦃Q⦄ into verification conditions, provided that all functions used in prog have specifications registered with @[spec].

Verification Conditions and specifications #

A verification condition is an entailment in the stateful logic of Std.Do.SPred in which the original program prog no longer occurs. Verification conditions are introduced by the mspec tactic; see the mspec tactic for what they look like. When there's no applicable mspec spec, mvcgen will try and rewrite an application prog = f a b c with the simp set registered via @[spec].

Features #

When used like mvcgen +noLetElim [foo_spec, bar_def, instBEqFloat], mvcgen will additionally

add a Hoare triple specification foo_spec : ... → ⦃P⦄ foo ... ⦃Q⦄ to spec set for a function foo occurring in prog,
unfold a definition def bar_def ... := ... in prog,
unfold any method of the instBEqFloat : BEq Float instance in prog.
it will no longer substitute away let-expressions that occur at most once in P, Q or prog.

Config options #

+noLetElim is just one config option of many. Check out Lean.Elab.Tactic.Do.VCGen.Config for all options. Of particular note is stepLimit = some 42, which is useful for bisecting bugs in mvcgen and tracing its execution.

Extended syntax #

Often, mvcgen will be used like this:

mvcgen [...]
case inv1 => by exact I1
case inv2 => by exact I2
all_goals (mleave; try grind)

There is special syntax for this:

mvcgen [...] invariants
· I1
· I2
with grind

When I1 and I2 need to refer to inaccessibles (mvcgen will introduce a lot of them for program variables), you can use case label syntax:

mvcgen [...] invariants
| inv1 _ acc _ => I1 acc
| _ => I2
with grind

This is more convenient than the equivalent · by rename_i _ acc _; exact I1 acc.

Invariant suggestions #

mvcgen will suggest invariants for you if you use the invariants? keyword.

mvcgen [...] invariants?

This is useful if you do not recall the exact syntax to construct invariants. Furthermore, it will suggest a concrete invariant encoding "this holds at the start of the loop and this must hold at the end of the loop" by looking at the corresponding VCs. Although the suggested invariant is a good starting point, it is too strong and requires users to interpolate it such that the inductive step can be proved. Example:

def mySum (l : List Nat) : Nat := Id.run do
  let mut acc := 0
  for x in l do
    acc := acc + x
  return acc

/--
info: Try this:
  invariants
    · ⇓⟨xs, letMuts⟩ => ⌜xs.prefix = [] ∧ letMuts = 0 ∨ xs.suffix = [] ∧ letMuts = l.sum⌝
-/
#guard_msgs (info) in
theorem mySum_suggest_invariant (l : List Nat) : mySum l = l.sum := by
  generalize h : mySum l = r
  apply Id.of_wp_run_eq h
  mvcgen invariants?
  all_goals admit

Tags:
Defined in module:: Init.Tactics

mvcgen?

A hint tactic that expands to mvcgen invariants?.

Tags:
Defined in module:: Std.Tactic.Do.Syntax

mvcgen_trivial

mvcgen_trivial is the tactic automatically called by mvcgen to discharge VCs. It tries to discharge the VC by applying (try mpure_intro); trivial and otherwise delegates to mvcgen_trivial_extensible. Users are encouraged to extend mvcgen_trivial_extensible instead of this tactic in order not to override the default (try mpure_intro); trivial behavior.

Tags:
Defined in module:: Std.Tactic.Do.Syntax

mvcgen_trivial_extensible

This tactic has no documentation.

Tags:
Defined in module:: Std.Tactic.Do.Syntax

native_decide

native_decide is a synonym for decide +native. It will attempt to prove a goal of type p by synthesizing an instance of Decidable p and then evaluating it to isTrue ... Unlike decide, this uses #eval to evaluate the decidability instance.

This should be used with care because it adds the entire lean compiler to the trusted part, and the axiom Lean.ofReduceBool will show up in #print axioms for theorems using this method or anything that transitively depends on them. Nevertheless, because it is compiled, this can be significantly more efficient than using decide, and for very large computations this is one way to run external programs and trust the result.

example : (List.range 1000).length = 1000 := by native_decide

Tags:
Defined in module:: Init.Tactics

next => tac focuses on the next goal and solves it using tac, or else fails. next x₁ ... xₙ => tac additionally renames the n most recent hypotheses with inaccessible names to the given names.

Tags:
Defined in module:: Init.Tactics

nlinarith

An extension of linarith with some preprocessing to allow it to solve some nonlinear arithmetic problems. (Based on Coq's nra tactic.) See linarith for the available syntax of options, which are inherited by nlinarith; that is, nlinarith! and nlinarith only [h1, h2] all work as in linarith. The preprocessing is as follows:

For every subterm a ^ 2 or a * a in a hypothesis or the goal, the assumption 0 ≤ a ^ 2 or 0 ≤ a * a is added to the context.
For every pair of hypotheses a1 R1 b1, a2 R2 b2 in the context, R1, R2 ∈ {<, ≤, =}, the assumption 0 R' (b1 - a1) * (b2 - a2) is added to the context (non-recursively), where R ∈ {<, ≤, =} is the appropriate comparison derived from R1, R2.

Tags:
Defined in module:: Mathlib.Tactic.Linarith.Frontend

nofun

The tactic nofun is shorthand for exact nofun: it introduces the assumptions, then performs an empty pattern match, closing the goal if the introduced pattern is impossible.

Tags:
Defined in module:: Init.Tactics

nomatch

The tactic nomatch h is shorthand for exact nomatch h.

Tags:
Defined in module:: Init.Tactics

noncomm_ring

A tactic for simplifying identities in not-necessarily-commutative rings.

An example:

example {R : Type*} [Ring R] (a b c : R) : a * (b + c + c - b) = 2 * a * c := by
  noncomm_ring

You can use noncomm_ring [h] to also simplify using h.

Tags:
Defined in module:: Mathlib.Tactic.NoncommRing

nontriviality

Attempts to generate a Nontrivial α hypothesis.

The tactic first checks to see that there is not already a Nontrivial α instance before trying to synthesize one using other techniques.

If the goal is an (in)equality, the type α is inferred from the goal. Otherwise, the type needs to be specified in the tactic invocation, as nontriviality α.

The nontriviality tactic will first look for strict inequalities amongst the hypotheses, and use these to derive the Nontrivial instance directly.

Otherwise, it will perform a case split on Subsingleton α ∨ Nontrivial α, and attempt to discharge the Subsingleton goal using simp [h₁, h₂, ..., hₙ, nontriviality], where [h₁, h₂, ..., hₙ] is a list of additional simp lemmas that can be passed to nontriviality using the syntax nontriviality α using h₁, h₂, ..., hₙ.

example {R : Type} [OrderedRing R] {a : R} (h : 0 < a) : 0 < a := by
  nontriviality -- There is now a `Nontrivial R` hypothesis available.
  assumption

example {R : Type} [CommRing R] {r s : R} : r * s = s * r := by
  nontriviality -- There is now a `Nontrivial R` hypothesis available.
  apply mul_comm

example {R : Type} [OrderedRing R] {a : R} (h : 0 < a) : (2 : ℕ) ∣ 4 := by
  nontriviality R -- there is now a `Nontrivial R` hypothesis available.
  dec_trivial

def myeq {α : Type} (a b : α) : Prop := a = b

example {α : Type} (a b : α) (h : a = b) : myeq a b := by
  success_if_fail nontriviality α -- Fails
  nontriviality α using myeq -- There is now a `Nontrivial α` hypothesis available
  assumption

Tags:
Defined in module:: Mathlib.Tactic.Nontriviality.Core

norm_cast

The norm_cast family of tactics is used to normalize certain coercions (casts) in expressions.

norm_cast normalizes casts in the target.
norm_cast at h normalizes casts in hypothesis h.

The tactic is basically a version of simp with a specific set of lemmas to move casts upwards in the expression. Therefore even in situations where non-terminal simp calls are discouraged (because of fragility), norm_cast is considered to be safe. It also has special handling of numerals.

For instance, given an assumption

a b : ℤ
h : ↑a + ↑b < (10 : ℚ)

writing norm_cast at h will turn h into

h : a + b < 10

There are also variants of basic tactics that use norm_cast to normalize expressions during their operation, to make them more flexible about the expressions they accept (we say that it is a tactic modulo the effects of norm_cast):

exact_mod_cast for exact and apply_mod_cast for apply. Writing exact_mod_cast h and apply_mod_cast h will normalize casts in the goal and h before using exact h or apply h.
rw_mod_cast for rw. It applies norm_cast between rewrites.
assumption_mod_cast for assumption. This is effectively norm_cast at *; assumption, but more efficient. It normalizes casts in the goal and, for every hypothesis h in the context, it will try to normalize casts in h and use exact h.

See also push_cast, which moves casts inwards rather than lifting them outwards.

Tags:
Defined in module:: Init.Tactics

norm_cast0

Implementation of norm_cast (the full norm_cast calls trivial afterwards).

Tags:
Defined in module:: Init.Tactics

norm_num

norm_num normalizes numerical expressions in the goal. By default, it supports the operations + - * / ⁻¹ ^ and % over types with (at least) an AddMonoidWithOne instance, such as ℕ, ℤ, ℚ, ℝ, ℂ. In addition to evaluating numerical expressions, norm_num will use simp to simplify the goal. If the goal has the form A = B, A ≠ B, A < B or A ≤ B, where A and B are numerical expressions, norm_num will try to close it. It also has a relatively simple primality prover.

This tactic is extensible. Extensions can allow norm_num to evaluate more kinds of expressions, or to prove more kinds of propositions. See the @[norm_num] attribute for further information on extending norm_num.

norm_num at l normalizes at location(s) l.
norm_num [h1, ...] adds the arguments h1, ... to the simp set in addition to the default simp set. All options for simp arguments are supported, in particular ←, ↑ and ↓.
norm_num only does not use the default simp set for simplification. norm_num only [h1, ...] uses only the arguments h1, ... in addition to the routines tagged @[norm_num]. norm_num only still performs post-processing steps, like simp only, use norm_num1 if you exclusively want to normalize numerical expressions.
norm_num (config := cfg) uses cfg as configuration for simp calls (see the simp tactic for further details).

Examples:

example : 43 ≤ 74 + (33 : ℤ) := by norm_num
example : ¬ (7-2)/(2*3) ≥ (1:ℝ) + 2/(3^2) := by norm_num

Tags:
Defined in module:: Mathlib.Tactic.NormNum.Core

norm_num1

norm_num1 normalizes numerical expressions in the goal. It is a basic version of norm_num that does not call simp.

By default, it supports the operations + - * / ⁻¹ ^ and % over types with (at least) an AddMonoidWithOne instance, such as ℕ, ℤ, ℚ, ℝ, ℂ. If the goal has the form A = B, A ≠ B, A < B or A ≤ B, where A and B are numerical expressions, norm_num1 will try to close it. It also has a relatively simple primality prover. :e This tactic is extensible. Extensions can allow norm_num1 to evaluate more kinds of expressions, or to prove more kinds of propositions. See the @[norm_num] attribute for further information on extending norm_num1.

norm_num1 at l normalizes at location(s) l.

Examples:

example : 43 ≤ 74 + (33 : ℤ) := by norm_num1
example : ¬ (7-2)/(2*3) ≥ (1:ℝ) + 2/(3^2) := by norm_num1

Tags:
Defined in module:: Mathlib.Tactic.NormNum.Core

nth_grewrite

nth_grewrite is just like nth_rewrite, but for grewrite.

Tags:
Defined in module:: Mathlib.Tactic.GRewrite.Elab

nth_grw

nth_grw is just like nth_rw, but for grw.

Tags:
Defined in module:: Mathlib.Tactic.GRewrite.Elab

nth_rewrite

nth_rewrite is a variant of rewrite that only changes the n₁, ..., nₖᵗʰ occurrence of the expression to be rewritten. nth_rewrite n₁ ... nₖ [eq₁, eq₂,..., eqₘ] will rewrite the n₁, ..., nₖᵗʰ occurrence of each of the m equalities eqᵢin that order. Occurrences are counted beginning with 1 in order of precedence.

For example,

example (h : a = 1) : a + a + a + a + a = 5 := by
  nth_rewrite 2 3 [h]
/-
a: ℕ
h: a = 1
⊢ a + 1 + 1 + a + a = 5
-/

Notice that the second and third occurrences of a from the left have been rewritten by nth_rewrite.

To understand the importance of order of precedence, consider the example below

example (a b c : Nat) : (a + b) + c = (b + a) + c := by
  nth_rewrite 2 [Nat.add_comm] -- ⊢ (b + a) + c = (b + a) + c

Here, although the occurrence parameter is 2, (a + b) is rewritten to (b + a). This happens because in order of precedence, the first occurrence of _ + _ is the one that adds a + b to c. The occurrence in a + b counts as the second occurrence.

If a term t is introduced by rewriting with eqᵢ, then this instance of t will be counted as an occurrence of t for all subsequent rewrites of t with eqⱼ for j > i. This behaviour is illustrated by the example below

example (h : a = a + b) : a + a + a + a + a = 0 := by
  nth_rewrite 3 [h, h]
/-
a b: ℕ
h: a = a + b
⊢ a + a + (a + b + b) + a + a = 0
-/

Here, the first nth_rewrite with h introduces an additional occurrence of a in the goal. That is, the goal state after the first rewrite looks like below

/-
a b: ℕ
h: a = a + b
⊢ a + a + (a + b) + a + a = 0
-/

This new instance of a also turns out to be the third occurrence of a. Therefore, the next nth_rewrite with h rewrites this a.

Tags:
Defined in module:: Mathlib.Tactic.NthRewrite

nth_rw

nth_rw is a variant of rw that only changes the n₁, ..., nₖᵗʰ occurrence of the expression to be rewritten. Like rw, and unlike nth_rewrite, it will try to close the goal by trying rfl afterwards. nth_rw n₁ ... nₖ [eq₁, eq₂,..., eqₘ] will rewrite the n₁, ..., nₖᵗʰ occurrence of each of the m equalities eqᵢin that order. Occurrences are counted beginning with 1 in order of precedence. For example,

example (h : a = 1) : a + a + a + a + a = 5 := by
  nth_rw 2 3 [h]
/-
a: ℕ
h: a = 1
⊢ a + 1 + 1 + a + a = 5
-/

Notice that the second and third occurrences of a from the left have been rewritten by nth_rw.

To understand the importance of order of precedence, consider the example below

example (a b c : Nat) : (a + b) + c = (b + a) + c := by
  nth_rewrite 2 [Nat.add_comm] -- ⊢ (b + a) + c = (b + a) + c

example (h : a = a + b) : a + a + a + a + a = 0 := by
  nth_rw 3 [h, h]
/-
a b: ℕ
h: a = a + b
⊢ a + a + (a + b + b) + a + a = 0
-/

Here, the first nth_rw with h introduces an additional occurrence of a in the goal. That is, the goal state after the first rewrite looks like below

/-
a b: ℕ
h: a = a + b
⊢ a + a + (a + b) + a + a = 0
-/

This new instance of a also turns out to be the third occurrence of a. Therefore, the next nth_rw with h rewrites this a.

Further, nth_rw will close the remaining goal with rfl if possible.

Tags:
Defined in module:: Mathlib.Tactic.NthRewrite

observe

observe hp : p asserts the proposition p as a hypothesis named hp, and tries to prove it using exact?. If no proof is found, the tactic fails. In other words, this tactic is equivalent to have hp : p := by exact?.

observe : p uses the name this for the new hypothesis.
observe? hp : p will emit a trace message of the form have hp : p := proof_term. This may be particularly useful to speed up proofs.

Tags:
Defined in module:: Mathlib.Tactic.Observe

obtain

The obtain tactic is a combination of have and rcases. See rcases for a description of supported patterns.

obtain ⟨patt⟩ : type := proof

is equivalent to

have h : type := proof
rcases h with ⟨patt⟩

If ⟨patt⟩ is omitted, rcases will try to infer the pattern.

If type is omitted, := proof is required.

Tags:
Defined in module:: Init.RCases

omega

The omega tactic, for resolving integer and natural linear arithmetic problems.

It is not yet a full decision procedure (no "dark" or "grey" shadows), but should be effective on many problems.

We handle hypotheses of the form x = y, x < y, x ≤ y, and k ∣ x for x y in Nat or Int (and k a literal), along with negations of these statements.

We decompose the sides of the inequalities as linear combinations of atoms.

If we encounter x / k or x % k for literal integers k we introduce new auxiliary variables and the relevant inequalities.

On the first pass, we do not perform case splits on natural subtraction. If omega fails, we recursively perform a case split on a natural subtraction appearing in a hypothesis, and try again.

The options

omega +splitDisjunctions +splitNatSub +splitNatAbs +splitMinMax

can be used to:

splitDisjunctions: split any disjunctions found in the context, if the problem is not otherwise solvable.
splitNatSub: for each appearance of ((a - b : Nat) : Int), split on a ≤ b if necessary.
splitNatAbs: for each appearance of Int.natAbs a, split on 0 ≤ a if necessary.
splitMinMax: for each occurrence of min a b, split on min a b = a ∨ min a b = b Currently, all of these are on by default.

Tags:
Defined in module:: Init.Tactics

on_goal

on_goal n => tacSeq creates a block scope for the n-th goal and tries the sequence of tactics tacSeq on it.

on_goal -n => tacSeq does the same, but the n-th goal is chosen by counting from the bottom.

The goal is not required to be solved and any resulting subgoals are inserted back into the list of goals, replacing the chosen goal.

Tags:
Defined in module:: Batteries.Tactic.PermuteGoals

open

open Foo in tacs (the tactic) acts like open Foo at command level, but it opens a namespace only within the tactics tacs.

Tags:
Defined in module:: Lean.Parser.Command

order

A finishing tactic for solving goals in arbitrary Preorder, PartialOrder, or LinearOrder. Supports ⊤, ⊥, and lattice operations.

Tags:
Defined in module:: Mathlib.Tactic.Order

order_core

order_core is the part of the order tactic that tries to find a contradiction.

Tags:
Defined in module:: Mathlib.Tactic.Order

peel

Peels matching quantifiers off of a given term and the goal and introduces the relevant variables.

peel e peels all quantifiers (at reducible transparency), using this for the name of the peeled hypothesis.
peel e with h is peel e but names the peeled hypothesis h. If h is _ then uses this for the name of the peeled hypothesis.
peel n e peels n quantifiers (at default transparency).
peel n e with x y z ... h peels n quantifiers, names the peeled hypothesis h, and uses x, y, z, and so on to name the introduced variables; these names may be _. If h is _ then uses this for the name of the peeled hypothesis. The length of the list of variables does not need to equal n.
peel e with x₁ ... xₙ h is peel n e with x₁ ... xₙ h.

There are also variants that apply to an iff in the goal:

peel n peels n quantifiers in an iff.
peel with x₁ ... xₙ peels n quantifiers in an iff and names them.

Given p q : ℕ → Prop, h : ∀ x, p x, and a goal ⊢ : ∀ x, q x, the tactic peel h with x h' will introduce x : ℕ, h' : p x into the context and the new goal will be ⊢ q x. This works with ∃, as well as ∀ᶠ and ∃ᶠ, and it can even be applied to a sequence of quantifiers. Note that this is a logically weaker setup, so using this tactic is not always feasible.

For a more complex example, given a hypothesis and a goal:

h : ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) < ε
⊢ ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) ≤ ε

(which differ only in </≤), applying peel h with ε hε N n hn h_peel will yield a tactic state:

h : ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) < ε
ε : ℝ
hε : 0 < ε
N n : ℕ
hn : N ≤ n
h_peel : 1 / (n + 1 : ℝ) < ε
⊢ 1 / (n + 1 : ℝ) ≤ ε

and the goal can be closed with exact h_peel.le. Note that in this example, h and the goal are logically equivalent statements, but peel cannot be immediately applied to show that the goal implies h.

In addition, peel supports goals of the form (∀ x, p x) ↔ ∀ x, q x, or likewise for any other quantifier. In this case, there is no hypothesis or term to supply, but otherwise the syntax is the same. So for such goals, the syntax is peel 1 or peel with x, and after which the resulting goal is p x ↔ q x. The congr! tactic can also be applied to goals of this form using congr! 1 with x. While congr! applies congruence lemmas in general, peel can be relied upon to only apply to outermost quantifiers.

Finally, the user may supply a term e via ... using e in order to close the goal immediately. In particular, peel h using e is equivalent to peel h; exact e. The using syntax may be paired with any of the other features of peel.

This tactic works by repeatedly applying lemmas such as forall_imp, Exists.imp, Filter.Eventually.mp, Filter.Frequently.mp, and Filter.Eventually.of_forall.

Tags:
Defined in module:: Mathlib.Tactic.Peel

pgame_wf_tac

Discharges proof obligations of the form ⊢ Subsequent .. arising in termination proofs of definitions using well-founded recursion on PGame.

Tags:
Defined in module:: Mathlib.SetTheory.PGame.Basic

pi_lower_bound

Create a proof of a < π for a fixed rational number a, given a witness, which is a sequence of rational numbers √2 < r 1 < r 2 < ... < r n < 2 satisfying the property that √(2 + r i) ≤ r(i+1), where r 0 = 0 and √(2 - r n) ≥ a/2^(n+1).

Tags:
Defined in module:: Mathlib.Analysis.Real.Pi.Bounds

pi_upper_bound

Create a proof of π < a for a fixed rational number a, given a witness, which is a sequence of rational numbers √2 < r 1 < r 2 < ... < r n < 2 satisfying the property that √(2 + r i) ≥ r(i+1), where r 0 = 0 and √(2 - r n) ≤ (a - 1/4^n) / 2^(n+1).

Tags:
Defined in module:: Mathlib.Analysis.Real.Pi.Bounds

pick_goal

pick_goal n will move the n-th goal to the front.

pick_goal -n will move the n-th goal (counting from the bottom) to the front.

See also Tactic.rotate_goals, which moves goals from the front to the back and vice-versa.

Tags:
Defined in module:: Batteries.Tactic.PermuteGoals

plausible

plausible considers a proof goal and tries to generate examples that would contradict the statement.

Let's consider the following proof goal.

xs : List Nat,
h : ∃ (x : Nat) (H : x ∈ xs), x < 3
⊢ ∀ (y : Nat), y ∈ xs → y < 5

The local constants will be reverted and an instance will be found for Testable (∀ (xs : List Nat), (∃ x ∈ xs, x < 3) → (∀ y ∈ xs, y < 5)). The Testable instance is supported by an instance of Sampleable (List Nat), Decidable (x < 3) and Decidable (y < 5).

Examples will be created in ascending order of size (more or less)

The first counter-examples found will be printed and will result in an error:

===================
Found problems!
xs := [1, 28]
x := 1
y := 28
-------------------

If plausible successfully tests 100 examples, it acts like admit. If it gives up or finds a counter-example, it reports an error.

For more information on writing your own Sampleable and Testable instances, see Testing.Plausible.Testable.

Optional arguments given with plausible (config : { ... })

numInst (default 100): number of examples to test properties with
maxSize (default 100): final size argument

Options:

set_option trace.plausible.decoration true: print the proposition with quantifier annotations
set_option trace.plausible.discarded true: print the examples discarded because they do not satisfy assumptions
set_option trace.plausible.shrink.steps true: trace the shrinking of counter-example
set_option trace.plausible.shrink.candidates true: print the lists of candidates considered when shrinking each variable
set_option trace.plausible.instance true: print the instances of testable being used to test the proposition
set_option trace.plausible.success true: print the tested samples that satisfy a property

Tags:
Defined in module:: Plausible.Tactic

pnat_positivity

For each x : PNat in the context, add the hypothesis 0 < (↑x : ℕ).

Tags:
Defined in module:: Mathlib.Tactic.PNatToNat

pnat_to_nat

pnat_to_nat shifts all PNats in the context to Nat, rewriting propositions about them. A typical use case is pnat_to_nat; lia.

Tags:
Defined in module:: Mathlib.Tactic.PNatToNat

polyrith

The polyrith tactic is no longer supported in Mathlib, because it relied on a defunct external service.

Attempts to prove polynomial equality goals through polynomial arithmetic on the hypotheses (and additional proof terms if the user specifies them). It proves the goal by generating an appropriate call to the tactic linear_combination. If this call succeeds, the call to linear_combination is suggested to the user.

polyrith will use all relevant hypotheses in the local context.
polyrith [t1, t2, t3] will add proof terms t1, t2, t3 to the local context.
polyrith only [h1, h2, h3, t1, t2, t3] will use only local hypotheses h1, h2, h3, and proofs t1, t2, t3. It will ignore the rest of the local context.

Notes:

This tactic only works with a working internet connection, since it calls Sage using the SageCell web API at https://sagecell.sagemath.org/. Many thanks to the Sage team and organization for allowing this use.
This tactic assumes that the user has curl available on path.

Tags:
Defined in module:: Mathlib.Tactic.Polyrith

positivity

Tactic solving goals of the form 0 ≤ x, 0 < x and x ≠ 0. The tactic works recursively according to the syntax of the expression x, if the atoms composing the expression all have numeric lower bounds which can be proved positive/nonnegative/nonzero by norm_num. This tactic either closes the goal or fails.

positivity [t₁, …, tₙ] first executes have := t₁; …; have := tₙ in the current goal, then runs positivity. This is useful when positivity needs derived premises such as 0 < y for division/reciprocal, or 0 ≤ x for real powers.

Examples:

example {a : ℤ} (ha : 3 < a) : 0 ≤ a ^ 3 + a := by positivity

example {a : ℤ} (ha : 1 < a) : 0 < |(3:ℤ) + a| := by positivity

example {b : ℤ} : 0 ≤ max (-3) (b ^ 2) := by positivity

example {a b c d : ℝ} (hab : 0 < a * b) (hb : 0 ≤ b) (hcd : c < d) :
    0 < a ^ c + 1 / (d - c) := by
  positivity [sub_pos_of_lt hcd, pos_of_mul_pos_left hab hb]

Tags:
Defined in module:: Mathlib.Tactic.Positivity.Core

pull

pull c rewrites the goal by pulling the constant c closer to the head of the expression. For instance, pull _ ∈ _ rewrites x ∈ y ∨ ¬ x ∈ z into x ∈ y ∪ zᶜ. More precisely, the pull tactic repeatedly rewrites an expression by applying lemmas of the form ... (c ...) = c ... (where c can appear 1 or more times on the left hand side). pull is the inverse tactic to push. To extend the pull tactic, you can tag a lemma with the @[push] attribute. pull works as both a tactic and a conv tactic.

A lemma is considered a pull lemma if its reverse direction is a push lemma that actually moves the given constant away from the head. For example

not_or : ¬ (p ∨ q) ↔ ¬ p ∧ ¬ q is a pull lemma, but not_not : ¬ ¬ p ↔ p is not.
log_mul : log (x * y) = log x + log y is a pull lemma, but log_abs : log |x| = log x is not.
Pi.mul_def : f * g = fun (i : ι) => f i * g i and Pi.one_def : 1 = fun (x : ι) => 1 are both pull lemmas for fun, because every push fun _ ↦ _ lemma is also considered a pull lemma.

TODO: define a @[pull] attribute for tagging pull lemmas that are not push lemmas.

pull _ ~ _ pulls the operator or relation ~.
pull c at l1 l2 ... rewrites at the given locations.
pull c at * rewrites at all hypotheses and the goal.
pull (disch := tac) c uses the tactic tac to discharge any hypotheses for @[push] lemmas.

Examples:

pull _ ∈ _ rewrites x ∈ y ∨ ¬ x ∈ z into x ∈ y ∪ zᶜ.
pull (disch := positivity) Real.log rewrites log a + 2 * log b into log (a * b ^ 2).
pull fun _ ↦ _ rewrites f ^ 2 + 5 into fun x => f x ^ 2 + 5 where f is a function.

Tags:
Defined in module:: Mathlib.Tactic.Push

pure_coherence

pure_coherence uses the coherence theorem for monoidal categories to prove the goal. It can prove any equality made up only of associators, unitors, and identities.

example {C : Type} [Category* C] [MonoidalCategory C] :
  (λ_ (𝟙_ C)).hom = (ρ_ (𝟙_ C)).hom := by
  pure_coherence

Users will typically just use the coherence tactic, which can also cope with identities of the form a ≫ f ≫ b ≫ g ≫ c = a' ≫ f ≫ b' ≫ g ≫ c' where a = a', b = b', and c = c' can be proved using pure_coherence

Tags:
Defined in module:: Mathlib.Tactic.CategoryTheory.Coherence

push

push c rewrites the goal by pushing the constant c deeper into an expression. For instance, push _ ∈ _ rewrites x ∈ {y} ∪ zᶜ into x = y ∨ ¬ x ∈ z. More precisely, the push tactic repeatedly rewrites an expression by applying lemmas of the form c ... = ... (c ...) (where c can appear 0 or more times on the right hand side). To extend the push tactic, you can tag a lemma of this form with the @[push] attribute.

To instead move a constant closer to the head of the expression, use the pull tactic.

push works as both a tactic and a conv tactic.

push _ ~ _ pushes the (binary) operator ~, push ~ _ pushes the (unary) operator ~.
push c at l1 l2 ... rewrites at the given locations.
push c at * rewrites at all hypotheses and the goal.
push (disch := tac) c uses the tactic tac to discharge any hypotheses for @[push] lemmas.

Examples:

push _ ∈ _ rewrites x ∈ {y} ∪ zᶜ into x = y ∨ ¬ x ∈ z.
push (disch := positivity) Real.log rewrites log (a * b ^ 2) into log a + 2 * log b.
push ¬ _ is the same as push_neg or push Not, and it rewrites ¬ ∀ ε > 0, ∃ δ > 0, δ < ε into ∃ ε > 0, ∀ δ > 0, ε ≤ δ.
push fun _ ↦ _ rewrites fun x => f x ^ 2 + 5 into f ^ 2 + 5
push ∀ _, _ rewrites ∀ a, p a ∧ q a into (∀ a, p a) ∧ (∀ a, q a).

Tags:
Defined in module:: Mathlib.Tactic.Push

push_cast

push_cast rewrites the goal to move certain coercions (casts) inward, toward the leaf nodes. This uses norm_cast lemmas in the forward direction. For example, ↑(a + b) will be written to ↑a + ↑b.

push_cast moves casts inward in the goal.
push_cast at h moves casts inward in the hypothesis h. It can be used with extra simp lemmas with, for example, push_cast [Int.add_zero].

Example:

example (a b : Nat)
    (h1 : ((a + b : Nat) : Int) = 10)
    (h2 : ((a + b + 0 : Nat) : Int) = 10) :
    ((a + b : Nat) : Int) = 10 := by
  /-
  h1 : ↑(a + b) = 10
  h2 : ↑(a + b + 0) = 10
  ⊢ ↑(a + b) = 10
  -/
  push_cast
  /- Now
  ⊢ ↑a + ↑b = 10
  -/
  push_cast at h1
  push_cast [Int.add_zero] at h2
  /- Now
  h1 h2 : ↑a + ↑b = 10
  -/
  exact h1

push_neg

push_neg rewrites the goal by pushing negations deeper into an expression. For instance, the goal ¬ ∀ x, ∃ y, x ≤ y will be transformed by push_neg into ∃ x, ∀ y, y < x. Binder names are preserved (contrary to what would happen with simp using the relevant lemmas). push_neg works as both a tactic and a conv tactic.

push_neg is a special case of the more general push tactic, namely push Not. The push tactic can be extended using the @[push] attribute. push has special-casing built in for push Not.

Tactics that introduce a negation usually have a version that automatically calls push_neg on that negation. These include by_cases!, contrapose! and by_contra!.

push_neg at l1 l2 ... rewrites at the given locations.
push_neg at * rewrites at each hypothesis and the goal.
push_neg +distrib rewrites ¬ (p ∧ q) into ¬ p ∨ ¬ q (by default, the tactic rewrites it into p → ¬ q instead).

Example:

example (h : ¬ ∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| ≤ δ → |f x - y₀| ≤ ε) :
    ∃ ε > 0, ∀ δ > 0, ∃ x, |x - x₀| ≤ δ ∧ ε < |f x - y₀| := by
  push_neg at h
  -- Now we have the hypothesis `h : ∃ ε > 0, ∀ δ > 0, ∃ x, |x - x₀| ≤ δ ∧ ε < |f x - y₀|`
  exact h

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Defined in module:: Mathlib.Tactic.Push

qify

The qify tactic is used to shift propositions from ℕ or ℤ to ℚ. This is often useful since ℚ has well-behaved division.

example (a b c x y z : ℕ) (h : ¬ x*y*z < 0) : c < a + 3*b := by
  qify
  qify at h
  /-
  h : ¬↑x * ↑y * ↑z < 0
  ⊢ ↑c < ↑a + 3 * ↑b
  -/
  sorry

qify can be given extra lemmas to use in simplification. This is especially useful in the presence of nat subtraction: passing ≤ arguments will allow push_cast to do more work.

example (a b c : ℤ) (h : a / b = c) (hab : b ∣ a) (hb : b ≠ 0) : a = c * b := by
  qify [hab] at h hb ⊢
  exact (div_eq_iff hb).1 h

qify makes use of the @[zify_simps] and @[qify_simps] attributes to move propositions, and the push_cast tactic to simplify the ℚ-valued expressions.

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Defined in module:: Mathlib.Tactic.Qify

rcases

rcases is a tactic that will perform cases recursively, according to a pattern. It is used to destructure hypotheses or expressions composed of inductive types like h1 : a ∧ b ∧ c ∨ d or h2 : ∃ x y, trans_rel R x y. Usual usage might be rcases h1 with ⟨ha, hb, hc⟩ | hd or rcases h2 with ⟨x, y, _ | ⟨z, hxz, hzy⟩⟩ for these examples.

Each element of an rcases pattern is matched against a particular local hypothesis (most of which are generated during the execution of rcases and represent individual elements destructured from the input expression). An rcases pattern has the following grammar:

A name like x, which names the active hypothesis as x.
A blank _, which does nothing (letting the automatic naming system used by cases name the hypothesis).
A hyphen -, which clears the active hypothesis and any dependents.
The keyword rfl, which expects the hypothesis to be h : a = b, and calls subst on the hypothesis (which has the effect of replacing b with a everywhere or vice versa).
A type ascription p : ty, which sets the type of the hypothesis to ty and then matches it against p. (Of course, ty must unify with the actual type of h for this to work.)
A tuple pattern ⟨p1, p2, p3⟩, which matches a constructor with many arguments, or a series of nested conjunctions or existentials. For example if the active hypothesis is a ∧ b ∧ c, then the conjunction will be destructured, and p1 will be matched against a, p2 against b and so on.
A @ before a tuple pattern as in @⟨p1, p2, p3⟩ will bind all arguments in the constructor, while leaving the @ off will only use the patterns on the explicit arguments.
An alternation pattern p1 | p2 | p3, which matches an inductive type with multiple constructors, or a nested disjunction like a ∨ b ∨ c.

A pattern like ⟨a, b, c⟩ | ⟨d, e⟩ will do a split over the inductive datatype, naming the first three parameters of the first constructor as a,b,c and the first two of the second constructor d,e. If the list is not as long as the number of arguments to the constructor or the number of constructors, the remaining variables will be automatically named. If there are nested brackets such as ⟨⟨a⟩, b | c⟩ | d then these will cause more case splits as necessary. If there are too many arguments, such as ⟨a, b, c⟩ for splitting on ∃ x, ∃ y, p x, then it will be treated as ⟨a, ⟨b, c⟩⟩, splitting the last parameter as necessary.

rcases also has special support for quotient types: quotient induction into Prop works like matching on the constructor quot.mk.

rcases h : e with PAT will do the same as rcases e with PAT with the exception that an assumption h : e = PAT will be added to the context.

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Defined in module:: Init.RCases

rcongr

Repeatedly apply congr and ext, using the given patterns as arguments for ext.

There are two ways this tactic stops:

congr fails (makes no progress), after having already applied ext.
congr canceled out the last usage of ext. In this case, the state is reverted to before the congr was applied.

For example, when the goal is

⊢ (fun x => f x + 3) '' s = (fun x => g x + 3) '' s

then rcongr x produces the goal

x : α ⊢ f x = g x

This gives the same result as congr; ext x; congr.

In contrast, congr would produce

⊢ (fun x => f x + 3) = (fun x => g x + 3)

and congr with x (or congr; ext x) would produce

x : α ⊢ f x + 3 = g x + 3

Tags:
Defined in module:: Batteries.Tactic.Congr

recover

recover tacs applies the tactic (sequence) tacs and then re-adds goals that were incorrectly marked as closed. This helps to debug issues where a tactic closes goals without solving them (i.e. goals were removed from the MetaM state without the metavariable being assigned), resulting in the error "(kernel) declaration has metavariables".

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Defined in module:: Mathlib.Tactic.Recover

reduce

reduce at loc completely reduces the given location. This also exists as a conv-mode tactic.

This does the same transformation as the #reduce command.

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Defined in module:: Mathlib.Tactic.DefEqTransformations

reduce_mod_char

The tactic reduce_mod_char looks for numeric expressions in characteristic p and reduces these to lie between 0 and p.

For example:

example : (5 : ZMod 4) = 1 := by reduce_mod_char
example : (X ^ 2 - 3 * X + 4 : (ZMod 4)[X]) = X ^ 2 + X := by reduce_mod_char

It also handles negation, turning it into multiplication by p - 1, and similarly subtraction.

This tactic uses the type of the subexpression to figure out if it is indeed of positive characteristic, for improved performance compared to trying to synthesise a CharP instance. The variant reduce_mod_char! also tries to use CharP R n hypotheses in the context. (Limitations of the typeclass system mean the tactic can't search for a CharP R n instance if n is not yet known; use have : CharP R n := inferInstance; reduce_mod_char! as a workaround.)

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Defined in module:: Mathlib.Tactic.ReduceModChar

refine

refine e behaves like exact e, except that named (?x) or unnamed (?_) holes in e that are not solved by unification with the main goal's target type are converted into new goals, using the hole's name, if any, as the goal case name.

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Defined in module:: Init.Tactics

refine'

refine' e behaves like refine e, except that unsolved placeholders (_) and implicit parameters are also converted into new goals.

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Defined in module:: Init.Tactics

refine_lift

Auxiliary macro for lifting have/suffices/let/... It makes sure the "continuation" ?_ is the main goal after refining.

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Defined in module:: Init.Tactics

refine_lift'

Similar to refine_lift, but using refine'

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Defined in module:: Init.Tactics

refold_let

refold_let x y z at loc looks for the bodies of local definitions x, y, and z at the given location and replaces them with x, y, or z. This is the inverse of "zeta reduction." This also exists as a conv-mode tactic.

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Defined in module:: Mathlib.Tactic.DefEqTransformations

rel

The rel tactic applies "generalized congruence" rules to solve a relational goal by "substitution". For example,

example {a b x c d : ℝ} (h1 : a ≤ b) (h2 : c ≤ d) :
    x ^ 2 * a + c ≤ x ^ 2 * b + d := by
  rel [h1, h2]

In this example we "substitute" the hypotheses a ≤ b and c ≤ d into the LHS x ^ 2 * a + c of the goal and obtain the RHS x ^ 2 * b + d, thus proving the goal.

The "generalized congruence" rules used are the library lemmas which have been tagged with the attribute @[gcongr]. For example, the first example constructs the proof term

add_le_add (mul_le_mul_of_nonneg_left h1 (pow_bit0_nonneg x 1)) h2

using the generalized congruence lemmas add_le_add and mul_le_mul_of_nonneg_left. If there are no applicable generalized congruence lemmas, the tactic fails.

The tactic attempts to discharge side goals to these "generalized congruence" lemmas (such as the side goal 0 ≤ x ^ 2 in the above application of mul_le_mul_of_nonneg_left) using the tactic gcongr_discharger, which wraps positivity but can also be extended. If the side goals cannot be discharged in this way, the tactic fails.

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Defined in module:: Mathlib.Tactic.GCongr.Core

rename

rename t => x renames the most recent hypothesis whose type matches t (which may contain placeholders) to x, or fails if no such hypothesis could be found.

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Defined in module:: Init.Tactics

rename'

rename' h => hnew renames the hypothesis named h to hnew. To rename several hypothesis, use rename' h₁ => h₁new, h₂ => h₂new. You can use rename' a => b, b => a to swap two variables.

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Defined in module:: Mathlib.Tactic.Rename

rename_bvar

rename_bvar old → new renames all bound variables named old to new in the target.
rename_bvar old → new at h does the same in hypothesis h.

example (P : ℕ → ℕ → Prop) (h : ∀ n, ∃ m, P n m) : ∀ l, ∃ m, P l m := by
  rename_bvar n → q at h -- h is now ∀ (q : ℕ), ∃ (m : ℕ), P q m,
  rename_bvar m → n -- target is now ∀ (l : ℕ), ∃ (n : ℕ), P k n,
  exact h -- Lean does not care about those bound variable names

Note: name clashes are resolved automatically.

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Defined in module:: Mathlib.Tactic.RenameBVar

rename_i

rename_i x_1 ... x_n renames the last n inaccessible names using the given names.

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Defined in module:: Init.Tactics

repeat

repeat tac repeatedly applies tac so long as it succeeds. The tactic tac may be a tactic sequence, and if tac fails at any point in its execution, repeat will revert any partial changes that tac made to the tactic state.

The tactic tac should eventually fail, otherwise repeat tac will run indefinitely.

repeat'

repeat' tac recursively applies tac on all of the goals so long as it succeeds. That is to say, if tac produces multiple subgoals, then repeat' tac is applied to each of them.

repeat1

repeat1 tac applies tac to main goal at least once. If the application succeeds, the tactic is applied recursively to the generated subgoals until it eventually fails.

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Defined in module:: Mathlib.Tactic.Core

repeat1'

repeat1' tac recursively applies to tac on all of the goals so long as it succeeds, but repeat1' tac fails if tac succeeds on none of the initial goals.

replace

Acts like have, but removes a hypothesis with the same name as this one if possible. For example, if the state is:

Then after replace h : β the state will be:

case h
f : α → β
h : α
⊢ β

f : α → β
h : β
⊢ goal

whereas have h : β would result in:

case h
f : α → β
h : α
⊢ β

f : α → β
h✝ : α
h : β
⊢ goal

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Defined in module:: Mathlib.Tactic.Replace

replace

Acts like have, but removes a hypothesis with the same name as this one if possible. For example, if the state is:

f : α → β
h : α
⊢ goal

Then after replace h := f h the state will be:

f : α → β
h : β
⊢ goal

whereas have h := f h would result in:

f : α → β
h† : α
h : β
⊢ goal

This can be used to simulate the specialize and apply at tactics of Coq.

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Defined in module:: Init.Tactics

restrict_tac

restrict_tac solves relations among subsets (copied from aesop cat)

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Defined in module:: Mathlib.Topology.Sheaves.Presheaf

restrict_tac?

restrict_tac? passes along Try this from aesop

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Defined in module:: Mathlib.Topology.Sheaves.Presheaf

revert

revert x... is the inverse of intro x...: it moves the given hypotheses into the main goal's target type.

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Defined in module:: Init.Tactics

rewrite

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 theorems 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.

Using rw (occs := .pos L) [e], where L : List Nat, you can control which "occurrences" are rewritten. (This option applies to each rule, so usually this will only be used with a single rule.) Occurrences count from 1. At each allowed occurrence, arguments of the rewrite rule e may be instantiated, restricting which later rewrites can be found. (Disallowed occurrences do not result in instantiation.) (occs := .neg L) allows skipping specified occurrences.

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Defined in module:: Init.Tactics

rewrite!

rewrite! is like rewrite, but can also insert casts to adjust types that depend on the LHS of a rewrite. It is available as an ordinary tactic and a conv tactic.

The sort of casts that are inserted is controlled by the castMode configuration option. By default, only proof terms are casted; by proof irrelevance, this adds no observable complexity.

With rewrite! +letAbs (castMode := .all), casts are inserted whenever necessary. This means that the 'motive is not type correct' error never occurs, at the expense of creating potentially complicated terms.

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Defined in module:: Mathlib.Tactic.DepRewrite

rfl

This tactic applies to a goal whose target has the form x ~ x, where ~ is equality, heterogeneous equality or any relation that has a reflexivity lemma tagged with the attribute @[refl].

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Defined in module:: Init.Tactics

rfl'

rfl' is similar to rfl, but disables smart unfolding and unfolds all kinds of definitions, theorems included (relevant for declarations defined by well-founded recursion).

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Defined in module:: Init.Tactics

rfl_cat

rfl_cat is a macro for intros; rfl which is attempted in aesop_cat before doing the more expensive aesop tactic.

This gives a speedup because simp (called by aesop) can be very slow. https://github.com/leanprover-community/mathlib4/pull/25475 contains measurements from June 2025.

Implementation notes:

refine id ?_: In some cases it is important that the type of the proof matches the expected type exactly. e.g. if the goal is 2 = 1 + 1, the rfl tactic will give a proof of type 2 = 2. Starting a proof with refine id ?_ is a trick to make sure that the proof has exactly the expected type, in this case 2 = 1 + 1. See also https://leanprover.zulipchat.com/#narrow/channel/270676-lean4/topic/changing.20a.20proof.20can.20break.20a.20later.20proof
apply_rfl: rfl is a macro that attempts both eq_refl and apply_rfl. Since apply_rfl subsumes eq_refl, we can use apply_rfl instead. This fails twice as fast as rfl.

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Defined in module:: Mathlib.CategoryTheory.Category.Basic

rify

The rify tactic is used to shift propositions from ℕ, ℤ or ℚ to ℝ. Although less useful than its cousins zify and qify, it can be useful when your goal or context already involves real numbers.

In the example below, assumption hn is about natural numbers, hk is about integers and involves casting a natural number to ℤ, and the conclusion is about real numbers. The proof uses rify to lift both assumptions to ℝ before calling linarith.

example {n : ℕ} {k : ℤ} (hn : 8 ≤ n) (hk : 2 * k ≤ n + 2) :
    (0 : ℝ) < n - k - 1 := by
  rify at hn hk /- Now have hn : 8 ≤ (n : ℝ)   hk : 2 * (k : ℝ) ≤ (n : ℝ) + 2 -/
  linarith

rify makes use of the @[zify_simps], @[qify_simps] and @[rify_simps] attributes to move propositions, and the push_cast tactic to simplify the ℝ-valued expressions.

rify can be given extra lemmas to use in simplification. This is especially useful in the presence of nat subtraction: passing ≤ arguments will allow push_cast to do more work.

example (a b c : ℕ) (h : a - b < c) (hab : b ≤ a) : a < b + c := by
  rify [hab] at h ⊢
  linarith

Note that zify or qify would work just as well in the above example (and zify is the natural choice since it is enough to get rid of the pathological ℕ subtraction).

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Defined in module:: Mathlib.Tactic.Rify

right

Applies the second constructor when the goal is an inductive type with exactly two constructors, or fails otherwise.

example {p q : Prop} (h : q) : p ∨ q := by
  right
  exact h

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Defined in module:: Init.Tactics

ring

ring solves equations in commutative (semi)rings, allowing for variables in the exponent. If the goal is not appropriate for ring (e.g. not an equality) ring_nf will be suggested. See also ring1, which fails if the goal is not an equality.

ring! will use a more aggressive reducibility setting to determine equality of atoms.

Examples:

example (n : ℕ) (m : ℤ) : 2^(n+1) * m = 2 * 2^n * m := by ring
example (a b : ℤ) (n : ℕ) : (a + b)^(n + 2) = (a^2 + b^2 + a * b + b * a) * (a + b)^n := by ring
example (x y : ℕ) : x + id y = y + id x := by ring!
example (x : ℕ) (h : x * 2 > 5): x + x > 5 := by ring; assumption -- suggests ring_nf

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Defined in module:: Mathlib.Tactic.Ring.RingNF

ring1

ring1 solves the goal when it is an equality in commutative (semi)rings, allowing variables in the exponent.

This version of ring fails if the target is not an equality.

ring1! uses a more aggressive reducibility setting to determine equality of atoms.

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Defined in module:: Mathlib.Tactic.Ring.Basic

ring_nf

ring_nf simplifies expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form, allowing variables in the exponents.

ring_nf works as both a tactic and a conv tactic.

See also the ring tactic for solving a goal which is an equation in the language of commutative (semi)rings.

ring_nf! will use a more aggressive reducibility setting to identify atoms.
ring_nf (config := cfg) allows for additional configuration (see RingNF.Config):
- red: the reducibility setting (overridden by !)
- zetaDelta: if true, local let variables can be unfolded (overridden by !)
- recursive: if true, ring_nf will also recurse into atoms
ring_nf at l1 l2 ... can be used to rewrite at the given locations.

Examples: This can be used non-terminally to normalize ring expressions in the goal such as ⊢ P (x + x + x) ~> ⊢ P (x * 3), as well as being able to prove some equations that ring cannot because they involve ring reasoning inside a subterm, such as sin (x + y) + sin (y + x) = 2 * sin (x + y).

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Defined in module:: Mathlib.Tactic.Ring.RingNF

rintro

The rintro tactic is a combination of the intros tactic with rcases to allow for destructuring patterns while introducing variables. See rcases for a description of supported patterns. For example, rintro (a | ⟨b, c⟩) ⟨d, e⟩ will introduce two variables, and then do case splits on both of them producing two subgoals, one with variables a d e and the other with b c d e.

rintro, unlike rcases, also supports the form (x y : ty) for introducing and type-ascripting multiple variables at once, similar to binders.

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Defined in module:: Init.RCases

rotate_left

rotate_left n rotates goals to the left by n. That is, rotate_left 1 takes the main goal and puts it to the back of the subgoal list. If n is omitted, it defaults to 1.

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Defined in module:: Init.Tactics

rotate_right

Rotate the goals to the right by n. That is, take the goal at the back and push it to the front n times. If n is omitted, it defaults to 1.

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Defined in module:: Init.Tactics

rsuffices

The rsuffices tactic is an alternative version of suffices, that allows the usage of any syntax that would be valid in an obtain block. This tactic just calls obtain on the expression, and then rotate_left.

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Defined in module:: Mathlib.Tactic.RSuffices

run_tac

The run_tac doSeq tactic executes code in TacticM Unit.

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Defined in module:: Init.Tactics

rw

rw is like rewrite, but also tries to close the goal by "cheap" (reducible) rfl afterwards.

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Defined in module:: Init.Tactics

rw!

rw! is like rewrite!, but also cleans up introduced refl-casts after every substitution. It is available as an ordinary tactic and a conv tactic.

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Defined in module:: Mathlib.Tactic.DepRewrite

rw?

rw? tries to find a lemma which can rewrite the goal.

rw? should not be left in proofs; it is a search tool, like apply?.

Suggestions are printed as rw [h] or rw [← h].

You can use rw? [-my_lemma, -my_theorem] to prevent rw? using the named lemmas.

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Defined in module:: Init.Tactics

rw??

rw?? is an interactive tactic that suggests rewrites for any expression selected by the user. To use it, shift-click an expression in the goal or a hypothesis that you want to rewrite. Clicking on one of the rewrite suggestions will paste the relevant rewrite tactic into the editor.

The rewrite suggestions are grouped and sorted by the pattern that the rewrite lemmas match with. Rewrites that don't change the goal and rewrites that create the same goal as another rewrite are filtered out, as well as rewrites that have new metavariables in the replacement expression. To see all suggestions, click on the filter button (▼) in the top right.

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Defined in module:: Mathlib.Tactic.Widget.LibraryRewrite

rw_mod_cast

Rewrites with the given rules, normalizing casts prior to each step.

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Defined in module:: Init.TacticsExtra

rw_search

rw_search has been removed from Mathlib.

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Defined in module:: Mathlib.Tactic.RewriteSearch

rwa

rwa is short-hand for rw; assumption.

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Defined in module:: Init.Tactics

saturate

This tactic has no documentation.

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Defined in module:: Aesop.Frontend.Saturate

saturate?

This tactic has no documentation.

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Defined in module:: Aesop.Frontend.Saturate

says

If you write X says, where X is a tactic that produces a "Try this: Y" message, then you will get a message "Try this: X says Y". Once you've clicked to replace X says with X says Y, afterwards X says Y will only run Y.

The typical usage case is:

simp? [X] says simp only [X, Y, Z]

If you use set_option says.verify true (set automatically during CI) then X says Y runs X and verifies that it still prints "Try this: Y".

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Defined in module:: Mathlib.Tactic.Says

set

set a := t with h is a variant of let a := t. It adds the hypothesis h : a = t to the local context and replaces t with a everywhere it can.

set a := t with ← h will add h : t = a instead.

set! a := t with h does not do any replacing.

example (x : Nat) (h : x + x - x = 3) : x + x - x = 3 := by
  set y := x with ← h2
  sorry
/-
x : Nat
y : Nat := x
h : y + y - y = 3
h2 : x = y
⊢ y + y - y = 3
-/

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Defined in module:: Mathlib.Tactic.Set

set_option

set_option opt val in tacs (the tactic) acts like set_option opt val at the command level, but it sets the option only within the tactics tacs.

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Defined in module:: Lean.Parser.Command

show

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.

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Defined in module:: Init.Tactics

show_term

show_term tac runs tac, then prints the generated term in the form "exact X Y Z" or "refine X ?_ Z" (prefixed by expose_names if necessary) if there are remaining subgoals.

(For some tactics, the printed term will not be human readable.)

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Defined in module:: Init.Tactics

simp

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 an hᵢ is a defined constant f, then f is unfolded. If f has equational lemmas associated with it (and is not a projection or a reducible definition), these are used to rewrite with f.
simp [*] simplifies the main goal target using the lemmas tagged with the attribute [simp] and all hypotheses.
simp only [h₁, h₂, ..., hₙ] is like simp [h₁, h₂, ..., hₙ] but does not use [simp] lemmas.
simp [-id₁, ..., -idₙ] 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 hypotheses h₁ : T₁ ... hₙ : Tₙ. If the target or another hypothesis depends on hᵢ, a new simplified hypothesis hᵢ is introduced, but the old one remains in the local context.
simp at * simplifies all the hypotheses and the target.
simp [*] at * simplifies target and all (propositional) hypotheses using the other hypotheses.

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Defined in module:: Init.Tactics

simp!

simp! is shorthand for simp with autoUnfold := true. This will unfold applications of functions defined by pattern matching, when one of the patterns applies. This can be used to partially evaluate many definitions.

Tags:
Defined in module:: Init.Meta

simp?

example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
  simp? -- prints "Try this: simp only [ite_true]"

This command can also be used in simp_all and dsimp.

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Defined in module:: Init.Tactics

simp?!

example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
  simp? -- prints "Try this: simp only [ite_true]"

This command can also be used in simp_all and dsimp.

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Defined in module:: Init.Tactics

simp_all

simp_all is a stronger version of simp [*] at * where the hypotheses and target are simplified multiple times until no simplification is applicable. Only non-dependent propositional hypotheses are considered.

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Defined in module:: Init.Tactics

simp_all!

simp_all! is shorthand for simp_all with autoUnfold := true. This will unfold applications of functions defined by pattern matching, when one of the patterns applies. This can be used to partially evaluate many definitions.

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Defined in module:: Init.Meta

simp_all?

example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
  simp? -- prints "Try this: simp only [ite_true]"

This command can also be used in simp_all and dsimp.

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Defined in module:: Init.Tactics

simp_all?!

example (x : Nat) : (if True then x + 2 else 3) = x + 2 := by
  simp? -- prints "Try this: simp only [ite_true]"

This command can also be used in simp_all and dsimp.

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Defined in module:: Init.Tactics

simp_all_arith

simp_all_arith has been deprecated. It was a shorthand for simp_all +arith +decide. Note that +decide is not needed for reducing arithmetic terms since simprocs have been added to Lean.

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Defined in module:: Init.Meta

simp_all_arith!

simp_all_arith! has been deprecated. It was a shorthand for simp_all! +arith +decide. Note that +decide is not needed for reducing arithmetic terms since simprocs have been added to Lean.

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Defined in module:: Init.Meta

simp_arith

simp_arith has been deprecated. It was a shorthand for simp +arith +decide. Note that +decide is not needed for reducing arithmetic terms since simprocs have been added to Lean.

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Defined in module:: Init.Meta

simp_arith!

simp_arith! has been deprecated. It was a shorthand for simp! +arith +decide. Note that +decide is not needed for reducing arithmetic terms since simprocs have been added to Lean.

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Defined in module:: Init.Meta

simp_intro

The simp_intro tactic is a combination of simp and intro: it will simplify the types of variables as it introduces them and uses the new variables to simplify later arguments and the goal.

simp_intro x y z introduces variables named x y z
simp_intro x y z .. introduces variables named x y z and then keeps introducing _ binders
simp_intro (config := cfg) (discharger := tac) x y .. only [h₁, h₂]: simp_intro takes the same options as simp (see simp)

example : x + 0 = y → x = z := by
  simp_intro h
  -- h: x = y ⊢ y = z
  sorry

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Defined in module:: Mathlib.Tactic.SimpIntro

simp_rw

simp_rw functions as a mix of simp and rw. Like rw, it applies each rewrite rule in the given order, but like simp it repeatedly applies these rules and also under binders like ∀ x, ..., ∃ x, ... and fun x ↦.... Usage:

simp_rw [lemma_1, ..., lemma_n] will rewrite the goal by applying the lemmas in that order. A lemma preceded by ← is applied in the reverse direction.
simp_rw [lemma_1, ..., lemma_n] at h₁ ... hₙ will rewrite the given hypotheses.
simp_rw [...] at * rewrites in the whole context: all hypotheses and the goal.

Lemmas passed to simp_rw must be expressions that are valid arguments to simp. For example, neither simp nor rw can solve the following, but simp_rw can:

example {a : ℕ}
    (h1 : ∀ a b : ℕ, a - 1 ≤ b ↔ a ≤ b + 1)
    (h2 : ∀ a b : ℕ, a ≤ b ↔ ∀ c, c < a → c < b) :
    (∀ b, a - 1 ≤ b) = ∀ b c : ℕ, c < a → c < b + 1 := by
  simp_rw [h1, h2]

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Defined in module:: Mathlib.Tactic.SimpRw

simp_wf

Unfold definitions commonly used in well founded relation definitions.

Since Lean 4.12, Lean unfolds these definitions automatically before presenting the goal to the user, and this tactic should no longer be necessary. Calls to simp_wf can be removed or replaced by plain calls to simp.

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Defined in module:: Init.WFTactics

simpa

This is a "finishing" tactic modification of simp. It has two forms.

simpa [rules, ⋯] using e will simplify the goal and the type of e using rules, then try to close the goal using e.

Simplifying the type of e makes it more likely to match the goal (which has also been simplified). This construction also tends to be more robust under changes to the simp lemma set.

simpa [rules, ⋯] will simplify the goal and the type of a hypothesis this if present in the context, then try to close the goal using the assumption tactic.

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Defined in module:: Init.Tactics

simpa!

This is a "finishing" tactic modification of simp. It has two forms.

simpa [rules, ⋯] using e will simplify the goal and the type of e using rules, then try to close the goal using e.

Simplifying the type of e makes it more likely to match the goal (which has also been simplified). This construction also tends to be more robust under changes to the simp lemma set.

simpa [rules, ⋯] will simplify the goal and the type of a hypothesis this if present in the context, then try to close the goal using the assumption tactic.

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Defined in module:: Init.Tactics

simpa?

This is a "finishing" tactic modification of simp. It has two forms.

simpa [rules, ⋯] using e will simplify the goal and the type of e using rules, then try to close the goal using e.

Simplifying the type of e makes it more likely to match the goal (which has also been simplified). This construction also tends to be more robust under changes to the simp lemma set.

simpa [rules, ⋯] will simplify the goal and the type of a hypothesis this if present in the context, then try to close the goal using the assumption tactic.

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Defined in module:: Init.Tactics

simpa?!

This is a "finishing" tactic modification of simp. It has two forms.

simpa [rules, ⋯] using e will simplify the goal and the type of e using rules, then try to close the goal using e.

Simplifying the type of e makes it more likely to match the goal (which has also been simplified). This construction also tends to be more robust under changes to the simp lemma set.

simpa [rules, ⋯] will simplify the goal and the type of a hypothesis this if present in the context, then try to close the goal using the assumption tactic.

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Defined in module:: Init.Tactics

sizeOf_list_dec

This tactic, added to the decreasing_trivial toolbox, proves that sizeOf a < sizeOf as when a ∈ as, which is useful for well founded recursions over a nested inductive like inductive T | mk : List T → T.

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Defined in module:: Init.Data.List.BasicAux

skip

skip does nothing.

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Defined in module:: Init.Tactics

sleep

The tactic sleep ms sleeps for ms milliseconds and does nothing. It is used for debugging purposes only.

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Defined in module:: Init.Tactics

sleep_heartbeats

do nothing for at least n heartbeats

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Defined in module:: Mathlib.Util.SleepHeartbeats

slice_lhs

slice_lhs a b => tac zooms to the left-hand side, uses associativity for categorical composition as needed, zooms in on the a-th through b-th morphisms, and invokes tac.

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Defined in module:: Mathlib.Tactic.CategoryTheory.Slice

slice_rhs

slice_rhs a b => tac zooms to the right-hand side, uses associativity for categorical composition as needed, zooms in on the a-th through b-th morphisms, and invokes tac.

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Defined in module:: Mathlib.Tactic.CategoryTheory.Slice

smul_tac

Solve equations for RatFunc K by applying RatFunc.induction_on.

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Defined in module:: Mathlib.FieldTheory.RatFunc.Basic

solve

Similar to first, but succeeds only if one the given tactics solves the current goal.

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Defined in module:: Init.NotationExtra

solve_by_elim

solve_by_elim calls apply on the main goal to find an assumption whose head matches and then repeatedly calls apply on the generated subgoals until no subgoals remain, performing at most maxDepth (defaults to 6) recursive steps.

solve_by_elim discharges the current goal or fails.

solve_by_elim performs backtracking if subgoals can not be solved.

By default, the assumptions passed to apply are the local context, rfl, trivial, congrFun and congrArg.

The assumptions can be modified with similar syntax as for simp:

solve_by_elim [h₁, h₂, ..., hᵣ] also applies the given expressions.
solve_by_elim only [h₁, h₂, ..., hᵣ] does not include the local context, rfl, trivial, congrFun, or congrArg unless they are explicitly included.
solve_by_elim [-h₁, ... -hₙ] removes the given local hypotheses.
solve_by_elim using [a₁, ...] uses all lemmas which have been labelled with the attributes aᵢ (these attributes must be created using register_label_attr).

solve_by_elim* tries to solve all goals together, using backtracking if a solution for one goal makes other goals impossible. (Adding or removing local hypotheses may not be well-behaved when starting with multiple goals.)

Optional arguments passed via a configuration argument as solve_by_elim (config := { ... })

maxDepth: number of attempts at discharging generated subgoals
symm: adds all hypotheses derived by symm (defaults to true).
exfalso: allow calling exfalso and trying again if solve_by_elim fails (defaults to true).
transparency: change the transparency mode when calling apply. Defaults to .default, but it is often useful to change to .reducible, so semireducible definitions will not be unfolded when trying to apply a lemma.

See also the doc-comment for Lean.Meta.Tactic.Backtrack.BacktrackConfig for the options proc, suspend, and discharge which allow further customization of solve_by_elim. Both apply_assumption and apply_rules are implemented via these hooks.

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Defined in module:: Init.Tactics

sorry

The sorry tactic is a temporary placeholder for an incomplete tactic proof, closing the main goal using exact sorry.

This is intended for stubbing-out incomplete parts of a proof while still having a syntactically correct proof skeleton. Lean will give a warning whenever a proof uses sorry, so you aren't likely to miss it, but you can double check if a theorem depends on sorry by looking for sorryAx in the output of the #print axioms my_thm command, the axiom used by the implementation of sorry.

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Defined in module:: Init.Tactics

sorry_if_sorry

Close the main goal with sorry if its type contains sorry, and fail otherwise.

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Defined in module:: Mathlib.CategoryTheory.Category.Basic

specialize

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 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.

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Defined in module:: Init.Tactics

specialize_all

specialize_all x runs specialize h x for all hypotheses h where this tactic succeeds.

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Defined in module:: Mathlib.Tactic.TautoSet

split

The split tactic is useful for breaking nested if-then-else and match expressions into separate cases. For a match expression with n cases, the split tactic generates at most n subgoals.

For example, given n : Nat, and a target if n = 0 then Q else R, split will generate one goal with hypothesis n = 0 and target Q, and a second goal with hypothesis ¬n = 0 and target R. Note that the introduced hypothesis is unnamed, and is commonly renamed using the case or next tactics.

split will split the goal (target).
split at h will split the hypothesis h.

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Defined in module:: Init.Tactics

split_ands

split_ands applies And.intro until it does not make progress.

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Defined in module:: Batteries.Tactic.Init

split_ifs

Splits all if-then-else-expressions into multiple goals. Given a goal of the form g (if p then x else y), split_ifs will produce two goals: p ⊢ g x and ¬p ⊢ g y. If there are multiple ite-expressions, then split_ifs will split them all, starting with a top-most one whose condition does not contain another ite-expression. split_ifs at * splits all ite-expressions in all hypotheses as well as the goal. split_ifs with h₁ h₂ h₃ overrides the default names for the hypotheses.

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Defined in module:: Mathlib.Tactic.SplitIfs

squeeze_scope

The squeeze_scope tactic allows aggregating multiple calls to simp coming from the same syntax but in different branches of execution, such as in cases x <;> simp. The reported simp call covers all simp lemmas used by this syntax.

@[simp] def bar (z : Nat) := 1 + z
@[simp] def baz (z : Nat) := 1 + z

@[simp] def foo : Nat → Nat → Nat
  | 0, z => bar z
  | _+1, z => baz z

example : foo x y = 1 + y := by
  cases x <;> simp? -- two printouts:
  -- "Try this: simp only [foo, bar]"
  -- "Try this: simp only [foo, baz]"

example : foo x y = 1 + y := by
  squeeze_scope
    cases x <;> simp -- only one printout: "Try this: simp only [foo, baz, bar]"

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Defined in module:: Batteries.Tactic.SqueezeScope

stop

stop is a helper tactic for "discarding" the rest of a proof: it is defined as repeat sorry. It is useful when working on the middle of a complex proofs, and less messy than commenting the remainder of the proof.

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Defined in module:: Init.Tactics

subsingleton

The subsingleton tactic tries to prove a goal of the form x = y or x ≍ y using the fact that the types involved are subsingletons (a type with exactly zero or one terms). To a first approximation, it does apply Subsingleton.elim. As a nicety, subsingleton first runs the intros tactic.

If the goal is an equality, it either closes the goal or fails.
subsingleton [inst1, inst2, ...] can be used to add additional Subsingleton instances to the local context. This can be more flexible than have := inst1; have := inst2; ...; subsingleton since the tactic does not require that all placeholders be solved for.

Techniques the subsingleton tactic can apply:

proof irrelevance
heterogeneous proof irrelevance (via proof_irrel_heq)
using Subsingleton (via Subsingleton.elim)
proving BEq instances are equal if they are both lawful (via lawful_beq_subsingleton)

Properties #

The tactic is careful not to accidentally specialize Sort _ to Prop, avoiding the following surprising behavior of apply Subsingleton.elim:

example (α : Sort _) (x y : α) : x = y := by apply Subsingleton.elim

The reason this example goes through is that it applies the ∀ (p : Prop), Subsingleton p instance, specializing the universe level metavariable in Sort _ to 0.

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Defined in module:: Mathlib.Tactic.Subsingleton

subst

subst x... substitutes each hypothesis x with a definition found in the local context, then eliminates the hypothesis.

If x is a local definition, then its definition is used.
Otherwise, if there is a hypothesis of the form x = e or e = x, then e is used for the definition of x.

If h : a = b, then subst h may be used if either a or b unfolds to a local hypothesis. This is similar to the cases h tactic.

See also: subst_vars for substituting all local hypotheses that have a defining equation.

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Defined in module:: Init.Tactics

subst_eqs

subst_eq repeatedly substitutes according to the equality proof hypotheses in the context, replacing the left side of the equality with the right, until no more progress can be made.

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Defined in module:: Init.Tactics

subst_hom_lift

subst_hom_lift p f φ tries to substitute f with p(φ) by using p.IsHomLift f φ

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Defined in module:: Mathlib.CategoryTheory.FiberedCategory.HomLift

subst_vars

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

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Defined in module:: Init.Tactics

substs

Applies the subst tactic to all given hypotheses from left to right.

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Defined in module:: Mathlib.Tactic.Substs

success_if_fail_with_msg

success_if_fail_with_msg msg tacs runs tacs and succeeds only if they fail with the message msg.

msg can be any term that evaluates to an explicit String.

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Defined in module:: Mathlib.Tactic.SuccessIfFailWithMsg

suffices

Given a main goal ctx ⊢ t, suffices h : t' from e replaces the main goal with ctx ⊢ t', e must have type t in the context ctx, h : t'.

The variant suffices h : t' by tac is a shorthand for suffices h : t' from by tac. If h : is omitted, the name this is used.

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Defined in module:: Init.Tactics

suggestions

#suggestions will suggest relevant theorems from the library for the current goal, using the currently registered library suggestion engine.

The suggestions are printed in the order of their confidence, from highest to lowest.

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Defined in module:: Init.Tactics

swap

swap is a shortcut for pick_goal 2, which interchanges the 1st and 2nd goals.

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Defined in module:: Batteries.Tactic.PermuteGoals

swap_var

swap_var swap_rule₁, swap_rule₂, ⋯ applies swap_rule₁ then swap_rule₂ then ⋯.

A swap_rule is of the form x y or x ↔ y, and "applying it" means swapping the variable name x by y and vice-versa on all hypotheses and the goal.

example {P Q : Prop} (q : P) (p : Q) : P ∧ Q := by
  swap_var p ↔ q
  exact ⟨p, q⟩

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Defined in module:: Mathlib.Tactic.SwapVar

symm

symm 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.
symm at h will rewrite a hypothesis h : t ~ u to h : u ~ t.

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Defined in module:: Init.Tactics

symm_saturate

For every hypothesis h : a ~ b where a @[symm] lemma is available, add a hypothesis h_symm : b ~ a.

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Defined in module:: Init.Tactics

tauto

tauto breaks down assumptions of the form _ ∧ _, _ ∨ _, _ ↔ _ and ∃ _, _ and splits a goal of the form _ ∧ _, _ ↔ _ or ∃ _, _ until it can be discharged using rfl or solve_by_elim. This is a finishing tactic: it either closes the goal or raises an error.

The Lean 3 version of this tactic by default attempted to avoid classical reasoning where possible. This Lean 4 version makes no such attempt. The itauto tactic is designed for that purpose.

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Defined in module:: Mathlib.Tactic.Tauto

tauto_set

tauto_set attempts to prove tautologies involving hypotheses and goals of the form X ⊆ Y or X = Y, where X, Y are expressions built using ∪, ∩, , and ᶜ from finitely many variables of type Set α. It also unfolds expressions of the form Disjoint A B and symmDiff A B.

Examples:

example {α} (A B C D : Set α) (h1 : A ⊆ B) (h2 : C ⊆ D) : C \ B ⊆ D \ A := by
  tauto_set

example {α} (A B C : Set α) (h1 : A ⊆ B ∪ C) : (A ∩ B) ∪ (A ∩ C) = A := by
  tauto_set

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Defined in module:: Mathlib.Tactic.TautoSet

tfae_finish

tfae_finish closes goals of the form TFAE [P₁, P₂, ...] once a sufficient collection of hypotheses of the form Pᵢ → Pⱼ or Pᵢ ↔ Pⱼ have been introduced to the local context.

tfae_have can be used to conveniently introduce these hypotheses; see tfae_have.

Example:

example : TFAE [P, Q, R] := by
  tfae_have 1 → 2 := sorry /- proof of P → Q -/
  tfae_have 2 → 1 := sorry /- proof of Q → P -/
  tfae_have 2 ↔ 3 := sorry /- proof of Q ↔ R -/
  tfae_finish

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Defined in module:: Mathlib.Tactic.TFAE

tfae_have

tfae_have i → j := t, where the goal is TFAE [P₁, P₂, ...] introduces a hypothesis tfae_i_to_j : Pᵢ → Pⱼ and proof t to the local context. Note that i and j are natural number literals (beginning at 1) used as indices to specify the propositions P₁, P₂, ... that appear in the goal.

Once sufficient hypotheses have been introduced by tfae_have, tfae_finish can be used to close the goal.

All features of have are supported by tfae_have, including naming, matching, destructuring, and goal creation.

tfae_have i ← j := t adds a hypothesis in the reverse direction, of type Pⱼ → Pᵢ.
tfae_have i ↔ j := t adds a hypothesis in the both directions, of type Pᵢ ↔ Pⱼ.
tfae_have hij : i → j := t names the introduced hypothesis hij instead of tfae_i_to_j.
tfae_have i j | p₁ => t₁ | ... matches on the assumption p : Pᵢ.
tfae_have ⟨hij, hji⟩ : i ↔ j := t destructures the bi-implication into hij : Pᵢ → Pⱼ and hji : Pⱼ → Pⱼ.
tfae_have i → j := t ?a creates a new goal for ?a.

Examples:

example (h : P → R) : TFAE [P, Q, R] := by
  tfae_have 1 → 3 := h
  -- The resulting context now includes `tfae_1_to_3 : P → R`.
  sorry

-- An example of `tfae_have` and `tfae_finish`:
example : TFAE [P, Q, R] := by
  tfae_have 1 → 2 := sorry /- proof of P → Q -/
  tfae_have 2 → 1 := sorry /- proof of Q → P -/
  tfae_have 2 ↔ 3 := sorry /- proof of Q ↔ R -/
  tfae_finish

-- All features of `have` are supported by `tfae_have`:
example : TFAE [P, Q] := by
  -- assert `tfae_1_to_2 : P → Q`:
  tfae_have 1 → 2 := sorry

  -- assert `hpq : P → Q`:
  tfae_have hpq : 1 → 2 := sorry

  -- match on `p : P` and prove `Q` via `f p`:
  tfae_have 1 → 2
  | p => f p

  -- assert `pq : P → Q`, `qp : Q → P`:
  tfae_have ⟨pq, qp⟩ : 1 ↔ 2 := sorry

  -- assert `h : P → Q`; `?a` is a new goal:
  tfae_have h : 1 → 2 := f ?a

  sorry

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Defined in module:: Mathlib.Tactic.TFAE

toFinite_tac

A tactic (for use in default params) that applies Set.toFinite to synthesize a Set.Finite term.

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Defined in module:: Mathlib.Data.Set.Card

to_encard_tac

A tactic useful for transferring proofs for encard to their corresponding card statements

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Defined in module:: Mathlib.Data.Set.Card

trace

Evaluates a term to a string (when possible), and prints it as a trace message.

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Defined in module:: Mathlib.Tactic.Trace

trace

trace msg displays msg in the info view.

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Defined in module:: Init.Tactics

trace_state

trace_state displays the current state in the info view.

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Defined in module:: Init.Tactics

trans

trans 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].

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

Additionally, trans also applies to a goal whose target has the form t → u, in which case it replaces the goal with t → s and s → u.

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Defined in module:: Batteries.Tactic.Trans

transitivity

Synonym for trans tactic.

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Defined in module:: Batteries.Tactic.Trans

triv

Deprecated variant of trivial.

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Defined in module:: Batteries.Tactic.Init

trivial

trivial tries different simple tactics (e.g., rfl, contradiction, ...) to close the current goal. You can use the command macro_rules to extend the set of tactics used. Example:

macro_rules | `(tactic| trivial) => `(tactic| simp)

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Defined in module:: Init.Tactics

try

try tac runs tac and succeeds even if tac failed.

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Defined in module:: Init.Tactics

try?

This tactic has no documentation.

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Defined in module:: Init.Try

try_suggestions

Helper internal tactic used to implement evalSuggest in try?

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Defined in module:: Init.Try

try_this

Produces the text Try this: <tac> with the given tactic, and then executes it.

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Defined in module:: Mathlib.Tactic.TryThis

type_check

Type check the given expression, and trace its type.

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Defined in module:: Mathlib.Tactic.TypeCheck

unfold

unfold id unfolds all occurrences of definition id in the target.
unfold id1 id2 ... is equivalent to unfold id1; unfold id2; ....
unfold id at h unfolds at the hypothesis h.

Definitions can be either global or local definitions.

For non-recursive global definitions, this tactic is identical to delta. For recursive global definitions, it uses the "unfolding lemma" id.eq_def, which is generated for each recursive definition, to unfold according to the recursive definition given by the user. Only one level of unfolding is performed, in contrast to simp only [id], which unfolds definition id recursively.

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Defined in module:: Init.Tactics

unfold?

Replace the selected expression with a definitional unfolding.

After each unfolding, we apply whnfCore to simplify the expression.
Explicit natural number expressions are evaluated.
Unfolds of class projections of instances marked with @[default_instance] are not shown. This is relevant for notational type classes like +: we don't want to suggest Add.add a b as an unfolding of a + b. Similarly for OfNat n : Nat which unfolds into n : Nat.

To use unfold?, shift-click an expression in the tactic state. This gives a list of rewrite suggestions for the selected expression. Click on a suggestion to replace unfold? by a tactic that performs this rewrite.

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Defined in module:: Mathlib.Tactic.Widget.InteractiveUnfold

unfold_projs

unfold_projs at loc unfolds projections of class instances at the given location. This also exists as a conv-mode tactic.

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Defined in module:: Mathlib.Tactic.DefEqTransformations

unhygienic

unhygienic tacs runs tacs with name hygiene disabled. This means that tactics that would normally create inaccessible names will instead make regular variables. Warning: Tactics may change their variable naming strategies at any time, so code that depends on autogenerated names is brittle. Users should try not to use unhygienic if possible.

example : ∀ x : Nat, x = x := by unhygienic
  intro            -- x would normally be intro'd as inaccessible
  exact Eq.refl x  -- refer to x

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Defined in module:: Init.Tactics

uniqueDiffWithinAt_Ici_Iic_univ

An auxiliary tactic closing goals UniqueDiffWithinAt ℝ s a where s ∈ {Iic a, Ici a, univ}.

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Defined in module:: Mathlib.MeasureTheory.Integral.IntervalIntegral.FundThmCalculus

unit_interval

A tactic that solves 0 ≤ ↑x, 0 ≤ 1 - ↑x, ↑x ≤ 1, and 1 - ↑x ≤ 1 for x : I.

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Defined in module:: Mathlib.Topology.UnitInterval

unreachable!

This tactic causes a panic when run (at compile time). (This is distinct from exact unreachable!, which inserts code which will panic at run time.)

It is intended for tests to assert that a tactic will never be executed, which is otherwise an unusual thing to do (and the unreachableTactic linter will give a warning if you do).

The unreachableTactic linter has a special exception for uses of unreachable!.

example : True := by trivial <;> unreachable!

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Defined in module:: Batteries.Tactic.Unreachable

use

use e₁, e₂, ⋯ is similar to exists, but unlike exists it is equivalent to applying the tactic refine ⟨e₁, e₂, ⋯, ?_, ⋯, ?_⟩ with any number of placeholders (rather than just one) and then trying to close goals associated to the placeholders with a configurable discharger (rather than just try trivial).

Examples:

example : ∃ x : Nat, x = x := by use 42

example : ∃ x : Nat, ∃ y : Nat, x = y := by use 42, 42

example : Nonempty Nat := by use 5

example : Nonempty (PNat ≃ Nat) := by
  use PNat.natPred, Nat.succPNat
  · exact PNat.succPNat_natPred
  · intro; rfl

use! e₁, e₂, ⋯ is similar but it applies constructors everywhere rather than just for goals that correspond to the last argument of a constructor. This gives the effect that nested constructors are being flattened out, with the supplied values being used along the leaves and nodes of the tree of constructors. With use! one can feed in each 42 one at a time:

example : ∃ n : {n : Nat // n % 2 = 0}, n.val > 10 := by use! 20; simp

example : ∃ p : Nat × Nat, p.1 = p.2 := by use! (42, 42)

The second line makes use of the fact that use! tries refining with the argument before applying a constructor. Also note that use/use! by default uses a tactic called use_discharger to discharge goals, so use! 42 will close the goal in this example since use_discharger applies rfl, which as a consequence solves for the other Nat metavariable.

These tactics take an optional discharger to handle remaining explicit Prop constructor arguments. By default it is use (discharger := try with_reducible use_discharger) e₁, e₂, ⋯. To turn off the discharger and keep all goals, use (discharger := skip). To allow "heavy refls", use (discharger := try use_discharger).

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Defined in module:: Mathlib.Tactic.Use

use_discharger

Default discharger to try to use for the use and use! tactics. This is similar to the trivial tactic but doesn't do things like contradiction or decide.

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Defined in module:: Mathlib.Tactic.Use

use_finite_instance

Try using Set.toFinite to dispatch a Set.Finite goal.

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Defined in module:: Mathlib.LinearAlgebra.Dual.Basis

valid

A wrapper for omega which prefaces it with some quick and useful attempts

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Defined in module:: Mathlib.CategoryTheory.ComposableArrows.Basic

volume_tac

The tactic exact volume, to be used in optional (autoParam) arguments.

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Defined in module:: Mathlib.MeasureTheory.Measure.MeasureSpaceDef

wait_for_unblock_async

Spawns a logSnapshotTask that waits for unblock to be called, which is expected to happen in a subsequent document version that does not invalidate this tactic. Complains if cancellation token was set before unblocking, i.e. if the tactic was invalidated after all.

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Defined in module:: Lean.Server.Test.Cancel

whisker_simps

Simp lemmas for rewriting a 2-morphism into a normal form.

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Defined in module:: Mathlib.Tactic.CategoryTheory.BicategoryCoherence

whnf

whnf at loc puts the given location into weak-head normal form. This also exists as a conv-mode tactic.

Weak-head normal form is when the outer-most expression has been fully reduced, the expression may contain subexpressions which have not been reduced.

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Defined in module:: Mathlib.Tactic.DefEqTransformations

with_panel_widgets

Display the selected panel widgets in the nested tactic script. For example, assuming we have written a GeometryDisplay component,

by with_panel_widgets [GeometryDisplay]
  simp
  rfl

will show the geometry display alongside the usual tactic state throughout the proof.

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Defined in module:: ProofWidgets.Component.Panel.Basic

with_reducible

with_reducible tacs executes tacs using the reducible transparency setting. In this setting only definitions tagged as [reducible] are unfolded.

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Defined in module:: Init.Tactics

with_reducible_and_instances

with_reducible_and_instances tacs executes tacs using the .instances transparency setting. In this setting only definitions tagged as [reducible] or type class instances are unfolded.

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Defined in module:: Init.Tactics

with_unfolding_all

with_unfolding_all tacs executes tacs using the .all transparency setting. In this setting all definitions that are not opaque are unfolded.

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Defined in module:: Init.Tactics

with_unfolding_none

with_unfolding_none tacs executes tacs using the .none transparency setting. In this setting no definitions are unfolded.

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Defined in module:: Init.Tactics

witt_truncateFun_tac

A macro tactic used to prove that truncateFun respects ring operations.

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Defined in module:: Mathlib.RingTheory.WittVector.Truncated

wlog

wlog h : P adds an assumption h : P to the main goal, and adds a side goal that requires showing that the case h : ¬ P can be reduced to the case where P holds (typically by symmetry). The side goal will be at the top of the stack. In this side goal, there will be two additional assumptions:

h : ¬ P: the assumption that P does not hold
this: which is the statement that in the old context P suffices to prove the goal. By default, the entire context is reverted to produce this.

wlog h : P with H gives the name H to the statement that P proves the goal.
wlog h : P generalizing x y ... reverts certain parts of the context before creating the new goal. In this way, the wlog-claim this can be applied to x and y in different orders (exploiting symmetry, which is the typical use case).
wlog! h : P also calls push_neg at the generated hypothesis h. wlog! h : P ∧ Q will transform ¬ (P ∧ Q) to P → ¬ Q
wlog! +distrib h : P also calls push_neg +distrib at the generated hypothesis h. wlog! +distrib h : P ∧ Q will transform ¬ (P ∧ Q) to ¬P ∨ ¬Q.

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Defined in module:: Mathlib.Tactic.WLOG

zify

The zify tactic is used to shift propositions from Nat to Int. This is often useful since Int has well-behaved subtraction.

example (a b c x y z : Nat) (h : ¬ x*y*z < 0) : c < a + 3*b := by
  zify
  zify at h
  /-
  h : ¬↑x * ↑y * ↑z < 0
  ⊢ ↑c < ↑a + 3 * ↑b
  -/

zify can be given extra lemmas to use in simplification. This is especially useful in the presence of nat subtraction: passing ≤ arguments will allow push_cast to do more work.

example (a b c : Nat) (h : a - b < c) (hab : b ≤ a) : false := by
  zify [hab] at h
  /- h : ↑a - ↑b < ↑c -/

zify makes use of the @[zify_simps] attribute to move propositions, and the push_cast tactic to simplify the Int-valued expressions. zify is in some sense dual to the lift tactic. lift (z : Int) to Nat will change the type of an integer z (in the supertype) to Nat (the subtype), given a proof that z ≥ 0; propositions concerning z will still be over Int. zify changes propositions about Nat (the subtype) to propositions about Int (the supertype), without changing the type of any variable.

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Defined in module:: Mathlib.Tactic.Zify

∎

∎ (typed as \qed) is a macro that expands to try? in tactic mode.

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Defined in module:: Init.Try

◾

This tactic has no documentation.

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Defined in module:: Archive.Examples.IfNormalization.Result