Lean version: leanprover/lean4:v4.10.0-rc1

#adaptation_note

Defined in: «tactic#adaptation_note_»

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

#check

Defined in: Mathlib.Tactic.«tactic#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.

#find

Defined in: Mathlib.Tactic.Find.«tactic#find_»

(

Defined in: Lean.Parser.Tactic.paren

(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 ;.

<;>

Defined in: Batteries.Tactic.seq_focus

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.

<;>

Defined in: Lean.Parser.Tactic.«tactic_<;>_»

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

_

Defined in: Batteries.Tactic.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.

abel

Defined in: Mathlib.Tactic.Abel.abel_term

Unsupported legacy syntax from mathlib3, which allowed passing additional terms to abel.

abel

Defined in: Mathlib.Tactic.Abel.abel

Tactic for evaluating expressions in abelian groups.

• abel! will use a more aggressive reducibility setting to determine equality of atoms.
• abel1 fails if the target is not an equality.

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

abel!

Defined in: Mathlib.Tactic.Abel.abel!_term

Unsupported legacy syntax from mathlib3, which allowed passing additional terms to abel!.

abel!

Defined in: Mathlib.Tactic.Abel.tacticAbel!

Tactic for evaluating expressions in abelian groups.

• abel! will use a more aggressive reducibility setting to determine equality of atoms.
• abel1 fails if the target is not an equality.

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

abel1

Defined in: Mathlib.Tactic.Abel.abel1

Tactic for solving equations in the language of additive, commutative monoids and groups. This version of abel fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.

abel1! will use a more aggressive reducibility setting to identify atoms. This can prove goals that abel cannot, but is more expensive.

abel1!

Defined in: Mathlib.Tactic.Abel.abel1!

Tactic for solving equations in the language of additive, commutative monoids and groups. This version of abel fails if the target is not an equality that is provable by the axioms of commutative monoids/groups.

abel1! will use a more aggressive reducibility setting to identify atoms. This can prove goals that abel cannot, but is more expensive.

abel_nf

Defined in: Mathlib.Tactic.Abel.abelNF

Simplification tactic for expressions in the language of abelian groups, which rewrites all group expressions into a normal form.

• abel_nf! will use a more aggressive reducibility setting to identify atoms.
• abel_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • recursive: if true, abel_nf will also recurse into atoms
• abel_nf works as both a tactic and a conv tactic. In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.

abel_nf!

Defined in: Mathlib.Tactic.Abel.tacticAbel_nf!__

Simplification tactic for expressions in the language of abelian groups, which rewrites all group expressions into a normal form.

• abel_nf! will use a more aggressive reducibility setting to identify atoms.
• abel_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • recursive: if true, abel_nf will also recurse into atoms
• abel_nf works as both a tactic and a conv tactic. In tactic mode, abel_nf at h can be used to rewrite in a hypothesis.

absurd

Defined in: Batteries.Tactic.tacticAbsurd_

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.

ac_change

Defined in: Mathlib.Tactic.acChange

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

ac_rfl

Defined in: Lean.Parser.Tactic.acRfl

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

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

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

admit

Defined in: Lean.Parser.Tactic.tacticAdmit

admit is a shorthand for exact sorry.

aesop

Defined in: Aesop.Frontend.Parser.aesopTactic

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 }).

aesop?

Defined in: Aesop.Frontend.Parser.aesopTactic?

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 }).

aesop_cat

Defined in: CategoryTheory.aesop_cat

A thin wrapper for aesop which adds the CategoryTheory rule set and allows aesop to look through semireducible definitions when calling intros. It also turns on zetaDelta in the simp config, allowing aesop_cat to unfold any lets. This tactic fails when it is unable to solve the goal, making it suitable for use in auto-params.

aesop_cat?

Defined in: CategoryTheory.aesop_cat?

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

aesop_cat_nonterminal

Defined in: CategoryTheory.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.

aesop_unfold

Defined in: Aesop.tacticAesop_unfold_

aesop_unfold

Defined in: Aesop.tacticAesop_unfold_At_

all_goals

Defined in: Lean.Parser.Tactic.allGoals

all_goals tac runs tac on each goal, concatenating the resulting goals, if any.

and_intros

Defined in: Lean.Parser.Tactic.tacticAnd_intros

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

any_goals

Defined in: Lean.Parser.Tactic.anyGoals

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

apply

Defined in: Mathlib.Tactic.tacticApply_At_

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

apply

Defined in: Lean.Parser.Tactic.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.

apply

Defined in: Mathlib.Tactic.applyWith

apply (config := cfg) e is like apply e but allows you to provide a configuration cfg : ApplyConfig to pass to the underlying apply operation.

apply?

Defined in: Lean.Parser.Tactic.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.

apply_assumption

Defined in: Lean.Parser.Tactic.applyAssumption

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).

apply_ext_theorem

Defined in: Lean.Elab.Tactic.Ext.applyExtTheorem

Apply a single extensionality theorem to the current goal.

apply_fun

Defined in: Mathlib.Tactic.applyFun

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
  intros 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.

apply_mod_cast

Defined in: Lean.Parser.Tactic.tacticApply_mod_cast_

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

apply_rfl

Defined in: Lean.Parser.Tactic.applyRfl

This tactic applies to a goal whose target has the form x ~ x, where ~ is a reflexive relation other than =, that is, a relation which has a reflexive lemma tagged with the attribute @[refl].

apply_rules

Defined in: Lean.Parser.Tactic.applyRules

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.

arith_mult

Defined in: ArithmeticFunction.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.

arith_mult?

Defined in: ArithmeticFunction.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.

array_get_dec

Defined in: Array.tacticArray_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.

array_mem_dec

Defined in: Array.tacticArray_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.

assumption

Defined in: Lean.Parser.Tactic.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.

assumption'

Defined in: Mathlib.Tactic.tacticAssumption'

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

assumption_mod_cast

Defined in: Lean.Parser.Tactic.tacticAssumption_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.

aux_group₁

Defined in: Mathlib.Tactic.Group.aux_group₁

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

aux_group₂

Defined in: Mathlib.Tactic.Group.aux_group₂

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

bddDefault

Defined in: tacticBddDefault

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).

beta_reduce

Defined in: Mathlib.Tactic.betaReduceStx

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.

bicategory_coherence

Defined in: Mathlib.Tactic.BicategoryCoherence.tacticBicategory_coherence

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

bitwise_assoc_tac

Defined in: Nat.tacticBitwise_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.

borelize

Defined in: Mathlib.Tactic.Borelize.tacticBorelize___

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.

bv_omega

Defined in: Lean.Parser.Tactic.tacticBv_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 [bv_toNat] at *. bv_toNat is a @[simp] attribute that you can (cautiously) add to more theorems.

by_cases

Defined in: «tacticBy_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.

by_contra

Defined in: Batteries.Tactic.byContra

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 _: ¬p will be anonymous.

by_contra!

Defined in: byContra!

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 negation normalized using push_neg. For instance, ¬ a < b will be changed to b ≤ a. 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

calc

Defined in: Lean.calcTactic

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.

cancel_denoms

Defined in: tacticCancel_denoms_

cancel_denoms

Defined in: cancelDenoms

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

case

Defined in: Batteries.Tactic.casePatt

• 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.

case

Defined in: Lean.Parser.Tactic.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).

case'

Defined in: Lean.Parser.Tactic.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.

case'

Defined in: Batteries.Tactic.casePatt'

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.

cases

Defined in: Lean.Parser.Tactic.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 hypothesis, where ... is the constructor instance for that particular case.

cases'

Defined in: Mathlib.Tactic.cases'

cases_type

Defined in: Mathlib.Tactic.casesType

• 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

cases_type!

Defined in: Mathlib.Tactic.casesType!

• 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

casesm

Defined in: Mathlib.Tactic.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.

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

casesm* _ ∨ _, _ ∧ _

cc

Defined in: Mathlib.Tactic.cc

The congruence closure tactic cc tries to solve the goal by chaining equalities from context and applying congruence (i.e. if a = b, then f a = f b). It is a finishing tactic, i.e. it is meant to close the current goal, not to make some inconclusive progress. A mostly trivial example would be:

example (a b c : ℕ) (f : ℕ → ℕ) (h: a = b) (h' : b = c) : f a = f c := by
  cc

As an example requiring some thinking to do by hand, consider:

example (f : ℕ → ℕ) (x : ℕ)
    (H1 : f (f (f x)) = x) (H2 : f (f (f (f (f x)))) = x) :
    f x = x := by
  cc

change

Defined in: Lean.Parser.Tactic.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

Defined in: Lean.Parser.Tactic.changeWith

• change a with b will change occurrences of a to b in the goal, assuming a and b are are definitionally equal.
• change a with b at h similarly changes a to b in the type of hypothesis h.

change?

Defined in: 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`

checkpoint

Defined in: Lean.Parser.Tactic.checkpoint

checkpoint tac acts the same as tac, but it caches the input and output of tac, and if the file is re-elaborated and the input matches, the tactic is not re-run and its effects are reapplied to the state. This is useful for improving responsiveness when working on a long tactic proof, by wrapping expensive tactics with checkpoint.

See the save tactic, which may be more convenient to use.

(TODO: do this automatically and transparently so that users don't have to use this combinator explicitly.)

choose

Defined in: Mathlib.Tactic.Choose.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.

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

choose!

Defined in: Mathlib.Tactic.Choose.tacticChoose!___Using_

• 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.

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

classical

Defined in: Batteries.Tactic.tacticClassical_

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

Note that (unlike lean 3) classical is a scoping tactic - it adds the instance only within the scope of the tactic.

classical!

Defined in: Batteries.Tactic.tacticClassical!

classical! has been removed; use classical instead

clean

Defined in: Mathlib.Tactic.tacticClean_

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

clear

Defined in: Lean.Elab.Tactic.clearExcept

Clears all hypotheses it can besides those provided

clear

Defined in: Lean.Parser.Tactic.clear

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

clear!

Defined in: Mathlib.Tactic.clear!

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

clear_

Defined in: Mathlib.Tactic.clear_

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

clear_aux_decl

Defined in: Mathlib.Tactic.clearAuxDecl

This tactic clears all auxiliary declarations from the context.

clear_value

Defined in: Mathlib.Tactic.clearValue

clear_value n₁ n₂ ... clears the bodies of the local definitions n₁, n₂ ..., changing them into regular hypotheses. A hypothesis n : α := t is changed to n : α.

The order of n₁ n₂ ... does not matter, and values will be cleared in reverse order of where they appear in the context.

coherence

Defined in: Mathlib.Tactic.Coherence.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.)

compareOfLessAndEq_rfl

Defined in: tacticCompareOfLessAndEq_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:

1. seeing if rfl works
2. 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
3. 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)

compute_degree

Defined in: Mathlib.Tactic.ComputeDegree.computeDegree

compute_degree is a tactic to solve goals of the form

• natDegree f = d,
• degree f = d,
• natDegree f ≤ d,
• degree f ≤ d,
• coeff f d = r, if d is the degree of f.

The tactic may leave goals of the form 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 clos them.

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

compute_degree!

Defined in: Mathlib.Tactic.ComputeDegree.tacticCompute_degree!

compute_degree is a tactic to solve goals of the form

• natDegree f = d,
• degree f = d,
• natDegree f ≤ d,
• degree f ≤ d,
• coeff f d = r, if d is the degree of f.

The tactic may leave goals of the form 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 clos them.

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

congr

Defined in: Lean.Parser.Tactic.congr

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

congr

Defined in: Batteries.Tactic.congrConfigWith

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

congr

Defined in: Batteries.Tactic.congrConfig

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

congr!

Defined in: 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! (config := {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.

congrm

Defined in: Mathlib.Tactic.congrM

congrm e is a tactic for proving goals of the form lhs = rhs, lhs ↔ rhs, HEq 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.

congrm?

Defined in: tacticCongrm?

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

constructor

Defined in: Lean.Parser.Tactic.constructor

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

constructorm

Defined in: Mathlib.Tactic.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

continuity

Defined in: tacticContinuity

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

continuity?

Defined in: tacticContinuity?

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

contradiction

Defined in: Lean.Parser.Tactic.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

contrapose

Defined in: Mathlib.Tactic.Contrapose.contrapose

Transforms the goal into its contrapositive.

• contrapose turns a goal P → Q into ¬ Q → ¬ P
• 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

contrapose!

Defined in: Mathlib.Tactic.Contrapose.contrapose!

Transforms the goal into its contrapositive and uses pushes negations inside P and Q. Usage matches contrapose

conv

Defined in: Lean.Parser.Tactic.Conv.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).

conv'

Defined in: Lean.Parser.Tactic.Conv.convTactic

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

conv?

Defined in: tacticConv?

Display a widget panel allowing to generate a conv call zooming to the subexpression selected in the goal.

conv_lhs

Defined in: Mathlib.Tactic.Conv.convLHS

conv_rhs

Defined in: Mathlib.Tactic.Conv.convRHS

convert

Defined in: Mathlib.Tactic.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.

convert_to

Defined in: Mathlib.Tactic.convertTo

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.

count_heartbeats

Defined in: Mathlib.CountHeartbeats.tacticCount_heartbeats_

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

count_heartbeats!

Defined in: Mathlib.CountHeartbeats.tacticCount_heartbeats!_In__

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.

dbg_trace

Defined in: Lean.Parser.Tactic.dbgTrace

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"

decide

Defined in: Lean.Parser.Tactic.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.

Limitations:

• The target is not allowed to contain local variables or metavariables. If there are local variables, you can try first using the revert tactic with these local variables to move them into the target, which may allow decide to succeed.
• Because this uses kernel reduction to evaluate the term, Decidable instances defined by well-founded recursion might not work, because evaluating them requires reducing proofs. The kernel can also get stuck reducing Decidable instances with Eq.rec terms for rewriting propositions. These can appear for instances defined using tactics (such as rw and simp). To avoid this, use 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

decreasing_tactic

Defined in: tacticDecreasing_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).

decreasing_trivial

Defined in: tacticDecreasing_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)

decreasing_trivial_pre_omega

Defined in: tacticDecreasing_trivial_pre_omega

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

decreasing_with

Defined in: tacticDecreasing_with_

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

delta

Defined in: Lean.Parser.Tactic.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.

done

Defined in: Lean.Parser.Tactic.done

done succeeds iff there are no remaining goals.

dsimp

Defined in: Lean.Parser.Tactic.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.

dsimp!

Defined in: Lean.Parser.Tactic.dsimpAutoUnfold

dsimp! is shorthand for dsimp with autoUnfold := true. This will rewrite with all equation lemmas, which can be used to partially evaluate many definitions.

dsimp?

Defined in: Lean.Parser.Tactic.dsimpTrace

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.

dsimp?!

Defined in: Lean.Parser.Tactic.tacticDsimp?!_

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.

eapply

Defined in: Batteries.Tactic.tacticEapply_

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'

econstructor

Defined in: tacticEconstructor

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.

elementwise

Defined in: Tactic.Elementwise.tacticElementwise___

elementwise!

Defined in: Tactic.Elementwise.tacticElementwise!___

else

Defined in: Lean.Parser.Tactic.tacDepIfThenElse

In tactic mode, 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 and tac1 and tac2 are the subproofs.

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.

else

Defined in: Lean.Parser.Tactic.tacIfThenElse

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 ?_.)

eq_refl

Defined in: Lean.Parser.Tactic.eqRefl

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

erw

Defined in: Lean.Parser.Tactic.tacticErw__

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

eta_expand

Defined in: Mathlib.Tactic.etaExpandStx

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.

eta_reduce

Defined in: Mathlib.Tactic.etaReduceStx

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.

eta_struct

Defined in: Mathlib.Tactic.etaStructStx

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.

exact

Defined in: Lean.Parser.Tactic.exact

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

exact?

Defined in: Lean.Parser.Tactic.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.

exact_mod_cast

Defined in: Lean.Parser.Tactic.tacticExact_mod_cast_

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

exacts

Defined in: Batteries.Tactic.exacts

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

exfalso

Defined in: Lean.Parser.Tactic.tacticExfalso

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

exists

Defined in: Lean.Parser.Tactic.«tacticExists_,,»

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

existsi

Defined in: Mathlib.Tactic.«tacticExistsi_,,»

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

ext

Defined in: Lean.Elab.Tactic.Ext.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.

ext1

Defined in: Lean.Elab.Tactic.Ext.tacticExt1___

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.

extract_goal

Defined in: Mathlib.Tactic.ExtractGoal.extractGoal

• 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.

extract_lets

Defined in: Mathlib.extractLets

The extract_lets at h tactic takes a local hypothesis of the form h : let x := v; b and introduces a new local definition x := v while changing h to be h : b. It can be thought of as being a cases tactic for let expressions. It can also be thought of as being like intros at h for let expressions.

For example, if h : let x := 1; x = x, then extract_lets x at h introduces x : Nat := 1 and changes h to h : x = x.

Just like intros, the extract_lets tactic either takes a list of names, in which case that specifies the number of let bindings that must be extracted, or it takes no names, in which case all the let bindings are extracted.

The tactic extract_lets (without at) or extract_lets at h ⊢ acts as a weaker form of intros on the goal that only introduces obvious lets.

fail

Defined in: Lean.Parser.Tactic.fail

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

fail_if_no_progress

Defined in: Mathlib.Tactic.failIfNoProgress

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.

fail_if_success

Defined in: Lean.Parser.Tactic.failIfSuccess

fail_if_success t fails if the tactic t succeeds.

false_or_by_contra

Defined in: Lean.Parser.Tactic.falseOrByContra

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.

fapply

Defined in: Batteries.Tactic.tacticFapply_

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

fconstructor

Defined in: tacticFconstructor

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

field_simp

Defined in: Mathlib.Tactic.FieldSimp.fieldSimp

The goal of field_simp is to reduce an expression in a field to an expression of the form n / d where neither n nor d contains any division symbol, just using the simplifier (with a carefully crafted simpset named field_simps) to reduce the number of division symbols whenever possible by iterating the following steps:

• write an inverse as a division
• in any product, move the division to the right
• if there are several divisions in a product, group them together at the end and write them as a single division
• reduce a sum to a common denominator

If the goal is an equality, this simpset will also clear the denominators, so that the proof can normally be concluded by an application of ring.

field_simp [hx, hy] is a short form for simp (disch := field_simp_discharge) [-one_div, -one_divp, -mul_eq_zero, hx, hy, field_simps]

Note that this naive algorithm will not try to detect common factors in denominators to reduce the complexity of the resulting expression. Instead, it relies on the ability of ring to handle complicated expressions in the next step.

As always with the simplifier, reduction steps will only be applied if the preconditions of the lemmas can be checked. This means that proofs that denominators are nonzero should be included. The fact that a product is nonzero when all factors are, and that a power of a nonzero number is nonzero, are included in the simpset, but more complicated assertions (especially dealing with sums) should be given explicitly. If your expression is not completely reduced by the simplifier invocation, check the denominators of the resulting expression and provide proofs that they are nonzero to enable further progress.

To check that denominators are nonzero, field_simp will look for facts in the context, and will try to apply norm_num to close numerical goals.

The invocation of field_simp removes the lemma one_div from the simpset, as this lemma works against the algorithm explained above. It also removes mul_eq_zero : x * y = 0 ↔ x = 0 ∨ y = 0, as norm_num can not work on disjunctions to close goals of the form 24 ≠ 0, and replaces it with mul_ne_zero : x ≠ 0 → y ≠ 0 → x * y ≠ 0 creating two goals instead of a disjunction.

For example,

example (a b c d x y : ℂ) (hx : x ≠ 0) (hy : y ≠ 0) :
    a + b / x + c / x^2 + d / x^3 = a + x⁻¹ * (y * b / y + (d / x + c) / x) := by
  field_simp
  ring

Moreover, the field_simp tactic can also take care of inverses of units in a general (commutative) monoid/ring and partial division /ₚ, see Algebra.Group.Units for the definition. Analogue to the case above, the lemma one_divp is removed from the simpset as this works against the algorithm. If you have objects with an IsUnit x instance like (x : R) (hx : IsUnit x), you should lift them with lift x to Rˣ using id hx; rw [IsUnit.unit_of_val_units] clear hx before using field_simp.

See also the cancel_denoms tactic, which tries to do a similar simplification for expressions that have numerals in denominators. The tactics are not related: cancel_denoms will only handle numeric denominators, and will try to entirely remove (numeric) division from the expression by multiplying by a factor.

field_simp_discharge

Defined in: Mathlib.Tactic.FieldSimp.tacticField_simp_discharge

Discharge strategy for the field_simp tactic.

filter_upwards

Defined in: Mathlib.Tactic.filterUpwards

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], intros 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.

fin_cases

Defined in: Lean.Elab.Tactic.finCases

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.

find

Defined in: Mathlib.Tactic.Find.tacticFind

first

Defined in: Lean.Parser.Tactic.first

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

focus

Defined in: Lean.Parser.Tactic.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.

forward

Defined in: Aesop.Frontend.tacticForward___

frac_tac

Defined in: RatFunc.tacticFrac_tac

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

fun_prop

Defined in: Mathlib.Meta.FunProp.funPropTacStx

Tactic to prove function properties

funext

Defined in: tacticFunext___

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.

gcongr

Defined in: Mathlib.Tactic.GCongr.tacticGcongr__With__

The gcongr tactic applies "generalized congruence" rules, reducing a relational goal between a LHS and RHS matching the same pattern to relational subgoals between the differing inputs to the pattern. 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 a 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 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
  linarith

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 _ (pow_bit0_nonneg x 1)) _

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?

Defined in: tacticGcongr?

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

gcongr_discharger

Defined in: Mathlib.Tactic.GCongr.tacticGcongr_discharger

generalize

Defined in: Lean.Parser.Tactic.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.

generalize'

Defined in: «tacticGeneralize'_:_=_»

Backwards compatibility shim for generalize.

generalize_proofs

Defined in: Mathlib.Tactic.generalizeProofsElab

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 (config := { abstract := false }). 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:

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

get_elem_tactic

Defined in: tacticGet_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_trivial and gives a diagnostic error message otherwise; users are encouraged to extend get_elem_tactic_trivial instead of this tactic.

get_elem_tactic_trivial

Defined in: tacticGet_elem_tactic_trivial

get_elem_tactic_trivial 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 just try trivial (which handles the case where i < arr.size is in the context) and simp_arith and omega (for doing linear arithmetic in the index).

group

Defined in: Mathlib.Tactic.Group.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

guard_expr

Defined in: Lean.Parser.Tactic.guardExpr

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.

guard_goal_nums

Defined in: guardGoalNums

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

guard_hyp

Defined in: Lean.Parser.Tactic.guardHyp

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.

guard_hyp_nums

Defined in: guardHypNums

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.

guard_target

Defined in: Lean.Parser.Tactic.guardTarget

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.

have

Defined in: Lean.Parser.Tactic.tacticHave_

The have tactic is for adding hypotheses to the local context of the main goal.

• 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.

have

Defined in: Mathlib.Tactic.tacticHave_

have!?

Defined in: Mathlib.Tactic.Propose.«tacticHave!?:_Using__»

• have? using a, b, c tries to find a lemma which makes use of each of the local hypotheses a, b, c, and reports any results via trace messages.
• have? : h using a, b, c only returns lemmas whose type matches h (which may contain _).
• have?! using a, b, c will also call have to add results to the local goal state.

Note that have? (unlike apply?) does not inspect the goal at all, only the types of the lemmas in the using clause.

have? should not be left in proofs; it is a search tool, like apply?.

Suggestions are printed as have := f a b c.

have'

Defined in: Lean.Parser.Tactic.tacticHave'_

Similar to have, but using refine'

have'

Defined in: Lean.Parser.Tactic.«tacticHave'_:=_»

Similar to have, but using refine'

have?

Defined in: Mathlib.Tactic.Propose.propose'

• have? using a, b, c tries to find a lemma which makes use of each of the local hypotheses a, b, c, and reports any results via trace messages.
• have? : h using a, b, c only returns lemmas whose type matches h (which may contain _).
• have?! using a, b, c will also call have to add results to the local goal state.

Note that have? (unlike apply?) does not inspect the goal at all, only the types of the lemmas in the using clause.

have? should not be left in proofs; it is a search tool, like apply?.

Suggestions are printed as have := f a b c.

have?!

Defined in: Mathlib.Tactic.Propose.«tacticHave?!:_Using__»

• have? using a, b, c tries to find a lemma which makes use of each of the local hypotheses a, b, c, and reports any results via trace messages.
• have? : h using a, b, c only returns lemmas whose type matches h (which may contain _).
• have?! using a, b, c will also call have to add results to the local goal state.

Note that have? (unlike apply?) does not inspect the goal at all, only the types of the lemmas in the using clause.

have? should not be left in proofs; it is a search tool, like apply?.

Suggestions are printed as have := f a b c.

haveI

Defined in: Lean.Parser.Tactic.tacticHaveI_

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

hint

Defined in: Mathlib.Tactic.Hint.hintStx

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

html!

Defined in: ProofWidgets.HtmlCommand.htmlTac

The html! tactic is deprecated and does nothing. If you have a use for it, please open an issue on https://github.com/leanprover-community/ProofWidgets4.

induction

Defined in: Lean.Parser.Tactic.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.

induction'

Defined in: Mathlib.Tactic.induction'

infer_instance

Defined in: Lean.Parser.Tactic.tacticInfer_instance

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

infer_param

Defined in: Mathlib.Tactic.inferOptParam

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

inhabit

Defined in: Lean.Elab.Tactic.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.

injection

Defined in: Lean.Parser.Tactic.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.

injections

Defined in: Lean.Parser.Tactic.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).

interval_cases

Defined in: Mathlib.Tactic.intervalCases

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.

intro

Defined in: Batteries.Tactic.introDot

The syntax intro. is deprecated in favor of nofun.

intro

Defined in: Lean.Parser.Tactic.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.
• intro x y introduces two hypotheses and names them. Individual hypotheses can be anonymized via _, or matched against a pattern:

  -- ... ⊢ α × β → ...
  intro (a, b)
  -- ..., a : α, b : β ⊢ ...
  

• Alternatively, intro can be combined with pattern matching much like fun:

  intro
  | n + 1, 0 => tac
  | ...
  

intro

Defined in: Lean.Parser.Tactic.introMatch

The tactic

intro
| pat1 => tac1
| pat2 => tac2

is the same as:

intro x
match x with
| pat1 => tac1
| pat2 => tac2

That is, intro can be followed by match arms and it introduces the values while doing a pattern match. This is equivalent to fun with match arms in term mode.

intros

Defined in: Lean.Parser.Tactic.intros

Introduces zero or more hypotheses, optionally naming them.

• intros is equivalent to repeatedly applying intro until the goal is not an obvious candidate for intro, which is to say that so long as the goal is a let or a pi type (e.g. an implication, function, or universal quantifier), the intros tactic will introduce an anonymous hypothesis. This tactic does not unfold definitions.

• intros x y ... is equivalent to intro x y ..., introducing hypotheses for each supplied argument and unfolding definitions as necessary. Each argument can be either an identifier or a _. An identifier indicates a name to use for the corresponding introduced hypothesis, and a _ indicates that the hypotheses should be introduced anonymously.

Examples

Basic properties:

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

-- Introduces the two obvious hypotheses automatically
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

-- Introduces exactly two hypotheses, naming only the first
example : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by
  intros g _
  /- Tactic state
     g : Nat → Nat
     a✝ : AllEven g
     ⊢ AllEven fun k => g (k + 1) -/
  sorry

-- Introduces exactly three hypotheses, which requires unfolding `AllEven`
example : ∀ (f : Nat → Nat), AllEven f → AllEven (fun k => f (k + 1)) := by
  intros f h n
  /- Tactic state
     f : Nat → Nat
     h : AllEven f
     n : Nat
     ⊢ (fun k => f (k + 1)) n % 2 = 0 -/
  apply h

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

introv

Defined in: Mathlib.Tactic.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

isBoundedDefault

Defined in: Filter.tacticIsBoundedDefault

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).

itauto

Defined in: Mathlib.Tactic.ITauto.itauto

A decision procedure for intuitionistic propositional logic. Unlike finish 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! will work as a classical SAT solver, but the algorithm is not very good in this situation.)

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

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, and itauto! * will case on all propositional atoms. Warning: This can blow up the proof search, so it should be used sparingly.

itauto!

Defined in: Mathlib.Tactic.ITauto.itauto!

A decision procedure for intuitionistic propositional logic. Unlike finish 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! will work as a classical SAT solver, but the algorithm is not very good in this situation.)

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

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, and itauto! * will case on all propositional atoms. Warning: This can blow up the proof search, so it should be used sparingly.

iterate

Defined in: Lean.Parser.Tactic.tacticIterate____

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
  ⋯

left

Defined in: Lean.Parser.Tactic.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

let

Defined in: Lean.Parser.Tactic.letrec

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

let

Defined in: Mathlib.Tactic.tacticLet_

let

Defined in: Lean.Parser.Tactic.tacticLet_

The let tactic is for adding definitions to the local context of the main goal.

• 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.

let'

Defined in: Lean.Parser.Tactic.tacticLet'_

Similar to let, but using refine'

letI

Defined in: Lean.Parser.Tactic.tacticLetI_

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

lift

Defined in: Mathlib.Tactic.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 remove n and 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.

lift_lets

Defined in: Mathlib.Tactic.lift_lets

Lift all the let bindings in the type of an expression as far out as possible.

When applied to the main goal, this gives one the ability to intro embedded let expressions. For example,

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

During the lifting process, let bindings are merged if they have the same type and value.

liftable_prefixes

Defined in: Mathlib.Tactic.Coherence.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).

linarith

Defined in: 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 LinearOrderedCommRing.

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.)
• 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.

linarith!

Defined in: tacticLinarith!_

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 LinearOrderedCommRing.

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.)
• 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.

linear_combination

Defined in: Mathlib.Tactic.LinearCombination.linearCombination

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 normalize field of the configuration is set to false, then the tactic will simply set the user up to prove their target using the linear combination instead of normalizing the subtraction.

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

linear_combination2

Defined in: Mathlib.Tactic.LinearCombination.tacticLinear_combination2____

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 normalize field of the configuration is set to false, then the tactic will simply set the user up to prove their target using the linear combination instead of normalizing the subtraction.

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

map_tacs

Defined in: Batteries.Tactic.«tacticMap_tacs[_;]»

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

match

Defined in: Lean.Parser.Tactic.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

match

Defined in: Batteries.Tactic.«tacticMatch_,,With.»

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

Both now support multiple discriminants.

match_target

Defined in: Mathlib.Tactic.tacticMatch_target_

measurability

Defined in: tacticMeasurability_

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.

measurability!

Defined in: measurability!

measurability!?

Defined in: measurability!?

measurability?

Defined in: tacticMeasurability?_

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.

mfld_set_tac

Defined in: Tactic.MfldSetTac.mfldSetTac

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

mod_cases

Defined in: Mathlib.Tactic.ModCases.«tacticMod_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.

monicity

Defined in: Mathlib.Tactic.ComputeDegree.monicityMacro

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.

monicity!

Defined in: Mathlib.Tactic.ComputeDegree.tacticMonicity!

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.

mono

Defined in: Mathlib.Tactic.Monotonicity.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

monoidal_coherence

Defined in: Mathlib.Tactic.Coherence.tacticMonoidal_coherence

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

monoidal_simps

Defined in: Mathlib.Tactic.Coherence.monoidal_simps

Simp lemmas for rewriting a hom in monoical categories into a normal form.

move_add

Defined in: Mathlib.MoveAdd.tacticMove_add_

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.

move_mul

Defined in: Mathlib.MoveAdd.tacticMove_mul_

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.

move_oper

Defined in: Mathlib.MoveAdd.moveOperTac

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.

native_decide

Defined in: Lean.Parser.Tactic.nativeDecide

native_decide 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 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

next

Defined in: Lean.Parser.Tactic.«tacticNext_=>_»

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.

nlinarith

Defined in: 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.

nlinarith!

Defined in: tacticNlinarith!_

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.

nofun

Defined in: Lean.Parser.Tactic.tacticNofun

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.

nomatch

Defined in: Lean.Parser.Tactic.«tacticNomatch_,,»

The tactic nomatch h is shorthand for exact nomatch h.

noncomm_ring

Defined in: Mathlib.Tactic.NoncommRing.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.

nontriviality

Defined in: Mathlib.Tactic.Nontriviality.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

norm_cast

Defined in: Lean.Parser.Tactic.tacticNorm_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.

norm_cast0

Defined in: Lean.Parser.Tactic.normCast0

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

norm_num

Defined in: Mathlib.Tactic.normNum

Normalize numerical expressions. Supports the operations + - * / ⁻¹ ^ and % over numerical types such as ℕ, ℤ, ℚ, ℝ, ℂ and some general algebraic types, and can prove goals of the form A = B, A ≠ B, A < B and A ≤ B, where A and B are numerical expressions. It also has a relatively simple primality prover.

norm_num1

Defined in: Mathlib.Tactic.normNum1

Basic version of norm_num that does not call simp.

nth_rewrite

Defined in: Mathlib.Tactic.nthRewriteSeq

nth_rewrite is a variant of rewrite that only changes the nᵗʰ occurrence of the expression to be rewritten. nth_rewrite n [eq₁, eq₂,..., eqₘ] will rewrite the nᵗʰ 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 [h]
/-
a: ℕ
h: a = 1
⊢ a + 1 + a + a + a = 5
-/

Notice that the second occurrence of a from the left has 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.

nth_rw

Defined in: Mathlib.Tactic.nthRwSeq

nth_rw is a variant of rw that only changes the nᵗʰ 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 n [eq₁, eq₂,..., eqₘ] will rewrite the nᵗʰ 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 [h]
/-
a: ℕ
h: a = 1
⊢ a + 1 + a + a + a = 5
-/

Notice that the second occurrence of a from the left has 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_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.

observe

Defined in: Mathlib.Tactic.LibrarySearch.observe

observe hp : p asserts the proposition p, 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?.

If hp is omitted, then the placeholder this is used.

The variant 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.

observe?

Defined in: Mathlib.Tactic.LibrarySearch.«tacticObserve?__:_Using__,,»

observe hp : p asserts the proposition p, 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?.

If hp is omitted, then the placeholder this is used.

The variant 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.

observe?

Defined in: Mathlib.Tactic.LibrarySearch.«tacticObserve?__:_»

observe hp : p asserts the proposition p, 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?.

If hp is omitted, then the placeholder this is used.

The variant 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.

obtain

Defined in: Lean.Parser.Tactic.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.

omega

Defined in: Lean.Parser.Tactic.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 (config :=
  { splitDisjunctions := true, splitNatSub := true, splitNatAbs := true, splitMinMax := true })

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.

on_goal

Defined in: Batteries.Tactic.«tacticOn_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.

open

Defined in: Lean.Parser.Tactic.open

open Foo in tacs (the tactic) acts like open Foo at command level, but it opens a namespace only within the tactics tacs.

peel

Defined in: Mathlib.Tactic.Peel.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.

pick_goal

Defined in: Batteries.Tactic.«tacticPick_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.

polyrith

Defined in: Mathlib.Tactic.Polyrith.«tacticPolyrithOnly[_]»

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 python3 installed and available on the path. (Test by opening a terminal and executing python3 --version.) It also assumes that the requests library is installed: python3 -m pip install requests.

Examples:

example (x y : ℚ) (h1 : x*y + 2*x = 1) (h2 : x = y) :
    x*y = -2*y + 1 := by
  polyrith
-- Try this: linear_combination h1 - 2 * h2

example (x y z w : ℚ) (hzw : z = w) : x*z + 2*y*z = x*w + 2*y*w := by
  polyrith
-- Try this: linear_combination (2 * y + x) * hzw

constant scary : ∀ a b : ℚ, a + b = 0

example (a b c d : ℚ) (h : a + b = 0) (h2: b + c = 0) : a + b + c + d = 0 := by
  polyrith only [scary c d, h]
-- Try this: linear_combination scary c d + h

positivity

Defined in: Mathlib.Tactic.Positivity.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.

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

pure_coherence

Defined in: Mathlib.Tactic.Coherence.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

push_cast

Defined in: Lean.Parser.Tactic.pushCast

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

See also norm_cast.

push_neg

Defined in: Mathlib.Tactic.PushNeg.tacticPush_neg_

Push negations into the conclusion of a hypothesis. For instance, a hypothesis h : ¬ ∀ x, ∃ y, x ≤ y will be transformed by push_neg at h into h : ∃ x, ∀ y, y < x. Variable names are conserved. This tactic pushes negations inside expressions. For instance, given a hypothesis

h : ¬ ∀ ε > 0, ∃ δ > 0, ∀ x, |x - x₀| ≤ δ → |f x - y₀| ≤ ε)

writing push_neg at h will turn h into

h : ∃ ε, ε > 0 ∧ ∀ δ, δ > 0 → (∃ x, |x - x₀| ≤ δ ∧ ε < |f x - y₀|),

(The pretty printer does not use the abbreviations ∀ δ > 0 and ∃ ε > 0 but this issue has nothing to do with push_neg).

Note that names are conserved by this tactic, contrary to what would happen with simp using the relevant lemmas. One can also use this tactic at the goal using push_neg, at every hypothesis and the goal using push_neg at * or at selected hypotheses and the goal using say push_neg at h h' ⊢ as usual.

This tactic has two modes: in standard mode, it transforms ¬(p ∧ q) into p → ¬q, whereas in distrib mode it produces ¬p ∨ ¬q. To use distrib mode, use set_option push_neg.use_distrib true.

qify

Defined in: Mathlib.Tactic.Qify.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.

rcases

Defined in: Lean.Parser.Tactic.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.

rcongr

Defined in: Batteries.Tactic.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

recover

Defined in: Mathlib.Tactic.tacticRecover_

Modifier recover for a tactic (sequence) to debug cases where goals are closed incorrectly. The tactic recover tacs for a tactic (sequence) tacs applies the tactics and then adds goals that are not closed starting from the original

reduce

Defined in: Mathlib.Tactic.tacticReduce__

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.

reduce_mod_char

Defined in: Tactic.ReduceModChar.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.)

reduce_mod_char!

Defined in: Tactic.ReduceModChar.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.)

refine

Defined in: Lean.Parser.Tactic.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.

refine'

Defined in: Lean.Parser.Tactic.refine'

refine' e behaves like refine e, except that unsolved placeholders (_) and implicit parameters are also converted into new goals.

refine_lift

Defined in: Lean.Parser.Tactic.tacticRefine_lift_

Auxiliary macro for lifting have/suffices/let/... It makes sure the "continuation" ?_ is the main goal after refining.

refine_lift'

Defined in: Lean.Parser.Tactic.tacticRefine_lift'_

Similar to refine_lift, but using refine'

refold_let

Defined in: Mathlib.Tactic.refoldLetStx

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.

rel

Defined in: Mathlib.Tactic.GCongr.«tacticRel[_]»

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.

rename

Defined in: Lean.Parser.Tactic.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.

rename'

Defined in: Mathlib.Tactic.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.

rename_bvar

Defined in: Mathlib.Tactic.«tacticRename_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.

rename_i

Defined in: Lean.Parser.Tactic.renameI

rename_i x_1 ... x_n renames the last n inaccessible names using the given names.

repeat

Defined in: Lean.Parser.Tactic.tacticRepeat_

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.

See also:

• try tac is like repeat tac but will apply tac at most once.
• repeat' tac recursively applies tac to each goal.
• first | tac1 | tac2 implements the backtracking used by repeat

repeat'

Defined in: Lean.Parser.Tactic.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.

See also:

• repeat tac simply repeatedly applies tac.
• repeat1' tac is repeat' tac but requires that tac succeed for some goal at least once.

repeat1

Defined in: Mathlib.Tactic.tacticRepeat1_

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.

repeat1'

Defined in: Lean.Parser.Tactic.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.

See also:

• repeat tac simply applies tac repeatedly.
• repeat' tac is like repeat1' tac but it does not require that tac succeed at least once.

replace

Defined in: Mathlib.Tactic.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

replace

Defined in: Lean.Parser.Tactic.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.

revert

Defined in: Lean.Parser.Tactic.revert

revert x... is the inverse of intro x...: it moves the given hypotheses into the main goal's target type.

rewrite

Defined in: Lean.Parser.Tactic.rewriteSeq

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 (config := {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.

rfl

Defined in: Lean.Parser.Tactic.tacticRfl

rfl tries to close the current goal using reflexivity. This is supposed to be an extensible tactic and users can add their own support for new reflexive relations.

Remark: rfl is an extensible tactic. We later add macro_rules to try different reflexivity theorems (e.g., Iff.rfl).

rfl'

Defined in: Lean.Parser.Tactic.tacticRfl'

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).

rify

Defined in: Mathlib.Tactic.Rify.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).

right

Defined in: Lean.Parser.Tactic.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

ring

Defined in: Mathlib.Tactic.RingNF.ring

Tactic for evaluating expressions in commutative (semi)rings, allowing for variables in the exponent.

• ring! will use a more aggressive reducibility setting to determine equality of atoms.
• ring1 fails if the target is not an equality.

For example:

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!

ring!

Defined in: Mathlib.Tactic.RingNF.tacticRing!

Tactic for evaluating expressions in commutative (semi)rings, allowing for variables in the exponent.

• ring! will use a more aggressive reducibility setting to determine equality of atoms.
• ring1 fails if the target is not an equality.

For example:

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!

ring1

Defined in: Mathlib.Tactic.Ring.ring1

Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

• This version of ring fails if the target is not an equality.
• The variant ring1! will use a more aggressive reducibility setting to determine equality of atoms.

ring1!

Defined in: Mathlib.Tactic.Ring.tacticRing1!

Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

• This version of ring fails if the target is not an equality.
• The variant ring1! will use a more aggressive reducibility setting to determine equality of atoms.

ring1_nf

Defined in: Mathlib.Tactic.RingNF.ring1NF

Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

• This version of ring1 uses ring_nf to simplify in atoms.
• The variant ring1_nf! will use a more aggressive reducibility setting to determine equality of atoms.

ring1_nf!

Defined in: Mathlib.Tactic.RingNF.tacticRing1_nf!_

Tactic for solving equations of commutative (semi)rings, allowing variables in the exponent.

• This version of ring1 uses ring_nf to simplify in atoms.
• The variant ring1_nf! will use a more aggressive reducibility setting to determine equality of atoms.

ring_nf

Defined in: Mathlib.Tactic.RingNF.ringNF

Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

• ring_nf! will use a more aggressive reducibility setting to identify atoms.
• ring_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • recursive: if true, ring_nf will also recurse into atoms
• ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.

ring_nf!

Defined in: Mathlib.Tactic.RingNF.tacticRing_nf!__

Simplification tactic for expressions in the language of commutative (semi)rings, which rewrites all ring expressions into a normal form.

• ring_nf! will use a more aggressive reducibility setting to identify atoms.
• ring_nf (config := cfg) allows for additional configuration:
  • red: the reducibility setting (overridden by !)
  • recursive: if true, ring_nf will also recurse into atoms
• ring_nf works as both a tactic and a conv tactic. In tactic mode, ring_nf at h can be used to rewrite in a hypothesis.

rintro

Defined in: Lean.Parser.Tactic.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.

rotate_left

Defined in: Lean.Parser.Tactic.rotateLeft

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.

rotate_right

Defined in: Lean.Parser.Tactic.rotateRight

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.

rsuffices

Defined in: 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.

run_tac

Defined in: Lean.Parser.Tactic.runTac

The run_tac doSeq tactic executes code in TacticM Unit.

rw

Defined in: Lean.Parser.Tactic.rwSeq

rw is like rewrite, but also tries to close the goal by "cheap" (reducible) rfl afterwards.

rw?

Defined in: Lean.Parser.Tactic.rewrites?

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.

rw_mod_cast

Defined in: Lean.Parser.Tactic.tacticRw_mod_cast___

Rewrites with the given rules, normalizing casts prior to each step.

rw_search

Defined in: Mathlib.Tactic.RewriteSearch.tacticRw_search_

rw_search attempts to solve an equality goal by repeatedly rewriting using lemmas from the library.

If no solution is found, the best sequence of rewrites found before maxHeartbeats elapses is returned.

The search is a best-first search, minimising the Levenshtein edit distance between the pretty-printed expressions on either side of the equality. (The strings are tokenized at spaces, separating delimiters (, ), [, ], and , into their own tokens.)

You can use rw_search [-my_lemma, -my_theorem] to prevent rw_search from using the names theorems.

rwa

Defined in: Lean.Parser.Tactic.tacticRwa__

rwa calls rw, then closes any remaining goals using assumption.

saturate

Defined in: Aesop.Frontend.tacticSaturate_____

save

Defined in: Lean.Parser.Tactic.save

save is defined to be the same as skip, but the elaborator has special handling for occurrences of save in tactic scripts and will transform by tac1; save; tac2 to by (checkpoint tac1); tac2, meaning that the effect of tac1 will be cached and replayed. This is useful for improving responsiveness when working on a long tactic proof, by using save after expensive tactics.

(TODO: do this automatically and transparently so that users don't have to use this combinator explicitly.)

says

Defined in: Mathlib.Tactic.Says.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".

set

Defined in: Mathlib.Tactic.setTactic

set!

Defined in: Mathlib.Tactic.tacticSet!_

set_option

Defined in: Lean.Parser.Tactic.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.

show

Defined in: Lean.Parser.Tactic.tacticShow_

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.

show_term

Defined in: Lean.Parser.Tactic.showTerm

show_term tac runs tac, then prints the generated term in the form "exact X Y Z" or "refine X ?_ Z" if there are remaining subgoals.

(For some tactics, the printed term will not be human readable.)

simp

Defined in: Lean.Parser.Tactic.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 the equational lemmas associated with f are used. This provides a convenient way to unfold f.
• simp [*] simplifies the main goal target using the lemmas tagged with the attribute [simp] and all hypotheses.
• simp 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.

simp!

Defined in: Lean.Parser.Tactic.simpAutoUnfold

simp! is shorthand for simp with autoUnfold := true. This will rewrite with all equation lemmas, which can be used to partially evaluate many definitions.

simp?

Defined in: Lean.Parser.Tactic.simpTrace

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.

simp?!

Defined in: Lean.Parser.Tactic.tacticSimp?!_

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.

simp_all

Defined in: Lean.Parser.Tactic.simpAll

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.

simp_all!

Defined in: Lean.Parser.Tactic.simpAllAutoUnfold

simp_all! is shorthand for simp_all with autoUnfold := true. This will rewrite with all equation lemmas, which can be used to partially evaluate many definitions.

simp_all?

Defined in: Lean.Parser.Tactic.simpAllTrace

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.

simp_all?!

Defined in: Lean.Parser.Tactic.tacticSimp_all?!_

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.

simp_all_arith

Defined in: Lean.Parser.Tactic.simpAllArith

simp_all_arith combines the effects of simp_all and simp_arith.

simp_all_arith!

Defined in: Lean.Parser.Tactic.simpAllArithAutoUnfold

simp_all_arith! combines the effects of simp_all, simp_arith and simp!.

simp_arith

Defined in: Lean.Parser.Tactic.simpArith

simp_arith is shorthand for simp with arith := true and decide := true. This enables the use of normalization by linear arithmetic.

simp_arith!

Defined in: Lean.Parser.Tactic.simpArithAutoUnfold

simp_arith! is shorthand for simp_arith with autoUnfold := true. This will rewrite with all equation lemmas, which can be used to partially evaluate many definitions.

simp_intro

Defined in: Mathlib.Tactic.«tacticSimp_intro_____..Only_»

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

simp_rw

Defined in: Mathlib.Tactic.tacticSimp_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]

simp_wf

Defined in: tacticSimp_wf

Unfold definitions commonly used in well founded relation definitions. This is primarily intended for internal use in decreasing_tactic.

simpa

Defined in: Lean.Parser.Tactic.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.

simpa!

Defined in: Lean.Parser.Tactic.tacticSimpa!_

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.

simpa?

Defined in: Lean.Parser.Tactic.tacticSimpa?_

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.

simpa?!

Defined in: Lean.Parser.Tactic.tacticSimpa?!_

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.

sizeOf_list_dec

Defined in: List.tacticSizeOf_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.

skip

Defined in: Lean.Parser.Tactic.skip

skip does nothing.

sleep

Defined in: Lean.Parser.Tactic.sleep

The tactic sleep ms sleeps for ms milliseconds and does nothing. It is used for debugging purposes only.

slice_lhs

Defined in: sliceLHS

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.

slice_rhs

Defined in: sliceRHS

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.

slim_check

Defined in: slimCheckSyntax

slim_check considers a proof goal and tries to generate examples that would contradict the statement.

Let's consider the following proof goal.

xs : List ℕ,
h : ∃ (x : ℕ) (H : x ∈ xs), x < 3
⊢ ∀ (y : ℕ), y ∈ xs → y < 5

The local constants will be reverted and an instance will be found for Testable (∀ (xs : List ℕ), (∃ x ∈ xs, x < 3) → (∀ y ∈ xs, y < 5)). The Testable instance is supported by an instance of Sampleable (List ℕ), 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 slim_check 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.SlimCheck.Testable.

Optional arguments given with slim_check (config : { ... })

• numInst (default 100): number of examples to test properties with
• maxSize (default 100): final size argument

Options:

• set_option trace.slim_check.decoration true: print the proposition with quantifier annotations
• set_option trace.slim_check.discarded true: print the examples discarded because they do not satisfy assumptions
• set_option trace.slim_check.shrink.steps true: trace the shrinking of counter-example
• set_option trace.slim_check.shrink.candidates true: print the lists of candidates considered when shrinking each variable
• set_option trace.slim_check.instance true: print the instances of testable being used to test the proposition
• set_option trace.slim_check.success true: print the tested samples that satisfy a property

smul_tac

Defined in: RatFunc.tacticSmul_tac

Solve equations for RatFunc K by applying RatFunc.induction_on.

solve

Defined in: Lean.solveTactic

Similar to first, but succeeds only if one the given tactics solves the current goal.

solve_by_elim

Defined in: Lean.Parser.Tactic.solveByElim

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.

sorry

Defined in: Lean.Parser.Tactic.tacticSorry

The sorry tactic closes the goal using sorryAx. 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 using #print axioms my_thm and looking for sorryAx in the axiom list.

sorry_if_sorry

Defined in: CategoryTheory.sorryIfSorry

Close the main goal with sorry if its type contains sorry, and fail otherwise.

specialize

Defined in: Lean.Parser.Tactic.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.

split

Defined in: Lean.Parser.Tactic.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 used the case or next tactics.

• split will split the goal (target).
• split at h will split the hypothesis h.

split_ands

Defined in: Batteries.Tactic.tacticSplit_ands

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

split_ifs

Defined in: Mathlib.Tactic.splitIfs

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.

squeeze_scope

Defined in: Batteries.Tactic.squeezeScope

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]"

stop

Defined in: Lean.Parser.Tactic.tacticStop_

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.

subsingleton

Defined in: Mathlib.Tactic.subsingletonStx

The subsingleton tactic tries to prove a goal of the form x = y or HEq 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
• heterogenous 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.

subst

Defined in: Lean.Parser.Tactic.subst

subst x... substitutes each x with e in the goal if there is a hypothesis of type x = e or e = x. If x is itself a hypothesis of type y = e or e = y, y is substituted instead.

subst_eqs

Defined in: Lean.Parser.Tactic.substEqs

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.

subst_vars

Defined in: Lean.Parser.Tactic.substVars

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

substs

Defined in: Mathlib.Tactic.Substs.substs

Applies the subst tactic to all given hypotheses from left to right.

success_if_fail_with_msg

Defined in: Mathlib.Tactic.successIfFailWithMsg

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.

suffices

Defined in: Lean.Parser.Tactic.tacticSuffices_

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.

suffices

Defined in: Mathlib.Tactic.tacticSuffices_

swap

Defined in: Batteries.Tactic.tacticSwap

swap is a shortcut for pick_goal 2, which interchanges the 1st and 2nd goals.

swap_var

Defined in: Mathlib.Tactic.«tacticSwap_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⟩

symm

Defined in: Lean.Parser.Tactic.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.

symm_saturate

Defined in: Lean.Parser.Tactic.symmSaturate

For every hypothesis h : a ~ b where a @[symm] lemma is available, add a hypothesis h_symm : b ~ a.

tauto

Defined in: Mathlib.Tactic.Tauto.tauto

tauto breaks down assumptions of the form _ ∧ _, _ ∨ _, _ ↔ _ and ∃ _, _ and splits a goal of the form _ ∧ _, _ ↔ _ or ∃ _, _ until it can be discharged using reflexivity 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.

tfae_finish

Defined in: Mathlib.Tactic.TFAE.tfaeFinish

tfae_finish is used to close 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
  · /- proof of P → Q -/
  tfae_have 2 → 1
  · /- proof of Q → P -/
  tfae_have 2 ↔ 3
  · /- proof of Q ↔ R -/
  tfae_finish

tfae_have

Defined in: Mathlib.Tactic.TFAE.tfaeHave

tfae_have introduces hypotheses for proving goals of the form TFAE [P₁, P₂, ...]. Specifically, tfae_have i arrow j introduces a hypothesis of type Pᵢ arrow Pⱼ to the local context, where arrow can be →, ←, or ↔. Note that i and j are natural number indices (beginning at 1) used to specify the propositions P₁, P₂, ... that appear in the TFAE goal list. A proof is required afterward, typically via a tactic block.

example (h : P → R) : TFAE [P, Q, R] := by
  tfae_have 1 → 3
  · exact h
  ...

The resulting context now includes tfae_1_to_3 : P → R.

The introduced hypothesis can be given a custom name, in analogy to have syntax:

tfae_have h : 2 ↔ 3

Once sufficient hypotheses have been introduced by tfae_have, tfae_finish can be used to close the goal.

example : TFAE [P, Q, R] := by
  tfae_have 1 → 2
  · /- proof of P → Q -/
  tfae_have 2 → 1
  · /- proof of Q → P -/
  tfae_have 2 ↔ 3
  · /- proof of Q ↔ R -/
  tfae_finish

trace

Defined in: Lean.Parser.Tactic.trace

Evaluates a term to a string (when possible), and prints it as a trace message.

trace

Defined in: Lean.Parser.Tactic.traceMessage

trace msg displays msg in the info view.

trace_state

Defined in: Lean.Parser.Tactic.traceState

trace_state displays the current state in the info view.

trans

Defined in: Mathlib.Tactic.tacticTrans___

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.

transitivity

Defined in: Mathlib.Tactic.tacticTransitivity___

triv

Defined in: Batteries.Tactic.triv

Deprecated variant of trivial.

trivial

Defined in: Lean.Parser.Tactic.tacticTrivial

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)

try

Defined in: Lean.Parser.Tactic.tacticTry_

try tac runs tac and succeeds even if tac failed.

try_this

Defined in: Mathlib.Tactic.tacticTry_this_

Produces the text Try this: <tac> with the given tactic, and then executes it.

type_check

Defined in: tacticType_check_

Type check the given expression, and trace its type.

unfold

Defined in: Lean.Parser.Tactic.unfold

• unfold id unfolds definition id.
• unfold id1 id2 ... is equivalent to unfold id1; unfold id2; ....

For non-recursive definitions, this tactic is identical to delta. For definitions by pattern matching, it uses "equation lemmas" which are autogenerated for each match arm.

unfold_let

Defined in: Mathlib.Tactic.unfoldLetStx

unfold_let x y z at loc unfolds the local definitions x, y, and z at the given location, which is known as "zeta reduction." This also exists as a conv-mode tactic.

If no local definitions are given, then all local definitions are unfolded. This variant also exists as the conv-mode tactic zeta.

This is similar to the unfold tactic, which instead is for unfolding global definitions.

unfold_projs

Defined in: Mathlib.Tactic.unfoldProjsStx

unfold_projs at loc unfolds projections of class instances at the given location. This also exists as a conv-mode tactic.

unhygienic

Defined in: Lean.Parser.Tactic.tacticUnhygienic_

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

uniqueDiffWithinAt_Ici_Iic_univ

Defined in: intervalIntegral.tacticUniqueDiffWithinAt_Ici_Iic_univ

An auxiliary tactic closing goals UniqueDiffWithinAt ℝ s a where s ∈ {Iic a, Ici a, univ}.

unit_interval

Defined in: Tactic.Interactive.tacticUnit_interval

A tactic that solves 0 ≤ ↑x, 0 ≤ 1 - ↑x, ↑x ≤ 1, and 1 - ↑x ≤ 1 for x : I.

unreachable!

Defined in: Batteries.Tactic.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!

use

Defined in: Mathlib.Tactic.useSyntax

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 : ∃ x : String × String, x.1 = x.2 := by use ("forty-two", "forty-two")

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 : ∃ p : Nat × Nat, p.1 = p.2 := by use! 42, 42

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).

use!

Defined in: Mathlib.Tactic.«tacticUse!___,,»

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 : ∃ x : String × String, x.1 = x.2 := by use ("forty-two", "forty-two")

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 : ∃ p : Nat × Nat, p.1 = p.2 := by use! 42, 42

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).

use_discharger

Defined in: Mathlib.Tactic.tacticUse_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.

use_finite_instance

Defined in: tacticUse_finite_instance

volume_tac

Defined in: MeasureTheory.tacticVolume_tac

The tactic exact volume, to be used in optional (autoParam) arguments.

whisker_simps

Defined in: Mathlib.Tactic.BicategoryCoherence.whisker_simps

Simp lemmas for rewriting a 2-morphism into a normal form.

whnf

Defined in: Mathlib.Tactic.tacticWhnf__

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.

with_panel_widgets

Defined in: ProofWidgets.withPanelWidgetsTacticStx

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.

with_reducible

Defined in: Lean.Parser.Tactic.withReducible

with_reducible tacs executes tacs using the reducible transparency setting. In this setting only definitions tagged as [reducible] are unfolded.

with_reducible_and_instances

Defined in: Lean.Parser.Tactic.withReducibleAndInstances

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.

with_unfolding_all

Defined in: Lean.Parser.Tactic.withUnfoldingAll

with_unfolding_all tacs executes tacs using the .all transparency setting. In this setting all definitions that are not opaque are unfolded.

wlog

Defined in: Mathlib.Tactic.wlog

wlog h : P will add an assumption h : P to the main goal, and add 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 name this is used, but the idiom with H can be added to specify the name: wlog h : P with H.

Typically, it is useful to use the variant wlog h : P generalizing x y, to revert 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).

By default, the entire context is reverted.

zify

Defined in: Mathlib.Tactic.Zify.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.

syntax ... [Lean.Parser.Tactic.nestedTactic]

syntax ... [Lean.Parser.Tactic.unknown]

syntax ... [Lean.cdot] · tac focuses on the main goal and tries to solve it using tac, or else fails.