def Lean.Parser.Tactic.withAnnotateState : Lean.ParserDescr

with_annotate_state stx t annotates the lexical range of stx : Syntax with the initial and final state of running tactic t.

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def Lean.Parser.Tactic.intro : Lean.ParserDescr

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
  | ...
  

def Lean.Parser.Tactic.intros : Lean.ParserDescr

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

def Lean.Parser.Tactic.rename : Lean.ParserDescr

rename t => x renames the most recent hypothesis whose type matches t (which may contain placeholders) to x, or fails if no such hypothesis could be found.

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def Lean.Parser.Tactic.revert : Lean.ParserDescr

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

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def Lean.Parser.Tactic.clear : Lean.ParserDescr

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

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def Lean.Parser.Tactic.subst : Lean.ParserDescr

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.

def Lean.Parser.Tactic.substVars : Lean.ParserDescr

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

Equations

• Lean.Parser.Tactic.substVars = Lean.ParserDescr.node `Lean.Parser.Tactic.substVars 1024 (Lean.ParserDescr.nonReservedSymbol "subst_vars" false)

def Lean.Parser.Tactic.assumption : Lean.ParserDescr

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.

Equations

• Lean.Parser.Tactic.assumption = Lean.ParserDescr.node `Lean.Parser.Tactic.assumption 1024 (Lean.ParserDescr.nonReservedSymbol "assumption" false)

def Lean.Parser.Tactic.contradiction : Lean.ParserDescr

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
  

Equations

• Lean.Parser.Tactic.contradiction = Lean.ParserDescr.node `Lean.Parser.Tactic.contradiction 1024 (Lean.ParserDescr.nonReservedSymbol "contradiction" false)

def Lean.Parser.Tactic.falseOrByContra : Lean.ParserDescr

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.

Equations

• Lean.Parser.Tactic.falseOrByContra = Lean.ParserDescr.node `Lean.Parser.Tactic.falseOrByContra 1024 (Lean.ParserDescr.nonReservedSymbol "false_or_by_contra" false)

def Lean.Parser.Tactic.apply : Lean.ParserDescr

apply e tries to match the current goal against the conclusion of e's type. If it succeeds, then the tactic returns as many subgoals as the number of premises that have not been fixed by type inference or type class resolution. Non-dependent premises are added before dependent ones.

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

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def Lean.Parser.Tactic.exact : Lean.ParserDescr

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

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def Lean.Parser.Tactic.refine : Lean.ParserDescr

refine e behaves like exact e, except that named (?x) or unnamed (?_) holes in e that are not solved by unification with the main goal's target type are converted into new goals, using the hole's name, if any, as the goal case name.

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def Lean.Parser.Tactic.refine' : Lean.ParserDescr

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

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def Lean.Parser.Tactic.tacticExfalso : Lean.ParserDescr

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

def Lean.Parser.Tactic.constructor : Lean.ParserDescr

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

def Lean.Parser.Tactic.left : Lean.ParserDescr

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

example : True ∨ False := by
  left
  trivial

Equations

• Lean.Parser.Tactic.left = Lean.ParserDescr.node `Lean.Parser.Tactic.left 1024 (Lean.ParserDescr.nonReservedSymbol "left" false)

def Lean.Parser.Tactic.right : Lean.ParserDescr

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

def Lean.Parser.Tactic.case : Lean.ParserDescr

• 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).

def Lean.Parser.Tactic.case' : Lean.ParserDescr

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

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def Lean.Parser.Tactic.«tacticNext_=>_» : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.allGoals : Lean.ParserDescr

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

See also any_goals tac.

def Lean.Parser.Tactic.anyGoals : Lean.ParserDescr

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

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

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def Lean.Parser.Tactic.focus : Lean.ParserDescr

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.

def Lean.Parser.Tactic.skip : Lean.ParserDescr

skip does nothing.

Equations

• Lean.Parser.Tactic.skip = Lean.ParserDescr.node `Lean.Parser.Tactic.skip 1024 (Lean.ParserDescr.nonReservedSymbol "skip" false)

def Lean.Parser.Tactic.done : Lean.ParserDescr

done succeeds iff there are no remaining goals.

def Lean.Parser.Tactic.traceState : Lean.ParserDescr

trace_state displays the current state in the info view.

Equations

• Lean.Parser.Tactic.traceState = Lean.ParserDescr.node `Lean.Parser.Tactic.traceState 1024 (Lean.ParserDescr.nonReservedSymbol "trace_state" false)

def Lean.Parser.Tactic.traceMessage : Lean.ParserDescr

trace msg displays msg in the info view.

def Lean.Parser.Tactic.failIfSuccess : Lean.ParserDescr

fail_if_success t fails if the tactic t succeeds.

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def Lean.Parser.Tactic.paren : Lean.ParserDescr

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

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def Lean.Parser.Tactic.withReducible : Lean.ParserDescr

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

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def Lean.Parser.Tactic.withReducibleAndInstances : Lean.ParserDescr

with_reducible_and_instances tacs executes tacs using the .instances transparency setting. In this setting only definitions tagged as [reducible] or type class instances are unfolded.

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def Lean.Parser.Tactic.withUnfoldingAll : Lean.ParserDescr

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

def Lean.Parser.Tactic.first : Lean.ParserDescr

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

def Lean.Parser.Tactic.rotateLeft : Lean.ParserDescr

rotate_left n rotates goals to the left by n. That is, rotate_left 1 takes the main goal and puts it to the back of the subgoal list. If n is omitted, it defaults to 1.

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def Lean.Parser.Tactic.rotateRight : Lean.ParserDescr

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.

def Lean.Parser.Tactic.tacticTry_ : Lean.ParserDescr

try tac runs tac and succeeds even if tac failed.

def Lean.Parser.Tactic.«tactic_<;>_» : Lean.TrailingParserDescr

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

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def Lean.Parser.Tactic.fail : Lean.ParserDescr

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

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def Lean.Parser.Tactic.eqRefl : Lean.ParserDescr

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

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• Lean.Parser.Tactic.eqRefl = Lean.ParserDescr.node `Lean.Parser.Tactic.eqRefl 1024 (Lean.ParserDescr.nonReservedSymbol "eq_refl" false)

def Lean.Parser.Tactic.tacticRfl : Lean.ParserDescr

This tactic applies to a goal whose target has the form x ~ x, where ~ is equality, heterogeneous equality or any relation that has a reflexivity lemma tagged with the attribute @[refl].

Equations

• Lean.Parser.Tactic.tacticRfl = Lean.ParserDescr.node `Lean.Parser.Tactic.tacticRfl 1024 (Lean.ParserDescr.nonReservedSymbol "rfl" false)

def Lean.Parser.Tactic.applyRfl : Lean.ParserDescr

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

Equations

• Lean.Parser.Tactic.applyRfl = Lean.ParserDescr.node `Lean.Parser.Tactic.applyRfl 1024 (Lean.ParserDescr.nonReservedSymbol "apply_rfl" false)

def Lean.Parser.Tactic.tacticRfl' : Lean.ParserDescr

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).

def Lean.Parser.Tactic.acRfl : Lean.ParserDescr

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

Equations

• Lean.Parser.Tactic.acRfl = Lean.ParserDescr.node `Lean.Parser.Tactic.acRfl 1024 (Lean.ParserDescr.nonReservedSymbol "ac_rfl" false)

def Lean.Parser.Tactic.tacticSorry : Lean.ParserDescr

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.

Equations

• Lean.Parser.Tactic.tacticSorry = Lean.ParserDescr.node `Lean.Parser.Tactic.tacticSorry 1024 (Lean.ParserDescr.nonReservedSymbol "sorry" false)

def Lean.Parser.Tactic.tacticAdmit : Lean.ParserDescr

admit is a shorthand for exact sorry.

Equations

• Lean.Parser.Tactic.tacticAdmit = Lean.ParserDescr.node `Lean.Parser.Tactic.tacticAdmit 1024 (Lean.ParserDescr.nonReservedSymbol "admit" false)

def Lean.Parser.Tactic.tacticInfer_instance : Lean.ParserDescr

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

def Lean.Parser.Tactic.posConfigItem : Lean.ParserDescr

+opt is short for (opt := true). It sets the opt configuration option to true.

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def Lean.Parser.Tactic.negConfigItem : Lean.ParserDescr

-opt is short for (opt := false). It sets the opt configuration option to false.

def Lean.Parser.Tactic.valConfigItem : Lean.ParserDescr

(opt := val) sets the opt configuration option to val.

As a special case, (config := ...) sets the entire configuration.

def Lean.Parser.Tactic.configItem : Lean.ParserDescr

A configuration item for a tactic configuration.

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def Lean.Parser.Tactic.optConfig : Lean.ParserDescr

Configuration options for tactics.

def Lean.Parser.Tactic.config : Lean.ParserDescr

Optional configuration option for tactics. (Deprecated. Replace (config)? with optConfig.)

def Lean.Parser.Tactic.locationWildcard : Lean.ParserDescr

The * location refers to all hypotheses and the goal.

def Lean.Parser.Tactic.locationHyp : Lean.ParserDescr

A hypothesis location specification consists of 1 or more hypothesis references and optionally ⊢ denoting the goal.

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def Lean.Parser.Tactic.location : Lean.ParserDescr

Location specifications are used by many tactics that can operate on either the hypotheses or the goal. It can have one of the forms:

• 'empty' is not actually present in this syntax, but most tactics use (location)? matchers. It means to target the goal only.
• at h₁ ... hₙ: target the hypotheses h₁, ..., hₙ
• at h₁ h₂ ⊢: target the hypotheses h₁ and h₂, and the goal
• at *: target all hypotheses and the goal

def Lean.Parser.Tactic.change : Lean.ParserDescr

• 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.

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def Lean.Parser.Tactic.changeWith : Lean.ParserDescr

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

def Lean.Parser.Tactic.rwRule : Lean.ParserDescr

If thm is a theorem a = b, then as a rewrite rule,

• thm means to replace a with b, and
• ← thm means to replace b with a.

def Lean.Parser.Tactic.rwRuleSeq : Lean.ParserDescr

A rwRuleSeq is a list of rwRule in brackets.

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def Lean.Parser.Tactic.rewriteSeq : Lean.ParserDescr

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

• rewrite [e₁, ..., eₙ] applies the given rules sequentially.
• rewrite [e] at l rewrites e at location(s) l, where l is either * or a list of hypotheses in the local context. In the latter case, a turnstile ⊢ or |- can also be used, to signify the target of the goal.

Using rw (occs := .pos L) [e], where L : List Nat, you can control which "occurrences" are rewritten. (This option applies to each rule, so usually this will only be used with a single rule.) Occurrences count from 1. At each allowed occurrence, arguments of the rewrite rule e may be instantiated, restricting which later rewrites can be found. (Disallowed occurrences do not result in instantiation.) (occs := .neg L) allows skipping specified occurrences.

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def Lean.Parser.Tactic.rwSeq : Lean.ParserDescr

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

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def Lean.Parser.Tactic.tacticRwa__ : Lean.ParserDescr

rwa is short-hand for rw; assumption.

def Lean.Parser.Tactic.injection : Lean.ParserDescr

The injection tactic is based on the fact that constructors of inductive data types are injections. That means that if c is a constructor of an inductive datatype, and if (c t₁) and (c t₂) are two terms that are equal then t₁ and t₂ are equal too. If q is a proof of a statement of conclusion t₁ = t₂, then injection applies injectivity to derive the equality of all arguments of t₁ and t₂ placed in the same positions. For example, from (a::b) = (c::d) we derive a=c and b=d. To use this tactic t₁ and t₂ should be constructor applications of the same constructor. Given h : a::b = c::d, the tactic injection h adds two new hypothesis with types a = c and b = d to the main goal. The tactic injection h with h₁ h₂ uses the names h₁ and h₂ to name the new hypotheses.

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def Lean.Parser.Tactic.injections : Lean.ParserDescr

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).

def Lean.Parser.Tactic.discharger : Lean.ParserDescr

The discharger clause of simp and related tactics. This is a tactic used to discharge the side conditions on conditional rewrite rules.

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def Lean.Parser.Tactic.simpPre : Lean.ParserDescr

Use this rewrite rule before entering the subterms

Equations

• Lean.Parser.Tactic.simpPre = Lean.ParserDescr.nodeWithAntiquot "simpPre" `Lean.Parser.Tactic.simpPre (Lean.ParserDescr.symbol "↓")

def Lean.Parser.Tactic.simpPost : Lean.ParserDescr

Use this rewrite rule after entering the subterms

Equations

• Lean.Parser.Tactic.simpPost = Lean.ParserDescr.nodeWithAntiquot "simpPost" `Lean.Parser.Tactic.simpPost (Lean.ParserDescr.symbol "↑")

def Lean.Parser.Tactic.simpLemma : Lean.ParserDescr

A simp lemma specification is:

• optional ↑ or ↓ to specify use before or after entering the subterm
• optional ← to use the lemma backward
• thm for the theorem to rewrite with

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def Lean.Parser.Tactic.simpErase : Lean.ParserDescr

An erasure specification -thm says to remove thm from the simp set

def Lean.Parser.Tactic.simpStar : Lean.ParserDescr

The simp lemma specification * means to rewrite with all hypotheses

Equations

• Lean.Parser.Tactic.simpStar = Lean.ParserDescr.nodeWithAntiquot "simpStar" `Lean.Parser.Tactic.simpStar (Lean.ParserDescr.symbol "*")

def Lean.Parser.Tactic.simp : Lean.ParserDescr

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

• simp simplifies the main goal target using lemmas tagged with the attribute [simp].
• simp [h₁, h₂, ..., hₙ] simplifies the main goal target using the lemmas tagged with the attribute [simp] and the given hᵢ's, where the hᵢ's are expressions.-
• If an hᵢ is a defined constant f, then f is unfolded. If f has equational lemmas associated with it (and is not a projection or a reducible definition), these are used to rewrite with f.
• simp [*] simplifies the main goal target using the lemmas tagged with the attribute [simp] and all hypotheses.
• simp only [h₁, h₂, ..., hₙ] is like simp [h₁, h₂, ..., hₙ] but does not use [simp] lemmas.
• simp [-id₁, ..., -idₙ] simplifies the main goal target using the lemmas tagged with the attribute [simp], but removes the ones named idᵢ.
• simp at h₁ h₂ ... hₙ simplifies the hypotheses h₁ : T₁ ... hₙ : Tₙ. If the target or another hypothesis depends on hᵢ, a new simplified hypothesis hᵢ is introduced, but the old one remains in the local context.
• simp at * simplifies all the hypotheses and the target.
• simp [*] at * simplifies target and all (propositional) hypotheses using the other hypotheses.

def Lean.Parser.Tactic.simpAll : Lean.ParserDescr

simp_all is a stronger version of simp [*] at * where the hypotheses and target are simplified multiple times until no simplification is applicable. Only non-dependent propositional hypotheses are considered.

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def Lean.Parser.Tactic.dsimp : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.simpArg : Lean.ParserDescr

A simpArg is either a *, -lemma or a simp lemma specification (which includes the ↑ ↓ ← specifications for pre, post, reverse rewriting).

def Lean.Parser.Tactic.simpArgs : Lean.ParserDescr

A simp args list is a list of simpArg. This is the main argument to simp.

def Lean.Parser.Tactic.dsimpArg : Lean.ParserDescr

A dsimpArg is similar to simpArg, but it does not have the simpStar form because it does not make sense to use hypotheses in dsimp.

Equations

• Lean.Parser.Tactic.dsimpArg = Lean.ParserDescr.binary `orelse Lean.Parser.Tactic.simpErase Lean.Parser.Tactic.simpLemma

def Lean.Parser.Tactic.dsimpArgs : Lean.ParserDescr

A dsimp args list is a list of dsimpArg. This is the main argument to dsimp.

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def Lean.Parser.Tactic.simpTraceArgsRest : Lean.ParserDescr

The common arguments of simp? and simp?!.

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def Lean.Parser.Tactic.simpTrace : Lean.ParserDescr

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.

def Lean.Parser.Tactic.tacticSimp?!_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.simpAllTraceArgsRest : Lean.ParserDescr

The common arguments of simp_all? and simp_all?!.

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def Lean.Parser.Tactic.simpAllTrace : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.tacticSimp_all?!_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.dsimpTraceArgsRest : Lean.ParserDescr

The common arguments of dsimp? and dsimp?!.

def Lean.Parser.Tactic.dsimpTrace : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.tacticDsimp?!_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.simpaArgsRest : Lean.ParserDescr

The arguments to the simpa family tactics.

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def Lean.Parser.Tactic.simpa : Lean.ParserDescr

This is a "finishing" tactic modification of simp. It has two forms.

• simpa [rules, ⋯] using e will simplify the goal and the type of e using rules, then try to close the goal using e.

Simplifying the type of e makes it more likely to match the goal (which has also been simplified). This construction also tends to be more robust under changes to the simp lemma set.

• simpa [rules, ⋯] will simplify the goal and the type of a hypothesis this if present in the context, then try to close the goal using the assumption tactic.

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def Lean.Parser.Tactic.tacticSimpa!_ : Lean.ParserDescr

This is a "finishing" tactic modification of simp. It has two forms.

• simpa [rules, ⋯] using e will simplify the goal and the type of e using rules, then try to close the goal using e.

Simplifying the type of e makes it more likely to match the goal (which has also been simplified). This construction also tends to be more robust under changes to the simp lemma set.

• simpa [rules, ⋯] will simplify the goal and the type of a hypothesis this if present in the context, then try to close the goal using the assumption tactic.

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def Lean.Parser.Tactic.tacticSimpa?_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.tacticSimpa?!_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.delta : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.unfold : Lean.ParserDescr

• unfold id unfolds all occurrences of definition id in the target.
• unfold id1 id2 ... is equivalent to unfold id1; unfold id2; ....
• unfold id at h unfolds at the hypothesis h.

Definitions can be either global or local definitions.

For non-recursive global definitions, this tactic is identical to delta. For recursive global definitions, it uses the "unfolding lemma" id.eq_def, which is generated for each recursive definition, to unfold according to the recursive definition given by the user. Only one level of unfolding is performed, in contrast to simp only [id], which unfolds definition id recursively.

def Lean.Parser.Tactic.tacticRefine_lift_ : Lean.ParserDescr

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

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def Lean.Parser.Tactic.tacticHave_ : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.tacticSuffices_ : Lean.ParserDescr

Given a main goal ctx ⊢ t, suffices h : t' from e replaces the main goal with ctx ⊢ t', e must have type t in the context ctx, h : t'.

The variant suffices h : t' by tac is a shorthand for suffices h : t' from by tac. If h : is omitted, the name this is used.

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def Lean.Parser.Tactic.tacticLet_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.tacticShow_ : Lean.ParserDescr

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

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def Lean.Parser.Tactic.letrec : Lean.ParserDescr

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

def Lean.Parser.Tactic.tacticRefine_lift'_ : Lean.ParserDescr

Similar to refine_lift, but using refine'

def Lean.Parser.Tactic.tacticHave'_ : Lean.ParserDescr

Similar to have, but using refine'

def Lean.Parser.Tactic.«tacticHave'_:=_» : Lean.ParserDescr

Similar to have, but using refine'

def Lean.Parser.Tactic.tacticLet'_ : Lean.ParserDescr

Similar to let, but using refine'

def Lean.Parser.Tactic.inductionAltLHS : Lean.ParserDescr

The left hand side of an induction arm, | foo a b c or | @foo a b c where foo is a constructor of the inductive type and a b c are the arguments to the constructor.

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def Lean.Parser.Tactic.inductionAlt : Lean.ParserDescr

In induction alternative, which can have 1 or more cases on the left and _, ?_, or a tactic sequence after the =>.

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def Lean.Parser.Tactic.inductionAlts : Lean.ParserDescr

After with, there is an optional tactic that runs on all branches, and then a list of alternatives.

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def Lean.Parser.Tactic.induction : Lean.ParserDescr

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

For example, given n : Nat and a goal with a hypothesis h : P n and target Q n, induction n produces one goal with hypothesis h : P 0 and target Q 0, and one goal with hypotheses h : P (Nat.succ a) and ih₁ : P a → Q a and target Q (Nat.succ a). Here the names a and ih₁ are chosen automatically and are not accessible. You can use with to provide the variables names for each constructor.

• induction e, where e is an expression instead of a variable, generalizes e in the goal, and then performs induction on the resulting variable.
• induction e using r allows the user to specify the principle of induction that should be used. Here r should be a term whose result type must be of the form C t, where C is a bound variable and t is a (possibly empty) sequence of bound variables
• induction e generalizing z₁ ... zₙ, where z₁ ... zₙ are variables in the local context, generalizes over z₁ ... zₙ before applying the induction but then introduces them in each goal. In other words, the net effect is that each inductive hypothesis is generalized.
• Given x : Nat, induction x with | zero => tac₁ | succ x' ih => tac₂ uses tactic tac₁ for the zero case, and tac₂ for the succ case.

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def Lean.Parser.Tactic.generalizeArg : Lean.ParserDescr

A generalize argument, of the form term = x or h : term = x.

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def Lean.Parser.Tactic.generalize : Lean.ParserDescr

• 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.

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def Lean.Parser.Tactic.casesTarget : Lean.ParserDescr

A cases argument, of the form e or h : e (where h asserts that e = cᵢ a b for each constructor cᵢ of the inductive).

def Lean.Parser.Tactic.cases : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.renameI : Lean.ParserDescr

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

def Lean.Parser.Tactic.tacticRepeat_ : Lean.ParserDescr

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

def Lean.Parser.Tactic.repeat' : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.repeat1' : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.tacticTrivial : Lean.ParserDescr

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)

Equations

• Lean.Parser.Tactic.tacticTrivial = Lean.ParserDescr.node `Lean.Parser.Tactic.tacticTrivial 1024 (Lean.ParserDescr.nonReservedSymbol "trivial" false)

def Lean.Parser.Tactic.classical : Lean.ParserDescr

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

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

def Lean.Parser.Tactic.split : Lean.ParserDescr

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.

def Lean.Parser.Tactic.dbgTrace : Lean.ParserDescr

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"

def Lean.Parser.Tactic.tacticStop_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.specialize : Lean.ParserDescr

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

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def Lean.Parser.Tactic.tacticUnhygienic_ : Lean.ParserDescr

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

def Lean.Parser.Tactic.checkpoint : Lean.ParserDescr

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.)

def Lean.Parser.Tactic.save : Lean.ParserDescr

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.)

Equations

• Lean.Parser.Tactic.save = Lean.ParserDescr.node `Lean.Parser.Tactic.save 1024 (Lean.ParserDescr.nonReservedSymbol "save" false)

def Lean.Parser.Tactic.sleep : Lean.ParserDescr

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

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def Lean.Parser.Tactic.«tacticExists_,,» : Lean.ParserDescr

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

def Lean.Parser.Tactic.congr : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.tacDepIfThenElse : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.tacIfThenElse : Lean.ParserDescr

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 ?_.)

def Lean.Parser.Tactic.tacticNofun : Lean.ParserDescr

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.

def Lean.Parser.Tactic.«tacticNomatch_,,» : Lean.ParserDescr

The tactic nomatch h is shorthand for exact nomatch h.

def Lean.Parser.Tactic.replace : Lean.ParserDescr

Acts like have, but removes a hypothesis with the same name as this one if possible. For example, if the state is:

f : α → β
h : α
⊢ goal

Then after replace h := f h the state will be:

f : α → β
h : β
⊢ goal

whereas have h := f h would result in:

f : α → β
h† : α
h : β
⊢ goal

This can be used to simulate the specialize and apply at tactics of Coq.

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def Lean.Parser.Tactic.tacticAnd_intros : Lean.ParserDescr

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

Equations

• Lean.Parser.Tactic.tacticAnd_intros = Lean.ParserDescr.node `Lean.Parser.Tactic.tacticAnd_intros 1024 (Lean.ParserDescr.nonReservedSymbol "and_intros" false)

def Lean.Parser.Tactic.substEqs : Lean.ParserDescr

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.

def Lean.Parser.Tactic.runTac : Lean.ParserDescr

The run_tac doSeq tactic executes code in TacticM Unit.

def Lean.Parser.Tactic.tacticHaveI_ : Lean.ParserDescr

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

def Lean.Parser.Tactic.tacticLetI_ : Lean.ParserDescr

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

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structure Lean.Parser.Tactic.DecideConfig : Type

Configuration for the decide tactic family.

• kernel : Bool

  If true (default: false), then use only kernel reduction when reducing the Decidable instance. This is more efficient, since the default mode reduces twice (once in the elaborator and again in the kernel), however kernel reduction ignores transparency settings.

• native : Bool

  If true (default: false), then uses the native code compiler to evaluate the Decidable instance, admitting the result via the axiom Lean.ofReduceBool. This can be significantly more efficient, but it is at the cost of increasing the trusted code base, namely the Lean compiler and all definitions with an @[implemented_by] attribute. The instance is only evaluated once. The native_decide tactic is a synonym for decide +native.

• zetaReduce : Bool

  If true (default: true), then when preprocessing the goal, do zeta reduction to attempt to eliminate free variables.

• revert : Bool

  If true (default: false), then when preprocessing, removes irrelevant variables and reverts the local context. A variable is relevant if it appears in the target, if it appears in a relevant variable, or if it is a proposition that refers to a relevant variable.

def Lean.Parser.Tactic.decide : Lean.ParserDescr

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

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

Options:

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

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

Examples

Proving inequalities:

example : 2 + 2 ≠ 5 := by decide

Trying to prove a false proposition:

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

Trying to prove a proposition whose Decidable instance fails to reduce

opaque unknownProp : Prop

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

Properties and relations

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

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

def Lean.Parser.Tactic.nativeDecide : Lean.ParserDescr

native_decide is a synonym for decide +native. It will attempt to prove a goal of type p by synthesizing an instance of Decidable p and then evaluating it to isTrue ... Unlike decide, this uses #eval to evaluate the decidability instance.

This should be used with care because it adds the entire lean compiler to the trusted part, and the axiom Lean.ofReduceBool will show up in #print axioms for theorems using this method or anything that transitively depends on them. Nevertheless, because it is compiled, this can be significantly more efficient than using decide, and for very large computations this is one way to run external programs and trust the result.

example : (List.range 1000).length = 1000 := by native_decide

def Lean.Parser.Tactic.omega : Lean.ParserDescr

The omega tactic, for resolving integer and natural linear arithmetic problems.

It is not yet a full decision procedure (no "dark" or "grey" shadows), but should be effective on many problems.

We handle hypotheses of the form x = y, x < y, x ≤ y, and k ∣ x for x y in Nat or Int (and k a literal), along with negations of these statements.

We decompose the sides of the inequalities as linear combinations of atoms.

If we encounter x / k or x % k for literal integers k we introduce new auxiliary variables and the relevant inequalities.

On the first pass, we do not perform case splits on natural subtraction. If omega fails, we recursively perform a case split on a natural subtraction appearing in a hypothesis, and try again.

The options

omega +splitDisjunctions +splitNatSub +splitNatAbs +splitMinMax

can be used to:

• splitDisjunctions: split any disjunctions found in the context, if the problem is not otherwise solvable.
• splitNatSub: for each appearance of ((a - b : Nat) : Int), split on a ≤ b if necessary.
• splitNatAbs: for each appearance of Int.natAbs a, split on 0 ≤ a if necessary.
• splitMinMax: for each occurrence of min a b, split on min a b = a ∨ min a b = b Currently, all of these are on by default.

def Lean.Parser.Tactic.tacticBv_omega : Lean.ParserDescr

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.

Equations

• Lean.Parser.Tactic.tacticBv_omega = Lean.ParserDescr.node `Lean.Parser.Tactic.tacticBv_omega 1024 (Lean.ParserDescr.nonReservedSymbol "bv_omega" false)

def Lean.Parser.Tactic.acNf0 : Lean.ParserDescr

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

def Lean.Parser.Tactic.normCast0 : Lean.ParserDescr

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

def Lean.Parser.Tactic.tacticAssumption_mod_cast : Lean.ParserDescr

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.

def Lean.Parser.Tactic.tacticNorm_cast_ : Lean.ParserDescr

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.

def Lean.Parser.Tactic.tacticAc_nf_ : Lean.ParserDescr

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

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

instance : 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_nf
 -- goal: a + (b + (c + d)) = a + (b + (c + d))

def Lean.Parser.Tactic.pushCast : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.normCastAddElim : Lean.ParserDescr

norm_cast_add_elim foo registers foo as an elim-lemma in norm_cast.

def Lean.Parser.Tactic.symm : Lean.ParserDescr

• 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.

def Lean.Parser.Tactic.symmSaturate : Lean.ParserDescr

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

Equations

• Lean.Parser.Tactic.symmSaturate = Lean.ParserDescr.node `Lean.Parser.Tactic.symmSaturate 1024 (Lean.ParserDescr.nonReservedSymbol "symm_saturate" false)

def Lean.Parser.Tactic.SolveByElim.erase : Lean.ParserDescr

Syntax for omitting a local hypothesis in solve_by_elim.

def Lean.Parser.Tactic.SolveByElim.star : Lean.ParserDescr

Syntax for including all local hypotheses in solve_by_elim.

Equations

• Lean.Parser.Tactic.SolveByElim.star = Lean.ParserDescr.nodeWithAntiquot "star" `Lean.Parser.Tactic.SolveByElim.star (Lean.ParserDescr.symbol "*")

def Lean.Parser.Tactic.SolveByElim.arg : Lean.ParserDescr

Syntax for adding or removing a term, or *, in solve_by_elim.

def Lean.Parser.Tactic.SolveByElim.args : Lean.ParserDescr

Syntax for adding and removing terms in solve_by_elim.

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def Lean.Parser.Tactic.SolveByElim.using_ : Lean.ParserDescr

Syntax for using all lemmas labelled with an attribute in solve_by_elim.

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def Lean.Parser.Tactic.solveByElim : Lean.ParserDescr

solve_by_elim calls apply on the main goal to find an assumption whose head matches and then repeatedly calls apply on the generated subgoals until no subgoals remain, performing at most maxDepth (defaults to 6) recursive steps.

solve_by_elim discharges the current goal or fails.

solve_by_elim performs backtracking if subgoals can not be solved.

By default, the assumptions passed to apply are the local context, rfl, trivial, congrFun and congrArg.

The assumptions can be modified with similar syntax as for simp:

• solve_by_elim [h₁, h₂, ..., hᵣ] also applies the given expressions.
• solve_by_elim only [h₁, h₂, ..., hᵣ] does not include the local context, rfl, trivial, congrFun, or congrArg unless they are explicitly included.
• solve_by_elim [-h₁, ... -hₙ] removes the given local hypotheses.
• solve_by_elim using [a₁, ...] uses all lemmas which have been labelled with the attributes aᵢ (these attributes must be created using register_label_attr).

solve_by_elim* tries to solve all goals together, using backtracking if a solution for one goal makes other goals impossible. (Adding or removing local hypotheses may not be well-behaved when starting with multiple goals.)

Optional arguments passed via a configuration argument as solve_by_elim (config := { ... })

• maxDepth: number of attempts at discharging generated subgoals
• symm: adds all hypotheses derived by symm (defaults to true).
• exfalso: allow calling exfalso and trying again if solve_by_elim fails (defaults to true).
• transparency: change the transparency mode when calling apply. Defaults to .default, but it is often useful to change to .reducible, so semireducible definitions will not be unfolded when trying to apply a lemma.

See also the doc-comment for Lean.Meta.Tactic.Backtrack.BacktrackConfig for the options proc, suspend, and discharge which allow further customization of solve_by_elim. Both apply_assumption and apply_rules are implemented via these hooks.

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def Lean.Parser.Tactic.applyAssumption : Lean.ParserDescr

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).

def Lean.Parser.Tactic.applyRules : Lean.ParserDescr

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.

def Lean.Parser.Tactic.exact? : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.apply? : Lean.ParserDescr

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.

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def Lean.Parser.Tactic.rewrites_forbidden : Lean.ParserDescr

Syntax for excluding some names, e.g. [-my_lemma, -my_theorem].

def Lean.Parser.Tactic.rewrites? : Lean.ParserDescr

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.

def Lean.Parser.Tactic.showTerm : Lean.ParserDescr

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.)

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def Lean.Parser.Tactic.showTermElab : Lean.ParserDescr

show_term e elaborates e, then prints the generated term.

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def Lean.Parser.Tactic.by? : Lean.ParserDescr

The command by? will print a suggestion for replacing the proof block with a proof term using show_term.

Equations

• Lean.Parser.Tactic.by? = Lean.ParserDescr.node `Lean.Parser.Tactic.by? 1022 (Lean.ParserDescr.binary `andthen (Lean.ParserDescr.symbol "by?") (Lean.ParserDescr.const `tacticSeq))

def Lean.Parser.Attr.simp : Lean.ParserDescr

Theorems tagged with the simp attribute are used by the simplifier (i.e., the simp tactic, and its variants) to simplify expressions occurring in your goals. We call theorems tagged with the simp attribute "simp theorems" or "simp lemmas". Lean maintains a database/index containing all active simp theorems. Here is an example of a simp theorem.

@[simp] theorem ne_eq (a b : α) : (a ≠ b) = Not (a = b) := rfl

This simp theorem instructs the simplifier to replace instances of the term a ≠ b (e.g. x + 0 ≠ y) with Not (a = b) (e.g., Not (x + 0 = y)). The simplifier applies simp theorems in one direction only: if A = B is a simp theorem, then simp replaces As with Bs, but it doesn't replace Bs with As. Hence a simp theorem should have the property that its right-hand side is "simpler" than its left-hand side. In particular, = and ↔ should not be viewed as symmetric operators in this situation. The following would be a terrible simp theorem (if it were even allowed):

@[simp] lemma mul_right_inv_bad (a : G) : 1 = a * a⁻¹ := ...

Replacing 1 with a * a⁻¹ is not a sensible default direction to travel. Even worse would be a theorem that causes expressions to grow without bound, causing simp to loop forever.

By default the simplifier applies simp theorems to an expression e after its sub-expressions have been simplified. We say it performs a bottom-up simplification. You can instruct the simplifier to apply a theorem before its sub-expressions have been simplified by using the modifier ↓. Here is an example

@[simp↓] theorem not_and_eq (p q : Prop) : (¬ (p ∧ q)) = (¬p ∨ ¬q) :=

You can instruct the simplifier to rewrite the lemma from right-to-left:

attribute @[simp ←] and_assoc

When multiple simp theorems are applicable, the simplifier uses the one with highest priority. The equational theorems of function are applied at very low priority (100 and below). If there are several with the same priority, it is uses the "most recent one". Example:

@[simp high] theorem cond_true (a b : α) : cond true a b = a := rfl
@[simp low+1] theorem or_true (p : Prop) : (p ∨ True) = True :=
  propext <| Iff.intro (fun _ => trivial) (fun _ => Or.inr trivial)
@[simp 100] theorem ite_self {d : Decidable c} (a : α) : ite c a a = a := by
  cases d <;> rfl

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def Lean.Parser.Attr.grind_norm : Lean.ParserDescr

Theorems tagged with the grind_norm attribute are used by the grind tactic normalizer/pre-processor.

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def Lean.Parser.Attr.grind_norm_proc : Lean.ParserDescr

Simplification procedures tagged with the grind_norm_proc attribute are used by the grind tactic normalizer/pre-processor.

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def Lean.Parser.Attr.normCastLabel : Lean.ParserDescr

The possible norm_cast kinds: elim, move, or squash.

def Lean.Parser.Attr.norm_cast : Lean.ParserDescr

The norm_cast attribute should be given to lemmas that describe the behaviour of a coercion with respect to an operator, a relation, or a particular function.

It only concerns equality or iff lemmas involving ↑, ⇑ and ↥, describing the behavior of the coercion functions. It does not apply to the explicit functions that define the coercions.

Examples:

@[norm_cast] theorem coe_nat_inj' {m n : ℕ} : (↑m : ℤ) = ↑n ↔ m = n

@[norm_cast] theorem coe_int_denom (n : ℤ) : (n : ℚ).denom = 1

@[norm_cast] theorem cast_id : ∀ n : ℚ, ↑n = n

@[norm_cast] theorem coe_nat_add (m n : ℕ) : (↑(m + n) : ℤ) = ↑m + ↑n

@[norm_cast] theorem cast_coe_nat (n : ℕ) : ((n : ℤ) : α) = n

@[norm_cast] theorem cast_one : ((1 : ℚ) : α) = 1

Lemmas tagged with @[norm_cast] are classified into three categories: move, elim, and squash. They are classified roughly as follows:

• elim lemma: LHS has 0 head coes and ≥ 1 internal coe
• move lemma: LHS has 1 head coe and 0 internal coes, RHS has 0 head coes and ≥ 1 internal coes
• squash lemma: LHS has ≥ 1 head coes and 0 internal coes, RHS has fewer head coes

norm_cast uses move and elim lemmas to factor coercions toward the root of an expression and to cancel them from both sides of an equation or relation. It uses squash lemmas to clean up the result.

It is typically not necessary to specify these categories, as norm_cast lemmas are automatically classified by default. The automatic classification can be overridden by giving an optional elim, move, or squash parameter to the attribute.

@[simp, norm_cast elim] lemma nat_cast_re (n : ℕ) : (n : ℂ).re = n := by
  rw [← of_real_nat_cast, of_real_re]

Don't do this unless you understand what you are doing.

def «term‹_›» : Lean.ParserDescr

‹t› resolves to an (arbitrary) hypothesis of type t. It is useful for referring to hypotheses without accessible names. t may contain holes that are solved by unification with the expected type; in particular, ‹_› is a shortcut for by assumption.

def tacticGet_elem_tactic_trivial : Lean.ParserDescr

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).

Equations

• tacticGet_elem_tactic_trivial = Lean.ParserDescr.node `tacticGet_elem_tactic_trivial 1024 (Lean.ParserDescr.nonReservedSymbol "get_elem_tactic_trivial" false)

def tacticGet_elem_tactic : Lean.ParserDescr

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.

def Lean.Parser.Syntax.exact? : Lean.ParserDescr

Searches environment for definitions or theorems that can be substituted in for exact?% to solve the goal.

@[reducible, inline]

abbrev autoParam (α : Sort u) (tactic : Lean.Syntax) : Sort u

Gadget for automatic parameter support. This is similar to the optParam gadget, but it uses the given tactic. Like optParam, this gadget only affects elaboration. For example, the tactic will not be invoked during type class resolution.

Equations

• autoParam α tactic = α