# Documentation

## Init.Tactics

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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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
| ...

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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

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

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

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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.

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

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

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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

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

• Other simple contradictions such as
example (x : Nat) (h : x ≠ x) : p := by contradiction

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

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

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

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

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

example : True ∨ False := by
left
trivial

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

example {p q : Prop} (h : q) : p ∨ q := by
right
exact h

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

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

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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.

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skip does nothing.

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done succeeds iff there are no remaining goals.

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trace_state displays the current state in the info view.

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trace msg displays msg in the info view.

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fail_if_success t fails if the tactic t succeeds.

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

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

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

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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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Rotate the goals to the right by n. That is, take the goal at the back and push it to the front n times. If n is omitted, it defaults to 1.

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try tac runs tac and succeeds even if tac failed.

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

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

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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).

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rfl' is similar to rfl, but disables smart unfolding and unfolds all kinds of definitions, theorems included (relevant for declarations defined by well-founded recursion).

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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

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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.

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admit is a shorthand for exact sorry.

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

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Optional configuration option for tactics

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The * location refers to all hypotheses and the goal.

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

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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
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• 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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• 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.
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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.
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A rwRuleSeq is a list of rwRule in brackets.

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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.

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

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

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

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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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Use this rewrite rule before entering the subterms

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Use this rewrite rule after entering the subterms

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

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The simp lemma specification * means to rewrite with all hypotheses

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

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

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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.

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

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The common arguments of simp? and simp?!.

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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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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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The common arguments of simp_all? and simp_all?!.

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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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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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The common arguments of dsimp? and dsimp?!.

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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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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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The arguments to the simpa family tactics.

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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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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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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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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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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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• 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.

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

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

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Similar to refine_lift, but using refine'

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Similar to have, but using refine'

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Similar to have, but using refine'

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Similar to let, but using refine'

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

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

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

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• 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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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).

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

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repeat tac repeatedly applies tac to the main goal until it fails. That is, if tac produces multiple subgoals, only subgoals up to the first failure will be visited. The Std library provides repeat' which repeats separately in each subgoal.

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trivial tries different simple tactics (e.g., rfl, contradiction, ...) to close the current goal. You can use the command macro_rules to extend the set of tactics used. Example:

macro_rules | (tactic| trivial) => (tactic| simp)

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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.
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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"

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stop is a helper tactic for "discarding" the rest of a proof: it is defined as repeat sorry. It is useful when working on the middle of a complex proofs, and less messy than commenting the remainder of the proof.

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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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unhygienic tacs runs tacs with name hygiene disabled. This means that tactics that would normally create inaccessible names will instead make regular variables. Warning: Tactics may change their variable naming strategies at any time, so code that depends on autogenerated names is brittle. Users should try not to use unhygienic if possible.

example : ∀ x : Nat, x = x := by unhygienic
intro            -- x would normally be intro'd as inaccessible
exact Eq.refl x  -- refer to x

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

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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.)

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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.)

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

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

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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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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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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 ?_.)

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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.

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The tactic nomatch h is shorthand for exact nomatch h.

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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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repeat' tac runs tac on all of the goals to produce a new list of goals, then runs tac again on all of those goals, and repeats until tac fails on all remaining goals.

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repeat1' tac applies tac to main goal at least once. If the application succeeds, the tactic is applied recursively to the generated subgoals until it eventually fails.

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

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subst_eq repeatedly substitutes according to the equality proof hypotheses in the context, replacing the left side of the equality with the right, until no more progress can be made.

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The run_tac doSeq tactic executes code in TacticM Unit.

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

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

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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.
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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.

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

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

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

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The norm_cast family of tactics is used to normalize casts inside expressions. It is basically a simp tactic 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 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 exact, apply, rw, and assumption that work modulo norm_cast - in other words, they apply norm_cast to make them more flexible. They are called exact_mod_cast, apply_mod_cast, rw_mod_cast, and assumption_mod_cast, respectively. 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. Writing assumption_mod_cast will normalize casts in the goal and, for every hypothesis h in the context, it will try to normalize casts in h and use exact h. rw_mod_cast acts like the rw tactic but it applies norm_cast between steps.

See also push_cast, which moves casts inwards rather than lifting them outwards.

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push_cast rewrites the goal to move 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. It is equivalent to simp only with push_cast. It can also be used at hypotheses with push_cast at h and with extra simp lemmas with push_cast [int.add_zero].

example (a b : ℕ) (h1 : ((a + b : ℕ) : ℤ) = 10) (h2 : ((a + b + 0 : ℕ) : ℤ) = 10) :
((a + b : ℕ) : ℤ) = 10 :=
begin
push_cast,
push_cast at h1,
end

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

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• symm applies to a goal whose target has the form t ~ u where ~ is a symmetric relation, that is, a relation which has a symmetry lemma tagged with the attribute [symm]. It replaces the target with u ~ t.
• symm at h will rewrite a hypothesis h : t ~ u to h : u ~ t.
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For every hypothesis h : a ~ b where a @[symm] lemma is available, add a hypothesis h_symm : b ~ a.

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Syntax for omitting a local hypothesis in solve_by_elim.

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Syntax for including all local hypotheses in solve_by_elim.

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

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Syntax for adding and removing terms in solve_by_elim.

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

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

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

apply_assumption will use consequences of local hypotheses obtained via symm.

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

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

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

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

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

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

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

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

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

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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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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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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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show_term e elaborates e, then prints the generated term.

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

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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) :=


When multiple simp theorems are applicable, the simplifier uses the one with highest priority. 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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The possible norm_cast kinds: elim, move, or squash.

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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.

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‹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.

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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).

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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.

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The syntax arr[i] gets the i'th element of the collection arr. If there are proof side conditions to the application, they will be automatically inferred by the get_elem_tactic tactic.

The actual behavior of this class is type-dependent, but here are some important implementations:

• arr[i] : α where arr : Array α and i : Nat or i : USize: does array indexing with no bounds check and a proof side goal i < arr.size.
• l[i] : α where l : List α and i : Nat: index into a list, with proof side goal i < l.length.
• stx[i] : Syntax where stx : Syntax and i : Nat: get a syntax argument, no side goal (returns .missing out of range)

There are other variations on this syntax:

• arr[i]: proves the proof side goal by get_elem_tactic
• arr[i]!: panics if the side goal is false
• arr[i]?: returns none if the side goal is false
• arr[i]'h: uses h to prove the side goal
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• One or more equations did not get rendered due to their size.
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The syntax arr[i] gets the i'th element of the collection arr. If there are proof side conditions to the application, they will be automatically inferred by the get_elem_tactic tactic.

The actual behavior of this class is type-dependent, but here are some important implementations:

• arr[i] : α where arr : Array α and i : Nat or i : USize: does array indexing with no bounds check and a proof side goal i < arr.size.
• l[i] : α where l : List α and i : Nat: index into a list, with proof side goal i < l.length.
• stx[i] : Syntax where stx : Syntax and i : Nat: get a syntax argument, no side goal (returns .missing out of range)

There are other variations on this syntax:

• arr[i]: proves the proof side goal by get_elem_tactic
• arr[i]!: panics if the side goal is false
• arr[i]?: returns none if the side goal is false
• arr[i]'h: uses h to prove the side goal
Equations
• One or more equations did not get rendered due to their size.
Instances For

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

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Instances For