lean4-htt/src/Init/Tactics.lean

/-
Copyright (c) 2022 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Mario Carneiro
-/
prelude
import Init.Notation
set_option linter.missingDocs true -- keep it documented

namespace Lean.Parser.Tactic
/--
`with_annotate_state stx t` annotates the lexical range of `stx : Syntax` with
the initial and final state of running tactic `t`.
-/
scoped syntax (name := withAnnotateState)
  "with_annotate_state " rawStx ppSpace tactic : tactic

/--
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:
  ```lean
  -- ... ⊢ α × β → ...
  intro (a, b)
  -- ..., a : α, b : β ⊢ ...
  ```
* Alternatively, `intro` can be combined with pattern matching much like `fun`:
  ```lean
  intro
  | n + 1, 0 => tac
  | ...
  ```
-/
syntax (name := intro) "intro" notFollowedBy("|") (ppSpace colGt term:max)* : tactic

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

Let bindings:
```lean
example : let n := 1; let k := 2; n + k = 3 := by
  intros
  /- n✝ : Nat := 1
     k✝ : Nat := 2
     ⊢ n✝ + k✝ = 3 -/
  rfl
```
-/
syntax (name := intros) "intros" (ppSpace colGt (ident <|> hole))* : tactic

/--
`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.
-/
syntax (name := rename) "rename " term " => " ident : tactic

/--
`revert x...` is the inverse of `intro x...`: it moves the given hypotheses
into the main goal's target type.
-/
syntax (name := revert) "revert" (ppSpace colGt term:max)+ : tactic

/--
`clear x...` removes the given hypotheses, or fails if there are remaining
references to a hypothesis.
-/
syntax (name := clear) "clear" (ppSpace colGt term:max)+ : tactic

/--
`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.
-/
syntax (name := subst) "subst" (ppSpace colGt term:max)+ : tactic

/--
Applies `subst` to all hypotheses of the form `h : x = t` or `h : t = x`.
-/
syntax (name := substVars) "subst_vars" : tactic

/--
`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`.
-/
syntax (name := assumption) "assumption" : tactic

/--
`contradiction` closes the main goal if its hypotheses are "trivially contradictory".
- Inductive type/family with no applicable constructors
```lean
example (h : False) : p := by contradiction
```
- Injectivity of constructors
```lean
example (h : none = some true) : p := by contradiction  --
```
- Decidable false proposition
```lean
example (h : 2 + 2 = 3) : p := by contradiction
```
- Contradictory hypotheses
```lean
example (h : p) (h' : ¬ p) : q := by contradiction
```
- Other simple contradictions such as
```lean
example (x : Nat) (h : x ≠ x) : p := by contradiction
```
-/
syntax (name := contradiction) "contradiction" : tactic

/--
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`.
-/
syntax (name := falseOrByContra) "false_or_by_contra" : tactic

/--
`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.
-/
syntax (name := apply) "apply " term : tactic

/--
`exact e` closes the main goal if its target type matches that of `e`.
-/
syntax (name := exact) "exact " term : tactic

/--
`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.
-/
syntax (name := refine) "refine " term : tactic

/--
`refine' e` behaves like `refine e`, except that unsolved placeholders (`_`)
and implicit parameters are also converted into new goals.
-/
syntax (name := refine') "refine' " term : tactic

/-- `exfalso` converts a goal `⊢ tgt` into `⊢ False` by applying `False.elim`. -/
macro "exfalso" : tactic => `(tactic| refine False.elim ?_)

/--
If the main goal's target type is an inductive type, `constructor` solves it with
the first matching constructor, or else fails.
-/
syntax (name := constructor) "constructor" : tactic

/--
Applies the first constructor when
the goal is an inductive type with exactly two constructors, or fails otherwise.
```
example : True ∨ False := by
  left
  trivial
```
-/
syntax (name := left) "left" : tactic

/--
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
```
-/
syntax (name := right) "right" : tactic

/--
* `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)`.
-/
syntax (name := case) "case " sepBy1(caseArg, " | ") " => " tacticSeq : tactic

/--
`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.
-/
syntax (name := case') "case' " sepBy1(caseArg, " | ") " => " tacticSeq : tactic

/--
`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.
-/
macro "next " args:binderIdent* " => " tac:tacticSeq : tactic => `(tactic| case _ $args* => $tac)

/-- `all_goals tac` runs `tac` on each goal, concatenating the resulting goals, if any. -/
syntax (name := allGoals) "all_goals " tacticSeq : tactic

/--
`any_goals tac` applies the tactic `tac` to every goal, and succeeds if at
least one application succeeds.
-/
syntax (name := anyGoals) "any_goals " tacticSeq : tactic

/--
`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.
-/
syntax (name := focus) "focus " tacticSeq : tactic

/-- `skip` does nothing. -/
syntax (name := skip) "skip" : tactic

/-- `done` succeeds iff there are no remaining goals. -/
syntax (name := done) "done" : tactic

/-- `trace_state` displays the current state in the info view. -/
syntax (name := traceState) "trace_state" : tactic

/-- `trace msg` displays `msg` in the info view. -/
syntax (name := traceMessage) "trace " str : tactic

/-- `fail_if_success t` fails if the tactic `t` succeeds. -/
syntax (name := failIfSuccess) "fail_if_success " tacticSeq : tactic

/--
`(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 `;`.
-/
syntax (name := paren) "(" withoutPosition(tacticSeq) ")" : tactic

/--
`with_reducible tacs` executes `tacs` using the reducible transparency setting.
In this setting only definitions tagged as `[reducible]` are unfolded.
-/
syntax (name := withReducible) "with_reducible " tacticSeq : tactic

/--
`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.
-/
syntax (name := withReducibleAndInstances) "with_reducible_and_instances " tacticSeq : tactic

/--
`with_unfolding_all tacs` executes `tacs` using the `.all` transparency setting.
In this setting all definitions that are not opaque are unfolded.
-/
syntax (name := withUnfoldingAll) "with_unfolding_all " tacticSeq : tactic

/-- `first | tac | ...` runs each `tac` until one succeeds, or else fails. -/
syntax (name := first) "first " withPosition((ppDedent(ppLine) colGe "| " tacticSeq)+) : tactic

/--
`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`.
-/
syntax (name := rotateLeft) "rotate_left" (ppSpace num)? : tactic

/--
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`.
-/
syntax (name := rotateRight) "rotate_right" (ppSpace num)? : tactic

/-- `try tac` runs `tac` and succeeds even if `tac` failed. -/
macro "try " t:tacticSeq : tactic => `(tactic| first | $t | skip)

/--
`tac <;> tac'` runs `tac` on the main goal and `tac'` on each produced goal,
concatenating all goals produced by `tac'`.
-/
macro:1 x:tactic tk:" <;> " y:tactic:2 : tactic => `(tactic|
  focus
    $x:tactic
    -- annotate token with state after executing `x`
    with_annotate_state $tk skip
    all_goals $y:tactic)

/-- `fail msg` is a tactic that always fails, and produces an error using the given message. -/
syntax (name := fail) "fail" (ppSpace str)? : tactic

/-- `eq_refl` is equivalent to `exact rfl`, but has a few optimizations. -/
syntax (name := eqRefl) "eq_refl" : tactic

/--
`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`).
-/
macro "rfl" : tactic => `(tactic| case' _ => fail "The rfl tactic failed. Possible reasons:
- The goal is not a reflexive relation (neither `=` nor a relation with a @[refl] lemma).
- The arguments of the relation are not equal.
Try using the reflexivitiy lemma for your relation explicitly, e.g. `exact Eq.rfl`.")

macro_rules | `(tactic| rfl) => `(tactic| eq_refl)
macro_rules | `(tactic| rfl) => `(tactic| exact HEq.rfl)

/--
This tactic applies to a goal whose target has the form `x ~ x`,
where `~` is a reflexive relation other than `=`,
that is, a relation which has a reflexive lemma tagged with the attribute @[refl].
-/
syntax (name := applyRfl) "apply_rfl" : tactic

macro_rules | `(tactic| rfl) => `(tactic| apply_rfl)

/--
`rfl'` is similar to `rfl`, but disables smart unfolding and unfolds all kinds of definitions,
theorems included (relevant for declarations defined by well-founded recursion).
-/
macro "rfl'" : tactic => `(tactic| set_option smartUnfolding false in with_unfolding_all rfl)

/--
`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
```
-/
syntax (name := acRfl) "ac_rfl" : tactic

/--
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.
-/
macro "sorry" : tactic => `(tactic| exact @sorryAx _ false)

/-- `admit` is a shorthand for `exact sorry`. -/
macro "admit" : tactic => `(tactic| exact @sorryAx _ false)

/--
`infer_instance` is an abbreviation for `exact inferInstance`.
It synthesizes a value of any target type by typeclass inference.
-/
macro "infer_instance" : tactic => `(tactic| exact inferInstance)

/-- Optional configuration option for tactics -/
syntax config := atomic(" (" &"config") " := " withoutPosition(term) ")"

/-- The `*` location refers to all hypotheses and the goal. -/
syntax locationWildcard := " *"

/--
A hypothesis location specification consists of 1 or more hypothesis references
and optionally `⊢` denoting the goal.
-/
syntax locationHyp := (ppSpace colGt term:max)+ patternIgnore(ppSpace (atomic("|" noWs "-") <|> "⊢"))?

/--
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
-/
syntax location := withPosition(" at" (locationWildcard <|> locationHyp))

/--
* `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.
-/
syntax (name := change) "change " term (location)? : tactic

/--
* `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`.
-/
syntax (name := changeWith) "change " term " with " term (location)? : tactic

/--
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`.
-/
syntax rwRule    := patternIgnore("← " <|> "<- ")? term
/-- A `rwRuleSeq` is a list of `rwRule` in brackets. -/
syntax rwRuleSeq := " [" withoutPosition(rwRule,*,?) "]"

/--
`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.
-/
syntax (name := rewriteSeq) "rewrite" (config)? rwRuleSeq (location)? : tactic

/--
`rw` is like `rewrite`, but also tries to close the goal by "cheap" (reducible) `rfl` afterwards.
-/
macro (name := rwSeq) "rw " c:(config)? s:rwRuleSeq l:(location)? : tactic =>
  match s with
  | `(rwRuleSeq| [$rs,*]%$rbrak) =>
    -- We show the `rfl` state on `]`
    `(tactic| (rewrite $(c)? [$rs,*] $(l)?; with_annotate_state $rbrak (try (with_reducible rfl))))
  | _ => Macro.throwUnsupported

/-- `rwa` calls `rw`, then closes any remaining goals using `assumption`. -/
macro "rwa " rws:rwRuleSeq loc:(location)? : tactic =>
  `(tactic| (rw $rws:rwRuleSeq $[$loc:location]?; assumption))

/--
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.
-/
syntax (name := injection) "injection " term (" with" (ppSpace colGt (ident <|> hole))+)? : tactic

/-- `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)`. -/
-- TODO: add with
syntax (name := injections) "injections" (ppSpace colGt (ident <|> hole))* : tactic

/--
The discharger clause of `simp` and related tactics.
This is a tactic used to discharge the side conditions on conditional rewrite rules.
-/
syntax discharger := atomic(" (" patternIgnore(&"discharger" <|> &"disch")) " := " withoutPosition(tacticSeq) ")"

/-- Use this rewrite rule before entering the subterms -/
syntax simpPre   := "↓"
/-- Use this rewrite rule after entering the subterms -/
syntax simpPost  := "↑"
/--
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
-/
syntax simpLemma := (simpPre <|> simpPost)? patternIgnore("← " <|> "<- ")? term
/-- An erasure specification `-thm` says to remove `thm` from the simp set -/
syntax simpErase := "-" term:max
/-- The simp lemma specification `*` means to rewrite with all hypotheses -/
syntax simpStar  := "*"
/--
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.
-/
syntax (name := simp) "simp" (config)? (discharger)? (&" only")?
  (" [" withoutPosition((simpStar <|> simpErase <|> simpLemma),*,?) "]")? (location)? : tactic
/--
`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.
-/
syntax (name := simpAll) "simp_all" (config)? (discharger)? (&" only")?
  (" [" withoutPosition((simpErase <|> simpLemma),*,?) "]")? : tactic

/--
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.
-/
syntax (name := dsimp) "dsimp" (config)? (discharger)? (&" only")?
  (" [" withoutPosition((simpErase <|> simpLemma),*,?) "]")? (location)? : tactic

/--
A `simpArg` is either a `*`, `-lemma` or a simp lemma specification
(which includes the `↑` `↓` `←` specifications for pre, post, reverse rewriting).
-/
def simpArg := simpStar.binary `orelse (simpErase.binary `orelse simpLemma)

/-- A simp args list is a list of `simpArg`. This is the main argument to `simp`. -/
syntax simpArgs := " [" simpArg,* "]"

/--
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`.
-/
def dsimpArg := simpErase.binary `orelse simpLemma

/-- A dsimp args list is a list of `dsimpArg`. This is the main argument to `dsimp`. -/
syntax dsimpArgs := " [" dsimpArg,* "]"

/-- The common arguments of `simp?` and `simp?!`. -/
syntax simpTraceArgsRest := (config)? (discharger)? (&" only")? (simpArgs)? (ppSpace location)?

/--
`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`.
-/
syntax (name := simpTrace) "simp?" "!"? simpTraceArgsRest : tactic

@[inherit_doc simpTrace]
macro tk:"simp?!" rest:simpTraceArgsRest : tactic => `(tactic| simp?%$tk ! $rest)

/-- The common arguments of `simp_all?` and `simp_all?!`. -/
syntax simpAllTraceArgsRest := (config)? (discharger)? (&" only")? (dsimpArgs)?

@[inherit_doc simpTrace]
syntax (name := simpAllTrace) "simp_all?" "!"? simpAllTraceArgsRest : tactic

@[inherit_doc simpTrace]
macro tk:"simp_all?!" rest:simpAllTraceArgsRest : tactic => `(tactic| simp_all?%$tk ! $rest)

/-- The common arguments of `dsimp?` and `dsimp?!`. -/
syntax dsimpTraceArgsRest := (config)? (&" only")? (dsimpArgs)? (ppSpace location)?

@[inherit_doc simpTrace]
syntax (name := dsimpTrace) "dsimp?" "!"? dsimpTraceArgsRest : tactic

@[inherit_doc simpTrace]
macro tk:"dsimp?!" rest:dsimpTraceArgsRest : tactic => `(tactic| dsimp?%$tk ! $rest)

/-- The arguments to the `simpa` family tactics. -/
syntax simpaArgsRest := (config)? (discharger)? &" only "? (simpArgs)? (" using " term)?

/--
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.
-/
syntax (name := simpa) "simpa" "?"? "!"? simpaArgsRest : tactic

@[inherit_doc simpa] macro "simpa!" rest:simpaArgsRest : tactic =>
  `(tactic| simpa ! $rest:simpaArgsRest)

@[inherit_doc simpa] macro "simpa?" rest:simpaArgsRest : tactic =>
  `(tactic| simpa ? $rest:simpaArgsRest)

@[inherit_doc simpa] macro "simpa?!" rest:simpaArgsRest : tactic =>
  `(tactic| simpa ?! $rest:simpaArgsRest)

/--
`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.
-/
syntax (name := delta) "delta" (ppSpace colGt ident)+ (location)? : tactic

/--
* `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.
-/
syntax (name := unfold) "unfold" (ppSpace colGt ident)+ (location)? : tactic

/--
Auxiliary macro for lifting have/suffices/let/...
It makes sure the "continuation" `?_` is the main goal after refining.
-/
macro "refine_lift " e:term : tactic => `(tactic| focus (refine no_implicit_lambda% $e; rotate_right))

/--
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`.
-/
macro "have " d:haveDecl : tactic => `(tactic| refine_lift have $d:haveDecl; ?_)

/--
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.
 -/
macro "suffices " d:sufficesDecl : tactic => `(tactic| refine_lift suffices $d; ?_)
/--
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`.
-/
macro "let " d:letDecl : tactic => `(tactic| refine_lift let $d:letDecl; ?_)
/--
`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`.
-/
macro "show " e:term : tactic => `(tactic| refine_lift show $e from ?_) -- TODO: fix, see comment
/-- `let rec f : t := e` adds a recursive definition `f` to the current goal.
The syntax is the same as term-mode `let rec`. -/
syntax (name := letrec) withPosition(atomic("let " &"rec ") letRecDecls) : tactic
macro_rules
  | `(tactic| let rec $d) => `(tactic| refine_lift let rec $d; ?_)

/-- Similar to `refine_lift`, but using `refine'` -/
macro "refine_lift' " e:term : tactic => `(tactic| focus (refine' no_implicit_lambda% $e; rotate_right))
/-- Similar to `have`, but using `refine'` -/
macro "have' " d:haveDecl : tactic => `(tactic| refine_lift' have $d:haveDecl; ?_)
/-- Similar to `have`, but using `refine'` -/
macro (priority := high) "have'" x:ident " := " p:term : tactic => `(tactic| have' $x:ident : _ := $p)
/-- Similar to `let`, but using `refine'` -/
macro "let' " d:letDecl : tactic => `(tactic| refine_lift' let $d:letDecl; ?_)

/--
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.
-/
syntax inductionAltLHS := "| " (("@"? ident) <|> hole) (ident <|> hole)*
/--
In induction alternative, which can have 1 or more cases on the left
and `_`, `?_`, or a tactic sequence after the `=>`.
-/
syntax inductionAlt  := ppDedent(ppLine) inductionAltLHS+ " => " (hole <|> syntheticHole <|> tacticSeq)
/--
After `with`, there is an optional tactic that runs on all branches, and
then a list of alternatives.
-/
syntax inductionAlts := " with" (ppSpace tactic)? withPosition((colGe inductionAlt)+)

/--
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.
-/
syntax (name := induction) "induction " term,+ (" using " term)?
  (" generalizing" (ppSpace colGt term:max)+)? (inductionAlts)? : tactic

/-- A `generalize` argument, of the form `term = x` or `h : term = x`. -/
syntax generalizeArg := atomic(ident " : ")? term:51 " = " ident

/--
* `generalize ([h :] e = x),+` replaces all occurrences `e`s in the main goal
  with a fresh hypothesis `x`s. 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.
-/
syntax (name := generalize) "generalize " generalizeArg,+ (location)? : tactic

/--
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).
-/
syntax casesTarget := atomic(ident " : ")? term
/--
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.
-/
syntax (name := cases) "cases " casesTarget,+ (" using " term)? (inductionAlts)? : tactic

/-- `rename_i x_1 ... x_n` renames the last `n` inaccessible names using the given names. -/
syntax (name := renameI) "rename_i" (ppSpace colGt binderIdent)+ : tactic

/--
`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 `Batteries` library provides `repeat'` which repeats separately in each subgoal.
-/
syntax "repeat " tacticSeq : tactic
macro_rules
  | `(tactic| repeat $seq) => `(tactic| first | ($seq); repeat $seq | skip)

/--
`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)
```
-/
syntax "trivial" : tactic

/--
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`.
-/
syntax (name := split) "split" (ppSpace colGt term)? (location)? : tactic

/-- `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"
```
-/
syntax (name := dbgTrace) "dbg_trace " str : tactic

/--
`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.
-/
macro "stop" tacticSeq : tactic => `(tactic| repeat sorry)

/--
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.
-/
syntax (name := specialize) "specialize " term : tactic

macro_rules | `(tactic| trivial) => `(tactic| assumption)
macro_rules | `(tactic| trivial) => `(tactic| rfl)
macro_rules | `(tactic| trivial) => `(tactic| contradiction)
macro_rules | `(tactic| trivial) => `(tactic| decide)
macro_rules | `(tactic| trivial) => `(tactic| apply True.intro)
macro_rules | `(tactic| trivial) => `(tactic| apply And.intro <;> trivial)

/--
`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
```
-/
macro "unhygienic " t:tacticSeq : tactic => `(tactic| set_option tactic.hygienic false in $t)

/--
`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.)
-/
syntax (name := checkpoint) "checkpoint " tacticSeq : tactic

/--
`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.)
-/
macro (name := save) "save" : tactic => `(tactic| skip)

/--
The tactic `sleep ms` sleeps for `ms` milliseconds and does nothing.
It is used for debugging purposes only.
-/
syntax (name := sleep) "sleep " num : tactic

/--
`exists e₁, e₂, ...` is shorthand for `refine ⟨e₁, e₂, ...⟩; try trivial`.
It is useful for existential goals.
-/
macro "exists " es:term,+ : tactic =>
  `(tactic| (refine ⟨$es,*, ?_⟩; try trivial))

/--
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`.
-/
syntax (name := congr) "congr" (ppSpace num)? : tactic


/--
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.
-/
syntax (name := tacDepIfThenElse)
  ppRealGroup(ppRealFill(ppIndent("if " binderIdent " : " term " then") ppSpace matchRhsTacticSeq)
    ppDedent(ppSpace) ppRealFill("else " matchRhsTacticSeq)) : tactic

/--
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 ?_`.)
-/
syntax (name := tacIfThenElse)
  ppRealGroup(ppRealFill(ppIndent("if " term " then") ppSpace matchRhsTacticSeq)
    ppDedent(ppSpace) ppRealFill("else " matchRhsTacticSeq)) : tactic

/--
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.
-/
macro "nofun" : tactic => `(tactic| exact nofun)

/--
The tactic `nomatch h` is shorthand for `exact nomatch h`.
-/
macro "nomatch " es:term,+ : tactic =>
  `(tactic| exact nomatch $es:term,*)

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

```lean
f : α → β
h : α
⊢ goal
```

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

```lean
f : α → β
h : β
⊢ goal
```

whereas `have h := f h` would result in:

```lean
f : α → β
h† : α
h : β
⊢ goal
```

This can be used to simulate the `specialize` and `apply at` tactics of Coq.
-/
syntax (name := replace) "replace" haveDecl : tactic

/--
`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.
-/
syntax (name := repeat') "repeat' " tacticSeq : tactic

/--
`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.
-/
syntax (name := repeat1') "repeat1' " tacticSeq : tactic

/-- `and_intros` applies `And.intro` until it does not make progress. -/
syntax "and_intros" : tactic
macro_rules | `(tactic| and_intros) => `(tactic| repeat' refine And.intro ?_ ?_)

/--
`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.
-/
syntax (name := substEqs) "subst_eqs" : tactic

/-- The `run_tac doSeq` tactic executes code in `TacticM Unit`. -/
syntax (name := runTac) "run_tac " doSeq : tactic

/-- `haveI` behaves like `have`, but inlines the value instead of producing a `let_fun` term. -/
macro "haveI" d:haveDecl : tactic => `(tactic| refine_lift haveI $d:haveDecl; ?_)

/-- `letI` behaves like `let`, but inlines the value instead of producing a `let_fun` term. -/
macro "letI" d:haveDecl : tactic => `(tactic| refine_lift letI $d:haveDecl; ?_)

/--
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.
-/
syntax (name := omega) "omega" (config)? : tactic

/--
`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.
-/
macro "bv_omega" : tactic => `(tactic| (try simp only [bv_toNat] at *) <;> omega)

/-- Implementation of `norm_cast` (the full `norm_cast` calls `trivial` afterwards). -/
syntax (name := normCast0) "norm_cast0" (location)? : tactic

/-- `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. -/
macro "assumption_mod_cast" : tactic => `(tactic| norm_cast0 at * <;> assumption)

/--
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
```lean
a b : ℤ
h : ↑a + ↑b < (10 : ℚ)
```
writing `norm_cast at h` will turn `h` into
```lean
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.
-/
macro "norm_cast" loc:(location)? : tactic =>
  `(tactic| norm_cast0 $[$loc]? <;> try trivial)

/--
`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:
```lean
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`.
-/
syntax (name := pushCast) "push_cast" (config)? (discharger)? (&" only")?
  (" [" (simpStar <|> simpErase <|> simpLemma),* "]")? (location)? : tactic

/--
`norm_cast_add_elim foo` registers `foo` as an elim-lemma in `norm_cast`.
-/
syntax (name := normCastAddElim) "norm_cast_add_elim" ident : command

/--
* `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`.
-/
syntax (name := symm) "symm" (location)? : tactic

/-- For every hypothesis `h : a ~ b` where a `@[symm]` lemma is available,
add a hypothesis `h_symm : b ~ a`. -/
syntax (name := symmSaturate) "symm_saturate" : tactic

namespace SolveByElim

/-- Syntax for omitting a local hypothesis in `solve_by_elim`. -/
syntax erase := "-" term:max
/-- Syntax for including all local hypotheses in `solve_by_elim`. -/
syntax star := "*"
/-- Syntax for adding or removing a term, or `*`, in `solve_by_elim`. -/
syntax arg := star <|> erase <|> term
/-- Syntax for adding and removing terms in `solve_by_elim`. -/
syntax args := " [" SolveByElim.arg,* "]"
/-- Syntax for using all lemmas labelled with an attribute in `solve_by_elim`. -/
syntax using_ := " using " ident,*

end SolveByElim

section SolveByElim
open SolveByElim (args using_)

/--
`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.
-/
syntax (name := solveByElim)
  "solve_by_elim" "*"? (config)? (&" only")? (args)? (using_)? : tactic

/--
`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`).
-/
syntax (name := applyAssumption)
  "apply_assumption" (config)? (&" only")? (args)? (using_)? : tactic

/--
`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.
-/
syntax (name := applyRules) "apply_rules" (config)? (&" only")? (args)? (using_)? : tactic
end SolveByElim

/--
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.
-/
syntax (name := exact?) "exact?" (" using " (colGt ident),+)? : tactic

/--
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.
-/
syntax (name := apply?) "apply?" (" using " (colGt term),+)? : tactic

/--
Syntax for excluding some names, e.g. `[-my_lemma, -my_theorem]`.
-/
syntax rewrites_forbidden := " [" (("-" ident),*,?) "]"

/--
`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.
-/
syntax (name := rewrites?) "rw?" (ppSpace location)? (rewrites_forbidden)? : tactic

/--
`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.)
-/
syntax (name := showTerm) "show_term " tacticSeq : tactic

/--
`show_term e` elaborates `e`, then prints the generated term.
-/
macro (name := showTermElab) tk:"show_term " t:term : term =>
  `(term| no_implicit_lambda% (show_term_elab%$tk $t))

/--
The command `by?` will print a suggestion for replacing the proof block with a proof term
using `show_term`.
-/
macro (name := by?) tk:"by?" t:tacticSeq : term => `(show_term%$tk by%$tk $t)

end Tactic

namespace Attr
/--
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.
```lean
@[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 `A`s with `B`s,
but it doesn't replace `B`s with `A`s. 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):
```lean
@[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
```lean
@[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:
```lean
@[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
```
-/
syntax (name := simp) "simp" (Tactic.simpPre <|> Tactic.simpPost)? (ppSpace prio)? : attr

/--
Theorems tagged with the `grind_norm` attribute are used by the `grind` tactic normalizer/pre-processor.
-/
syntax (name := grind_norm) "grind_norm" (Tactic.simpPre <|> Tactic.simpPost)? (ppSpace prio)? : attr

/--
Simplification procedures tagged with the `grind_norm_proc` attribute are used by the `grind` tactic normalizer/pre-processor.
-/
syntax (name := grind_norm_proc) "grind_norm_proc" (Tactic.simpPre <|> Tactic.simpPost)? : attr


/-- The possible `norm_cast` kinds: `elim`, `move`, or `squash`. -/
syntax normCastLabel := &"elim" <|> &"move" <|> &"squash"

/--
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:
```lean
@[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.

```lean
@[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.
-/
syntax (name := norm_cast) "norm_cast" (ppSpace normCastLabel)? (ppSpace num)? : attr

end Attr

end Parser
end Lean

/--
`‹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`.
-/
syntax "‹" withoutPosition(term) "›" : term
macro_rules | `(‹$type›) => `((by assumption : $type))

/--
`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).
-/
syntax "get_elem_tactic_trivial" : tactic

macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| omega)
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| simp (config := { arith := true }); done)
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| trivial)

/--
`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.
-/
macro "get_elem_tactic" : tactic =>
  `(tactic| first
      /-
      Recall that `macro_rules` are tried in reverse order.
      We want `assumption` to be tried first.
      This is important for theorems such as
      ```
      [simp] theorem getElem_pop (a : Array α) (i : Nat) (hi : i < a.pop.size) :
      a.pop[i] = a[i]'(Nat.lt_of_lt_of_le (a.size_pop ▸ hi) (Nat.sub_le _ _)) :=
      ```
      There is a proof embedded in the right-hand-side, and we want it to be just `hi`.
      If `omega` is used to "fill" this proof, we will have a more complex proof term that
      cannot be inferred by unification.
      We hardcoded `assumption` here to ensure users cannot accidentaly break this IF
      they add new `macro_rules` for `get_elem_tactic_trivial`.

      TODO: Implement priorities for `macro_rules`.
      TODO: Ensure we have a **high-priority** macro_rules for `get_elem_tactic_trivial` which is just `assumption`.
      -/
    | assumption
    | get_elem_tactic_trivial
    | fail "failed to prove index is valid, possible solutions:
  - Use `have`-expressions to prove the index is valid
  - Use `a[i]!` notation instead, runtime check is perfomed, and 'Panic' error message is produced if index is not valid
  - Use `a[i]?` notation instead, result is an `Option` type
  - Use `a[i]'h` notation instead, where `h` is a proof that index is valid"
   )

/--
Searches environment for definitions or theorems that can be substituted in
for `exact?%` to solve the goal.
 -/
syntax (name := Lean.Parser.Syntax.exact?) "exact?%" : term

set_option linter.unusedVariables.funArgs false in
/--
  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. -/
abbrev autoParam.{u} (α : Sort u) (tactic : Lean.Syntax) : Sort u := α