doc: documentation for Init.Tactics
This commit is contained in:
Mario Carneiro
Leonardo de Moura
parent
70ef3875d1
commit
8182f83929
3 changed files with 425 additions and 157 deletions
|
|
@ -2414,7 +2414,26 @@ The proof side-condition `dom xs i` is automatically dispatched by the
|
|||
`get_elem_tactic_trivial`.
|
||||
-/
|
||||
class GetElem (cont : Type u) (idx : Type v) (elem : outParam (Type w)) (dom : outParam (cont → idx → Prop)) where
|
||||
/-- The implementation of `xs[i]`. `h` is discharged by `get_elem_tactic`. -/
|
||||
/--
|
||||
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
|
||||
-/
|
||||
getElem (xs : cont) (i : idx) (h : dom xs i) : elem
|
||||
|
||||
export GetElem (getElem)
|
||||
|
|
|
|||
|
|
@ -1,25 +1,37 @@
|
|||
/-
|
||||
Copyright (c) 2022 Microsoft Corporation. All rights reserved.
|
||||
Released under Apache 2.0 license as described in the file LICENSE.
|
||||
Authors: Leonardo de Moura
|
||||
Authors: Leonardo de Moura, Mario Carneiro
|
||||
-/
|
||||
prelude
|
||||
import Init.Notation
|
||||
set_option linter.missingDocs true -- keep it documented
|
||||
|
||||
namespace Lean
|
||||
|
||||
/--
|
||||
`binderIdent` matches an `ident` or a `_`. It is used for identifiers in binding
|
||||
position, where `_` means that the value should be left unnamed and inaccessible.
|
||||
-/
|
||||
syntax binderIdent := ident <|> hole
|
||||
|
||||
namespace 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
|
||||
/--
|
||||
`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
|
||||
|
||||
/--
|
||||
Introduce 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:
|
||||
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)
|
||||
|
|
@ -33,29 +45,49 @@ For each hypothesis to be introduced, the remaining main goal's target type must
|
|||
```
|
||||
-/
|
||||
syntax (name := intro) "intro " notFollowedBy("|") (colGt term:max)* : tactic
|
||||
/-- `intros x...` behaves like `intro x...`, but then keeps introducing (anonymous) hypotheses until goal is not of a function type. -/
|
||||
|
||||
/--
|
||||
`intros x...` behaves like `intro x...`, but then keeps introducing (anonymous)
|
||||
hypotheses until goal is not of a function type.
|
||||
-/
|
||||
syntax (name := intros) "intros " (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. -/
|
||||
`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. -/
|
||||
|
||||
/--
|
||||
`revert x...` is the inverse of `intro x...`: it moves the given hypotheses
|
||||
into the main goal's target type.
|
||||
-/
|
||||
syntax (name := revert) "revert " (colGt term:max)+ : tactic
|
||||
/-- `clear x...` removes the given hypotheses, or fails if there are remaining references to a hypothesis. -/
|
||||
|
||||
/--
|
||||
`clear x...` removes the given hypotheses, or fails if there are remaining
|
||||
references to a hypothesis.
|
||||
-/
|
||||
syntax (name := clear) "clear " (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. -/
|
||||
`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 " (colGt term:max)+ : tactic
|
||||
|
||||
/--
|
||||
Apply `subst` to all hypotheses of the form `h : x = t` or `h : t = x`.
|
||||
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`. -/
|
||||
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
|
||||
|
|
@ -80,87 +112,142 @@ example (x : Nat) (h : x ≠ x) : p := by contradiction
|
|||
```
|
||||
-/
|
||||
syntax (name := contradiction) "contradiction" : 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.
|
||||
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.
|
||||
`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
|
||||
/-- 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
|
||||
|
||||
/--
|
||||
`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. -/
|
||||
`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
|
||||
|
||||
/--
|
||||
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
|
||||
|
||||
/--
|
||||
* `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.
|
||||
-/
|
||||
syntax (name := case) "case " binderIdent binderIdent* " => " 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
|
||||
`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' " binderIdent binderIdent* " => " tacticSeq : tactic
|
||||
|
||||
/--
|
||||
`next => tac` focuses on the next goal 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. -/
|
||||
`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. -/
|
||||
|
||||
/--
|
||||
`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. -/
|
||||
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
|
||||
|
||||
/--
|
||||
`(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 `;`. -/
|
||||
can be either separated by newlines or `;`.
|
||||
-/
|
||||
syntax (name := paren) "(" tacticSeq ")" : tactic
|
||||
/-- `with_reducible tacs` excutes `tacs` using the reducible transparency setting.
|
||||
In this setting only definitions tagged as `[reducible]` are unfolded. -/
|
||||
|
||||
/--
|
||||
`with_reducible tacs` excutes `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` excutes `tacs` using the `.instances` transparency setting.
|
||||
In this setting only definitions tagged as `[reducible]` or type class instances are unfolded. -/
|
||||
|
||||
/--
|
||||
`with_reducible_and_instances tacs` excutes `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` excutes `tacs` using the `.all` transparency setting.
|
||||
In this setting all definitions that are not opaque are unfolded. -/
|
||||
|
||||
/--
|
||||
`with_unfolding_all tacs` excutes `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((colGe "|" tacticSeq)+) : tactic
|
||||
/-- `rotate_left n` rotates goals to the left by `n`. That is, `rotate_left 1`
|
||||
|
||||
/--
|
||||
`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`. -/
|
||||
If `n` is omitted, it defaults to `1`.
|
||||
-/
|
||||
syntax (name := rotateLeft) "rotate_left" (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`. -/
|
||||
|
||||
/--
|
||||
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" (num)? : tactic
|
||||
|
||||
/-- `try tac` runs `tac` and succeeds even if `tac` failed. -/
|
||||
macro "try " t:tacticSeq : tactic => `(first | $t | skip)
|
||||
/-- `tac <;> tac'` runs `tac` on the main goal and `tac'` on each produced goal, concatenating all goals produced by `tac'`. -/
|
||||
|
||||
/--
|
||||
`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:0 : tactic => `(tactic|
|
||||
focus
|
||||
$x:tactic
|
||||
|
|
@ -179,11 +266,13 @@ for new reflexive relations.
|
|||
macro "rfl" : tactic => `(eq_refl)
|
||||
|
||||
/--
|
||||
`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). -/
|
||||
`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 => `(set_option smartUnfolding false in with_unfolding_all rfl)
|
||||
|
||||
/-- `ac_rfl` proves equalities up to application of an associative and commutative operator.
|
||||
/--
|
||||
`ac_rfl` proves equalities up to application of an associative and commutative operator.
|
||||
```
|
||||
instance : IsAssociative (α := Nat) (.+.) := ⟨Nat.add_assoc⟩
|
||||
instance : IsCommutative (α := Nat) (.+.) := ⟨Nat.add_comm⟩
|
||||
|
|
@ -193,19 +282,46 @@ 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 => `(exact @sorryAx _ false)
|
||||
|
||||
/-- `admit` is a shorthand for `exact sorry`. -/
|
||||
macro "admit" : tactic => `(exact @sorryAx _ false)
|
||||
/-- The `sorry` tactic is a shorthand for `exact sorry`. -/
|
||||
macro "sorry" : tactic => `(exact @sorryAx _ false)
|
||||
/-- `infer_instance` is an abbreviation for `exact inferInstance` -/
|
||||
|
||||
/--
|
||||
`infer_instance` is an abbreviation for `exact inferInstance`.
|
||||
It synthesizes a value of any target type by typeclass inference.
|
||||
-/
|
||||
macro "infer_instance" : tactic => `(exact inferInstance)
|
||||
|
||||
/-- Optional configuration option for tactics -/
|
||||
syntax config := atomic("(" &"config") " := " term ")"
|
||||
|
||||
/-- The `*` location refers to all hypotheses and the goal. -/
|
||||
syntax locationWildcard := "*"
|
||||
syntax locationHyp := (colGt term:max)+ ("⊢" <|> "|-")?
|
||||
syntax location := withPosition(" at " (locationWildcard <|> locationHyp))
|
||||
|
||||
/--
|
||||
A hypothesis location specification consists of 1 or more hypothesis references
|
||||
and optionally `⊢` denoting the goal.
|
||||
-/
|
||||
syntax locationHyp := (colGt term:max)+ ("⊢" <|> "|-")?
|
||||
|
||||
/--
|
||||
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'`,
|
||||
|
|
@ -222,15 +338,24 @@ syntax (name := change) "change " term (location)? : tactic
|
|||
-/
|
||||
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 := ("← " <|> "<- ")? term
|
||||
/-- A `rwRuleSeq` is a list of `rwRule` in brackets. -/
|
||||
syntax rwRuleSeq := "[" 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`.
|
||||
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.
|
||||
- `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.
|
||||
-/
|
||||
syntax (name := rewriteSeq) "rewrite " (config)? rwRuleSeq (location)? : tactic
|
||||
|
||||
|
|
@ -245,12 +370,16 @@ macro (name := rwSeq) "rw " c:(config)? s:rwRuleSeq l:(location)? : tactic =>
|
|||
| _ => Macro.throwUnsupported
|
||||
|
||||
/--
|
||||
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 `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 " (colGt (ident <|> hole))+)? : tactic
|
||||
|
|
@ -261,62 +390,100 @@ constructor equalities like `(a::b::c) = (d::e::f)`. -/
|
|||
-- TODO: add with
|
||||
syntax (name := injections) "injections" : 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("(" (&"discharger" <|> &"disch")) " := " 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)? ("← " <|> "<- ")? 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.
|
||||
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 [h₁, h₂, ..., hₙ]` simplifies the main goal target using the lemmas tagged
|
||||
with the attribute `[simp]` and the given `hᵢ`'s, where the `hᵢ`'s are expressions.
|
||||
If an `hᵢ` is a defined constant `f`, then the equational lemmas associated with
|
||||
`f` are used. This provides a convenient way to unfold `f`.
|
||||
- `simp [*]` simplifies the main goal target using the lemmas tagged with the
|
||||
attribute `[simp]` and all hypotheses.
|
||||
- `simp only [h₁, h₂, ..., hₙ]` is like `simp [h₁, h₂, ..., hₙ]` but does not use `[simp]` lemmas.
|
||||
- `simp [-id₁, ..., -idₙ]` simplifies the main goal target using the lemmas tagged
|
||||
with the attribute `[simp]`, but removes the ones named `idᵢ`.
|
||||
- `simp at h₁ h₂ ... hₙ` simplifies the hypotheses `h₁ : T₁` ... `hₙ : Tₙ`. If
|
||||
the target or another hypothesis depends on `hᵢ`, a new simplified hypothesis
|
||||
`hᵢ` is introduced, but the old one remains in the local context.
|
||||
- `simp at *` simplifies all the hypotheses and the target.
|
||||
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the other hypotheses.
|
||||
- `simp [*] at *` simplifies target and all (propositional) hypotheses using the
|
||||
other hypotheses.
|
||||
-/
|
||||
syntax (name := simp) "simp " (config)? (discharger)? (&"only ")? ("[" (simpStar <|> simpErase <|> simpLemma),* "]")? (location)? : tactic
|
||||
syntax (name := simp) "simp " (config)? (discharger)? (&"only ")?
|
||||
("[" (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 simplication is applicable.
|
||||
`simp_all` is a stronger version of `simp [*] at *` where the hypotheses and target
|
||||
are simplified multiple times until no simplication is applicable.
|
||||
Only non-dependent propositional hypotheses are considered.
|
||||
-/
|
||||
syntax (name := simpAll) "simp_all " (config)? (discharger)? (&"only ")? ("[" (simpErase <|> simpLemma),* "]")? : tactic
|
||||
syntax (name := simpAll) "simp_all " (config)? (discharger)? (&"only ")?
|
||||
("[" (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.
|
||||
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 ")? ("[" (simpErase <|> simpLemma),* "]")? (location)? : tactic
|
||||
syntax (name := dsimp) "dsimp " (config)? (discharger)? (&"only ")?
|
||||
("[" (simpErase <|> simpLemma),* "]")? (location)? : tactic
|
||||
|
||||
/--
|
||||
`delta id` delta-expands the definition `id`.
|
||||
This is a low-level tactic, it will expose how recursive definitions have been compiled by Lean. -/
|
||||
`delta id` delta-expands the definition `id`.
|
||||
This is a low-level tactic, it will expose how recursive definitions have been
|
||||
compiled by Lean.
|
||||
-/
|
||||
syntax (name := delta) "delta " ident (location)? : tactic
|
||||
|
||||
/--
|
||||
`unfold id,+` unfolds definition `id`. For non-recursive definitions, this tactic is identical to `delta`.
|
||||
For recursive definitions, it hides the encoding tricks used by the Lean frontend to convince the
|
||||
kernel that the definition terminates. -/
|
||||
`unfold id,+` unfolds definition `id`. 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 " ident,+ (location)? : tactic
|
||||
|
||||
/-- Auxiliary macro for lifting have/suffices/let/...
|
||||
It makes sure the "continuation" `?_` is the main goal after refining. -/
|
||||
/--
|
||||
Auxiliary macro for lifting have/suffices/let/...
|
||||
It makes sure the "continuation" `?_` is the main goal after refining.
|
||||
-/
|
||||
macro "refine_lift " e:term : tactic => `(focus (refine no_implicit_lambda% $e; rotate_right))
|
||||
|
||||
/--
|
||||
`have h : t := e` adds the hypothesis `h : t` to the current goal if `e` a term of type `t`. If `t` is omitted, it will be inferred.
|
||||
If `h` is omitted, the name `this` is used.
|
||||
The variant `have pattern := e` is equivalent to `match e with | pattern => _`, and it is convenient for types that have only applicable constructor.
|
||||
Example: given `h : p ∧ q ∧ r`, `have ⟨h₁, h₂, h₃⟩ := h` produces the hypotheses `h₁ : p`, `h₂ : q`, and `h₃ : r`.
|
||||
`have h : t := e` adds the hypothesis `h : t` to the current goal if `e` a term
|
||||
of type `t`.
|
||||
* If `t` is omitted, it will be inferred.
|
||||
* If `h` is omitted, the name `this` is used.
|
||||
* The variant `have pattern := e` is equivalent to `match e with | pattern => _`,
|
||||
and 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 => `(refine_lift have $d:haveDecl; ?_)
|
||||
|
||||
/--
|
||||
Given a main goal `ctx |- t`, `suffices h : t' from e` replaces the main goal with `ctx |- t'`,
|
||||
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`.
|
||||
|
|
@ -326,8 +493,10 @@ macro "suffices " d:sufficesDecl : tactic => `(refine_lift suffices $d; ?_)
|
|||
/--
|
||||
`let h : t := e` adds the hypothesis `h : t := e` to the current goal if `e` a term of type `t`.
|
||||
If `t` is omitted, it will be inferred.
|
||||
The variant `let pattern := e` is equivalent to `match e with | pattern => _`, and it is convenient for types that have only applicable constructor.
|
||||
Example: given `h : p ∧ q ∧ r`, `let ⟨h₁, h₂, h₃⟩ := h` produces the hypotheses `h₁ : p`, `h₂ : q`, and `h₃ : r`.
|
||||
The variant `let pattern := e` is equivalent to `match e with | pattern => _`,
|
||||
and it is convenient for types that have only applicable constructor.
|
||||
Example: given `h : p ∧ q ∧ r`, `let ⟨h₁, h₂, h₃⟩ := h` produces the hypotheses
|
||||
`h₁ : p`, `h₂ : q`, and `h₃ : r`.
|
||||
-/
|
||||
macro "let " d:letDecl : tactic => `(refine_lift let $d:letDecl; ?_)
|
||||
/--
|
||||
|
|
@ -350,42 +519,88 @@ macro (priority := high) "have'" x:ident " := " p:term : tactic => `(have' $x :
|
|||
/-- Similar to `let`, but using `refine'` -/
|
||||
macro "let' " d:letDecl : tactic => `(refine_lift' let $d:letDecl; ?_)
|
||||
|
||||
syntax inductionAltLHS := "| " (("@"? ident) <|> hole) (ident <|> hole)*
|
||||
syntax inductionAlt := ppDedent(ppLine) inductionAltLHS+ " => " (hole <|> syntheticHole <|> tacticSeq)
|
||||
syntax inductionAlts := "with " (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 theorem 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.
|
||||
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 contstructor.
|
||||
-/
|
||||
syntax (name := induction) "induction " term,+ (" using " ident)? ("generalizing " (colGt term:max)+)? (inductionAlts)? : tactic
|
||||
|
||||
syntax generalizeArg := atomic(ident " : ")? term:51 " = " ident
|
||||
syntax inductionAltLHS := "| " (("@"? ident) <|> hole) (ident <|> hole)*
|
||||
/--
|
||||
`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. -/
|
||||
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 " (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 theorem 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 " ident)?
|
||||
("generalizing " (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.
|
||||
-/
|
||||
syntax (name := generalize) "generalize " generalizeArg,+ : 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 are 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.
|
||||
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 " ident)? (inductionAlts)? : tactic
|
||||
|
||||
|
|
@ -401,7 +616,8 @@ 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.
|
||||
`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)
|
||||
|
|
@ -426,15 +642,21 @@ example : False ∨ True := by
|
|||
-/
|
||||
syntax (name := dbgTrace) "dbg_trace " str : tactic
|
||||
|
||||
/-- `stop` is a helper tactic for "discarding" the rest of a proof. It is useful when working on the middle of a complex proofs,
|
||||
and less messy than commenting the remainder of the proof. -/
|
||||
/--
|
||||
`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 => `(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 either 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.
|
||||
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
|
||||
|
||||
|
|
@ -445,7 +667,8 @@ 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.
|
||||
/--
|
||||
`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.
|
||||
|
|
@ -458,10 +681,11 @@ example : ∀ x : Nat, x = x := by unhygienic
|
|||
-/
|
||||
macro "unhygienic " t:tacticSeq : tactic => `(set_option tactic.hygienic false in $t)
|
||||
|
||||
/-- `fail msg` is a tactic that always fail and produces an error using the given message. -/
|
||||
/-- `fail msg` is a tactic that always fails, and produces an error using the given message. -/
|
||||
syntax (name := fail) "fail " (str)? : tactic
|
||||
|
||||
/-- `checkpoint tac` acts the same as `tac`, but it caches the input and output of `tac`,
|
||||
/--
|
||||
`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`.
|
||||
|
|
@ -469,31 +693,41 @@ when working on a long tactic proof, by wrapping expensive tactics with `checkpo
|
|||
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.) -/
|
||||
this combinator explicitly.)
|
||||
-/
|
||||
syntax (name := checkpoint) "checkpoint " tacticSeq : tactic
|
||||
|
||||
/-- `save` is defined to be the same as `skip`, but the elaborator has
|
||||
/--
|
||||
`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.) -/
|
||||
this combinator explicitly.)
|
||||
-/
|
||||
macro (name := save) "save" : tactic => `(skip)
|
||||
|
||||
/-- The tactic `sleep ms` sleeps for `ms` milliseconds and does nothing. It is used for debugging purposes only. -/
|
||||
/--
|
||||
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. -/
|
||||
/--
|
||||
`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`,
|
||||
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 " (num)? : tactic
|
||||
|
|
@ -502,40 +736,46 @@ end Tactic
|
|||
|
||||
namespace Attr
|
||||
/--
|
||||
Theorems tagged with the `simp` attribute are by the simplifier (i.e., the `simp` tactic, and its variants) to simplify
|
||||
expressions occurring in your goals.
|
||||
Theorems tagged with the `simp` attribute are 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
|
||||
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.
|
||||
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
|
||||
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:
|
||||
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
|
||||
@[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)? (prio)? : attr
|
||||
|
|
@ -545,26 +785,33 @@ 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`. -/
|
||||
`‹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`.
|
||||
-/
|
||||
macro "‹" type:term "›" : term => `((by assumption : $type))
|
||||
|
||||
/-- `get_elem_tactic_trivial` is an extensible tactic automatically called
|
||||
/--
|
||||
`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`
|
||||
(for doing linear arithmetic in the index). -/
|
||||
(for doing linear arithmetic in the index).
|
||||
-/
|
||||
syntax "get_elem_tactic_trivial" : tactic
|
||||
|
||||
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| trivial)
|
||||
macro_rules | `(tactic| get_elem_tactic_trivial) => `(tactic| simp (config := { arith := true }); done)
|
||||
|
||||
/-- `get_elem_tactic` is the tactic automatically called by the notation `arr[i]`
|
||||
/--
|
||||
`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. -/
|
||||
users are encouraged to extend `get_elem_tactic_trivial` instead of this tactic.
|
||||
-/
|
||||
macro "get_elem_tactic" : tactic =>
|
||||
`(first
|
||||
| get_elem_tactic_trivial
|
||||
|
|
@ -575,10 +822,12 @@ macro "get_elem_tactic" : tactic =>
|
|||
- Use `a[i]'h` notation instead, where `h` is a proof that index is valid"
|
||||
)
|
||||
|
||||
@[inheritDoc getElem]
|
||||
macro:max x:term noWs "[" i:term "]" : term => `(getElem $x $i (by get_elem_tactic))
|
||||
|
||||
/-- Helper declaration for the unexpander -/
|
||||
@[inline] def getElem' [GetElem cont idx elem dom] (xs : cont) (i : idx) (h : dom xs i) : elem :=
|
||||
getElem xs i h
|
||||
|
||||
@[inheritDoc getElem]
|
||||
macro x:term noWs "[" i:term "]'" h:term:max : term => `(getElem' $x $i $h)
|
||||
|
|
|
|||
|
|
@ -4,5 +4,5 @@
|
|||
{"start": {"line": 1, "character": 2}, "end": {"line": 1, "character": 12}},
|
||||
"contents":
|
||||
{"value":
|
||||
"`rewrite [e]` applies identity `e` as a rewrite rule to the target of the main goal.\nIf `e` is preceded by left arrow (`←` or `<-`), the rewrite is applied in the reverse direction.\nIf `e` is a defined constant, then the equational theorems associated with `e` are used. This provides a convenient way to unfold `e`.\n- `rewrite [e₁, ..., eₙ]` applies the given rules sequentially.\n- `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.\n",
|
||||
"`rewrite [e]` applies identity `e` as a rewrite rule to the target of the main goal.\nIf `e` is preceded by left arrow (`←` or `<-`), the rewrite is applied in the reverse direction.\nIf `e` is a defined constant, then the equational theorems associated with `e` are used.\nThis provides a convenient way to unfold `e`.\n- `rewrite [e₁, ..., eₙ]` applies the given rules sequentially.\n- `rewrite [e] at l` rewrites `e` at location(s) `l`, where `l` is either `*` or a\n list of hypotheses in the local context. In the latter case, a turnstile `⊢` or `|-`\n can also be used, to signify the target of the goal.\n",
|
||||
"kind": "markdown"}}
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue