/- 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 /-- `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 /-- 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("|") (colGt term:max)* : tactic /-- `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. -/ 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 " (colGt term:max)+ : tactic /-- `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. -/ syntax (name := subst) "subst " (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 /-- `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 /-- 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 /-- A case tag argument has the form `tag x₁ ... xₙ`; it refers to tag `tag` and renames the last `n` hypotheses to `x₁ ... xₙ`. -/ syntax caseArg := binderIdent binderIdent* /-- * `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) "(" tacticSeq ")" : tactic /-- `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. -/ 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. -/ 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` 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" (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" (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'`. -/ macro:1 x:tactic tk:" <;> " y:tactic:0 : tactic => `(tactic| focus $x:tactic -- annotate token with state after executing `x` with_annotate_state $tk skip all_goals $y:tactic) /-- `eq_refl` is equivalent to `exact rfl`, but has a few optimizations. -/ syntax (name := refl) "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. -/ 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). -/ 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. ``` instance : IsAssociative (α := Nat) (.+.) := ⟨Nat.add_assoc⟩ instance : IsCommutative (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 => `(exact @sorryAx _ false) /-- `admit` is a shorthand for `exact sorry`. -/ macro "admit" : 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 => `(exact inferInstance) /-- Optional configuration option for tactics -/ syntax config := atomic("(" &"config") " := " 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 := (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'`, 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 := ("← " <|> "<- ")? 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`. - `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. -/ 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 /-- 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 /-- `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" (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("(" (&"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: - `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 ")? ("[" (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. Only non-dependent propositional hypotheses are considered. -/ 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. -/ syntax (name := dsimp) "dsimp " (config)? (discharger)? (&"only ")? ("[" (simpErase <|> simpLemma),* "]")? (location)? : tactic /-- `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 " (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 " (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 => `(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 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'`, `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 => `(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`. -/ macro "let " d:letDecl : 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 => `(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 => `(focus (refine' no_implicit_lambda% $e; rotate_right)) /-- Similar to `have`, but using `refine'` -/ macro "have' " d:haveDecl : tactic => `(refine_lift' have $d:haveDecl; ?_) /-- Similar to `have`, but using `refine'` -/ macro (priority := high) "have'" x:ident " := " p:term : tactic => `(have' $x : _ := $p) /-- Similar to `let`, but using `refine'` -/ macro "let' " d:letDecl : 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 contstructor. -/ 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 " (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 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 /-- `rename_i x_1 ... x_n` renames the last `n` inaccessible names using the given names. -/ syntax (name := renameI) "rename_i " (colGt binderIdent)+ : tactic /-- `repeat tac` applies `tac` to main goal. If the application succeeds, the tactic is applied recursively to the generated subgoals until it eventually fails. -/ 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 in cases. For a `match` expression with `n` cases, the `split` tactic generates at most `n` subgoals -/ syntax (name := split) "split " (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 => `(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 => `(set_option tactic.hygienic false in $t) /-- `fail msg` is a tactic that always fails, and produces an error using the given message. -/ syntax (name := fail) "fail " (str)? : tactic /-- `checkpoint tac` acts the same as `tac`, but it caches the input and output of `tac`, and if the file is re-elaborated and the input matches, the tactic is not re-run and its effects are reapplied to the state. This is useful for improving responsiveness when working on a long tactic proof, by wrapping expensive tactics with `checkpoint`. See the `save` tactic, which may be more convenient to use. (TODO: do this automatically and transparently so that users don't have to use this combinator explicitly.) -/ syntax (name := checkpoint) "checkpoint " tacticSeq : tactic /-- `save` is defined to be the same as `skip`, but the elaborator has special handling for occurrences of `save` in tactic scripts and will transform `by tac1; save; tac2` to `by (checkpoint tac1); tac2`, meaning that the effect of `tac1` will be cached and replayed. This is useful for improving responsiveness when working on a long tactic proof, by using `save` after expensive tactics. (TODO: do this automatically and transparently so that users don't have to use this combinator explicitly.) -/ macro (name := save) "save" : tactic => `(skip) /-- The tactic `sleep ms` sleeps for `ms` milliseconds and does nothing. It is used for debugging purposes only. -/ syntax (name := sleep) "sleep" num : tactic /-- `exists e₁, e₂, ...` is shorthand for `refine ⟨e₁, e₂, ...⟩; try trivial`. It is useful for existential goals. -/ macro "exists " es:term,+ : tactic => `(tactic| (refine ⟨$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 " (num)? : tactic 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. 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)? (prio)? : 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`. -/ macro "‹" type:term "›" : term => `((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` (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]` 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 => `(first | 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" ) @[inheritDoc getElem] macro:max x:term noWs "[" i:term "]" : term => `(getElem $x $i (by get_elem_tactic)) @[inheritDoc getElem] macro x:term noWs "[" i:term "]'" h:term:max : term => `(getElem $x $i $h)