feat: add peel tactic (#7685)

As requested on [Zulip](https://leanprover.zulipchat.com/#narrow/stream/287929-mathlib4/topic/Logical.20congruence.20or.20weakening.20tactic).

Given `p q : ℕ → Prop`, `h : ∀ x, p x`, and a goal `⊢ : ∀ x, q x`, the tactic `peel h with h' x` will introduce `x : ℕ`, `h' : p x` into the context and the new goal will be `⊢ q x`. This works with `∃`, as well as `∀ᶠ` and `∃ᶠ`, and it can even be applied to a sequence of quantifiers.

For a more complex example, given a hypothesis and a goal:
```
h : ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) < ε
⊢ ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) ≤ ε
```
(which differ only in `<`/`≤`), applying `peel h with h_peel ε hε N n hn` will yield a tactic state:
```
h : ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) < ε
ε : ℝ
hε : 0 < ε
N n : ℕ
hn : N ≤ n
h_peel : 1 / (n + 1 : ℝ) < ε
⊢ 1 / (n + 1 : ℝ) ≤ ε
```
and the goal can be closed with `exact h_peel.le`.

Co-authored-by: Kyle Miller <kmill31415@gmail.com>



Co-authored-by: Patrick Massot <patrickmassot@free.fr>
Co-authored-by: Kyle Miller <kmill31415@gmail.com>

Mathlib.lean

@@ -3215,6 +3215,7 @@ import Mathlib.Tactic.NormNum.Result
 import Mathlib.Tactic.NthRewrite
 import Mathlib.Tactic.Observe
 import Mathlib.Tactic.PPWithUniv
+import Mathlib.Tactic.Peel
 import Mathlib.Tactic.PermuteGoals
 import Mathlib.Tactic.Polyrith
 import Mathlib.Tactic.Positivity

Mathlib/Tactic.lean

@@ -115,6 +115,7 @@ import Mathlib.Tactic.NormNum.Result
 import Mathlib.Tactic.NthRewrite
 import Mathlib.Tactic.Observe
 import Mathlib.Tactic.PPWithUniv
+import Mathlib.Tactic.Peel
 import Mathlib.Tactic.PermuteGoals
 import Mathlib.Tactic.Polyrith
 import Mathlib.Tactic.Positivity

Mathlib/Tactic/Peel.lean

@@ -0,0 +1,244 @@
+/-
+Copyright (c) 2023 Jireh Loreaux. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Jireh Loreaux
+-/
+import Mathlib.Order.Filter.Basic
+import Mathlib.Tactic.Basic
+
+/-!
+# The `peel` tactic
+
+`peel h with h' idents*` tries to apply `forall_imp` (or `Exists.imp`, or `Filter.Eventually.mp`,
+`Filter.Frequently.mp` and `Filter.eventually_of_forall`) with the argument `h` and uses `idents*`
+to introduce variables with the supplied names, giving the "peeled" argument the name `h'`.
+
+One can provide a numeric argument as in `peel 4 h` which will peel 4 quantifiers off
+the expressions automatically name any variables not specifically named in the `with` clause.
+
+In addition, the user may supply a term `e` via `... using e` in order to close the goal
+immediately. In particular, `peel h using e` is equivalent to `peel h; exact e`. The `using` syntax
+may be paired with any of the other features of `peel`.
+-/
+
+namespace Mathlib.Tactic.Peel
+open Lean Expr Meta Elab Tactic
+
+/--
+Peels matching quantifiers off of a given term and the goal and introduces the relevant variables.
+
+- `peel e` peels all quantifiers (at reducible transparency),
+  using `this` for the name of the peeled hypothesis.
+- `peel e with h` is `peel e` but names the peeled hypothesis `h`.
+  If `h` is `_` then uses `this` for the name of the peeled hypothesis.
+- `peel n e` peels `n` quantifiers (at default transparency).
+- `peel n e with h x y z ...` peels `n` quantifiers, names the peeled hypothesis `h`,
+  and uses `x`, `y`, `z`, and so on to name the introduced variables; these names may be `_`.
+  If `h` is `_` then uses `this` for the name of the peeled hypothesis.
+  The length of the list of variables does not need to equal `n`.
+- `peel e with h x₁ ... xₙ` is `peel n e with h x₁ ... xₙ`.
+
+There are also variants that apply to an iff in the goal:
+- `peel n` peels `n` quantifiers in an iff.
+- `peel with x₁ ... xₙ` peels `n` quantifiers in an iff and names them.
+
+Given `p q : ℕ → Prop`, `h : ∀ x, p x`, and a goal `⊢ : ∀ x, q x`, the tactic `peel h with h' x`
+will introduce `x : ℕ`, `h' : p x` into the context and the new goal will be `⊢ q x`. This works
+with `∃`, as well as `∀ᶠ` and `∃ᶠ`, and it can even be applied to a sequence of quantifiers. Note
+that this is a logically weaker setup, so using this tactic is not always feasible.
+
+For a more complex example, given a hypothesis and a goal:
+```
+h : ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) < ε
+⊢ ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) ≤ ε
+```
+(which differ only in `<`/`≤`), applying `peel h with h_peel ε hε N n hn` will yield a tactic state:
+```
+h : ∀ ε > (0 : ℝ), ∃ N : ℕ, ∀ n ≥ N, 1 / (n + 1 : ℝ) < ε
+ε : ℝ
+hε : 0 < ε
+N n : ℕ
+hn : N ≤ n
+h_peel : 1 / (n + 1 : ℝ) < ε
+⊢ 1 / (n + 1 : ℝ) ≤ ε
+```
+and the goal can be closed with `exact h_peel.le`.
+Note that in this example, `h` and the goal are logically equivalent statements, but `peel`
+*cannot* be immediately applied to show that the goal implies `h`.
+
+In addition, `peel` supports goals of the form `(∀ x, p x) ↔ ∀ x, q x`, or likewise for any
+other quantifier. In this case, there is no hypothesis or term to supply, but otherwise the syntax
+is the same. So for such goals, the syntax is `peel 1` or `peel with x`, and after which the
+resulting goal is `p x ↔ q x`. The `congr!` tactic can also be applied to goals of this form using
+`congr! 1 with x`. While `congr!` applies congruence lemmas in general, `peel` can be relied upon
+to only apply to outermost quantifiers.
+
+Finally, the user may supply a term `e` via `... using e` in order to close the goal
+immediately. In particular, `peel h using e` is equivalent to `peel h; exact e`. The `using` syntax
+may be paired with any of the other features of `peel`.
+
+This tactic works by repeatedly applying lemmas such as `forall_imp`, `Exists.imp`,
+`Filter.Eventually.mp`, `Filter.Frequently.mp`, and `Filter.eventually_of_forall`.
+-/
+syntax (name := peel)
+  "peel" (num)? (ppSpace colGt term)?
+  (" with" (ppSpace colGt (ident <|> hole))+)? (usingArg)? : tactic
+
+private lemma and_imp_left_of_imp_imp {p q r : Prop} (h : r → p → q) : r ∧ p → r ∧ q := by tauto
+
+private theorem eventually_imp {α : Type*} {p q : α → Prop} {f : Filter α}
+    (hq : ∀ (x : α), p x → q x) (hp : ∀ᶠ (x : α) in f, p x) : ∀ᶠ (x : α) in f, q x :=
+  Filter.Eventually.mp hp (Filter.eventually_of_forall hq)
+
+private theorem frequently_imp {α : Type*} {p q : α → Prop} {f : Filter α}
+    (hq : ∀ (x : α), p x → q x) (hp : ∃ᶠ (x : α) in f, p x) : ∃ᶠ (x : α) in f, q x :=
+  Filter.Frequently.mp hp (Filter.eventually_of_forall hq)
+
+private theorem eventually_congr {α : Type*} {p q : α → Prop} {f : Filter α}
+    (hq : ∀ (x : α), p x ↔ q x) : (∀ᶠ (x : α) in f, p x) ↔ ∀ᶠ (x : α) in f, q x := by
+  congr! 2; exact hq _
+
+private theorem frequently_congr {α : Type*} {p q : α → Prop} {f : Filter α}
+    (hq : ∀ (x : α), p x ↔ q x) : (∃ᶠ (x : α) in f, p x) ↔ ∃ᶠ (x : α) in f, q x := by
+  congr! 2; exact hq _
+
+/-- The list of constants that are regarded as being quantifiers. -/
+def quantifiers : List Name :=
+  [``Exists, ``And, ``Filter.Eventually, ``Filter.Frequently]
+
+/-- If `unfold` is false then do `whnfR`, otherwise unfold everything that's not a quantifier,
+according to the `quantifiers` list. -/
+def whnfQuantifier (p : Expr) (unfold : Bool) : MetaM Expr := do
+  if unfold then
+    whnfHeadPred p fun e =>
+      if let .const n .. := e.getAppFn then
+        return !(n ∈ quantifiers)
+      else
+        return false
+  else
+    whnfR p
+
+/-- Throws an error saying `ty` and `target` could not be matched up. -/
+def throwPeelError {α : Type} (ty target : Expr) : MetaM α :=
+  throwError "Tactic 'peel' could not match quantifiers in{indentD ty}\nand{indentD target}"
+
+/-- If `f` is a lambda then use its binding name to generate a new hygienic name,
+and otherwise choose a new hygienic name. -/
+def mkFreshBinderName (f : Expr) : MetaM Name :=
+  mkFreshUserName (if let .lam n .. := f then n else `a)
+
+/-- Applies a "peel theorem" with two main arguments, where the first is the new goal
+and the second can be filled in using `e`. Then it intros two variables with the
+provided names.
+
+If, for example, `goal : ∃ y : α, q y` and `thm := Exists.imp`, the metavariable returned has
+type `q x` where `x : α` has been introduced into the context. -/
+def applyPeelThm (thm : Name) (goal : MVarId)
+    (e : Expr) (ty target : Expr) (n : Name) (n' : Name) :
+    MetaM (FVarId × List MVarId) := do
+  let new_goal :: ge :: _ ← goal.applyConst thm <|> throwPeelError ty target
+    | throwError "peel: internal error"
+  ge.assignIfDefeq e <|> throwPeelError ty target
+  let (fvars, new_goal) ← new_goal.introN 2 [n, n']
+  return (fvars[1]!, [new_goal])
+
+/-- This is the core to the `peel` tactic.
+
+It tries to match `e` and `goal` as quantified statements (using `∀` and the quantifiers in
+the `quantifiers` list), then applies "peel theorems" using `applyPeelThm`.
+
+We treat `∧` as a quantifier for sake of dealing with quantified statements
+like `∃ δ > (0 : ℝ), q δ`, which is notation for `∃ δ, δ > (0 : ℝ) ∧ q δ`. -/
+def peelCore (goal : MVarId) (e : Expr) (n? : Option Name) (n' : Name) (unfold : Bool) :
+    MetaM (FVarId × List MVarId) := goal.withContext do
+  let ty ← whnfQuantifier (← inferType e) unfold
+  let target ← whnfQuantifier (← goal.getType) unfold
+  if ty.isForall && target.isForall then
+    applyPeelThm ``forall_imp goal e ty target (← n?.getDM (mkFreshUserName target.bindingName!)) n'
+  else if ty.getAppFn.isConst
+            && ty.getAppNumArgs == target.getAppNumArgs
+            && ty.getAppFn == target.getAppFn then
+    match target.getAppFnArgs with
+    | (``Exists, #[_, p]) =>
+      applyPeelThm ``Exists.imp goal e ty target (← n?.getDM (mkFreshBinderName p)) n'
+    | (``And, #[_, _]) =>
+      applyPeelThm ``and_imp_left_of_imp_imp goal e ty target (← n?.getDM (mkFreshUserName `p)) n'
+    | (``Filter.Eventually, #[_, p, _]) =>
+      applyPeelThm ``eventually_imp goal e ty target (← n?.getDM (mkFreshBinderName p)) n'
+    | (``Filter.Frequently, #[_, p, _]) =>
+      applyPeelThm ``frequently_imp goal e ty target (← n?.getDM (mkFreshBinderName p)) n'
+    | _ => throwPeelError ty target
+  else
+    throwPeelError ty target
+
+/-- Given a list `l` of names, this peels `num` quantifiers off of the expression `e` and
+the main goal and introduces variables with the provided names until the list of names is exhausted.
+Note: the name `n?` (with default `this`) is used for the name of the expression `e` with
+quantifiers peeled. -/
+def peelArgs (e : Expr) (num : Nat) (l : List Name) (n? : Option Name) (unfold : Bool := true) :
+    TacticM Unit := do
+  match num with
+    | 0 => return
+    | num + 1 =>
+      let fvarId ← liftMetaTacticAux (peelCore · e l.head? (n?.getD `this) unfold)
+      peelArgs (.fvar fvarId) num l.tail n?
+      unless num == 0 do
+        if let some mvarId ← observing? do (← getMainGoal).clear fvarId then
+          replaceMainGoal [mvarId]
+
+/-- Similar to `peelArgs` but peels arbitrarily many quantifiers. Returns whether or not
+any quantifiers were peeled. -/
+partial def peelUnbounded (e : Expr) (n? : Option Name) (unfold : Bool := false) :
+    TacticM Bool := do
+  let fvarId? ← observing? <| liftMetaTacticAux (peelCore · e none (n?.getD `this) unfold)
+  if let some fvarId := fvarId? then
+    let peeled ← peelUnbounded (.fvar fvarId) n?
+    if peeled then
+      if let some mvarId ← observing? do (← getMainGoal).clear fvarId then
+        replaceMainGoal [mvarId]
+    return true
+  else
+    return false
+
+/-- Peel off a single quantifier from an `↔`. -/
+def peelIffAux : TacticM Unit := do
+  evalTactic (← `(tactic| focus
+    first | apply forall_congr'
+          | apply exists_congr
+          | apply eventually_congr
+          | apply frequently_congr
+          | fail "failed to apply a quantifier congruence lemma."))
+
+/-- Peel off quantifiers from an `↔` and assign the names given in `l` to the introduced
+variables. -/
+def peelArgsIff (l : List Name) : TacticM Unit := withMainContext do
+  match l with
+    | [] => pure ()
+    | h :: hs =>
+      peelIffAux
+      let goal ← getMainGoal
+      let (_, new_goal) ← goal.intro h
+      replaceMainGoal [new_goal]
+      peelArgsIff hs
+
+elab_rules : tactic
+  | `(tactic| peel $[$num?:num]? $e:term $[with $n? $l?*]?) => withMainContext do
+    /- we use `elabTermForApply` instead of `elabTerm` so that terms passed to `peel` can contain
+    quantifiers with implicit bound variables without causing errors or requiring `@`.  -/
+    let e ← elabTermForApply e false
+    let n? := n?.bind fun n => if n.raw.isIdent then pure n.raw.getId else none
+    let l := (l?.getD #[]).map getNameOfIdent' |>.toList
+    -- If num is not present and if there are any provided variable names,
+    -- use the number of variable names.
+    let num? := num?.map (·.getNat) <|> if l.isEmpty then none else l.length
+    if let some num := num? then
+      peelArgs e num l n?
+    else
+      unless ← peelUnbounded e n? do
+        throwPeelError (← inferType e) (← getMainTarget)
+  | `(tactic| peel $n:num) => peelArgsIff <| .replicate n.getNat `_
+  | `(tactic| peel with $args*) => peelArgsIff (args.map getNameOfIdent').toList
+
+macro_rules
+  | `(tactic| peel $[$n:num]? $[$e:term]? $[with $h*]? using $u:term) =>
+    `(tactic| peel $[$n:num]? $[$e:term]? $[with $h*]?; exact $u)