feat: improve case-split heuristic in grind

This PR improves the case-split heuristics in `grind`.
In this PR, we do not increment the number of case splits in the first
case. The idea is to leverage non-chronological backtracking: if the first case
is solved using a proof that doesn't depend on the case hypothesis, we backtrack and close
the original goal directly. In this scenario, the case-split was "free", it didn't contribute
to the proof. By not counting it, we allow deeper exploration when case-splits turn out to be
irrelevant.
The new heuristic addresses the second example in #11545

src/Lean/Meta/Tactic/Grind/Split.lean

@@ -185,13 +185,10 @@ where
     match cs with
     | [] =>
       modify fun s => { s with split.candidates := cs'.reverse }
-      if let .some _ numCases isRec _ := c? then
-        let numSplits := (← get).split.num
-        -- We only increase the number of splits if there is more than one case or it is recursive.
-        let numSplits := if numCases > 1 || isRec then numSplits + 1 else numSplits
+      if let .some .. := c? then
         -- Remark: we reset `numEmatch` after each case split.
         -- We should consider other strategies in the future.
-        modify fun s => { s with split.num := numSplits, ematch.num := 0 }
+        modify fun s => { s with ematch.num := 0 }
       return c?
     | c::cs =>
     if !(← checkAnchorRefs c) then
@@ -422,10 +419,24 @@ def splitCore (c : SplitInfo) (numCases : Nat) (isRec : Bool)
       pure 0
     return (mvarIds, numDigits)
   let numSubgoals := mvarIds.length
-  let subgoals := mvarIds.mapIdx fun i mvarId => { goal with
-    mvarId
-    split.trace := { expr := cExpr, i, num := numSubgoals, source := c.source } :: goal.split.trace
-  }
+  /-
+  **Split counter heuristic**: We do not increment `numSplits` for the first case (`i = 0`)
+  of a non-recursive split. This leverages non-chronological backtracking: if the first case
+  is solved using a proof that doesn't depend on the case hypothesis, we backtrack and close
+  the original goal directly. In this scenario, the case-split was "free", it didn't contribute
+  to the proof. By not counting it, we allow deeper exploration when case-splits turn out to be
+  irrelevant.
+
+  For recursive types or subsequent cases (`i > 0`), we always increment the counter since
+  these represent genuine branches in the proof search.
+  -/
+  let subgoals := mvarIds.mapIdx fun i mvarId =>
+    let numSplits := goal.split.num
+    let numSplits := if i > 0 || isRec then numSplits + 1 else numSplits
+    { goal with
+      mvarId
+      split.num := numSplits
+      split.trace := { expr := cExpr, i, num := numSubgoals, source := c.source } :: goal.split.trace }
   let mut seqNew : Array (List (TSyntax `grind)) := #[]
   let mut stuckNew : Array Goal := #[]
   for subgoal in subgoals do

tests/lean/run/grind_11081.lean

@@ -25,7 +25,7 @@ open List
 
 /--
 error: `grind` failed
-case grind.1.1.1.1.1.1.1.1.1
+case grind.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1.1
 α : Type
 inst : DecidableEq α
 l₁ l₂ : List α
@@ -66,6 +66,44 @@ left_8 : l₁ ~ l₁.diff l₂
 right_8 : ∀ (a : α), count a l₁ = count a (l₁.diff l₂)
 left_9 : l₁ ~ l₂
 right_9 : ∀ (a : α), count a l₁ = count a l₂
+left_10 : filter p l₁ ~ filter p (l₁.diff l₂ ++ l₂)
+right_10 : ∀ (a : α), count a (filter p l₁) = count a (filter p (l₁.diff l₂ ++ l₂))
+left_11 : filter p (l₁.diff l₂ ++ l₂) ~ filter p l₁
+right_11 : ∀ (a : α), count a (filter p (l₁.diff l₂ ++ l₂)) = count a (filter p l₁)
+left_12 : l₁.diff l₂ ++ l₂ ~ l₂ ++ (l₁.diff l₂ ++ l₂)
+right_12 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₂ ++ (l₁.diff l₂ ++ l₂))
+left_13 : l₁.diff l₂ ++ l₂ ~ l₁.diff l₂ ++ l₂ ++ l₁
+right_13 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₁.diff l₂ ++ l₂ ++ l₁)
+left_14 : l₁.diff l₂ ++ l₂ ~ l₁.diff l₂ ++ l₂ ++ (l₁.diff l₂ ++ l₂)
+right_14 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₁.diff l₂ ++ l₂ ++ (l₁.diff l₂ ++ l₂))
+left_15 : l₁.diff l₂ ++ l₂ ~ l₁.diff l₂ ++ l₂ ++ l₂
+right_15 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₁.diff l₂ ++ l₂ ++ l₂)
+left_16 : l₁.diff l₂ ++ l₂ ~ l₁.diff l₂ ++ l₂ ++ l₁.diff l₂
+right_16 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₁.diff l₂ ++ l₂ ++ l₁.diff l₂)
+left_17 : l₁.diff l₂ ++ l₂ ~ l₁ ++ (l₁.diff l₂ ++ l₂)
+right_17 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₁ ++ (l₁.diff l₂ ++ l₂))
+left_18 : filter p (l₁.diff l₂ ++ l₂) ~ filter p (l₁.diff l₂)
+right_18 : ∀ (a : α), count a (filter p (l₁.diff l₂ ++ l₂)) = count a (filter p (l₁.diff l₂))
+left_19 : filter p (l₁.diff l₂) ~ filter p (l₁.diff l₂ ++ l₂)
+right_19 : ∀ (a : α), count a (filter p (l₁.diff l₂)) = count a (filter p (l₁.diff l₂ ++ l₂))
+left_20 : (filter p (l₁.diff l₂ ++ l₂)).Subperm (filter p l₁)
+right_20 : (filter p (l₁.diff l₂ ++ l₂)).Subperm (filter p (l₁.diff l₂ ++ l₂))
+left_21 : l₁.diff l₂ ++ l₂ ++ l₁.diff l₂ ~ l₁.diff l₂ ++ l₂
+right_21 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂ ++ l₁.diff l₂) = count a (l₁.diff l₂ ++ l₂)
+left_22 : l₁.diff l₂ ++ l₂ ++ l₁ ~ l₁.diff l₂ ++ l₂
+right_22 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂ ++ l₁) = count a (l₁.diff l₂ ++ l₂)
+left_23 : l₁.diff l₂ ++ l₂ ++ l₂ ~ l₁.diff l₂ ++ l₂
+right_23 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂ ++ l₂) = count a (l₁.diff l₂ ++ l₂)
+left_24 : l₁.diff l₂ ++ l₂ ++ (l₁.diff l₂ ++ l₂) ~ l₁.diff l₂ ++ l₂
+right_24 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂ ++ (l₁.diff l₂ ++ l₂)) = count a (l₁.diff l₂ ++ l₂)
+left_25 : l₁ ++ (l₁.diff l₂ ++ l₂) ~ l₁.diff l₂ ++ l₂
+right_25 : ∀ (a : α), count a (l₁ ++ (l₁.diff l₂ ++ l₂)) = count a (l₁.diff l₂ ++ l₂)
+left_26 : l₂ ++ (l₁.diff l₂ ++ l₂) ~ l₁.diff l₂ ++ l₂
+right_26 : ∀ (a : α), count a (l₂ ++ (l₁.diff l₂ ++ l₂)) = count a (l₁.diff l₂ ++ l₂)
+left_27 : l₁.diff l₂ ++ (l₁.diff l₂ ++ l₂) ~ l₁.diff l₂ ++ l₂
+right_27 : ∀ (a : α), count a (l₁.diff l₂ ++ (l₁.diff l₂ ++ l₂)) = count a (l₁.diff l₂ ++ l₂)
+left_28 : l₁.diff l₂ ++ l₂ ~ l₁.diff l₂ ++ (l₁.diff l₂ ++ l₂)
+right_28 : ∀ (a : α), count a (l₁.diff l₂ ++ l₂) = count a (l₁.diff l₂ ++ (l₁.diff l₂ ++ l₂))
 ⊢ False
 -/
 #guard_msgs in

tests/lean/run/grind_11539_2.lean

@@ -0,0 +1,5 @@
+example (a b : Nat) (f g : Nat → Nat)
+    (hf : (∀ i ≤ a, f i ≤ f (i + 1)) ∧ f 0 = 0)
+    (hg : (∀ i ≤ b, g i ≤ g (i + 1)) ∧ g 0 = 0 ∧ g b = 0) :
+    g (a + b - a) = 0 := by
+  grind