diff --git a/src/Init/Grind/Tactics.lean b/src/Init/Grind/Tactics.lean
index 478b29fdd329..708d19ceaa21 100644
--- a/src/Init/Grind/Tactics.lean
+++ b/src/Init/Grind/Tactics.lean
@@ -22,7 +22,8 @@ syntax grindFwd    := "→ "
 syntax grindUsr    := &"usr "
 syntax grindCases  := &"cases "
 syntax grindCasesEager := atomic(&"cases" &"eager ")
-syntax grindMod := grindEqBoth <|> grindEqRhs <|> grindEq <|> grindEqBwd <|> grindBwd <|> grindFwd <|> grindUsr <|> grindCasesEager <|> grindCases
+syntax grindIntro  := &"intro "
+syntax grindMod := grindEqBoth <|> grindEqRhs <|> grindEq <|> grindEqBwd <|> grindBwd <|> grindFwd <|> grindUsr <|> grindCasesEager <|> grindCases <|> grindIntro
 syntax (name := grind) "grind" (grindMod)? : attr
 end Attr
 end Lean.Parser
diff --git a/src/Lean/Elab/Tactic/Grind.lean b/src/Lean/Elab/Tactic/Grind.lean
index a9a12c3f074c..91cd97085674 100644
--- a/src/Lean/Elab/Tactic/Grind.lean
+++ b/src/Lean/Elab/Tactic/Grind.lean
@@ -58,7 +58,7 @@ def elabGrindParams (params : Grind.Params) (ps :  TSyntaxArray ``Parser.Tactic.
     match p with
     | `(Parser.Tactic.grindParam| - $id:ident) =>
       let declName ← realizeGlobalConstNoOverloadWithInfo id
-      if (← Grind.isCasesAttrCandidate declName false) then
+      if let some declName ← Grind.isCasesAttrCandidate? declName false then
         Grind.ensureNotBuiltinCases declName
         params := { params with casesTypes := (← params.casesTypes.eraseDecl declName) }
       else
@@ -82,8 +82,14 @@ def elabGrindParams (params : Grind.Params) (ps :  TSyntaxArray ``Parser.Tactic.
       | .cases eager =>
         withRef p <| Grind.validateCasesAttr declName eager
         params := { params with casesTypes := params.casesTypes.insert declName eager }
+      | .intro =>
+        if let some info ← Grind.isCasesAttrPredicateCandidate? declName false then
+          for ctor in info.ctors do
+            params ← withRef p <| addEMatchTheorem params ctor .default
+        else
+          throwError "invalid use of `intro` modifier, `{declName}` is not an inductive predicate"
       | .infer =>
-        if (← Grind.isCasesAttrCandidate declName false) then
+        if let some declName ← Grind.isCasesAttrCandidate? declName false then
           params := { params with casesTypes := params.casesTypes.insert declName false }
           if let some info ← isInductivePredicate? declName then
             -- If it is an inductive predicate,
diff --git a/src/Lean/Meta/Tactic/Grind/Attr.lean b/src/Lean/Meta/Tactic/Grind/Attr.lean
index ca5cd40870ed..505b5ae3dce4 100644
--- a/src/Lean/Meta/Tactic/Grind/Attr.lean
+++ b/src/Lean/Meta/Tactic/Grind/Attr.lean
@@ -12,6 +12,7 @@ namespace Lean.Meta.Grind
 inductive AttrKind where
   | ematch (k : EMatchTheoremKind)
   | cases (eager : Bool)
+  | intro
   | infer
 
 /-- Return theorem kind for `stx` of the form `Attr.grindThmMod` -/
@@ -26,6 +27,7 @@ def getAttrKindCore (stx : Syntax) : CoreM AttrKind := do
   | `(Parser.Attr.grindMod| usr) => return .ematch .user
   | `(Parser.Attr.grindMod| cases) => return .cases false
   | `(Parser.Attr.grindMod| cases eager) => return .cases true
+  | `(Parser.Attr.grindMod| intro) => return .intro
   | _ => throwError "unexpected `grind` theorem kind: `{stx}`"
 
 /-- Return theorem kind for `stx` of the form `(Attr.grindMod)?` -/
@@ -64,8 +66,14 @@ builtin_initialize
       | .ematch .user => throwInvalidUsrModifier
       | .ematch k => addEMatchAttr declName attrKind k
       | .cases eager => addCasesAttr declName eager attrKind
+      | .intro =>
+        if let some info ← isCasesAttrPredicateCandidate? declName false then
+          for ctor in info.ctors do
+            addEMatchAttr ctor attrKind .default
+        else
+          throwError "invalid `[grind intro]`, `{declName}` is not an inductive predicate"
       | .infer =>
-        if (← isCasesAttrCandidate declName false) then
+        if let some declName ← isCasesAttrCandidate? declName false then
           addCasesAttr declName false attrKind
           if let some info ← isInductivePredicate? declName then
             -- If it is an inductive predicate,
diff --git a/src/Lean/Meta/Tactic/Grind/Cases.lean b/src/Lean/Meta/Tactic/Grind/Cases.lean
index e5c3d954f5e2..2a3f8517237b 100644
--- a/src/Lean/Meta/Tactic/Grind/Cases.lean
+++ b/src/Lean/Meta/Tactic/Grind/Cases.lean
@@ -73,14 +73,21 @@ private def getAlias? (value : Expr) : MetaM (Option Name) :=
     else
       return none
 
-partial def isCasesAttrCandidate (declName : Name) (eager : Bool) : CoreM Bool := do
+partial def isCasesAttrCandidate? (declName : Name) (eager : Bool) : CoreM (Option Name) := do
   match (← getConstInfo declName) with
-  | .inductInfo info => return !info.isRec || !eager
+  | .inductInfo info => if !info.isRec || !eager then return some declName else return none
   | .defnInfo info =>
     let some declName ← getAlias? info.value |>.run' {} {}
-      | return false
-    isCasesAttrCandidate declName eager
-  | _ => return false
+      | return none
+    isCasesAttrCandidate? declName eager
+  | _ => return none
+
+def isCasesAttrCandidate (declName : Name) (eager : Bool) : CoreM Bool := do
+  return (← isCasesAttrCandidate? declName eager).isSome
+
+def isCasesAttrPredicateCandidate? (declName : Name) (eager : Bool) : MetaM (Option InductiveVal) := do
+  let some declName ← isCasesAttrCandidate? declName eager | return none
+  isInductivePredicate? declName
 
 def validateCasesAttr (declName : Name) (eager : Bool) : CoreM Unit := do
   unless (← isCasesAttrCandidate declName eager) do
diff --git a/tests/lean/run/grind_constProp.lean b/tests/lean/run/grind_constProp.lean
index 4c5d74241ae8..4e0726a49805 100644
--- a/tests/lean/run/grind_constProp.lean
+++ b/tests/lean/run/grind_constProp.lean
@@ -80,20 +80,38 @@ def State.get (σ : State) (x : Var) : Val :=
 section
 attribute [local grind] State.update State.find? State.get State.erase
 
-@[simp, grind =] theorem State.find?_update_self (σ : State) (x : Var) (v : Val) : (σ.update x v).find? x = some v := by
+@[simp, grind =] theorem State.find?_nil (x : Var) : find? [] x = none := by
+  grind
+
+@[simp] theorem State.find?_update_self (σ : State) (x : Var) (v : Val) : (σ.update x v).find? x = some v := by
   induction σ, x, v using State.update.induct <;> grind
 
-@[simp, grind =] theorem State.find?_update (σ : State) (v : Val) (h : x ≠ z) : (σ.update x v).find? z = σ.find? z := by
+@[simp] theorem State.find?_update (σ : State) (v : Val) (h : x ≠ z) : (σ.update x v).find? z = σ.find? z := by
   induction σ, x, v using State.update.induct <;> grind
 
+@[grind =] theorem State.find?_update_eq (σ : State) (v : Val)
+    : (σ.update x v).find? z = if x = z then some v else σ.find? z := by
+  grind only [= find?_update_self, = find?_update, cases Or]
+
 @[grind] theorem State.get_of_find? {σ : State} (h : σ.find? x = some v) : σ.get x = v := by
   grind
 
-@[simp, grind =] theorem State.find?_erase_self (σ : State) (x : Var) : (σ.erase x).find? x = none := by
+@[simp] theorem State.find?_erase_self (σ : State) (x : Var) : (σ.erase x).find? x = none := by
   induction σ, x using State.erase.induct <;> grind
 
-@[simp, grind =] theorem State.find?_erase (σ : State) (h : x ≠ z) : (σ.erase x).find? z = σ.find? z := by
+@[simp] theorem State.find?_erase (σ : State) (h : x ≠ z) : (σ.erase x).find? z = σ.find? z := by
   induction σ, x using State.erase.induct <;> grind
+
+@[simp, grind =] theorem State.find?_erase_eq (σ : State)
+    : (σ.erase x).find? z = if x = z then none else σ.find? z := by
+  grind only [= find?_erase_self, = find?_erase, cases Or]
+
+@[grind] theorem State.length_erase_le (σ : State) (x : Var) : (σ.erase x).length ≤ σ.length := by
+  induction σ, x using erase.induct <;> grind
+
+def State.length_erase_lt (σ : State) (x : Var) : (σ.erase x).length < σ.length.succ := by
+  grind
+
 end
 
 syntax ident " ↦ " term : term
@@ -206,9 +224,7 @@ def evalExpr (e : Expr) : EvalM Val := do
     | c' => .while c' b.simplify
 
 theorem Stmt.simplify_correct (h : (σ, s) ⇓ σ') : (σ, s.simplify) ⇓ σ' := by
-  -- TODO: we need a mechanism for saying we just want the intro rules
-  induction h <;> grind [=_ Expr.eval_simplify, Bigstep.skip, Bigstep.assign,
-    Bigstep.seq, Bigstep.whileFalse, Bigstep.whileTrue, Bigstep.ifTrue, Bigstep.ifFalse]
+  induction h <;> grind [=_ Expr.eval_simplify, intro Bigstep]
 
 @[simp, grind =] def Expr.constProp (e : Expr) (σ : State) : Expr :=
   match e with
@@ -220,13 +236,7 @@ theorem Stmt.simplify_correct (h : (σ, s) ⇓ σ') : (σ, s.simplify) ⇓ σ' :
   | una op arg => una op (arg.constProp σ)
 
 @[simp, grind =] theorem Expr.constProp_nil (e : Expr) : e.constProp [] = e := by
-  induction e <;> grind [State.find?] -- TODO add missing theorem(s) to avoid unfolding `find?`
-
-@[grind] theorem State.length_erase_le (σ : State) (x : Var) : (σ.erase x).length ≤ σ.length := by
-  induction σ, x using erase.induct <;> grind [State.erase] -- TODO add missing theorem(s)
-
-def State.length_erase_lt (σ : State) (x : Var) : (σ.erase x).length < σ.length.succ := by
-  grind
+  induction e <;> grind
 
 @[simp, grind =] def State.join (σ₁ σ₂ : State) : State :=
   match σ₁ with
@@ -308,25 +318,11 @@ theorem State.erase_le_of_le_cons (h : σ' ≼ (x, v) :: σ) : σ'.erase x ≼ 
   grind
 
 @[grind] theorem State.erase_le_update (h : σ' ≼ σ) : σ'.erase x ≼ σ.update x v := by
-  intro y w hf'
-  -- TODO: can we avoid this hint?
-  by_cases hxy : x = y <;> grind
+  grind
 
 @[grind] theorem State.update_le_update (h : σ' ≼ σ) : σ'.update x v ≼ σ.update x v := by
-  intro y w hf
-  induction σ generalizing σ' hf with
-  | nil  => grind
-  | cons zw' σ ih =>
-    have (z, w') := zw'; simp
-    have : σ'.erase z ≼ σ := erase_le_of_le_cons h
-    have ih := ih this
-    revert ih hf
-    split <;> simp [*] <;> by_cases hyz : y = z <;> simp (config := { contextual := true }) [*]
-    next => grind
-    next => grind
-    sorry
+  grind
 
--- TODO: we are missing theorems here, and cannot seal State functions
 @[grind] theorem Expr.eval_constProp_of_sub (e : Expr) (h : σ' ≼ σ) : (e.constProp σ').eval σ = e.eval σ := by
   induction e <;> grind