diff --git a/src/Std/Data/DTreeMap/Internal/Lemmas.lean b/src/Std/Data/DTreeMap/Internal/Lemmas.lean
index 1f30f26f41..dbd6560a85 100644
--- a/src/Std/Data/DTreeMap/Internal/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Internal/Lemmas.lean
@@ -97,7 +97,8 @@ private def queryMap : Std.DHashMap Name (fun _ => Name × Array (MacroM (TSynta
      ⟨`forIn, (``forIn_eq_forIn_toListModel, #[])⟩,
      ⟨`forM, (``forM_eq_forM, #[])⟩,
      ⟨`minKey?, (``minKey?_eq_minKey?, #[``(minKey?_of_perm _)])⟩,
-     ⟨`minKey, (``minKey_eq_minKey, #[``(minKey_of_perm _)])⟩]
+     ⟨`minKey, (``minKey_eq_minKey, #[``(minKey_of_perm _)])⟩,
+     ⟨`minKey!, (``minKey!_eq_minKey!, #[``(minKey!_of_perm _)])⟩]
 
 /-- Internal implementation detail of the tree map -/
 scoped syntax "simp_to_model" (" [" (ident,*) "]")? ("using" term)? : tactic
@@ -172,6 +173,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransOrd α] (h : t.WF) :
     t.isEmpty = false ↔ ∃ a, a ∈ t := by
   simpa only [mem_iff_contains] using isEmpty_eq_false_iff_exists_contains_eq_true h
 
+theorem isEmpty_eq_false_of_contains [TransOrd α] (h : t.WF) {a : α} : (hc : t.contains a = true) →
+    t.isEmpty = false := by
+  simp_to_model [isEmpty, contains] using List.isEmpty_eq_false_of_containsKey
+
 theorem isEmpty_iff_forall_contains [TransOrd α] (h : t.WF) :
     t.isEmpty = true ↔ ∀ a, t.contains a = false := by
   simp_to_model [isEmpty, contains] using List.isEmpty_iff_forall_containsKey
@@ -276,14 +281,14 @@ theorem isEmpty_erase! [TransOrd α] (h : t.WF) {k : α} :
     (t.erase! k).isEmpty = (t.isEmpty || (t.size = 1 && t.contains k)) := by
   simpa only [erase_eq_erase!] using isEmpty_erase h
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransOrd α] (h : t.WF) (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransOrd α] (h : t.WF) (k : α) :
     t.isEmpty = ((t.erase k h.balanced).impl.isEmpty && !(t.contains k)) := by
   simp_to_model [erase, isEmpty, contains] using
     List.isEmpty_eq_isEmpty_eraseKey_and_not_containsKey
 
 theorem isEmpty_eq_isEmpty_erase!_and_not_containsKey [TransOrd α] (h : t.WF) (k : α) :
     t.isEmpty = ((t.erase! k).isEmpty && !(t.contains k)) := by
-  simpa [erase_eq_erase!] using isEmpty_eq_isEmpty_erase_and_not_containsKey h k
+  simpa [erase_eq_erase!] using isEmpty_eq_isEmpty_erase_and_not_contains h k
 
 theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransOrd α] (h : t.WF) {k : α} :
     (he : (t.erase k h.balanced).impl.isEmpty = false) →
@@ -4505,7 +4510,7 @@ theorem minKey_eq_some_iff_mem_and_forall [TransOrd α] [LawfulEqOrd α] (h : t.
 
 theorem minKey_insert [TransOrd α] (h : t.WF) {k v} :
     (t.insert k v h.balanced).impl.minKey (isEmpty_insert h) =
-      (t.minKey?).elim k fun k' => if compare k k'|>.isLE then k else k' := by
+      t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k' := by
   simp_to_model [insert, minKey, minKey?] using List.minKey_insertEntry
 
 theorem minKey_insert_le_minKey [TransOrd α] (h : t.WF) {k v he} :
@@ -4620,6 +4625,212 @@ theorem minKey_alter_eq_self [TransOrd α] (h : t.WF) {k f he} :
 
 end Const
 
+theorem minKey_eq_minKey! [TransOrd α] [Inhabited α] (h : t.WF) {he} :
+    t.minKey he = t.minKey! := by
+  simp_to_model [minKey, minKey!] using List.minKey_eq_minKey!
+
+theorem minKey?_eq_some_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → t.minKey? = some t.minKey! := by
+  simp_to_model [minKey?, minKey!, isEmpty] using List.minKey?_eq_some_minKey!
+
+theorem minKey!_eq_default [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty) → t.minKey! = default := by
+  simp_to_model [minKey!, isEmpty] using List.minKey!_eq_default
+
+theorem minKey!_eq_iff_getKey?_eq_self_and_forall [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {km},
+    t.minKey! = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (compare km k).isLE := by
+  simp_to_model [minKey!, getKey?, contains, isEmpty] using
+    List.minKey!_eq_iff_getKey?_eq_self_and_forall
+
+theorem minKey!_eq_some_iff_mem_and_forall [TransOrd α]
+    [LawfulEqOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {km},
+    t.minKey! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (compare km k).isLE := by
+  simp_to_model [minKey!, contains, isEmpty] using List.minKey!_eq_some_iff_mem_and_forall
+
+theorem minKey!_insert [TransOrd α] [Inhabited α] (h : t.WF) {k v} :
+    (t.insert k v h.balanced |>.impl.minKey!) =
+      (t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simp_to_model [minKey!, minKey?, insert] using List.minKey!_insertEntry
+
+theorem minKey!_insert! [TransOrd α] [Inhabited α] (h : t.WF) {k v} :
+    (t.insert! k v |>.minKey!) =
+      (t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simpa [insert_eq_insert!] using minKey!_insert h
+
+theorem minKey!_insert_le_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v},
+    compare (t.insert k v h.balanced |>.impl.minKey!) t.minKey! |>.isLE := by
+  simp_to_model [minKey!, isEmpty, insert] using List.minKey!_insertEntry_le_minKey!
+
+theorem minKey!_insert!_le_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v},
+    compare (t.insert! k v |>.minKey!) t.minKey! |>.isLE := by
+  simpa only [insert_eq_insert!] using minKey!_insert_le_minKey! h
+
+theorem minKey!_insert_le_self [TransOrd α] [Inhabited α] (h : t.WF) {k v} :
+    compare (t.insert k v h.balanced |>.impl.minKey!) k |>.isLE := by
+  simp_to_model [minKey!, insert] using List.minKey!_insertEntry_le_self
+
+theorem minKey!_insert!_le_self [TransOrd α] [Inhabited α] (h : t.WF) {k v} :
+    compare (t.insert! k v |>.minKey!) k |>.isLE := by
+  simpa only [insert_eq_insert!] using minKey!_insert_le_self h
+
+theorem contains_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → t.contains t.minKey! := by
+  simp_to_model [minKey!, isEmpty, contains] using List.containsKey_minKey!
+
+theorem minKey!_mem [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → t.minKey! ∈ t :=
+  contains_minKey! h
+
+theorem minKey!_le_of_contains [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k}, (hc : t.contains k) →
+    compare t.minKey! k |>.isLE := by
+  simp_to_model [minKey!, contains] using List.minKey!_le_of_containsKey
+
+theorem minKey!_le_of_mem [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k}, (hc : k ∈ t) →
+    compare t.minKey! k |>.isLE :=
+  minKey!_le_of_contains h
+
+theorem le_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k},
+    (compare k t.minKey!).isLE ↔ (∀ k', k' ∈ t → (compare k k').isLE) := by
+  simp_to_model [minKey!, contains, isEmpty] using List.le_minKey!
+
+theorem getKey?_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) →
+    t.getKey? t.minKey! = some t.minKey! := by
+  simp_to_model [minKey!, getKey?, isEmpty] using List.getKey?_minKey!
+
+theorem getKey_minKey! [TransOrd α] [Inhabited α] (h : t.WF) : ∀ {he},
+    t.getKey t.minKey! he = t.minKey! := by
+  simp_to_model [minKey!, contains, isEmpty, getKey] using List.getKey_minKey!
+
+theorem getKey_minKey!_eq_minKey [TransOrd α] [Inhabited α] (h : t.WF) : ∀ {hc},
+    t.getKey t.minKey! hc = t.minKey (isEmpty_eq_false_of_contains h hc) := by
+  simp_to_model [minKey!, minKey, contains, isEmpty, getKey] using List.getKey_minKey!_eq_minKey
+
+theorem getKey!_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → t.getKey! t.minKey! = t.minKey! := by
+  simp_to_model [minKey!, isEmpty, getKey!] using List.getKey!_minKey!
+
+theorem getKeyD_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {fallback},
+    t.getKeyD t.minKey! fallback = t.minKey! := by
+  simp_to_model [minKey!, getKeyD, isEmpty] using List.getKeyD_minKey!
+
+theorem minKey!_erase_eq_iff_not_compare_minKey!_eq [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k}, (he : (t.erase k h.balanced).impl.isEmpty = false) →
+    (t.erase k h.balanced |>.impl.minKey!) = t.minKey! ↔
+      ¬ compare k t.minKey! = .eq := by
+  simp_to_model [minKey!, isEmpty, erase] using List.minKey!_eraseKey_eq_iff_beq_minKey!_eq_false
+
+theorem minKey!_erase!_eq_iff_not_compare_minKey!_eq [TransOrd α] [Inhabited α] (h : t.WF) {k} :
+    (he : (t.erase! k).isEmpty = false) →
+    (t.erase! k |>.minKey!) = t.minKey! ↔
+      ¬ compare k t.minKey! = .eq := by
+  simpa only [erase_eq_erase!] using minKey!_erase_eq_iff_not_compare_minKey!_eq h
+
+theorem minKey!_erase_eq_of_not_compare_minKey!_eq [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k}, (he : (t.erase k h.balanced).impl.isEmpty = false) → (heq : ¬ compare k t.minKey! = .eq) →
+    (t.erase k h.balanced |>.impl.minKey!) = t.minKey! := by
+  simp_to_model [minKey!, isEmpty, erase] using
+    List.minKey!_eraseKey_eq_of_beq_minKey!_eq_false
+
+theorem minKey!_erase!_eq_of_not_compare_minKey!_eq [TransOrd α] [Inhabited α] (h : t.WF) {k} :
+    (he : (t.erase! k).isEmpty = false) → (heq : ¬ compare k t.minKey! = .eq) →
+    (t.erase! k |>.minKey!) = t.minKey! := by
+  simpa only [erase_eq_erase!] using minKey!_erase_eq_of_not_compare_minKey!_eq h
+
+theorem minKey!_le_minKey!_erase [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k}, (he : (t.erase k h.balanced).impl.isEmpty = false) →
+    compare t.minKey! (t.erase k h.balanced |>.impl.minKey!) |>.isLE := by
+  simp_to_model [minKey!, isEmpty, erase] using List.minKey!_le_minKey!_erase
+
+theorem minKey!_le_minKey!_erase! [TransOrd α] [Inhabited α] (h : t.WF) {k} :
+    (he : (t.erase! k).isEmpty = false) →
+    compare t.minKey! (t.erase! k |>.minKey!) |>.isLE := by
+  simpa only [erase_eq_erase!] using minKey!_le_minKey!_erase h
+
+theorem minKey!_insertIfNew [TransOrd α] [Inhabited α] (h : t.WF) : ∀ {k v},
+    (t.insertIfNew k v h.balanced |>.impl.minKey!) =
+      t.minKey?.elim k fun k' => if compare k k' = .lt then k else k' := by
+  simp_to_model [minKey!, minKey?, insertIfNew] using List.minKey!_insertEntryIfNew
+
+theorem minKey!_insertIfNew! [TransOrd α] [Inhabited α] (h : t.WF) {k v} :
+    (t.insertIfNew! k v |>.minKey!) =
+      t.minKey?.elim k fun k' => if compare k k' = .lt then k else k' := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKey!_insertIfNew h
+
+theorem minKey!_insertIfNew_le_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v},
+    compare (t.insertIfNew k v h.balanced |>.impl.minKey!) t.minKey! |>.isLE := by
+  simp_to_model [minKey!, isEmpty, insertIfNew] using List.minKey!_insertEntryIfNew_le_minKey!
+
+theorem minKey!_insertIfNew!_le_minKey! [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v},
+    compare (t.insertIfNew! k v |>.minKey!) t.minKey! |>.isLE := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKey!_insertIfNew_le_minKey! h
+
+theorem minKey!_insertIfNew_le_self [TransOrd α] [Inhabited α] (h : t.WF) : ∀ {k v},
+    compare (t.insertIfNew k v h.balanced |>.impl.minKey!) k |>.isLE := by
+  simp_to_model [minKey!, insertIfNew] using List.minKey!_insertEntryIfNew_le_self
+
+theorem minKey!_insertIfNew!_le_self [TransOrd α] [Inhabited α] (h : t.WF) {k v} :
+    compare (t.insertIfNew! k v |>.minKey!) k |>.isLE := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKey!_insertIfNew_le_self h
+
+theorem minKey!_modify [TransOrd α] [LawfulEqOrd α] [Inhabited α] (h : t.WF) : ∀ {k f},
+    (t.modify k f |>.minKey!) = t.minKey! := by
+  simp_to_model [minKey!, modify] using List.minKey!_modifyKey
+
+theorem minKey!_alter_eq_self [TransOrd α] [LawfulEqOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k f}, (he : (t.alter k f h.balanced).impl.isEmpty = false) →
+    (t.alter k f h.balanced |>.impl.minKey!) = k ↔
+      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simp_to_model [minKey!, alter, isEmpty, contains, get?] using List.minKey!_alterKey_eq_self
+
+theorem minKey!_alter!_eq_self [TransOrd α] [LawfulEqOrd α] [Inhabited α] (h : t.WF) {k f} :
+    (he : (t.alter! k f).isEmpty = false) →
+    (t.alter! k f |>.minKey!) = k ↔
+      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simpa only [alter_eq_alter!] using minKey!_alter_eq_self h
+
+namespace Const
+
+variable {β : Type v} {t : Impl α β}
+
+theorem minKey!_modify [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k f}, (he : (modify k f t).isEmpty = false) →
+    (modify k f t |> minKey!) = if compare t.minKey! k = .eq then k else t.minKey! := by
+  simp_to_model [minKey!, minKey, isEmpty, Const.modify] using List.Const.minKey!_modifyKey
+
+theorem minKey!_modify_eq_minKey! [TransOrd α] [LawfulEqOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k f}, (modify k f t |> minKey!) = t.minKey! := by
+  simp_to_model [minKey!, Const.modify] using List.Const.minKey!_modifyKey_eq_minKey!
+
+theorem compare_minKey!_modify_eq [TransOrd α] [Inhabited α] (h : t.WF) : ∀ {k f},
+    compare (Const.modify k f t |> minKey!) t.minKey! = .eq := by
+  simp_to_model [minKey!, Const.modify] using List.Const.minKey!_modifyKey_beq
+
+theorem minKey!_alter_eq_self [TransOrd α] [Inhabited α] (h : t.WF) :
+    ∀ {k f}, (he : (alter k f t h.balanced).impl.isEmpty = false) →
+    (alter k f t h.balanced |>.impl.minKey!) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simp_to_model [minKey!, Const.alter, contains, isEmpty, Const.get?] using
+    List.Const.minKey!_alterKey_eq_self
+
+theorem minKey!_alter!_eq_self [TransOrd α] [Inhabited α] (h : t.WF) {k f} :
+    (he : (alter! k f t).isEmpty = false) →
+    (alter! k f t |>.minKey!) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simpa only [alter_eq_alter!] using minKey!_alter_eq_self h
+
+end Const
+
 end Min
 
 end Std.DTreeMap.Internal.Impl
diff --git a/src/Std/Data/DTreeMap/Internal/Model.lean b/src/Std/Data/DTreeMap/Internal/Model.lean
index cc870c64f2..edd53d326a 100644
--- a/src/Std/Data/DTreeMap/Internal/Model.lean
+++ b/src/Std/Data/DTreeMap/Internal/Model.lean
@@ -476,6 +476,10 @@ theorem minEntry_eq_get_minEntry? [Ord α] {l : Impl α β} {he} :
 theorem minKey_eq_minEntry_fst [Ord α] {l : Impl α β} {he} : l.minKey he = (l.minEntry he).fst := by
   induction l, he using minKey.induct <;> simp only [minKey, minEntry] <;> trivial
 
+theorem minKey!_eq_get!_minKey? [Ord α] [Inhabited α] {l : Impl α β} :
+    l.minKey! = l.minKey?.get! := by
+  induction l using minKey!.induct <;> simp_all only [minKey!, minKey?] <;> rfl
+
 theorem balanceL_eq_balance {k : α} {v : β k} {l r : Impl α β} {hlb hrb hlr} :
     balanceL k v l r hlb hrb hlr = balance k v l r hlb hrb (Or.inl hlr.erase) := by
   rw [balanceL_eq_balanceLErase, balanceLErase_eq_balanceL!,
diff --git a/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean b/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
index 24ef498f42..f188f4f7bb 100644
--- a/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
@@ -1833,4 +1833,8 @@ theorem minKey_eq_minKey [Ord α] [TransOrd α] {l : Impl α β} (hlo : l.Ordere
   simp [minKey_eq_get_minKey?, minKey_eq_minEntry_fst, minEntry_eq_get_minEntry?,
     minEntry?_eq_minEntry? hlo, List.minKey?, Option.get_map]
 
+theorem minKey!_eq_minKey! [Ord α] [TransOrd α] [Inhabited α] {l : Impl α β} (hlo : l.Ordered) :
+    l.minKey! = List.minKey! l.toListModel := by
+  simp [Impl.minKey!_eq_get!_minKey?, List.minKey!_eq_get!_minKey?, minKey?_eq_minKey? hlo]
+
 end Std.DTreeMap.Internal.Impl
diff --git a/src/Std/Data/DTreeMap/Lemmas.lean b/src/Std/Data/DTreeMap/Lemmas.lean
index 3ee76d9d09..88ccf8b036 100644
--- a/src/Std/Data/DTreeMap/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Lemmas.lean
@@ -72,6 +72,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransCmp cmp] :
     t.isEmpty = false ↔ ∃ a, a ∈ t :=
   Impl.isEmpty_eq_false_iff_exists_mem t.wf
 
+theorem isEmpty_eq_false_of_contains [TransCmp cmp] {a : α} (hc : t.contains a = true) :
+    t.isEmpty = false :=
+  Impl.isEmpty_eq_false_of_contains t.wf hc
+
 theorem isEmpty_iff_forall_contains [TransCmp cmp] :
     t.isEmpty = true ↔ ∀ a, t.contains a = false :=
   Impl.isEmpty_iff_forall_contains t.wf
@@ -145,9 +149,9 @@ theorem isEmpty_erase [TransCmp cmp] {k : α} :
     (t.erase k).isEmpty = (t.isEmpty || (t.size == 1 && t.contains k)) :=
   Impl.isEmpty_erase t.wf
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransCmp cmp] (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransCmp cmp] (k : α) :
     t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
-  Impl.isEmpty_eq_isEmpty_erase_and_not_containsKey t.wf k
+  Impl.isEmpty_eq_isEmpty_erase_and_not_contains t.wf k
 
 theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] {k : α}
     (he : (t.erase k).isEmpty = false) :
@@ -2977,7 +2981,7 @@ theorem minKey_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] {he k
 
 theorem minKey_insert [TransCmp cmp] {k v} :
     (t.insert k v).minKey isEmpty_insert =
-      (t.minKey?).elim k fun k' => if cmp k k'|>.isLE then k else k' :=
+      t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k' :=
   Impl.minKey_insert t.wf
 
 theorem minKey_insert_le_minKey [TransCmp cmp] {k v he} :
@@ -3087,6 +3091,7 @@ theorem minKey_modify_eq_minKey [TransCmp cmp] [LawfulEqCmp cmp] {k f he} :
     (modify t k f).minKey he = t.minKey (cast (congrArg (· = false) isEmpty_modify) he) :=
   Impl.Const.minKey_modify_eq_minKey t.wf
 
+@[simp]
 theorem compare_minKey_modify_eq [TransCmp cmp] {k f he} :
     cmp (modify t k f |>.minKey he)
       (t.minKey <| cast (congrArg (· = false) isEmpty_modify) he) = .eq :=
@@ -3099,6 +3104,138 @@ theorem minKey_alter_eq_self [TransCmp cmp] {k f he} :
 
 end Const
 
+theorem minKey?_eq_some_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.minKey? = some t.minKey! :=
+  Impl.minKey?_eq_some_minKey! t.wf he
+
+theorem minKey!_eq_default [TransCmp cmp] [Inhabited α] (he : t.isEmpty) :
+    t.minKey! = default :=
+  Impl.minKey!_eq_default t.wf he
+
+theorem minKey!_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] [Inhabited α]
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKey!_eq_iff_getKey?_eq_self_and_forall t.wf he
+
+theorem minKey!_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α]
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKey!_eq_some_iff_mem_and_forall t.wf he
+
+theorem minKey!_insert [TransCmp cmp] [Inhabited α] {k v} :
+    (t.insert k v |>.minKey!) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  Impl.minKey!_insert t.wf
+
+theorem minKey!_insert_le_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k v} :
+    cmp (t.insert k v |>.minKey!) t.minKey! |>.isLE :=
+  Impl.minKey!_insert_le_minKey! t.wf he
+
+theorem minKey!_insert_le_self [TransCmp cmp] [Inhabited α] {k v} :
+    cmp (t.insert k v |>.minKey!) k |>.isLE :=
+  Impl.minKey!_insert_le_self t.wf
+
+theorem contains_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.contains t.minKey! :=
+  Impl.contains_minKey! t.wf he
+
+theorem minKey!_mem [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.minKey! ∈ t :=
+  Impl.minKey!_mem t.wf he
+
+theorem minKey!_le_of_contains [TransCmp cmp] [Inhabited α] {k} (hc : t.contains k) :
+    cmp t.minKey! k |>.isLE :=
+  Impl.minKey!_le_of_contains t.wf hc
+
+theorem minKey!_le_of_mem [TransCmp cmp] [Inhabited α] {k} (hc : k ∈ t) :
+    cmp t.minKey! k |>.isLE :=
+  Impl.minKey!_le_of_mem t.wf hc
+
+theorem le_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k} :
+    (cmp k t.minKey!).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  Impl.le_minKey! t.wf he
+
+theorem getKey?_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.getKey? t.minKey! = some t.minKey! :=
+  Impl.getKey?_minKey! t.wf he
+
+theorem getKey_minKey! [TransCmp cmp] [Inhabited α] {hc} :
+    t.getKey t.minKey! hc = t.minKey! :=
+  Impl.getKey_minKey! t.wf
+
+@[simp]
+theorem getKey_minKey!_eq_minKey [TransCmp cmp] [Inhabited α] {hc} :
+    t.getKey t.minKey! hc = t.minKey (isEmpty_eq_false_of_contains hc) :=
+  Impl.getKey_minKey!_eq_minKey t.wf
+
+theorem getKey!_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.getKey! t.minKey! = t.minKey! :=
+  Impl.getKey!_minKey! t.wf he
+
+theorem getKeyD_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback} :
+    t.getKeyD t.minKey! fallback = t.minKey! :=
+  Impl.getKeyD_minKey! t.wf he
+
+theorem minKey!_erase_eq_of_not_compare_minKey!_eq [TransCmp cmp] [Inhabited α] {k}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k t.minKey! = .eq) :
+    (t.erase k |>.minKey!) = t.minKey! :=
+  Impl.minKey!_erase_eq_of_not_compare_minKey!_eq t.wf he heq
+
+theorem minKey!_le_minKey!_erase [TransCmp cmp] [Inhabited α] {k}
+    (he : (t.erase k).isEmpty = false) :
+    cmp t.minKey! (t.erase k |>.minKey!) |>.isLE :=
+  Impl.minKey!_le_minKey!_erase t.wf he
+
+theorem minKey!_insertIfNew [TransCmp cmp] [Inhabited α] {k v} :
+    (t.insertIfNew k v |>.minKey!) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  Impl.minKey!_insertIfNew t.wf
+
+theorem minKey!_insertIfNew_le_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) t.minKey! |>.isLE :=
+  Impl.minKey!_insertIfNew_le_minKey! t.wf he
+
+theorem minKey!_insertIfNew_le_self [TransCmp cmp] [Inhabited α] {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) k |>.isLE :=
+  Impl.minKey!_insertIfNew_le_self t.wf
+
+@[simp]
+theorem minKey!_modify [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k f} :
+    (t.modify k f |>.minKey!) = t.minKey! :=
+  Impl.minKey!_modify t.wf
+
+theorem minKey!_alter_eq_self [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k f}
+    (he : (t.alter k f).isEmpty = false) :
+    (t.alter k f |>.minKey!) = k ↔ (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.minKey!_alter_eq_self t.wf he
+
+namespace Const
+
+variable {β : Type v} {t : DTreeMap α β cmp}
+
+theorem minKey!_modify [TransCmp cmp] [Inhabited α] {k f}
+    (he : (modify t k f).isEmpty = false) :
+    (modify t k f |> minKey!) = if cmp t.minKey! k = .eq then k else t.minKey! :=
+  Impl.Const.minKey!_modify t.wf he
+
+@[simp]
+theorem minKey!_modify_eq_minKey! [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k f} :
+    (modify t k f |>.minKey!) = t.minKey! :=
+  Impl.Const.minKey!_modify_eq_minKey! t.wf
+
+@[simp]
+theorem compare_minKey!_modify_eq [TransCmp cmp] [Inhabited α] {k f} :
+    cmp (modify t k f |> minKey!) t.minKey! = .eq :=
+  Impl.Const.compare_minKey!_modify_eq t.wf (instOrd := ⟨cmp⟩)
+
+theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] {k f}
+    (he : (alter t k f).isEmpty = false) :
+    (alter t k f |>.minKey!) = k ↔
+      (f (Const.get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.Const.minKey!_alter_eq_self t.wf he
+
+end Const
+
 end Min
 
 end Std.DTreeMap
diff --git a/src/Std/Data/DTreeMap/Raw/Lemmas.lean b/src/Std/Data/DTreeMap/Raw/Lemmas.lean
index 1fe9236a72..e09b98421f 100644
--- a/src/Std/Data/DTreeMap/Raw/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Raw/Lemmas.lean
@@ -73,6 +73,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransCmp cmp] (h : t.WF) :
     t.isEmpty = false ↔ ∃ a, a ∈ t :=
   Impl.isEmpty_eq_false_iff_exists_mem h
 
+theorem isEmpty_eq_false_of_contains [TransCmp cmp] (h : t.WF) {a : α} (hc : t.contains a = true) :
+    t.isEmpty = false :=
+  Impl.isEmpty_eq_false_of_contains h hc
+
 theorem isEmpty_iff_forall_contains [TransCmp cmp] (h : t.WF) :
     t.isEmpty = true ↔ ∀ a, t.contains a = false :=
   Impl.isEmpty_iff_forall_contains h
@@ -146,7 +150,7 @@ theorem isEmpty_erase [TransCmp cmp] (h : t.WF) {k : α} :
     (t.erase k).isEmpty = (t.isEmpty || (t.size == 1 && t.contains k)) :=
   Impl.isEmpty_erase! h
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransCmp cmp] (h : t.WF) (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransCmp cmp] (h : t.WF) (k : α) :
     t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
   Impl.isEmpty_eq_isEmpty_erase!_and_not_containsKey h k
 
@@ -2964,6 +2968,135 @@ theorem minKey?_alter_eq_self [TransCmp cmp] (h : t.WF) {k f} :
 
 end Const
 
+theorem minKey?_eq_some_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.minKey? = some t.minKey! :=
+  Impl.minKey?_eq_some_minKey! h he
+
+theorem minKey!_eq_default [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty) :
+    t.minKey! = default :=
+  Impl.minKey!_eq_default h he
+
+theorem minKey!_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKey!_eq_iff_getKey?_eq_self_and_forall h he
+
+theorem minKey!_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKey!_eq_some_iff_mem_and_forall h he
+
+theorem minKey!_insert [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    (t.insert k v |>.minKey!) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  Impl.minKey!_insert! h
+
+theorem minKey!_insert_le_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false)
+    {k v} :
+    cmp (t.insert k v |>.minKey!) t.minKey! |>.isLE :=
+  Impl.minKey!_insert!_le_minKey! h he (instOrd := ⟨cmp⟩)
+
+theorem minKey!_insert_le_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    cmp (t.insert k v |>.minKey!) k |>.isLE :=
+  Impl.minKey!_insert!_le_self h (instOrd := ⟨cmp⟩)
+
+theorem contains_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.contains t.minKey! :=
+  Impl.contains_minKey! h he
+
+theorem minKey!_mem [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.minKey! ∈ t :=
+  Impl.minKey!_mem h he
+
+theorem minKey!_le_of_contains [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (hc : t.contains k) :
+    cmp t.minKey! k |>.isLE :=
+  Impl.minKey!_le_of_contains h hc
+
+theorem minKey!_le_of_mem [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (hc : k ∈ t) :
+    cmp t.minKey! k |>.isLE :=
+  Impl.minKey!_le_of_mem h hc
+
+theorem le_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {k} :
+    (cmp k t.minKey!).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  Impl.le_minKey! h he (instOrd := ⟨cmp⟩)
+
+theorem getKey?_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.getKey? t.minKey! = some t.minKey! :=
+  Impl.getKey?_minKey! h he
+
+theorem getKey_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) {hc} :
+    t.getKey t.minKey! hc = t.minKey! :=
+  Impl.getKey_minKey! h
+
+theorem getKey!_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.getKey! t.minKey! = t.minKey! :=
+  Impl.getKey!_minKey! h he
+
+theorem getKeyD_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getKeyD t.minKey! fallback = t.minKey! :=
+  Impl.getKeyD_minKey! h he
+
+theorem minKey!_erase_eq_of_not_compare_minKey!_eq [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k t.minKey! = .eq) :
+    (t.erase k |>.minKey!) = t.minKey! :=
+  Impl.minKey!_erase!_eq_of_not_compare_minKey!_eq h he heq
+
+theorem minKey!_le_minKey!_erase [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) :
+    cmp t.minKey! (t.erase k |>.minKey!) |>.isLE :=
+  Impl.minKey!_le_minKey!_erase! h he
+
+theorem minKey!_insertIfNew [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    (t.insertIfNew k v |>.minKey!) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  Impl.minKey!_insertIfNew! h
+
+theorem minKey!_insertIfNew_le_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) t.minKey! |>.isLE :=
+  Impl.minKey!_insertIfNew!_le_minKey! h he (instOrd := ⟨cmp⟩)
+
+theorem minKey!_insertIfNew_le_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) k |>.isLE :=
+  Impl.minKey!_insertIfNew!_le_self h (instOrd := ⟨cmp⟩)
+
+@[simp]
+theorem minKey!_modify [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF) {k f} :
+    (t.modify k f |>.minKey!) = t.minKey! :=
+  Impl.minKey!_modify h
+
+theorem minKey!_alter_eq_self [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF) {k f}
+    (he : (t.alter k f).isEmpty = false) :
+    (t.alter k f |>.minKey!) = k ↔ (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.minKey!_alter!_eq_self h he
+
+namespace Const
+
+variable {β : Type v} {t : Raw α β cmp}
+
+theorem minKey!_modify [TransCmp cmp] [Inhabited α] (h : t.WF) {k f}
+    (he : (modify t k f).isEmpty = false) :
+    (modify t k f |> minKey!) = if cmp t.minKey! k = .eq then k else t.minKey! :=
+  Impl.Const.minKey!_modify h he
+
+@[simp]
+theorem minKey!_modify_eq_minKey! [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF) {k f} :
+    (modify t k f |>.minKey!) = t.minKey! :=
+  Impl.Const.minKey!_modify_eq_minKey! h
+
+@[simp]
+theorem compare_minKey!_modify_eq [TransCmp cmp] [Inhabited α] (h : t.WF) {k f} :
+    cmp (modify t k f |> minKey!) t.minKey! = .eq :=
+  Impl.Const.compare_minKey!_modify_eq h (instOrd := ⟨cmp⟩)
+
+theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k f}
+    (he : (alter t k f).isEmpty = false) :
+    (alter t k f |>.minKey!) = k ↔
+      (f (Const.get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.Const.minKey!_alter!_eq_self h he
+
+end Const
+
 end Min
 
 end Std.DTreeMap.Raw
diff --git a/src/Std/Data/Internal/List/Associative.lean b/src/Std/Data/Internal/List/Associative.lean
index a5bf07591e..9c03ce3e41 100644
--- a/src/Std/Data/Internal/List/Associative.lean
+++ b/src/Std/Data/Internal/List/Associative.lean
@@ -4945,6 +4945,192 @@ theorem minKey_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α
 
 end Const
 
+/-- Returns the smallest key in an associative list or panics if the list is empty. -/
+def minKey! [Ord α] [Inhabited α] (xs : List ((a : α) × β a)) : α :=
+  minKey? xs |>.get!
+
+theorem minKey!_of_perm [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l l' : List ((a : α) × β a)} (hd : DistinctKeys l) (hp : l.Perm l') :
+    minKey! l = minKey! l' := by
+  simp [minKey!, minKey?_of_perm hd hp]
+
+theorem minKey!_eq_get!_minKey? [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} :
+    minKey! l = (minKey? l).get! :=
+  rfl
+
+theorem minKey_eq_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} {he} :
+    minKey l he = minKey! l := by
+  simp [minKey_eq_get_minKey?, minKey!_eq_get!_minKey?, Option.get_eq_get!]
+
+theorem minKey?_eq_some_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (he : l.isEmpty = false) :
+    minKey? l = some (minKey! l) := by
+  simp [← minKey_eq_minKey! (he := he), minKey_eq_get_minKey?]
+
+theorem minKey!_eq_default [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (h : l.isEmpty) :
+    minKey! l = default := by
+  simp [minKey!, minKey?_eq_none_iff_isEmpty.mpr h]
+
+theorem minKey!_eq_iff_getKey?_eq_self_and_forall [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km} :
+    minKey! l = km ↔ getKey? km l = some km ∧ ∀ k, containsKey k l → (compare km k).isLE := by
+  simpa [minKey_eq_minKey!] using minKey_eq_iff_getKey?_eq_self_and_forall hd (he := he)
+
+theorem minKey!_eq_some_iff_mem_and_forall [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [LawfulEqOrd α] [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l)
+    (he : l.isEmpty = false) {km} :
+    minKey! l = km ↔ containsKey km l ∧ ∀ k, containsKey k l → (compare km k).isLE := by
+  simpa [minKey_eq_minKey!] using minKey_eq_some_iff_mem_and_forall hd (he := he)
+
+theorem minKey!_insertEntry [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    (insertEntry k v l |> minKey!) =
+      ((minKey? l).elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simpa [minKey_eq_minKey!] using minKey_insertEntry hd
+
+theorem minKey!_insertEntry_le_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k v} :
+    compare (insertEntry k v l |> minKey!) (minKey! l) |>.isLE := by
+  simpa [minKey_eq_minKey!] using minKey_insertEntry_le_minKey hd (he := he)
+
+theorem minKey!_insertEntry_le_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    compare (insertEntry k v l |> minKey!) k |>.isLE := by
+  simpa [minKey_eq_minKey!] using minKey_insertEntry_le_self hd
+
+theorem containsKey_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) :
+    containsKey (minKey! l) l := by
+  simpa [minKey_eq_minKey!] using containsKey_minKey hd (he := he)
+
+theorem minKey!_le_of_containsKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k} (hc : containsKey k l) :
+    compare (minKey! l) k |>.isLE := by
+   simpa [minKey_eq_minKey!] using minKey_le_of_containsKey hd hc
+
+theorem le_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k} :
+    (compare k (minKey! l)).isLE ↔ (∀ k', containsKey k' l → (compare k k').isLE) := by
+  simpa [minKey_eq_minKey!] using le_minKey hd (he := he)
+
+theorem getKey?_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) :
+    getKey? (minKey! l) l = some (minKey! l) := by
+  simpa [minKey_eq_minKey!] using getKey?_minKey hd (he := he)
+
+theorem getKey_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he} :
+    getKey (minKey! l) l he = minKey! l := by
+  simpa [minKey_eq_minKey!] using getKey_minKey hd (he := isEmpty_eq_false_of_containsKey he)
+
+theorem getKey_minKey!_eq_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he} :
+    getKey (minKey! l) l he = minKey l (isEmpty_eq_false_of_containsKey he) := by
+  simpa [minKey_eq_minKey!] using getKey_minKey hd (he := isEmpty_eq_false_of_containsKey he)
+
+theorem getKey!_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) :
+    getKey! (minKey! l) l = minKey! l := by
+  simpa [minKey_eq_minKey!] using getKey!_minKey hd (he := he)
+
+theorem getKeyD_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback} :
+    getKeyD (minKey! l) l fallback = minKey! l := by
+  simpa [minKey_eq_minKey!] using getKeyD_minKey hd (he := he)
+
+theorem minKey!_eraseKey_eq_iff_beq_minKey_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k}
+    (he : (eraseKey k l).isEmpty = false) :
+    (eraseKey k l |> minKey!) = minKey! l ↔ (k == (minKey! l)) = false := by
+  simpa [minKey_eq_minKey!] using minKey_eraseKey_eq_iff_beq_minKey_eq_false hd (he := he)
+
+theorem minKey!_eraseKey_eq_iff_beq_minKey!_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k}
+    (he : (eraseKey k l).isEmpty = false) :
+    (eraseKey k l |> minKey!) = minKey! l ↔ (k == (minKey! l)) = false := by
+  simpa [minKey_eq_minKey!] using minKey_eraseKey_eq_iff_beq_minKey_eq_false hd (he := he)
+
+theorem minKey!_eraseKey_eq_of_beq_minKey!_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k}
+    (he : (eraseKey k l).isEmpty = false) : (heq : (k == minKey! l) = false) →
+    (eraseKey k l |> minKey!) = minKey! l := by
+  simpa only [minKey_eq_minKey!] using minKey_eraseKey_eq_of_beq_minKey_eq_false hd (he := he)
+
+theorem minKey!_le_minKey!_erase [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k} (he : (eraseKey k l).isEmpty = false) :
+    compare (minKey! l) (eraseKey k l |> minKey!) |>.isLE := by
+  simpa only [minKey_eq_minKey!] using minKey_le_minKey_erase hd (he := he)
+
+theorem minKey!_insertEntryIfNew [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    (insertEntryIfNew k v l |> minKey!) =
+      (minKey? l).elim k fun k' => if compare k k' = .lt then k else k' := by
+  simpa only [minKey_eq_minKey!] using minKey_insertEntryIfNew hd
+
+theorem minKey!_insertEntryIfNew_le_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k v} :
+    compare (insertEntryIfNew k v l |> minKey!) (minKey! l) |>.isLE := by
+  simpa only [minKey_eq_minKey!] using minKey_insertEntryIfNew_le_minKey hd (he := he)
+
+theorem minKey!_insertEntryIfNew_le_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    compare (insertEntryIfNew k v l |> minKey!) k |>.isLE := by
+  simpa only [minKey_eq_minKey!] using minKey_insertEntryIfNew_le_self hd
+
+theorem minKey!_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k f} :
+    (modifyKey k f l |> minKey!) = minKey! l := by
+  cases he : l.isEmpty
+  · have := minKey_modifyKey hd (he := isEmpty_modifyKey k f l ▸ he)
+    -- fails after inlining `this`
+    simpa [minKey_eq_minKey!] using this
+  · simp_all [modifyKey]
+
+theorem minKey!_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    [Inhabited α] {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k f}
+    (he : (alterKey k f l).isEmpty = false) :
+    (alterKey k f l |> minKey!) = k ↔
+      (f (getValueCast? k l)).isSome ∧ ∀ k', containsKey k' l → (compare k k').isLE := by
+  simpa only [minKey_eq_minKey!] using minKey_alterKey_eq_self hd (he := he)
+
+namespace Const
+
+variable {β : Type v}
+
+theorem minKey!_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f} (he : (modifyKey k f l).isEmpty = false) :
+    (modifyKey k f l |> minKey!) = if (minKey! l) == k then k else (minKey! l) := by
+  simpa only [minKey_eq_minKey!] using minKey_modifyKey hd (he := he)
+
+theorem minKey!_modifyKey_eq_minKey! [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    [Inhabited α] {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f} :
+    (modifyKey k f l |> minKey!) = minKey! l := by
+  cases he : l.isEmpty
+  · have := minKey_modifyKey_eq_minKey hd (he := isEmpty_modifyKey k f l ▸ he)
+    -- fails after inlining `this`
+    simpa [minKey_eq_minKey!] using this
+  · simp_all [modifyKey]
+
+theorem minKey!_modifyKey_beq [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f} :
+    (modifyKey k f l |> minKey!) == (minKey! l) := by
+  cases he : l.isEmpty
+  · have := minKey_modifyKey_beq hd (he := isEmpty_modifyKey k f l ▸ he)
+    -- fails after inlining `this`
+    simpa [minKey_eq_minKey!] using this
+  · simp_all [modifyKey]
+
+theorem minKey!_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f} (he : (alterKey k f l).isEmpty = false) :
+    (alterKey k f l |> minKey!) = k ↔
+      (f (getValue? k l)).isSome ∧ ∀ k', containsKey k' l → (compare k k').isLE := by
+  simpa only [minKey_eq_minKey!] using minKey_alterKey_eq_self hd (he := he)
+
+end Const
+
 end Min
 
 end Std.Internal.List
diff --git a/src/Std/Data/TreeMap/Lemmas.lean b/src/Std/Data/TreeMap/Lemmas.lean
index ff9abb064c..536f077e02 100644
--- a/src/Std/Data/TreeMap/Lemmas.lean
+++ b/src/Std/Data/TreeMap/Lemmas.lean
@@ -69,6 +69,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransCmp cmp] :
     t.isEmpty = false ↔ ∃ a, a ∈ t :=
   DTreeMap.isEmpty_eq_false_iff_exists_mem
 
+theorem isEmpty_eq_false_of_contains [TransCmp cmp] {a : α} (hc : t.contains a = true) :
+    t.isEmpty = false :=
+  DTreeMap.isEmpty_eq_false_of_contains hc
+
 theorem isEmpty_iff_forall_contains [TransCmp cmp] :
     t.isEmpty = true ↔ ∀ a, t.contains a = false :=
   DTreeMap.isEmpty_iff_forall_contains
@@ -142,9 +146,9 @@ theorem isEmpty_erase [TransCmp cmp] {k : α} :
     (t.erase k).isEmpty = (t.isEmpty || (t.size == 1 && t.contains k)) :=
   DTreeMap.isEmpty_erase
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransCmp cmp] (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransCmp cmp] (k : α) :
     t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
-  DTreeMap.isEmpty_eq_isEmpty_erase_and_not_containsKey k
+  DTreeMap.isEmpty_eq_isEmpty_erase_and_not_contains k
 
 theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] {k : α}
     (he : (t.erase k).isEmpty = false) :
@@ -1930,7 +1934,7 @@ theorem minKey_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] {he k
 
 theorem minKey_insert [TransCmp cmp] {k v} :
     (t.insert k v).minKey isEmpty_insert =
-      (t.minKey?).elim k fun k' => if cmp k k'|>.isLE then k else k' :=
+      t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k' :=
   DTreeMap.minKey_insert
 
 theorem minKey_insert_le_minKey [TransCmp cmp] {k v he} :
@@ -2036,6 +2040,122 @@ theorem minKey_alter_eq_self [TransCmp cmp] {k f he} :
       (f t[k]?).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
   DTreeMap.Const.minKey_alter_eq_self
 
+theorem minKey?_eq_some_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.minKey? = some t.minKey! :=
+  DTreeMap.minKey?_eq_some_minKey! he
+
+theorem minKey!_eq_default [TransCmp cmp] [Inhabited α] (he : t.isEmpty) :
+    t.minKey! = default :=
+  DTreeMap.minKey!_eq_default he
+
+theorem minKey!_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] [Inhabited α]
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.minKey!_eq_iff_getKey?_eq_self_and_forall he
+
+theorem minKey!_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α]
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.minKey!_eq_some_iff_mem_and_forall he
+
+theorem minKey!_insert [TransCmp cmp] [Inhabited α] {k v} :
+    (t.insert k v |>.minKey!) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  DTreeMap.minKey!_insert
+
+theorem minKey!_insert_le_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k v} :
+    cmp (t.insert k v |>.minKey!) t.minKey! |>.isLE :=
+  DTreeMap.minKey!_insert_le_minKey! he
+
+theorem minKey!_insert_le_self [TransCmp cmp] [Inhabited α] {k v} :
+    cmp (t.insert k v |>.minKey!) k |>.isLE :=
+  DTreeMap.minKey!_insert_le_self
+
+theorem contains_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.contains t.minKey! :=
+  DTreeMap.contains_minKey! he
+
+theorem minKey!_mem [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.minKey! ∈ t :=
+  DTreeMap.minKey!_mem he
+
+theorem minKey!_le_of_contains [TransCmp cmp] [Inhabited α] {k} (hc : t.contains k) :
+    cmp t.minKey! k |>.isLE :=
+  DTreeMap.minKey!_le_of_contains hc
+
+theorem minKey!_le_of_mem [TransCmp cmp] [Inhabited α] {k} (hc : k ∈ t) :
+    cmp t.minKey! k |>.isLE :=
+  DTreeMap.minKey!_le_of_mem hc
+
+theorem le_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k} :
+    (cmp k t.minKey!).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.le_minKey! he
+
+theorem getKey?_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.getKey? t.minKey! = some t.minKey! :=
+  DTreeMap.getKey?_minKey! he
+
+theorem getKey_minKey! [TransCmp cmp] [Inhabited α] {hc} :
+    t.getKey t.minKey! hc = t.minKey! :=
+  DTreeMap.getKey_minKey!
+
+@[simp]
+theorem getKey_minKey!_eq_minKey [TransCmp cmp] [Inhabited α] {hc} :
+    t.getKey t.minKey! hc = t.minKey (isEmpty_eq_false_of_contains hc) :=
+  DTreeMap.getKey_minKey!_eq_minKey
+
+theorem getKey!_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.getKey! t.minKey! = t.minKey! :=
+  DTreeMap.getKey!_minKey! he
+
+theorem getKeyD_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback} :
+    t.getKeyD t.minKey! fallback = t.minKey! :=
+  DTreeMap.getKeyD_minKey! he
+
+theorem minKey!_erase_eq_of_not_compare_minKey!_eq [TransCmp cmp] [Inhabited α] {k}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k t.minKey! = .eq) :
+    (t.erase k |>.minKey!) = t.minKey! :=
+  DTreeMap.minKey!_erase_eq_of_not_compare_minKey!_eq he heq
+
+theorem minKey!_le_minKey!_erase [TransCmp cmp] [Inhabited α] {k}
+    (he : (t.erase k).isEmpty = false) :
+    cmp t.minKey! (t.erase k |>.minKey!) |>.isLE :=
+  DTreeMap.minKey!_le_minKey!_erase he
+
+theorem minKey!_insertIfNew [TransCmp cmp] [Inhabited α] {k v} :
+    (t.insertIfNew k v |>.minKey!) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.minKey!_insertIfNew
+
+theorem minKey!_insertIfNew_le_minKey! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) t.minKey! |>.isLE :=
+  DTreeMap.minKey!_insertIfNew_le_minKey! he
+
+theorem minKey!_insertIfNew_le_self [TransCmp cmp] [Inhabited α] {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) k |>.isLE :=
+  DTreeMap.minKey!_insertIfNew_le_self
+
+theorem minKey!_modify [TransCmp cmp] [Inhabited α] {k f}
+    (he : (modify t k f).isEmpty = false) :
+    (modify t k f |> minKey!) = if cmp t.minKey! k = .eq then k else t.minKey! :=
+  DTreeMap.Const.minKey!_modify he
+
+@[simp]
+theorem minKey!_modify_eq_minKey! [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] {k f} :
+    (modify t k f |>.minKey!) = t.minKey! :=
+  DTreeMap.Const.minKey!_modify_eq_minKey!
+
+@[simp]
+theorem compare_minKey!_modify_eq [TransCmp cmp] [Inhabited α] {k f} :
+    cmp (modify t k f |> minKey!) t.minKey! = .eq :=
+  DTreeMap.Const.compare_minKey!_modify_eq
+
+theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] {k f}
+    (he : (alter t k f).isEmpty = false) :
+    (alter t k f |>.minKey!) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  DTreeMap.Const.minKey!_alter_eq_self he
+
 end Min
 
 end Std.TreeMap
diff --git a/src/Std/Data/TreeMap/Raw/Lemmas.lean b/src/Std/Data/TreeMap/Raw/Lemmas.lean
index 8ca8ee261e..217a3cc176 100644
--- a/src/Std/Data/TreeMap/Raw/Lemmas.lean
+++ b/src/Std/Data/TreeMap/Raw/Lemmas.lean
@@ -70,6 +70,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransCmp cmp] (h : t.WF) :
     t.isEmpty = false ↔ ∃ a, a ∈ t :=
   DTreeMap.Raw.isEmpty_eq_false_iff_exists_mem h
 
+theorem isEmpty_eq_false_of_contains [TransCmp cmp] (h : t.WF) {a : α} (hc : t.contains a = true) :
+    t.isEmpty = false :=
+  DTreeMap.Raw.isEmpty_eq_false_of_contains h hc
+
 theorem isEmpty_iff_forall_contains [TransCmp cmp] (h : t.WF) :
     t.isEmpty = true ↔ ∀ a, t.contains a = false :=
   DTreeMap.Raw.isEmpty_iff_forall_contains h
@@ -143,9 +147,9 @@ theorem isEmpty_erase [TransCmp cmp] (h : t.WF) {k : α} :
     (t.erase k).isEmpty = (t.isEmpty || (t.size == 1 && t.contains k)) :=
   DTreeMap.Raw.isEmpty_erase h
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransCmp cmp] (h : t.WF) (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransCmp cmp] (h : t.WF) (k : α) :
     t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
-  DTreeMap.Raw.isEmpty_eq_isEmpty_erase_and_not_containsKey h k
+  DTreeMap.Raw.isEmpty_eq_isEmpty_erase_and_not_contains h k
 
 theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] (h : t.WF) {k : α}
     (he : (t.erase k).isEmpty = false) :
@@ -1921,6 +1925,119 @@ theorem minKey?_alter_eq_self [TransCmp cmp] (h : t.WF) {k f} :
       (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
   DTreeMap.Raw.Const.minKey?_alter_eq_self h
 
+theorem minKey?_eq_some_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.minKey? = some t.minKey! :=
+  DTreeMap.Raw.minKey?_eq_some_minKey! h he
+
+theorem minKey!_eq_default [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty) :
+    t.minKey! = default :=
+  DTreeMap.Raw.minKey!_eq_default h he
+
+theorem minKey!_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKey!_eq_iff_getKey?_eq_self_and_forall h he
+
+theorem minKey!_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {km} :
+    t.minKey! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKey!_eq_some_iff_mem_and_forall h he
+
+theorem minKey!_insert [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    (t.insert k v |>.minKey!) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  DTreeMap.Raw.minKey!_insert h
+
+theorem minKey!_insert_le_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false)
+    {k v} :
+    cmp (t.insert k v |>.minKey!) t.minKey! |>.isLE :=
+  DTreeMap.Raw.minKey!_insert_le_minKey! h he
+
+theorem minKey!_insert_le_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    cmp (t.insert k v |>.minKey!) k |>.isLE :=
+  DTreeMap.Raw.minKey!_insert_le_self h
+
+theorem contains_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.contains t.minKey! :=
+  DTreeMap.Raw.contains_minKey! h he
+
+theorem minKey!_mem [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.minKey! ∈ t :=
+  DTreeMap.Raw.minKey!_mem h he
+
+theorem minKey!_le_of_contains [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (hc : t.contains k) :
+    cmp t.minKey! k |>.isLE :=
+  DTreeMap.Raw.minKey!_le_of_contains h hc
+
+theorem minKey!_le_of_mem [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (hc : k ∈ t) :
+    cmp t.minKey! k |>.isLE :=
+  DTreeMap.Raw.minKey!_le_of_mem h hc
+
+theorem le_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {k} :
+    (cmp k t.minKey!).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.Raw.le_minKey! h he
+
+theorem getKey?_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.getKey? t.minKey! = some t.minKey! :=
+  DTreeMap.Raw.getKey?_minKey! h he
+
+theorem getKey_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) {hc} :
+    t.getKey t.minKey! hc = t.minKey! :=
+  DTreeMap.Raw.getKey_minKey! h
+
+theorem getKey!_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.getKey! t.minKey! = t.minKey! :=
+  DTreeMap.Raw.getKey!_minKey! h he
+
+theorem getKeyD_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getKeyD t.minKey! fallback = t.minKey! :=
+  DTreeMap.Raw.getKeyD_minKey! h he
+
+theorem minKey!_erase_eq_of_not_compare_minKey!_eq [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k t.minKey! = .eq) :
+    (t.erase k |>.minKey!) = t.minKey! :=
+  DTreeMap.Raw.minKey!_erase_eq_of_not_compare_minKey!_eq h he heq
+
+theorem minKey!_le_minKey!_erase [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) :
+    cmp t.minKey! (t.erase k |>.minKey!) |>.isLE :=
+  DTreeMap.Raw.minKey!_le_minKey!_erase h he
+
+theorem minKey!_insertIfNew [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    (t.insertIfNew k v |>.minKey!) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.Raw.minKey!_insertIfNew h
+
+theorem minKey!_insertIfNew_le_minKey! [TransCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) t.minKey! |>.isLE :=
+  DTreeMap.Raw.minKey!_insertIfNew_le_minKey! h he
+
+theorem minKey!_insertIfNew_le_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} :
+    cmp (t.insertIfNew k v |>.minKey!) k |>.isLE :=
+  DTreeMap.Raw.minKey!_insertIfNew_le_self h
+
+theorem minKey!_modify [TransCmp cmp] [Inhabited α] (h : t.WF) {k f}
+    (he : (modify t k f).isEmpty = false) :
+    (modify t k f |> minKey!) = if cmp t.minKey! k = .eq then k else t.minKey! :=
+  DTreeMap.Raw.Const.minKey!_modify h he
+
+@[simp]
+theorem minKey!_modify_eq_minKey! [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF) {k f} :
+    (modify t k f |>.minKey!) = t.minKey! :=
+  DTreeMap.Raw.Const.minKey!_modify_eq_minKey! h
+
+@[simp]
+theorem compare_minKey!_modify_eq [TransCmp cmp] [Inhabited α] (h : t.WF) {k f} :
+    cmp (modify t k f |> minKey!) t.minKey! = .eq :=
+  DTreeMap.Raw.Const.compare_minKey!_modify_eq h
+
+theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k f}
+    (he : (alter t k f).isEmpty = false) :
+    (alter t k f |>.minKey!) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  DTreeMap.Raw.Const.minKey!_alter_eq_self h he
+
 end Min
 
 end Std.TreeMap.Raw
diff --git a/src/Std/Data/TreeSet/Lemmas.lean b/src/Std/Data/TreeSet/Lemmas.lean
index 6eee9f52b4..c141b7c0fe 100644
--- a/src/Std/Data/TreeSet/Lemmas.lean
+++ b/src/Std/Data/TreeSet/Lemmas.lean
@@ -69,6 +69,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransCmp cmp] :
     t.isEmpty = false ↔ ∃ a, a ∈ t :=
   DTreeMap.isEmpty_eq_false_iff_exists_mem
 
+theorem isEmpty_eq_false_of_contains [TransCmp cmp] {a : α} (hc : t.contains a = true) :
+    t.isEmpty = false :=
+  DTreeMap.isEmpty_eq_false_of_contains hc
+
 theorem isEmpty_iff_forall_contains [TransCmp cmp] :
     t.isEmpty = true ↔ ∀ a, t.contains a = false :=
   DTreeMap.isEmpty_iff_forall_contains
@@ -148,9 +152,9 @@ theorem isEmpty_erase [TransCmp cmp] {k : α} :
     (t.erase k).isEmpty = (t.isEmpty || (t.size == 1 && t.contains k)) :=
   TreeMap.isEmpty_erase
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransCmp cmp] (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransCmp cmp] (k : α) :
     t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
-  DTreeMap.isEmpty_eq_isEmpty_erase_and_not_containsKey k
+  DTreeMap.isEmpty_eq_isEmpty_erase_and_not_contains k
 
 theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] {k : α}
     (he : (t.erase k).isEmpty = false) :
@@ -868,6 +872,88 @@ theorem min_le_min_erase [TransCmp cmp] {k he} :
       (t.erase k |>.min he) |>.isLE :=
   DTreeMap.minKey_le_minKey_erase
 
+theorem min?_eq_some_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.min? = some t.min! :=
+  DTreeMap.minKey?_eq_some_minKey! he
+
+theorem min!_eq_default [TransCmp cmp] [Inhabited α] (he : t.isEmpty) :
+    t.min! = default :=
+  DTreeMap.minKey!_eq_default he
+
+theorem min!_eq_iff_get?_eq_self_and_forall [TransCmp cmp] [Inhabited α]
+    (he : t.isEmpty = false) {km} :
+    t.min! = km ↔ t.get? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.minKey!_eq_iff_getKey?_eq_self_and_forall he
+
+theorem min!_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α]
+    (he : t.isEmpty = false) {km} :
+    t.min! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.minKey!_eq_some_iff_mem_and_forall he
+
+theorem min!_insert [TransCmp cmp] [Inhabited α] {k} :
+    (t.insert k |>.min!) =
+      t.min?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.minKey!_insertIfNew
+
+theorem min!_insert_le_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k} :
+    cmp (t.insert k |>.min!) t.min! |>.isLE :=
+  DTreeMap.minKey!_insertIfNew_le_minKey! he
+
+theorem min!_insert_le_self [TransCmp cmp] [Inhabited α] {k} :
+    cmp (t.insert k |>.min!) k |>.isLE :=
+  DTreeMap.minKey!_insertIfNew_le_self
+
+theorem contains_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.contains t.min! :=
+  DTreeMap.contains_minKey! he
+
+theorem min!_mem [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.min! ∈ t :=
+  DTreeMap.minKey!_mem he
+
+theorem min!_le_of_contains [TransCmp cmp] [Inhabited α] {k} (hc : t.contains k) :
+    cmp t.min! k |>.isLE :=
+  DTreeMap.minKey!_le_of_contains hc
+
+theorem min!_le_of_mem [TransCmp cmp] [Inhabited α] {k} (hc : k ∈ t) :
+    cmp t.min! k |>.isLE :=
+  DTreeMap.minKey!_le_of_mem hc
+
+theorem le_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {k} :
+    (cmp k t.min!).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.le_minKey! he
+
+theorem get?_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.get? t.min! = some t.min! :=
+  DTreeMap.getKey?_minKey! he
+
+theorem get_min! [TransCmp cmp] [Inhabited α] {hc} :
+    t.get t.min! hc = t.min! :=
+  DTreeMap.getKey_minKey!
+
+@[simp]
+theorem get_min!_eq_min [TransCmp cmp] [Inhabited α] {hc} :
+    t.get t.min! hc = t.min (isEmpty_eq_false_of_contains hc) :=
+  DTreeMap.getKey_minKey!_eq_minKey
+
+theorem get!_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) :
+    t.get! t.min! = t.min! :=
+  DTreeMap.getKey!_minKey! he
+
+theorem getD_min! [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback} :
+    t.getD t.min! fallback = t.min! :=
+  DTreeMap.getKeyD_minKey! he
+
+theorem min!_erase_eq_of_not_compare_min!_eq [TransCmp cmp] [Inhabited α] {k}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k t.min! = .eq) :
+    (t.erase k |>.min!) = t.min! :=
+  DTreeMap.minKey!_erase_eq_of_not_compare_minKey!_eq he heq
+
+theorem min!_le_min!_erase [TransCmp cmp] [Inhabited α] {k}
+    (he : (t.erase k).isEmpty = false) :
+    cmp t.min! (t.erase k |>.min!) |>.isLE :=
+  DTreeMap.minKey!_le_minKey!_erase he
+
 end Min
 
 end Std.TreeSet
diff --git a/src/Std/Data/TreeSet/Raw/Lemmas.lean b/src/Std/Data/TreeSet/Raw/Lemmas.lean
index 9a5a3fd5e1..0f5d42147d 100644
--- a/src/Std/Data/TreeSet/Raw/Lemmas.lean
+++ b/src/Std/Data/TreeSet/Raw/Lemmas.lean
@@ -70,6 +70,10 @@ theorem isEmpty_eq_false_iff_exists_mem [TransCmp cmp] (h : t.WF) :
     t.isEmpty = false ↔ ∃ a, a ∈ t :=
   DTreeMap.Raw.isEmpty_eq_false_iff_exists_mem h
 
+theorem isEmpty_eq_false_of_contains [TransCmp cmp] (h : t.WF) {a : α} (hc : t.contains a = true) :
+    t.isEmpty = false :=
+  TreeMap.Raw.isEmpty_eq_false_of_contains h hc
+
 theorem isEmpty_iff_forall_contains [TransCmp cmp] (h : t.WF) :
     t.isEmpty = true ↔ ∀ a, t.contains a = false :=
   DTreeMap.Raw.isEmpty_iff_forall_contains h
@@ -149,9 +153,9 @@ theorem isEmpty_erase [TransCmp cmp] (h : t.WF) {k : α} :
     (t.erase k).isEmpty = (t.isEmpty || (t.size == 1 && t.contains k)) :=
   TreeMap.Raw.isEmpty_erase h
 
-theorem isEmpty_eq_isEmpty_erase_and_not_containsKey [TransCmp cmp] (h : t.WF) (k : α) :
+theorem isEmpty_eq_isEmpty_erase_and_not_contains [TransCmp cmp] (h : t.WF) (k : α) :
     t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
-  TreeMap.Raw.isEmpty_eq_isEmpty_erase_and_not_containsKey h k
+  TreeMap.Raw.isEmpty_eq_isEmpty_erase_and_not_contains h k
 
 theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] (h : t.WF) {k : α}
     (he : (t.erase k).isEmpty = false) :
@@ -778,6 +782,84 @@ theorem min?_le_min?_erase [TransCmp cmp] (h : t.WF) {k km kme} :
     cmp km kme |>.isLE :=
   TreeMap.Raw.minKey?_le_minKey?_erase h
 
+theorem min?_eq_some_min! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.min? = some t.min! :=
+  DTreeMap.Raw.minKey?_eq_some_minKey! h he
+
+theorem min!_eq_default [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty) :
+    t.min! = default :=
+  DTreeMap.Raw.minKey!_eq_default h he
+
+theorem min!_eq_iff_get?_eq_self_and_forall [TransCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {km} :
+    t.min! = km ↔ t.get? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKey!_eq_iff_getKey?_eq_self_and_forall h he
+
+theorem min!_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {km} :
+    t.min! = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKey!_eq_some_iff_mem_and_forall h he
+
+theorem min!_insert [TransCmp cmp] [Inhabited α] (h : t.WF) {k} :
+    (t.insert k |>.min!) =
+      t.min?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.Raw.minKey!_insertIfNew h
+
+theorem min!_insert_le_min! [TransCmp cmp] [Inhabited α] (h : t.WF)
+    (he : t.isEmpty = false) {k} :
+    cmp (t.insert k |>.min!) t.min! |>.isLE :=
+  DTreeMap.Raw.minKey!_insertIfNew_le_minKey! h he
+
+theorem min!_insert_le_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k} :
+    cmp (t.insert k |>.min!) k |>.isLE :=
+  DTreeMap.Raw.minKey!_insertIfNew_le_self h
+
+theorem contains_min! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.contains t.min! :=
+  DTreeMap.Raw.contains_minKey! h he
+
+theorem min!_mem [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.min! ∈ t :=
+  DTreeMap.Raw.minKey!_mem h he
+
+theorem min!_le_of_contains [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (hc : t.contains k) :
+    cmp t.min! k |>.isLE :=
+  DTreeMap.Raw.minKey!_le_of_contains h hc
+
+theorem min!_le_of_mem [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (hc : k ∈ t) :
+    cmp t.min! k |>.isLE :=
+  DTreeMap.Raw.minKey!_le_of_mem h hc
+
+theorem le_min! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {k} :
+    (cmp k t.min!).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.Raw.le_minKey! h he
+
+theorem get?_min! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.get? t.min! = some t.min! :=
+  DTreeMap.Raw.getKey?_minKey! h he
+
+theorem get_min! [TransCmp cmp] [Inhabited α] (h : t.WF) {hc} :
+    t.get t.min! hc = t.min! :=
+  DTreeMap.Raw.getKey_minKey! h
+
+theorem get!_min! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) :
+    t.get! t.min! = t.min! :=
+  DTreeMap.Raw.getKey!_minKey! h he
+
+theorem getD_min! [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getD t.min! fallback = t.min! :=
+  DTreeMap.Raw.getKeyD_minKey! h he
+
+theorem min!_erase_eq_of_not_compare_min!_eq [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k t.min! = .eq) :
+    (t.erase k |>.min!) = t.min! :=
+  DTreeMap.Raw.minKey!_erase_eq_of_not_compare_minKey!_eq h he heq
+
+theorem min!_le_min!_erase [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) :
+    cmp t.min! (t.erase k |>.min!) |>.isLE :=
+  DTreeMap.Raw.minKey!_le_minKey!_erase h he
+
 end Min
 
 end Std.TreeSet.Raw