From 18ac05258bf27e41f1126bf0ff6a9ff842baec6e Mon Sep 17 00:00:00 2001
From: Paul Reichert <6992158+datokrat@users.noreply.github.com>
Date: Tue, 25 Mar 2025 09:18:49 +0100
Subject: [PATCH] feat: tree map lemmas for minKeyD (#7626)

This PR provides lemmas for the tree map function `minKeyD` and its
interations with other functions for which lemmas already exist.

---------

Co-authored-by: Paul Reichert <datokrat@users.noreply.github.com>
---
 src/Std/Data/DTreeMap/Internal/Lemmas.lean    | 217 +++++++++++++++++-
 src/Std/Data/DTreeMap/Internal/Model.lean     |   4 +
 src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean |   5 +
 src/Std/Data/DTreeMap/Lemmas.lean             | 134 +++++++++++
 src/Std/Data/DTreeMap/Raw/Lemmas.lean         | 134 +++++++++++
 src/Std/Data/Internal/List/Associative.lean   | 206 +++++++++++++++++
 src/Std/Data/TreeMap/Lemmas.lean              | 118 ++++++++++
 src/Std/Data/TreeMap/Raw/Lemmas.lean          | 118 ++++++++++
 src/Std/Data/TreeSet/Lemmas.lean              |  82 +++++++
 src/Std/Data/TreeSet/Raw/Lemmas.lean          |  82 +++++++
 10 files changed, 1099 insertions(+), 1 deletion(-)

diff --git a/src/Std/Data/DTreeMap/Internal/Lemmas.lean b/src/Std/Data/DTreeMap/Internal/Lemmas.lean
index dbd6560a85..fe4d3d3251 100644
--- a/src/Std/Data/DTreeMap/Internal/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Internal/Lemmas.lean
@@ -98,7 +98,8 @@ private def queryMap : Std.DHashMap Name (fun _ => Name × Array (MacroM (TSynta
      ⟨`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 _)])⟩,
+     ⟨`minKeyD, (``minKeyD_eq_minKeyD, #[``(minKeyD_of_perm _)])⟩]
 
 /-- Internal implementation detail of the tree map -/
 scoped syntax "simp_to_model" (" [" (ident,*) "]")? ("using" term)? : tactic
@@ -4831,6 +4832,220 @@ theorem minKey!_alter!_eq_self [TransOrd α] [Inhabited α] (h : t.WF) {k f} :
 
 end Const
 
+theorem minKey_eq_minKeyD [TransOrd α] (h : t.WF) {he fallback} :
+    t.minKey he = t.minKeyD fallback := by
+  simp_to_model [minKey, minKeyD] using List.minKey_eq_minKeyD
+
+theorem minKey?_eq_some_minKeyD [TransOrd α] (h : t.WF) {fallback} :
+    (he : t.isEmpty = false) → t.minKey? = some (t.minKeyD fallback) := by
+  simp_to_model [minKey?, minKeyD, isEmpty] using List.minKey?_eq_some_minKeyD
+
+theorem minKeyD_eq_fallback [TransOrd α] (h : t.WF) {fallback} :
+    (he : t.isEmpty) → t.minKeyD fallback = fallback := by
+  simp_to_model [minKeyD, isEmpty] using List.minKeyD_eq_fallback
+
+theorem minKey!_eq_minKeyD_default [TransOrd α] [Inhabited α] (h : t.WF) :
+    t.minKey! = t.minKeyD default := by
+  simp_to_model [minKey!, minKeyD] using List.minKey!_eq_minKeyD_default
+
+theorem minKeyD_eq_iff_getKey?_eq_self_and_forall [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {km fallback},
+    t.minKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (compare km k).isLE := by
+  simp_to_model [minKeyD, getKey?, contains, isEmpty] using
+    List.minKeyD_eq_iff_getKey?_eq_self_and_forall
+
+theorem minKeyD_eq_some_iff_mem_and_forall [TransOrd α]
+    [LawfulEqOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {km fallback},
+    t.minKeyD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (compare km k).isLE := by
+  simp_to_model [minKeyD, contains, isEmpty] using List.minKeyD_eq_some_iff_mem_and_forall
+
+theorem minKeyD_insert [TransOrd α] (h : t.WF) {k v fallback} :
+    (t.insert k v h.balanced |>.impl.minKeyD <| fallback) =
+      (t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simp_to_model [minKeyD, minKey?, insert] using List.minKeyD_insertEntry
+
+theorem minKeyD_insert! [TransOrd α] (h : t.WF) {k v fallback} :
+    (t.insert! k v |>.minKeyD fallback) =
+      (t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simpa [insert_eq_insert!] using minKeyD_insert h
+
+theorem minKeyD_insert_le_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v fallback},
+    compare (t.insert k v h.balanced |>.impl.minKeyD <| fallback) (t.minKeyD fallback) |>.isLE := by
+  simp_to_model [minKeyD, isEmpty, insert] using List.minKeyD_insertEntry_le_minKeyD
+
+theorem minKeyD_insert!_le_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v fallback},
+    compare (t.insert! k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE := by
+  simpa only [insert_eq_insert!] using minKeyD_insert_le_minKeyD h
+
+theorem minKeyD_insert_le_self [TransOrd α] (h : t.WF) {k v fallback} :
+    compare (t.insert k v h.balanced |>.impl.minKeyD <| fallback) k |>.isLE := by
+  simp_to_model [minKeyD, insert] using List.minKeyD_insertEntry_le_self
+
+theorem minKeyD_insert!_le_self [TransOrd α] (h : t.WF) {k v fallback} :
+    compare (t.insert! k v |>.minKeyD fallback) k |>.isLE := by
+  simpa only [insert_eq_insert!] using minKeyD_insert_le_self h
+
+theorem contains_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {fallback}, t.contains (t.minKeyD fallback) := by
+  simp_to_model [minKeyD, isEmpty, contains] using List.containsKey_minKeyD
+
+theorem minKeyD_mem [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {fallback}, (t.minKeyD fallback) ∈ t :=
+  contains_minKeyD h
+
+theorem minKeyD_le_of_contains [TransOrd α] (h : t.WF) :
+    ∀ {k}, (hc : t.contains k) → ∀ {fallback},
+    compare (t.minKeyD fallback) k |>.isLE := by
+  simp_to_model [minKeyD, contains] using List.minKeyD_le_of_containsKey
+
+theorem minKeyD_le_of_mem [TransOrd α] (h : t.WF) :
+    ∀ {k}, (hc : k ∈ t) → ∀ {fallback},
+    compare (t.minKeyD fallback) k |>.isLE :=
+  minKeyD_le_of_contains h
+
+theorem le_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k fallback},
+    (compare k (t.minKeyD fallback)).isLE ↔ (∀ k', k' ∈ t → (compare k k').isLE) := by
+  simp_to_model [minKeyD, contains, isEmpty] using List.le_minKeyD
+
+theorem getKey?_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {fallback},
+    t.getKey? (t.minKeyD fallback) = some (t.minKeyD fallback) := by
+  simp_to_model [minKeyD, getKey?, isEmpty] using List.getKey?_minKeyD
+
+theorem getKey_minKeyD [TransOrd α] (h : t.WF) : ∀ {fallback he},
+    t.getKey (t.minKeyD fallback) he = (t.minKeyD fallback) := by
+  simp_to_model [minKeyD, contains, isEmpty, getKey] using List.getKey_minKeyD
+
+theorem getKey_minKeyD_eq_minKey [TransOrd α] (h : t.WF) : ∀ {fallback hc},
+    t.getKey (t.minKeyD fallback) hc = t.minKey (isEmpty_eq_false_of_contains h hc) := by
+  simp_to_model [minKeyD, minKey, contains, isEmpty, getKey] using List.getKey_minKeyD_eq_minKey
+
+theorem getKey!_minKeyD [TransOrd α] [Inhabited α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {fallback},
+    t.getKey! (t.minKeyD fallback) = (t.minKeyD fallback) := by
+  simp_to_model [minKeyD, isEmpty, getKey!] using List.getKey!_minKeyD
+
+theorem getKeyD_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {fallback fallback'},
+    t.getKeyD (t.minKeyD fallback) fallback' = t.minKeyD fallback := by
+  simp_to_model [minKeyD, getKeyD, isEmpty] using List.getKeyD_minKeyD
+
+theorem minKeyD_erase_eq_iff_not_compare_minKeyD_eq [TransOrd α] (h : t.WF) :
+    ∀ {k fallback}, (he : (t.erase k h.balanced).impl.isEmpty = false) →
+    (t.erase k h.balanced |>.impl.minKeyD <| fallback) = t.minKeyD fallback ↔
+      ¬ compare k (t.minKeyD fallback) = .eq := by
+  simp_to_model [minKeyD, isEmpty, erase] using List.minKeyD_eraseKey_eq_iff_beq_minKeyD_eq_false
+
+theorem minKeyD_erase!_eq_iff_not_compare_minKeyD_eq [TransOrd α] (h : t.WF) {k fallback} :
+    (he : (t.erase! k).isEmpty = false) →
+    (t.erase! k |>.minKeyD fallback) = t.minKeyD fallback ↔
+      ¬ compare k (t.minKeyD fallback) = .eq := by
+  simpa only [erase_eq_erase!] using minKeyD_erase_eq_iff_not_compare_minKeyD_eq h
+
+theorem minKeyD_erase_eq_of_not_compare_minKeyD_eq [TransOrd α] (h : t.WF) :
+    ∀ {k fallback}, (he : (t.erase k h.balanced).impl.isEmpty = false) →
+    (heq : ¬ compare k (t.minKeyD fallback) = .eq) →
+    (t.erase k h.balanced |>.impl.minKeyD <| fallback) = t.minKeyD fallback := by
+  simp_to_model [minKeyD, isEmpty, erase] using
+    List.minKeyD_eraseKey_eq_of_beq_minKeyD_eq_false
+
+theorem minKeyD_erase!_eq_of_not_compare_minKeyD_eq [TransOrd α] (h : t.WF) {k fallback} :
+    (he : (t.erase! k).isEmpty = false) → (heq : ¬ compare k (t.minKeyD fallback) = .eq) →
+    (t.erase! k |>.minKeyD fallback) = t.minKeyD fallback := by
+  simpa only [erase_eq_erase!] using minKeyD_erase_eq_of_not_compare_minKeyD_eq h
+
+theorem minKeyD_le_minKeyD_erase [TransOrd α] (h : t.WF) :
+    ∀ {k}, (he : (t.erase k h.balanced).impl.isEmpty = false) → ∀ {fallback},
+    compare (t.minKeyD fallback) (t.erase k h.balanced |>.impl.minKeyD <| fallback) |>.isLE := by
+  simp_to_model [minKeyD, isEmpty, erase] using List.minKeyD_le_minKeyD_erase
+
+theorem minKeyD_le_minKeyD_erase! [TransOrd α] (h : t.WF) {k} :
+    (he : (t.erase! k).isEmpty = false) → ∀ {fallback},
+    compare (t.minKeyD fallback) (t.erase! k |>.minKeyD fallback) |>.isLE := by
+  simpa only [erase_eq_erase!] using minKeyD_le_minKeyD_erase h
+
+theorem minKeyD_insertIfNew [TransOrd α] (h : t.WF) : ∀ {k v fallback},
+    (t.insertIfNew k v h.balanced |>.impl.minKeyD <| fallback) =
+      t.minKey?.elim k fun k' => if compare k k' = .lt then k else k' := by
+  simp_to_model [minKeyD, minKey?, insertIfNew] using List.minKeyD_insertEntryIfNew
+
+theorem minKeyD_insertIfNew! [TransOrd α] (h : t.WF) {k v fallback} :
+    (t.insertIfNew! k v |>.minKeyD fallback) =
+      t.minKey?.elim k fun k' => if compare k k' = .lt then k else k' := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKeyD_insertIfNew h
+
+theorem minKeyD_insertIfNew_le_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v fallback},
+    compare (t.insertIfNew k v h.balanced |>.impl.minKeyD <| fallback)
+      (t.minKeyD fallback) |>.isLE := by
+  simp_to_model [minKeyD, isEmpty, insertIfNew] using List.minKeyD_insertEntryIfNew_le_minKeyD
+
+theorem minKeyD_insertIfNew!_le_minKeyD [TransOrd α] (h : t.WF) :
+    (he : t.isEmpty = false) → ∀ {k v fallback},
+    compare (t.insertIfNew! k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKeyD_insertIfNew_le_minKeyD h
+
+theorem minKeyD_insertIfNew_le_self [TransOrd α] (h : t.WF) : ∀ {k v fallback},
+    compare (t.insertIfNew k v h.balanced |>.impl.minKeyD <| fallback) k |>.isLE := by
+  simp_to_model [minKeyD, insertIfNew] using List.minKeyD_insertEntryIfNew_le_self
+
+theorem minKeyD_insertIfNew!_le_self [TransOrd α] (h : t.WF) {k v fallback} :
+    compare (t.insertIfNew! k v |>.minKeyD fallback) k |>.isLE := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKeyD_insertIfNew_le_self h
+
+theorem minKeyD_modify [TransOrd α] [LawfulEqOrd α] (h : t.WF) : ∀ {k f fallback},
+    (t.modify k f |>.minKeyD fallback) = t.minKeyD fallback := by
+  simp_to_model [minKeyD, modify] using List.minKeyD_modifyKey
+
+theorem minKeyD_alter_eq_self [TransOrd α] [LawfulEqOrd α] (h : t.WF) :
+    ∀ {k f fallback}, (he : (t.alter k f h.balanced).impl.isEmpty = false) →
+    (t.alter k f h.balanced |>.impl.minKeyD <| fallback) = k ↔
+      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simp_to_model [minKeyD, alter, isEmpty, contains, get?] using List.minKeyD_alterKey_eq_self
+
+theorem minKeyD_alter!_eq_self [TransOrd α] [LawfulEqOrd α] (h : t.WF) {k f fallback} :
+    (he : (t.alter! k f).isEmpty = false) →
+    (t.alter! k f |>.minKeyD fallback) = k ↔
+      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simpa only [alter_eq_alter!] using minKeyD_alter_eq_self h
+
+namespace Const
+
+variable {β : Type v} {t : Impl α β}
+
+theorem minKeyD_modify [TransOrd α] (h : t.WF) :
+    ∀ {k f}, (he : (modify k f t).isEmpty = false) → ∀ {fallback},
+    (modify k f t |>.minKeyD fallback) =
+      if compare (t.minKeyD fallback) k = .eq then k else t.minKeyD fallback := by
+  simp_to_model [minKeyD, minKey, isEmpty, Const.modify] using List.Const.minKeyD_modifyKey
+
+theorem minKeyD_modify_eq_minKeyD [TransOrd α] [LawfulEqOrd α] (h : t.WF) :
+    ∀ {k f fallback}, (modify k f t |>.minKeyD fallback) = t.minKeyD fallback := by
+  simp_to_model [minKeyD, Const.modify] using List.Const.minKeyD_modifyKey_eq_minKeyD
+
+theorem compare_minKeyD_modify_eq [TransOrd α] (h : t.WF) : ∀ {k f fallback},
+    compare (Const.modify k f t |>.minKeyD fallback) (t.minKeyD fallback) = .eq := by
+  simp_to_model [minKeyD, Const.modify] using List.Const.minKeyD_modifyKey_beq
+
+theorem minKeyD_alter_eq_self [TransOrd α] (h : t.WF) :
+    ∀ {k f}, (he : (alter k f t h.balanced).impl.isEmpty = false) → ∀ {fallback},
+    (alter k f t h.balanced |>.impl.minKeyD <| fallback) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simp_to_model [minKeyD, Const.alter, contains, isEmpty, Const.get?] using
+    List.Const.minKeyD_alterKey_eq_self
+
+theorem minKeyD_alter!_eq_self [TransOrd α] (h : t.WF) {k f} :
+    (he : (alter! k f t).isEmpty = false) → ∀ {fallback},
+    (alter! k f t |>.minKeyD fallback) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simpa only [alter_eq_alter!] using minKeyD_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 edd53d326a..7b2c30495a 100644
--- a/src/Std/Data/DTreeMap/Internal/Model.lean
+++ b/src/Std/Data/DTreeMap/Internal/Model.lean
@@ -480,6 +480,10 @@ 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 minKeyD_eq_getD_minKey? [Ord α] {l : Impl α β} {fallback} :
+    l.minKeyD fallback = l.minKey?.getD fallback := by
+  induction l, fallback using minKeyD.induct <;> simp_all only [minKeyD, 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 f188f4f7bb..ced33037cd 100644
--- a/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
@@ -1837,4 +1837,9 @@ theorem minKey!_eq_minKey! [Ord α] [TransOrd α] [Inhabited α] {l : Impl α β
     l.minKey! = List.minKey! l.toListModel := by
   simp [Impl.minKey!_eq_get!_minKey?, List.minKey!_eq_get!_minKey?, minKey?_eq_minKey? hlo]
 
+theorem minKeyD_eq_minKeyD [Ord α] [TransOrd α] {l : Impl α β} (hlo : l.Ordered)
+    {fallback} :
+    l.minKeyD fallback = List.minKeyD l.toListModel fallback := by
+  simp [Impl.minKeyD_eq_getD_minKey?, List.minKeyD_eq_getD_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 88ccf8b036..e1b6ed073e 100644
--- a/src/Std/Data/DTreeMap/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Lemmas.lean
@@ -3236,6 +3236,140 @@ theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] {k f}
 
 end Const
 
+theorem minKey?_eq_some_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.minKey? = some (t.minKeyD fallback) :=
+  Impl.minKey?_eq_some_minKeyD t.wf he
+
+theorem minKeyD_eq_fallback [TransCmp cmp] (he : t.isEmpty) {fallback} :
+    t.minKeyD fallback = fallback :=
+  Impl.minKeyD_eq_fallback t.wf he
+
+theorem minKey!_eq_minKeyD_default [TransCmp cmp] [Inhabited α] :
+    t.minKey! = t.minKeyD default :=
+  Impl.minKey!_eq_minKeyD_default t.wf
+
+theorem minKeyD_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp]
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKeyD_eq_iff_getKey?_eq_self_and_forall t.wf he
+
+theorem minKeyD_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp]
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKeyD_eq_some_iff_mem_and_forall t.wf he
+
+theorem minKeyD_insert [TransCmp cmp] {k v fallback} :
+    (t.insert k v |>.minKeyD fallback) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  Impl.minKeyD_insert t.wf
+
+theorem minKeyD_insert_le_minKeyD [TransCmp cmp] (he : t.isEmpty = false)
+    {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  Impl.minKeyD_insert_le_minKeyD t.wf he (instOrd := ⟨cmp⟩)
+
+theorem minKeyD_insert_le_self [TransCmp cmp] {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_insert_le_self t.wf (instOrd := ⟨cmp⟩)
+
+theorem contains_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.contains (t.minKeyD fallback) :=
+  Impl.contains_minKeyD t.wf he
+
+theorem minKeyD_mem [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.minKeyD fallback ∈ t :=
+  Impl.minKeyD_mem t.wf he
+
+theorem minKeyD_le_of_contains [TransCmp cmp] {k} (hc : t.contains k) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_le_of_contains t.wf hc
+
+theorem minKeyD_le_of_mem [TransCmp cmp] {k} (hc : k ∈ t) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_le_of_mem t.wf hc
+
+theorem le_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {k fallback} :
+    (cmp k (t.minKeyD fallback)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  Impl.le_minKeyD t.wf he (instOrd := ⟨cmp⟩)
+
+theorem getKey?_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.getKey? (t.minKeyD fallback) = some (t.minKeyD fallback) :=
+  Impl.getKey?_minKeyD t.wf he
+
+theorem getKey_minKeyD [TransCmp cmp] {fallback hc} :
+    t.getKey (t.minKeyD fallback) hc = t.minKeyD fallback :=
+  Impl.getKey_minKeyD t.wf
+
+theorem getKey!_minKeyD [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback} :
+    t.getKey! (t.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.getKey!_minKeyD t.wf he
+
+theorem getKeyD_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback fallback'} :
+    t.getKeyD (t.minKeyD fallback) fallback' = t.minKeyD fallback :=
+  Impl.getKeyD_minKeyD t.wf he
+
+theorem minKeyD_erase_eq_of_not_compare_minKeyD_eq [TransCmp cmp] {k fallback}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k (t.minKeyD fallback) = .eq) :
+    (t.erase k |>.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.minKeyD_erase_eq_of_not_compare_minKeyD_eq t.wf he heq
+
+theorem minKeyD_le_minKeyD_erase [TransCmp cmp] {k}
+    (he : (t.erase k).isEmpty = false) {fallback} :
+    cmp (t.minKeyD fallback) (t.erase k |>.minKeyD fallback) |>.isLE :=
+  Impl.minKeyD_le_minKeyD_erase t.wf he
+
+theorem minKeyD_insertIfNew [TransCmp cmp] {k v fallback} :
+    (t.insertIfNew k v |>.minKeyD fallback) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  Impl.minKeyD_insertIfNew t.wf
+
+theorem minKeyD_insertIfNew_le_minKeyD [TransCmp cmp]
+    (he : t.isEmpty = false) {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  Impl.minKeyD_insertIfNew_le_minKeyD t.wf he (instOrd := ⟨cmp⟩)
+
+theorem minKeyD_insertIfNew_le_self [TransCmp cmp] {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_insertIfNew_le_self t.wf (instOrd := ⟨cmp⟩)
+
+@[simp]
+theorem minKeyD_modify [TransCmp cmp] [LawfulEqCmp cmp] {k f fallback} :
+    (t.modify k f |>.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.minKeyD_modify t.wf
+
+theorem minKeyD_alter_eq_self [TransCmp cmp] [LawfulEqCmp cmp] {k f}
+    (he : (t.alter k f).isEmpty = false) {fallback} :
+    (t.alter k f |>.minKeyD fallback) = k ↔ (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.minKeyD_alter_eq_self t.wf he
+
+namespace Const
+
+variable {β : Type v} {t : DTreeMap α β cmp}
+
+theorem minKeyD_modify [TransCmp cmp] {k f}
+    (he : (modify t k f).isEmpty = false) {fallback} :
+    (modify t k f |>.minKeyD fallback) =
+      if cmp (t.minKeyD fallback) k = .eq then k else (t.minKeyD fallback) :=
+  Impl.Const.minKeyD_modify t.wf he
+
+@[simp]
+theorem minKeyD_modify_eq_minKeyD [TransCmp cmp] [LawfulEqCmp cmp] {k f fallback} :
+    (modify t k f |>.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.Const.minKeyD_modify_eq_minKeyD t.wf
+
+@[simp]
+theorem compare_minKeyD_modify_eq [TransCmp cmp] {k f fallback} :
+    cmp (modify t k f |>.minKeyD fallback) (t.minKeyD fallback) = .eq :=
+  Impl.Const.compare_minKeyD_modify_eq t.wf (instOrd := ⟨cmp⟩)
+
+theorem minKeyD_alter_eq_self [TransCmp cmp] {k f}
+    (he : (alter t k f).isEmpty = false) {fallback} :
+    (alter t k f |>.minKeyD fallback) = k ↔
+      (f (Const.get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.Const.minKeyD_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 e09b98421f..3a0b06318c 100644
--- a/src/Std/Data/DTreeMap/Raw/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Raw/Lemmas.lean
@@ -3097,6 +3097,140 @@ theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k f}
 
 end Const
 
+theorem minKey?_eq_some_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.minKey? = some (t.minKeyD fallback) :=
+  Impl.minKey?_eq_some_minKeyD h he
+
+theorem minKeyD_eq_fallback [TransCmp cmp] (h : t.WF) (he : t.isEmpty) {fallback} :
+    t.minKeyD fallback = fallback :=
+  Impl.minKeyD_eq_fallback h he
+
+theorem minKey!_eq_minKeyD_default [TransCmp cmp] [Inhabited α] (h : t.WF) :
+    t.minKey! = t.minKeyD default :=
+  Impl.minKey!_eq_minKeyD_default h
+
+theorem minKeyD_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKeyD_eq_iff_getKey?_eq_self_and_forall h he
+
+theorem minKeyD_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  Impl.minKeyD_eq_some_iff_mem_and_forall h he
+
+theorem minKeyD_insert [TransCmp cmp] (h : t.WF) {k v fallback} :
+    (t.insert k v |>.minKeyD fallback) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  Impl.minKeyD_insert! h
+
+theorem minKeyD_insert_le_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false)
+    {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  Impl.minKeyD_insert!_le_minKeyD h he (instOrd := ⟨cmp⟩)
+
+theorem minKeyD_insert_le_self [TransCmp cmp] (h : t.WF) {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_insert!_le_self h (instOrd := ⟨cmp⟩)
+
+theorem contains_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.contains (t.minKeyD fallback) :=
+  Impl.contains_minKeyD h he
+
+theorem minKeyD_mem [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.minKeyD fallback ∈ t :=
+  Impl.minKeyD_mem h he
+
+theorem minKeyD_le_of_contains [TransCmp cmp] (h : t.WF) {k} (hc : t.contains k) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_le_of_contains h hc
+
+theorem minKeyD_le_of_mem [TransCmp cmp] (h : t.WF) {k} (hc : k ∈ t) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_le_of_mem h hc
+
+theorem le_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {k fallback} :
+    (cmp k (t.minKeyD fallback)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  Impl.le_minKeyD h he (instOrd := ⟨cmp⟩)
+
+theorem getKey?_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getKey? (t.minKeyD fallback) = some (t.minKeyD fallback) :=
+  Impl.getKey?_minKeyD h he
+
+theorem getKey_minKeyD [TransCmp cmp] (h : t.WF) {fallback hc} :
+    t.getKey (t.minKeyD fallback) hc = t.minKeyD fallback :=
+  Impl.getKey_minKeyD h
+
+theorem getKey!_minKeyD [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getKey! (t.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.getKey!_minKeyD h he
+
+theorem getKeyD_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback fallback'} :
+    t.getKeyD (t.minKeyD fallback) fallback' = t.minKeyD fallback :=
+  Impl.getKeyD_minKeyD h he
+
+theorem minKeyD_erase_eq_of_not_compare_minKeyD_eq [TransCmp cmp] (h : t.WF) {k fallback}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k (t.minKeyD fallback) = .eq) :
+    (t.erase k |>.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.minKeyD_erase!_eq_of_not_compare_minKeyD_eq h he heq
+
+theorem minKeyD_le_minKeyD_erase [TransCmp cmp] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) {fallback} :
+    cmp (t.minKeyD fallback) (t.erase k |>.minKeyD fallback) |>.isLE :=
+  Impl.minKeyD_le_minKeyD_erase! h he
+
+theorem minKeyD_insertIfNew [TransCmp cmp] (h : t.WF) {k v fallback} :
+    (t.insertIfNew k v |>.minKeyD fallback) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  Impl.minKeyD_insertIfNew! h
+
+theorem minKeyD_insertIfNew_le_minKeyD [TransCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  Impl.minKeyD_insertIfNew!_le_minKeyD h he (instOrd := ⟨cmp⟩)
+
+theorem minKeyD_insertIfNew_le_self [TransCmp cmp] (h : t.WF) {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) k |>.isLE :=
+  Impl.minKeyD_insertIfNew!_le_self h (instOrd := ⟨cmp⟩)
+
+@[simp]
+theorem minKeyD_modify [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF) {k f fallback} :
+    (t.modify k f |>.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.minKeyD_modify h
+
+theorem minKeyD_alter_eq_self [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF) {k f}
+    (he : (t.alter k f).isEmpty = false) {fallback} :
+    (t.alter k f |>.minKeyD fallback) = k ↔ (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.minKeyD_alter!_eq_self h he
+
+namespace Const
+
+variable {β : Type v} {t : Raw α β cmp}
+
+theorem minKeyD_modify [TransCmp cmp] (h : t.WF) {k f}
+    (he : (modify t k f).isEmpty = false) {fallback} :
+    (modify t k f |>.minKeyD fallback) =
+      if cmp (t.minKeyD fallback) k = .eq then k else (t.minKeyD fallback) :=
+  Impl.Const.minKeyD_modify h he
+
+@[simp]
+theorem minKeyD_modify_eq_minKeyD [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF) {k f fallback} :
+    (modify t k f |>.minKeyD fallback) = t.minKeyD fallback :=
+  Impl.Const.minKeyD_modify_eq_minKeyD h
+
+@[simp]
+theorem compare_minKeyD_modify_eq [TransCmp cmp] (h : t.WF) {k f fallback} :
+    cmp (modify t k f |>.minKeyD fallback) (t.minKeyD fallback) = .eq :=
+  Impl.Const.compare_minKeyD_modify_eq h (instOrd := ⟨cmp⟩)
+
+theorem minKeyD_alter_eq_self [TransCmp cmp] (h : t.WF) {k f}
+    (he : (alter t k f).isEmpty = false) {fallback} :
+    (alter t k f |>.minKeyD fallback) = k ↔
+      (f (Const.get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.Const.minKeyD_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 9c03ce3e41..fd8db51dfe 100644
--- a/src/Std/Data/Internal/List/Associative.lean
+++ b/src/Std/Data/Internal/List/Associative.lean
@@ -5131,6 +5131,212 @@ theorem minKey!_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd 
 
 end Const
 
+/-- Returns the smallest key in an associative list or `fallback` if the list is empty. -/
+def minKeyD [Ord α] (xs : List ((a : α) × β a)) (fallback : α) : α :=
+  minKey? xs |>.getD fallback
+
+theorem minKeyD_of_perm [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l l' : List ((a : α) × β a)} {fallback} (hd : DistinctKeys l) (hp : l.Perm l') :
+    minKeyD l fallback = minKeyD l' fallback := by
+  simp [minKeyD, minKey?_of_perm hd hp]
+
+theorem minKeyD_eq_getD_minKey? [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} {fallback} :
+    minKeyD l fallback = (minKey? l).getD fallback :=
+  rfl
+
+theorem minKey_eq_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} {he fallback} :
+    minKey l he = minKeyD l fallback := by
+  simp [minKey_eq_get_minKey?, minKeyD_eq_getD_minKey?, Option.get_eq_getD (fallback := fallback)]
+
+theorem minKey?_eq_some_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} {fallback} (he : l.isEmpty = false) :
+    minKey? l = some (minKeyD l fallback) := by
+  simp [← minKey_eq_minKeyD (he := he), minKey_eq_get_minKey?]
+
+theorem minKey!_eq_minKeyD_default [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} :
+    minKey! l = minKeyD l default := by
+  simp [minKey!_eq_get!_minKey?, minKeyD_eq_getD_minKey?, Option.get!_eq_getD]
+
+theorem minKeyD_eq_fallback [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} {fallback} (h : l.isEmpty) :
+    minKeyD l fallback = fallback := by
+  simp [minKeyD, minKey?_eq_none_iff_isEmpty.mpr h]
+
+theorem minKeyD_eq_iff_getKey?_eq_self_and_forall [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {km fallback} :
+    minKeyD l fallback = km ↔
+      getKey? km l = some km ∧ ∀ k, containsKey k l → (compare km k).isLE := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_eq_iff_getKey?_eq_self_and_forall hd (he := he)
+
+theorem minKeyD_eq_some_iff_mem_and_forall [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    [LawfulEqOrd α] {l : List ((a : α) × β a)} (hd : DistinctKeys l)
+    (he : l.isEmpty = false) {km fallback} :
+    minKeyD l fallback = km ↔ containsKey km l ∧ ∀ k, containsKey k l → (compare km k).isLE := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_eq_some_iff_mem_and_forall hd (he := he)
+
+theorem minKeyD_insertEntry [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v fallback} :
+    (insertEntry k v l |> minKeyD <| fallback) =
+      ((minKey? l).elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using minKey_insertEntry hd
+
+theorem minKeyD_insertEntry_le_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k v fallback} :
+    compare (insertEntry k v l |> minKeyD <| fallback) (minKeyD l fallback) |>.isLE := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using minKey_insertEntry_le_minKey hd (he := he)
+
+theorem minKeyD_insertEntry_le_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v fallback} :
+    compare (insertEntry k v l |> minKeyD <| fallback) k |>.isLE := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using minKey_insertEntry_le_self hd
+
+theorem containsKey_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback} :
+    containsKey (minKeyD l fallback) l := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using containsKey_minKey hd (he := he)
+
+theorem minKeyD_le_of_containsKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k} (hc : containsKey k l) {fallback} :
+    compare (minKeyD l fallback) k |>.isLE := by
+   simpa [minKey_eq_minKeyD (fallback := fallback)] using minKey_le_of_containsKey hd hc
+
+theorem le_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k fallback} :
+    (compare k (minKeyD l fallback)).isLE ↔ (∀ k', containsKey k' l → (compare k k').isLE) := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using le_minKey hd (he := he)
+
+theorem getKey?_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback} :
+    getKey? (minKeyD l fallback) l = some (minKeyD l fallback) := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using getKey?_minKey hd (he := he)
+
+theorem getKey_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {fallback he} :
+    getKey (minKeyD l fallback) l he = minKeyD l fallback := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using
+    getKey_minKey hd (he := isEmpty_eq_false_of_containsKey he)
+
+theorem getKey_minKeyD_eq_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {fallback he} :
+    getKey (minKeyD l fallback) l he = minKey l (isEmpty_eq_false_of_containsKey he) := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using
+    getKey_minKey hd (he := isEmpty_eq_false_of_containsKey he)
+
+theorem getKey!_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback} :
+    getKey! (minKeyD l fallback) l = minKeyD l fallback := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using getKey!_minKey hd (he := he)
+
+theorem getKeyD_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {fallback fallback'} :
+    getKeyD (minKeyD l fallback) l fallback' = minKeyD l fallback := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using getKeyD_minKey hd (he := he)
+
+theorem minKeyD_eraseKey_eq_iff_beq_minKey_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback}
+    (he : (eraseKey k l).isEmpty = false) :
+    (eraseKey k l |> minKeyD <| fallback) = minKeyD l fallback ↔
+      (k == (minKeyD l fallback)) = false := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_eraseKey_eq_iff_beq_minKey_eq_false hd (he := he)
+
+theorem minKeyD_eraseKey_eq_iff_beq_minKeyD_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback}
+    (he : (eraseKey k l).isEmpty = false) :
+    (eraseKey k l |> minKeyD <| fallback) = minKeyD l fallback ↔
+      (k == (minKeyD l fallback)) = false := by
+  simpa [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_eraseKey_eq_iff_beq_minKey_eq_false hd (he := he)
+
+theorem minKeyD_eraseKey_eq_of_beq_minKeyD_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k fallback}
+    (he : (eraseKey k l).isEmpty = false) : (heq : (k == minKeyD l fallback) = false) →
+    (eraseKey k l |> minKeyD <| fallback) = minKeyD l fallback:= by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_eraseKey_eq_of_beq_minKey_eq_false hd (he := he)
+
+theorem minKeyD_le_minKeyD_erase [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k} (he : (eraseKey k l).isEmpty = false)
+    {fallback} :
+    compare (minKeyD l fallback) (eraseKey k l |> minKeyD <| fallback) |>.isLE := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_le_minKey_erase hd (he := he)
+
+theorem minKeyD_insertEntryIfNew [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v fallback} :
+    (insertEntryIfNew k v l |> minKeyD <| fallback) =
+      (minKey? l).elim k fun k' => if compare k k' = .lt then k else k' := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using minKey_insertEntryIfNew hd
+
+theorem minKeyD_insertEntryIfNew_le_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (he : l.isEmpty = false) {k v fallback} :
+    compare (insertEntryIfNew k v l |> minKeyD <| fallback) (minKeyD l fallback) |>.isLE := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using
+    minKey_insertEntryIfNew_le_minKey hd (he := he)
+
+theorem minKeyD_insertEntryIfNew_le_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v fallback} :
+    compare (insertEntryIfNew k v l |> minKeyD <| fallback) k |>.isLE := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using minKey_insertEntryIfNew_le_self hd
+
+theorem minKeyD_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k f fallback} :
+    (modifyKey k f l |> minKeyD <| fallback) = minKeyD l fallback := by
+  cases he : l.isEmpty
+  · have := minKey_modifyKey hd (he := isEmpty_modifyKey k f l ▸ he)
+    -- fails after inlining `this`
+    simpa [minKey_eq_minKeyD (fallback := fallback)] using this
+  · simp_all [modifyKey]
+
+theorem minKeyD_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k f fallback}
+    (he : (alterKey k f l).isEmpty = false) :
+    (alterKey k f l |> minKeyD <| fallback) = k ↔
+      (f (getValueCast? k l)).isSome ∧ ∀ k', containsKey k' l → (compare k k').isLE := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using minKey_alterKey_eq_self hd (he := he)
+
+namespace Const
+
+variable {β : Type v}
+
+theorem minKeyD_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f} (he : (modifyKey k f l).isEmpty = false)
+    {fallback} :
+    (modifyKey k f l |> minKeyD <| fallback) = if (minKeyD l fallback) == k then k else (minKeyD l fallback) := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] using minKey_modifyKey hd (he := he)
+
+theorem minKeyD_modifyKey_eq_minKeyD [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f fallback} :
+    (modifyKey k f l |> minKeyD <| fallback) = minKeyD l fallback := 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_minKeyD (fallback := fallback)] using this
+  · simp_all [modifyKey]
+
+theorem minKeyD_modifyKey_beq [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f fallback} :
+    (modifyKey k f l |> minKeyD <| fallback) == (minKeyD l fallback) := by
+  cases he : l.isEmpty
+  · have := minKey_modifyKey_beq hd (he := isEmpty_modifyKey k f l ▸ he)
+    -- fails after inlining `this`
+    simpa [minKey_eq_minKeyD (fallback := fallback)] using this
+  · simp_all [modifyKey]
+
+theorem minKeyD_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f} (he : (alterKey k f l).isEmpty = false)
+    {fallback} :
+    (alterKey k f l |> minKeyD <| fallback) = k ↔
+      (f (getValue? k l)).isSome ∧ ∀ k', containsKey k' l → (compare k k').isLE := by
+  simpa only [minKey_eq_minKeyD (fallback := fallback)] 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 536f077e02..1d265c3339 100644
--- a/src/Std/Data/TreeMap/Lemmas.lean
+++ b/src/Std/Data/TreeMap/Lemmas.lean
@@ -2156,6 +2156,124 @@ theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] {k f}
       (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
   DTreeMap.Const.minKey!_alter_eq_self he
 
+theorem minKey?_eq_some_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.minKey? = some (t.minKeyD fallback) :=
+  DTreeMap.minKey?_eq_some_minKeyD he
+
+theorem minKeyD_eq_fallback [TransCmp cmp] (he : t.isEmpty) {fallback} :
+    t.minKeyD fallback = fallback :=
+  DTreeMap.minKeyD_eq_fallback he
+
+theorem minKey!_eq_minKeyD_default [TransCmp cmp] [Inhabited α] :
+    t.minKey! = t.minKeyD default :=
+  DTreeMap.minKey!_eq_minKeyD_default
+
+theorem minKeyD_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp]
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.minKeyD_eq_iff_getKey?_eq_self_and_forall he
+
+theorem minKeyD_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp]
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.minKeyD_eq_some_iff_mem_and_forall he
+
+theorem minKeyD_insert [TransCmp cmp] {k v fallback} :
+    (t.insert k v |>.minKeyD fallback) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  DTreeMap.minKeyD_insert
+
+theorem minKeyD_insert_le_minKeyD [TransCmp cmp] (he : t.isEmpty = false)
+    {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  DTreeMap.minKeyD_insert_le_minKeyD he
+
+theorem minKeyD_insert_le_self [TransCmp cmp] {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) k |>.isLE :=
+  DTreeMap.minKeyD_insert_le_self
+
+theorem contains_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.contains (t.minKeyD fallback) :=
+  DTreeMap.contains_minKeyD he
+
+theorem minKeyD_mem [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.minKeyD fallback ∈ t :=
+  DTreeMap.minKeyD_mem he
+
+theorem minKeyD_le_of_contains [TransCmp cmp] {k} (hc : t.contains k) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  DTreeMap.minKeyD_le_of_contains hc
+
+theorem minKeyD_le_of_mem [TransCmp cmp] {k} (hc : k ∈ t) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  DTreeMap.minKeyD_le_of_mem hc
+
+theorem le_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {k fallback} :
+    (cmp k (t.minKeyD fallback)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.le_minKeyD he
+
+theorem getKey?_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.getKey? (t.minKeyD fallback) = some (t.minKeyD fallback) :=
+  DTreeMap.getKey?_minKeyD he
+
+theorem getKey_minKeyD [TransCmp cmp] {fallback hc} :
+    t.getKey (t.minKeyD fallback) hc = t.minKeyD fallback :=
+  DTreeMap.getKey_minKeyD
+
+theorem getKey!_minKeyD [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback} :
+    t.getKey! (t.minKeyD fallback) = t.minKeyD fallback :=
+  DTreeMap.getKey!_minKeyD he
+
+theorem getKeyD_minKeyD [TransCmp cmp] (he : t.isEmpty = false) {fallback fallback'} :
+    t.getKeyD (t.minKeyD fallback) fallback' = t.minKeyD fallback :=
+  DTreeMap.getKeyD_minKeyD he
+
+theorem minKeyD_erase_eq_of_not_compare_minKeyD_eq [TransCmp cmp] {k fallback}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k (t.minKeyD fallback) = .eq) :
+    (t.erase k |>.minKeyD fallback) = t.minKeyD fallback :=
+  DTreeMap.minKeyD_erase_eq_of_not_compare_minKeyD_eq he heq
+
+theorem minKeyD_le_minKeyD_erase [TransCmp cmp] {k}
+    (he : (t.erase k).isEmpty = false) {fallback} :
+    cmp (t.minKeyD fallback) (t.erase k |>.minKeyD fallback) |>.isLE :=
+  DTreeMap.minKeyD_le_minKeyD_erase he
+
+theorem minKeyD_insertIfNew [TransCmp cmp] {k v fallback} :
+    (t.insertIfNew k v |>.minKeyD fallback) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.minKeyD_insertIfNew
+
+theorem minKeyD_insertIfNew_le_minKeyD [TransCmp cmp]
+    (he : t.isEmpty = false) {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  DTreeMap.minKeyD_insertIfNew_le_minKeyD he
+
+theorem minKeyD_insertIfNew_le_self [TransCmp cmp] {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) k |>.isLE :=
+  DTreeMap.minKeyD_insertIfNew_le_self
+
+theorem minKeyD_modify [TransCmp cmp] {k f}
+    (he : (modify t k f).isEmpty = false) {fallback} :
+    (modify t k f |>.minKeyD fallback) =
+      if cmp (t.minKeyD fallback) k = .eq then k else (t.minKeyD fallback) :=
+  DTreeMap.Const.minKeyD_modify he
+
+@[simp]
+theorem minKeyD_modify_eq_minKeyD [TransCmp cmp] [LawfulEqCmp cmp] {k f fallback} :
+    (modify t k f |>.minKeyD fallback) = t.minKeyD fallback :=
+  DTreeMap.Const.minKeyD_modify_eq_minKeyD
+
+@[simp]
+theorem compare_minKeyD_modify_eq [TransCmp cmp] {k f fallback} :
+    cmp (modify t k f |>.minKeyD fallback) (t.minKeyD fallback) = .eq :=
+  DTreeMap.Const.compare_minKeyD_modify_eq
+
+theorem minKeyD_alter_eq_self [TransCmp cmp] {k f}
+    (he : (alter t k f).isEmpty = false) {fallback} :
+    (alter t k f |>.minKeyD fallback) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  DTreeMap.Const.minKeyD_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 217a3cc176..59d35177bb 100644
--- a/src/Std/Data/TreeMap/Raw/Lemmas.lean
+++ b/src/Std/Data/TreeMap/Raw/Lemmas.lean
@@ -2038,6 +2038,124 @@ theorem minKey!_alter_eq_self [TransCmp cmp] [Inhabited α] (h : t.WF) {k f}
       (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
   DTreeMap.Raw.Const.minKey!_alter_eq_self h he
 
+theorem minKey?_eq_some_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.minKey? = some (t.minKeyD fallback) :=
+  DTreeMap.Raw.minKey?_eq_some_minKeyD h he
+
+theorem minKeyD_eq_fallback [TransCmp cmp] (h : t.WF) (he : t.isEmpty) {fallback} :
+    t.minKeyD fallback = fallback :=
+  DTreeMap.Raw.minKeyD_eq_fallback h he
+
+theorem minKey!_eq_minKeyD_default [TransCmp cmp] [Inhabited α] (h : t.WF) :
+    t.minKey! = t.minKeyD default :=
+  DTreeMap.Raw.minKey!_eq_minKeyD_default h
+
+theorem minKeyD_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ t.getKey? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKeyD_eq_iff_getKey?_eq_self_and_forall h he
+
+theorem minKeyD_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {km fallback} :
+    t.minKeyD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKeyD_eq_some_iff_mem_and_forall h he
+
+theorem minKeyD_insert [TransCmp cmp] (h : t.WF) {k v fallback} :
+    (t.insert k v |>.minKeyD fallback) =
+      (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
+  DTreeMap.Raw.minKeyD_insert h
+
+theorem minKeyD_insert_le_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false)
+    {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  DTreeMap.Raw.minKeyD_insert_le_minKeyD h he
+
+theorem minKeyD_insert_le_self [TransCmp cmp] (h : t.WF) {k v fallback} :
+    cmp (t.insert k v |>.minKeyD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_insert_le_self h
+
+theorem contains_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.contains (t.minKeyD fallback) :=
+  DTreeMap.Raw.contains_minKeyD h he
+
+theorem minKeyD_mem [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.minKeyD fallback ∈ t :=
+  DTreeMap.Raw.minKeyD_mem h he
+
+theorem minKeyD_le_of_contains [TransCmp cmp] (h : t.WF) {k} (hc : t.contains k) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_le_of_contains h hc
+
+theorem minKeyD_le_of_mem [TransCmp cmp] (h : t.WF) {k} (hc : k ∈ t) {fallback} :
+    cmp (t.minKeyD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_le_of_mem h hc
+
+theorem le_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {k fallback} :
+    (cmp k (t.minKeyD fallback)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.Raw.le_minKeyD h he
+
+theorem getKey?_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getKey? (t.minKeyD fallback) = some (t.minKeyD fallback) :=
+  DTreeMap.Raw.getKey?_minKeyD h he
+
+theorem getKey_minKeyD [TransCmp cmp] (h : t.WF) {fallback hc} :
+    t.getKey (t.minKeyD fallback) hc = t.minKeyD fallback :=
+  DTreeMap.Raw.getKey_minKeyD h
+
+theorem getKey!_minKeyD [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.getKey! (t.minKeyD fallback) = t.minKeyD fallback :=
+  DTreeMap.Raw.getKey!_minKeyD h he
+
+theorem getKeyD_minKeyD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback fallback'} :
+    t.getKeyD (t.minKeyD fallback) fallback' = t.minKeyD fallback :=
+  DTreeMap.Raw.getKeyD_minKeyD h he
+
+theorem minKeyD_erase_eq_of_not_compare_minKeyD_eq [TransCmp cmp] (h : t.WF) {k fallback}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k (t.minKeyD fallback) = .eq) :
+    (t.erase k |>.minKeyD fallback) = t.minKeyD fallback :=
+  DTreeMap.Raw.minKeyD_erase_eq_of_not_compare_minKeyD_eq h he heq
+
+theorem minKeyD_le_minKeyD_erase [TransCmp cmp] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) {fallback} :
+    cmp (t.minKeyD fallback) (t.erase k |>.minKeyD fallback) |>.isLE :=
+  DTreeMap.Raw.minKeyD_le_minKeyD_erase h he
+
+theorem minKeyD_insertIfNew [TransCmp cmp] (h : t.WF) {k v fallback} :
+    (t.insertIfNew k v |>.minKeyD fallback) =
+      t.minKey?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.Raw.minKeyD_insertIfNew h
+
+theorem minKeyD_insertIfNew_le_minKeyD [TransCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) (t.minKeyD fallback) |>.isLE :=
+  DTreeMap.Raw.minKeyD_insertIfNew_le_minKeyD h he
+
+theorem minKeyD_insertIfNew_le_self [TransCmp cmp] (h : t.WF) {k v fallback} :
+    cmp (t.insertIfNew k v |>.minKeyD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_insertIfNew_le_self h
+
+theorem minKeyD_modify [TransCmp cmp] (h : t.WF) {k f}
+    (he : (modify t k f).isEmpty = false) {fallback} :
+    (modify t k f |>.minKeyD fallback) =
+      if cmp (t.minKeyD fallback) k = .eq then k else (t.minKeyD fallback) :=
+  DTreeMap.Raw.Const.minKeyD_modify h he
+
+@[simp]
+theorem minKeyD_modify_eq_minKeyD [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF) {k f fallback} :
+    (modify t k f |>.minKeyD fallback) = t.minKeyD fallback :=
+  DTreeMap.Raw.Const.minKeyD_modify_eq_minKeyD h
+
+@[simp]
+theorem compare_minKeyD_modify_eq [TransCmp cmp] (h : t.WF) {k f fallback} :
+    cmp (modify t k f |>.minKeyD fallback) (t.minKeyD fallback) = .eq :=
+  DTreeMap.Raw.Const.compare_minKeyD_modify_eq h
+
+theorem minKeyD_alter_eq_self [TransCmp cmp] (h : t.WF) {k f}
+    (he : (alter t k f).isEmpty = false) {fallback} :
+    (alter t k f |>.minKeyD fallback) = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  DTreeMap.Raw.Const.minKeyD_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 c141b7c0fe..b8b581ea40 100644
--- a/src/Std/Data/TreeSet/Lemmas.lean
+++ b/src/Std/Data/TreeSet/Lemmas.lean
@@ -954,6 +954,88 @@ theorem min!_le_min!_erase [TransCmp cmp] [Inhabited α] {k}
     cmp t.min! (t.erase k |>.min!) |>.isLE :=
   DTreeMap.minKey!_le_minKey!_erase he
 
+theorem min?_eq_some_minD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.min? = some (t.minD fallback) :=
+  TreeMap.minKey?_eq_some_minKeyD he
+
+theorem minD_eq_fallback [TransCmp cmp] (he : t.isEmpty) {fallback} :
+    t.minD fallback = fallback :=
+  TreeMap.minKeyD_eq_fallback he
+
+theorem min!_eq_minD_default [TransCmp cmp] [Inhabited α] :
+    t.min! = t.minD default :=
+  TreeMap.minKey!_eq_minKeyD_default
+
+theorem minD_eq_iff_get?_eq_self_and_forall [TransCmp cmp]
+    (he : t.isEmpty = false) {km fallback} :
+    t.minD fallback = km ↔ t.get? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  TreeMap.minKeyD_eq_iff_getKey?_eq_self_and_forall he
+
+theorem minD_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp]
+    (he : t.isEmpty = false) {km fallback} :
+    t.minD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  TreeMap.minKeyD_eq_some_iff_mem_and_forall he
+
+theorem minD_insert [TransCmp cmp] {k fallback} :
+    (t.insert k |>.minD fallback) =
+      t.min?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  TreeMap.minKeyD_insertIfNew
+
+theorem minD_insert_le_minD [TransCmp cmp]
+    (he : t.isEmpty = false) {k fallback} :
+    cmp (t.insert k |>.minD fallback) (t.minD fallback) |>.isLE :=
+  TreeMap.minKeyD_insertIfNew_le_minKeyD he
+
+theorem minD_insert_le_self [TransCmp cmp] {k fallback} :
+    cmp (t.insert k |>.minD fallback) k |>.isLE :=
+  TreeMap.minKeyD_insertIfNew_le_self
+
+theorem contains_minD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.contains (t.minD fallback) :=
+  TreeMap.contains_minKeyD he
+
+theorem minD_mem [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.minD fallback ∈ t :=
+  TreeMap.minKeyD_mem he
+
+theorem minD_le_of_contains [TransCmp cmp] {k} (hc : t.contains k) {fallback} :
+    cmp (t.minD fallback) k |>.isLE :=
+  TreeMap.minKeyD_le_of_contains hc
+
+theorem minD_le_of_mem [TransCmp cmp] {k} (hc : k ∈ t) {fallback} :
+    cmp (t.minD fallback) k |>.isLE :=
+  TreeMap.minKeyD_le_of_mem hc
+
+theorem le_minD [TransCmp cmp] (he : t.isEmpty = false) {k fallback} :
+    (cmp k (t.minD fallback)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  TreeMap.le_minKeyD he
+
+theorem get?_minD [TransCmp cmp] (he : t.isEmpty = false) {fallback} :
+    t.get? (t.minD fallback) = some (t.minD fallback) :=
+  TreeMap.getKey?_minKeyD he
+
+theorem get_minD [TransCmp cmp] {fallback hc} :
+    t.get (t.minD fallback) hc = t.minD fallback :=
+  TreeMap.getKey_minKeyD
+
+theorem get!_minD [TransCmp cmp] [Inhabited α] (he : t.isEmpty = false) {fallback} :
+    t.get! (t.minD fallback) = t.minD fallback :=
+  TreeMap.getKey!_minKeyD he
+
+theorem getD_minD [TransCmp cmp] (he : t.isEmpty = false) {fallback fallback'} :
+    t.getD (t.minD fallback) fallback' = t.minD fallback :=
+  TreeMap.getKeyD_minKeyD he
+
+theorem minD_erase_eq_of_not_compare_minD_eq [TransCmp cmp] {k fallback}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k (t.minD fallback) = .eq) :
+    (t.erase k |>.minD fallback) = t.minD fallback :=
+  TreeMap.minKeyD_erase_eq_of_not_compare_minKeyD_eq he heq
+
+theorem minD_le_minD_erase [TransCmp cmp] {k}
+    (he : (t.erase k).isEmpty = false) {fallback} :
+    cmp (t.minD fallback) (t.erase k |>.minD fallback) |>.isLE :=
+  TreeMap.minKeyD_le_minKeyD_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 0f5d42147d..bbd60c32af 100644
--- a/src/Std/Data/TreeSet/Raw/Lemmas.lean
+++ b/src/Std/Data/TreeSet/Raw/Lemmas.lean
@@ -860,6 +860,88 @@ theorem min!_le_min!_erase [TransCmp cmp] [Inhabited α] (h : t.WF) {k}
     cmp t.min! (t.erase k |>.min!) |>.isLE :=
   DTreeMap.Raw.minKey!_le_minKey!_erase h he
 
+theorem min?_eq_some_minD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.min? = some (t.minD fallback) :=
+  DTreeMap.Raw.minKey?_eq_some_minKeyD h he
+
+theorem minD_eq_fallback [TransCmp cmp] (h : t.WF) (he : t.isEmpty) {fallback} :
+    t.minD fallback = fallback :=
+  DTreeMap.Raw.minKeyD_eq_fallback h he
+
+theorem min!_eq_minD_default [TransCmp cmp] [Inhabited α] (h : t.WF) :
+    t.min! = t.minD default :=
+  DTreeMap.Raw.minKey!_eq_minKeyD_default h
+
+theorem minD_eq_iff_get?_eq_self_and_forall [TransCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {km fallback} :
+    t.minD fallback = km ↔ t.get? km = some km ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKeyD_eq_iff_getKey?_eq_self_and_forall h he
+
+theorem minD_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {km fallback} :
+    t.minD fallback = km ↔ km ∈ t ∧ ∀ k, k ∈ t → (cmp km k).isLE :=
+  DTreeMap.Raw.minKeyD_eq_some_iff_mem_and_forall h he
+
+theorem minD_insert [TransCmp cmp] (h : t.WF) {k fallback} :
+    (t.insert k |>.minD fallback) =
+      t.min?.elim k fun k' => if cmp k k' = .lt then k else k' :=
+  DTreeMap.Raw.minKeyD_insertIfNew h
+
+theorem minD_insert_le_minD [TransCmp cmp] (h : t.WF)
+    (he : t.isEmpty = false) {k fallback} :
+    cmp (t.insert k |>.minD fallback) (t.minD fallback) |>.isLE :=
+  DTreeMap.Raw.minKeyD_insertIfNew_le_minKeyD h he
+
+theorem minD_insert_le_self [TransCmp cmp] (h : t.WF) {k fallback} :
+    cmp (t.insert k |>.minD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_insertIfNew_le_self h
+
+theorem contains_minD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.contains (t.minD fallback) :=
+  DTreeMap.Raw.contains_minKeyD h he
+
+theorem minD_mem [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.minD fallback ∈ t :=
+  DTreeMap.Raw.minKeyD_mem h he
+
+theorem minD_le_of_contains [TransCmp cmp] (h : t.WF) {k} (hc : t.contains k) {fallback} :
+    cmp (t.minD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_le_of_contains h hc
+
+theorem minD_le_of_mem [TransCmp cmp] (h : t.WF) {k} (hc : k ∈ t) {fallback} :
+    cmp (t.minD fallback) k |>.isLE :=
+  DTreeMap.Raw.minKeyD_le_of_mem h hc
+
+theorem le_minD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {k fallback} :
+    (cmp k (t.minD fallback)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.Raw.le_minKeyD h he
+
+theorem get?_minD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.get? (t.minD fallback) = some (t.minD fallback) :=
+  DTreeMap.Raw.getKey?_minKeyD h he
+
+theorem get_minD [TransCmp cmp] (h : t.WF) {fallback hc} :
+    t.get (t.minD fallback) hc = t.minD fallback :=
+  DTreeMap.Raw.getKey_minKeyD h
+
+theorem get!_minD [TransCmp cmp] [Inhabited α] (h : t.WF) (he : t.isEmpty = false) {fallback} :
+    t.get! (t.minD fallback) = t.minD fallback :=
+  DTreeMap.Raw.getKey!_minKeyD h he
+
+theorem getD_minD [TransCmp cmp] (h : t.WF) (he : t.isEmpty = false) {fallback fallback'} :
+    t.getD (t.minD fallback) fallback' = t.minD fallback :=
+  DTreeMap.Raw.getKeyD_minKeyD h he
+
+theorem minD_erase_eq_of_not_compare_minD_eq [TransCmp cmp] (h : t.WF) {k fallback}
+    (he : (t.erase k).isEmpty = false) (heq : ¬ cmp k (t.minD fallback) = .eq) :
+    (t.erase k |>.minD fallback) = t.minD fallback :=
+  DTreeMap.Raw.minKeyD_erase_eq_of_not_compare_minKeyD_eq h he heq
+
+theorem minD_le_minD_erase [TransCmp cmp] (h : t.WF) {k}
+    (he : (t.erase k).isEmpty = false) {fallback} :
+    cmp (t.minD fallback) (t.erase k |>.minD fallback) |>.isLE :=
+  DTreeMap.Raw.minKeyD_le_minKeyD_erase h he
+
 end Min
 
 end Std.TreeSet.Raw