From 0a96b4cf72402350591cd0b7339946250a7febda Mon Sep 17 00:00:00 2001
From: Paul Reichert <6992158+datokrat@users.noreply.github.com>
Date: Mon, 24 Mar 2025 14:08:19 +0100
Subject: [PATCH] feat: tree map lemmas for minKey (#7660)

This PR provides lemmas for the tree map function `minKey` 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    | 165 +++++++++++++-
 src/Std/Data/DTreeMap/Internal/Model.lean     |  11 +
 src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean |   5 +
 src/Std/Data/DTreeMap/Lemmas.lean             | 153 +++++++++++++
 src/Std/Data/DTreeMap/Raw/Lemmas.lean         |  13 ++
 src/Std/Data/Internal/List/Associative.lean   | 206 +++++++++++++++++-
 src/Std/Data/TreeMap/Lemmas.lean              | 144 +++++++++++-
 src/Std/Data/TreeMap/Raw/Lemmas.lean          |  18 ++
 src/Std/Data/TreeSet/Lemmas.lean              | 101 +++++++++
 src/Std/Data/TreeSet/Raw/Lemmas.lean          |  13 ++
 10 files changed, 823 insertions(+), 6 deletions(-)

diff --git a/src/Std/Data/DTreeMap/Internal/Lemmas.lean b/src/Std/Data/DTreeMap/Internal/Lemmas.lean
index ad0f46aba6..1f30f26f41 100644
--- a/src/Std/Data/DTreeMap/Internal/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Internal/Lemmas.lean
@@ -96,7 +96,8 @@ private def queryMap : Std.DHashMap Name (fun _ => Name × Array (MacroM (TSynta
      ⟨`foldr, (``foldr_eq_foldr, #[])⟩,
      ⟨`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 _)])⟩]
 
 /-- Internal implementation detail of the tree map -/
 scoped syntax "simp_to_model" (" [" (ident,*) "]")? ("using" term)? : tactic
@@ -275,6 +276,25 @@ 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) :
+    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
+
+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) →
+    t.isEmpty = false := by
+  simp_to_model [erase, isEmpty, contains] using List.isEmpty_eq_false_of_isEmpty_eraseKey_eq_false
+
+theorem isEmpty_eq_false_of_isEmpty_erase!_eq_false [TransOrd α] (h : t.WF) {k : α} :
+    (he : (t.erase! k).isEmpty = false) →
+    t.isEmpty = false := by
+  simpa [erase_eq_erase!] using isEmpty_eq_false_of_isEmpty_erase_eq_false h
+
 theorem contains_erase [TransOrd α] (h : t.WF) {k a : α} :
     (t.erase k h.balanced).impl.contains a = (!(compare k a == .eq) && t.contains a) := by
   simp_to_model [erase, contains] using List.containsKey_eraseKey
@@ -4243,14 +4263,18 @@ theorem isSome_minKey?_eq_not_isEmpty [TransOrd α] (h : t.WF) :
     t.minKey?.isSome = !t.isEmpty := by
   simp_to_model using List.isSome_minKey?_eq_not_isEmpty
 
+theorem isSome_minKey?_iff_isEmpty_eq_false [TransOrd α] (h : t.WF) :
+    t.minKey?.isSome ↔ t.isEmpty = false := by
+  simp [isSome_minKey?_eq_not_isEmpty h]
+
 theorem minKey?_insert [TransOrd 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
+      some (t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k') := by
   simp_to_model [insert] using List.minKey?_insertEntry
 
 theorem minKey?_insert! [TransOrd 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
+      some (t.minKey?.elim k fun k' => if compare k k'|>.isLE then k else k') := by
   simpa only [insert_eq_insert!] using minKey?_insert h
 
 theorem isSome_minKey?_insert [TransOrd α] (h : t.WF) {k v} :
@@ -4434,7 +4458,7 @@ theorem minKey?_modify [TransOrd α] (h : t.WF) {k f} :
 
 theorem minKey?_modify_eq_minKey? [TransOrd α] [LawfulEqOrd α] (h : t.WF) {k f} :
     (Const.modify k f t).minKey? = t.minKey? := by
-  simp_to_model [Const.modify] using List.Const.minKey?_modifyKey_of_lawfulEqOrd
+  simp_to_model [Const.modify] using List.Const.minKey?_modifyKey_eq_minKey?
 
 theorem isSome_minKey?_modify [TransOrd α] {k f}  (h : t.WF) :
     (Const.modify k f t).minKey?.isSome = !t.isEmpty := by
@@ -4463,6 +4487,139 @@ theorem minKey?_alter!_eq_self [TransOrd α] (h : t.WF) {k f} :
 
 end Const
 
+theorem minKey_eq_get_minKey? [TransOrd α] (h : t.WF) {he} :
+    t.minKey he = t.minKey?.get (isSome_minKey?_iff_isEmpty_eq_false h |>.mpr he) := by
+  simp_to_model [minKey, minKey?] using List.minKey_eq_get_minKey?
+
+theorem minKey?_eq_some_minKey [TransOrd α] (h : t.WF) {he} :
+    t.minKey? = some (t.minKey he) := by
+  simp_to_model [minKey, minKey?] using List.minKey?_eq_some_minKey
+
+theorem minKey_eq_iff_getKey?_eq_self_and_forall [TransOrd α] (h : t.WF) {he km} :
+    t.minKey he = km ↔ t.getKey? km = some km ∧ ∀ k ∈ t, (compare km k).isLE := by
+  simp_to_model [minKey, getKey?, contains] using List.minKey_eq_iff_getKey?_eq_self_and_forall
+
+theorem minKey_eq_some_iff_mem_and_forall [TransOrd α] [LawfulEqOrd α] (h : t.WF) {he km} :
+    t.minKey he = km ↔ km ∈ t ∧ ∀ k ∈ t, (compare km k).isLE := by
+  simp_to_model [minKey, contains] using List.minKey_eq_some_iff_mem_and_forall
+
+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
+  simp_to_model [insert, minKey, minKey?] using List.minKey_insertEntry
+
+theorem minKey_insert_le_minKey [TransOrd α] (h : t.WF) {k v he} :
+    compare (t.insert k v h.balanced |>.impl.minKey <| isEmpty_insert h) (t.minKey he) |>.isLE := by
+  simp_to_model [minKey, insert] using List.minKey_insertEntry_le_minKey
+
+theorem minKey_insert_le_self [TransOrd α] (h : t.WF) {k v} :
+    compare (t.insert k v h.balanced |>.impl.minKey <| isEmpty_insert h) k |>.isLE := by
+  simp_to_model [minKey, insert] using List.minKey_insertEntry_le_self
+
+theorem contains_minKey [TransOrd α] (h : t.WF) {he} :
+    t.contains (t.minKey he) := by
+  simp_to_model [minKey, contains] using List.containsKey_minKey
+
+theorem minKey_mem [TransOrd α] (h : t.WF) {he} :
+    t.minKey he ∈ t :=
+  contains_minKey h
+
+theorem minKey_le_of_contains [TransOrd α] (h : t.WF) {k} :
+    (hc : t.contains k) →
+    compare (t.minKey <| (isEmpty_eq_false_iff_exists_contains_eq_true h).mpr ⟨k, hc⟩) k |>.isLE := by
+   simp_to_model [minKey, contains] using minKey_le_of_containsKey
+
+theorem minKey_le_of_mem [TransOrd α] (h : t.WF) {k} (hc : k ∈ t) :
+    compare (t.minKey <| (isEmpty_eq_false_iff_exists_contains_eq_true h).mpr ⟨k, hc⟩) k |>.isLE :=
+  minKey_le_of_contains h hc
+
+theorem le_minKey [TransOrd α] (h : t.WF) {k he} :
+    (compare k (t.minKey he)).isLE ↔ (∀ k', k' ∈ t → (compare k k').isLE) := by
+  simp_to_model [minKey, contains] using List.le_minKey
+
+theorem getKey?_minKey [TransOrd α] (h : t.WF) {he} :
+    t.getKey? (t.minKey he) = some (t.minKey he) := by
+  simp_to_model [getKey?, minKey] using List.getKey?_minKey
+
+theorem getKey_minKey [TransOrd α] (h : t.WF) {he hc} :
+    t.getKey (t.minKey he) hc = t.minKey he := by
+  simp_to_model [getKey, minKey] using List.getKey_minKey
+
+theorem getKey!_minKey [TransOrd α] [Inhabited α] (h : t.WF) {he} :
+    t.getKey! (t.minKey he) = t.minKey he := by
+  simp_to_model [getKey!, minKey] using List.getKey!_minKey
+
+theorem getKeyD_minKey [TransOrd α] (h : t.WF) {he fallback} :
+    t.getKeyD (t.minKey he) fallback = t.minKey he := by
+  simp_to_model [getKeyD, minKey] using List.getKeyD_minKey
+
+theorem minKey_erase_eq_iff_not_compare_eq_minKey [TransOrd α] (h : t.WF) {k he} :
+    (t.erase k h.balanced |>.impl.minKey he) =
+        t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false h he) ↔
+      ¬ compare k (t.minKey <| isEmpty_eq_false_of_isEmpty_erase_eq_false h he) = .eq := by
+  simp_to_model [minKey, erase] using List.minKey_eraseKey_eq_iff_beq_minKey_eq_false
+
+theorem minKey_erase_eq_of_not_compare_eq_minKey [TransOrd α] (h : t.WF) {k he} :
+    (hc : ¬ compare k (t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false h he)) = .eq) →
+    (t.erase k h.balanced |>.impl.minKey he) =
+      t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false h he) := by
+  simp_to_model [minKey, erase] using List.minKey_eraseKey_eq_of_beq_minKey_eq_false
+
+theorem minKey_le_minKey_erase [TransOrd α] (h : t.WF) {k he} :
+    compare (t.minKey <| isEmpty_eq_false_of_isEmpty_erase_eq_false h he)
+      (t.erase k h.balanced |>.impl.minKey he) |>.isLE := by
+  simp_to_model [minKey, erase] using List.minKey_le_minKey_erase
+
+theorem minKey_insertIfNew [TransOrd α] (h : t.WF) {k v} :
+    (t.insertIfNew k v h.balanced).impl.minKey (isEmpty_insertIfNew h) =
+      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_le_minKey [TransOrd α] (h : t.WF) {k v he} :
+    compare (t.insertIfNew k v h.balanced |>.impl.minKey <| isEmpty_insertIfNew h)
+      (t.minKey he) |>.isLE := by
+  simp_to_model [minKey, insertIfNew] using List.minKey_insertEntryIfNew_le_minKey
+
+theorem minKey_insertIfNew_le_self [TransOrd α] (h : t.WF) {k v} :
+    compare (t.insertIfNew k v h.balanced |>.impl.minKey <| isEmpty_insertIfNew h) k |>.isLE := by
+  simp_to_model [minKey, insertIfNew] using List.minKey_insertEntryIfNew_le_self
+
+theorem minKey_modify [TransOrd α] [LawfulEqOrd α] (h : t.WF) {k f he} :
+    (t.modify k f).minKey he = t.minKey (isEmpty_modify h ▸ he):= by
+  simp_to_model [minKey, modify] using List.minKey_modifyKey
+
+theorem minKey_alter_eq_self [TransOrd α] [LawfulEqOrd α] (h : t.WF) {k f he} :
+    (t.alter k f h.balanced).impl.minKey he = k ↔
+      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simp_to_model [minKey, alter, get?, contains] using List.minKey_alterKey_eq_self
+
+namespace Const
+
+variable {β : Type v} {t : Impl α β}
+
+theorem minKey_modify [TransOrd α] (h : t.WF) {k f he} :
+    (modify k f t).minKey he =
+      if compare (t.minKey <| isEmpty_modify h ▸ he) k = .eq then
+        k
+      else
+        (t.minKey <| Const.isEmpty_modify h ▸ he) := by
+  simp_to_model [minKey, Const.modify] using List.Const.minKey_modifyKey
+
+theorem minKey_modify_eq_minKey [TransOrd α] [LawfulEqOrd α] (h : t.WF) {k f he} :
+    (modify k f t).minKey he = t.minKey (isEmpty_modify h ▸ he) := by
+  simp_to_model [minKey, Const.modify] using List.Const.minKey_modifyKey_eq_minKey
+
+theorem compare_minKey_modify_eq [TransOrd α] (h : t.WF) {k f he} :
+    compare (modify k f t |>.minKey he) (t.minKey <| isEmpty_modify h ▸ he) = .eq := by
+  simp_to_model [minKey, Const.modify] using List.Const.minKey_modifyKey_beq
+
+theorem minKey_alter_eq_self [TransOrd α] (h : t.WF) {k f he} :
+    (alter k f t h.balanced).impl.minKey he = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (compare k k').isLE := by
+  simp_to_model [minKey, Const.alter, contains, Const.get?] using List.Const.minKey_alterKey_eq_self
+
+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 ac282b74ac..cc870c64f2 100644
--- a/src/Std/Data/DTreeMap/Internal/Model.lean
+++ b/src/Std/Data/DTreeMap/Internal/Model.lean
@@ -465,6 +465,17 @@ theorem minEntry?_eq_minEntry?ₘ [Ord α] {l : Impl α β} : l.minEntry? = l.mi
 theorem minKey?_eq_minEntry?_map_fst [Ord α] {l : Impl α β} : l.minKey? = l.minEntry?.map Sigma.fst := by
   induction l using minKey?.induct <;> simp only [minKey?, minEntry?] <;> trivial
 
+theorem some_minEntry_eq_minEntry? [Ord α] {l : Impl α β} {he} :
+    some (l.minEntry he) = l.minEntry? := by
+  induction l, he using minEntry.induct <;> simp_all [minEntry, minEntry?]
+
+theorem minEntry_eq_get_minEntry? [Ord α] {l : Impl α β} {he} :
+    l.minEntry he = l.minEntry?.get (by simp [← some_minEntry_eq_minEntry? (he := he)]) := by
+  simp [← some_minEntry_eq_minEntry? (he := 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 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 9e96c93dce..24ef498f42 100644
--- a/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Internal/WF/Lemmas.lean
@@ -1828,4 +1828,9 @@ theorem minKey?_eq_minKey? [Ord α] [TransOrd α] {l : Impl α β} (hlo : l.Orde
     l.minKey? = List.minKey? l.toListModel := by
   simp only [minKey?_eq_minEntry?_map_fst, minEntry?_eq_minEntry? hlo, List.minKey?]
 
+theorem minKey_eq_minKey [Ord α] [TransOrd α] {l : Impl α β} (hlo : l.Ordered) {he} :
+    l.minKey he = List.minKey l.toListModel (isEmpty_eq_isEmpty ▸ he) := by
+  simp [minKey_eq_get_minKey?, minKey_eq_minEntry_fst, minEntry_eq_get_minEntry?,
+    minEntry?_eq_minEntry? hlo, List.minKey?, Option.get_map]
+
 end Std.DTreeMap.Internal.Impl
diff --git a/src/Std/Data/DTreeMap/Lemmas.lean b/src/Std/Data/DTreeMap/Lemmas.lean
index d7392400c5..3ee76d9d09 100644
--- a/src/Std/Data/DTreeMap/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Lemmas.lean
@@ -145,6 +145,15 @@ 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) :
+    t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
+  Impl.isEmpty_eq_isEmpty_erase_and_not_containsKey t.wf k
+
+theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] {k : α}
+    (he : (t.erase k).isEmpty = false) :
+    t.isEmpty = false :=
+  Impl.isEmpty_eq_false_of_isEmpty_erase_eq_false t.wf he
+
 @[simp]
 theorem contains_erase [TransCmp cmp] {k a : α} :
     (t.erase k).contains a = (cmp k a != .eq && t.contains a) :=
@@ -2797,6 +2806,10 @@ theorem isSome_minKey?_eq_not_isEmpty [TransCmp cmp] :
     t.minKey?.isSome = !t.isEmpty :=
   Impl.isSome_minKey?_eq_not_isEmpty t.wf
 
+theorem isSome_minKey?_iff_isEmpty_eq_false [TransCmp cmp] :
+    t.minKey?.isSome ↔ t.isEmpty = false :=
+  Impl.isSome_minKey?_iff_isEmpty_eq_false t.wf
+
 theorem minKey?_insert [TransCmp cmp] {k v} :
     (t.insert k v).minKey? =
       some (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
@@ -2946,6 +2959,146 @@ theorem minKey?_alter_eq_self [TransCmp cmp] {k f} :
 
 end Const
 
+theorem minKey_eq_get_minKey? [TransCmp cmp] {he} :
+    t.minKey he = t.minKey?.get (isSome_minKey?_iff_isEmpty_eq_false.mpr he) :=
+  Impl.minKey_eq_get_minKey? t.wf
+
+theorem minKey?_eq_some_minKey [TransCmp cmp] {he} :
+    t.minKey? = some (t.minKey he) :=
+  Impl.minKey?_eq_some_minKey t.wf
+
+theorem minKey_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] {he km} :
+    t.minKey he = km ↔ t.getKey? km = some km ∧ ∀ k ∈ t, (cmp km k).isLE :=
+  Impl.minKey_eq_iff_getKey?_eq_self_and_forall t.wf
+
+theorem minKey_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] {he km} :
+    t.minKey he = km ↔ km ∈ t ∧ ∀ k ∈ t, (cmp km k).isLE :=
+  Impl.minKey_eq_some_iff_mem_and_forall t.wf
+
+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' :=
+  Impl.minKey_insert t.wf
+
+theorem minKey_insert_le_minKey [TransCmp cmp] {k v he} :
+    cmp (t.insert k v |>.minKey isEmpty_insert) (t.minKey he) |>.isLE :=
+  Impl.minKey_insert_le_minKey t.wf
+
+theorem minKey_insert_le_self [TransCmp cmp] {k v} :
+    cmp (t.insert k v |>.minKey isEmpty_insert) k |>.isLE :=
+  Impl.minKey_insert_le_self t.wf
+
+theorem contains_minKey [TransCmp cmp] {he} :
+    t.contains (t.minKey he) :=
+  Impl.contains_minKey t.wf
+
+theorem minKey_mem [TransCmp cmp] {he} :
+    t.minKey he ∈ t :=
+  Impl.minKey_mem t.wf
+
+theorem minKey_le_of_contains [TransCmp cmp] {k} (hc : t.contains k) :
+    cmp (t.minKey <| isEmpty_eq_false_iff_exists_contains_eq_true.mpr ⟨k, hc⟩) k |>.isLE :=
+  Impl.minKey_le_of_contains t.wf hc
+
+theorem minKey_le_of_mem [TransCmp cmp] {k} (hc : k ∈ t) :
+    cmp (t.minKey <| isEmpty_eq_false_iff_exists_contains_eq_true.mpr ⟨k, hc⟩) k |>.isLE :=
+  Impl.minKey_le_of_mem t.wf hc
+
+theorem le_minKey [TransCmp cmp] {k he} :
+    (cmp k (t.minKey he)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  Impl.le_minKey t.wf
+
+@[simp]
+theorem getKey?_minKey [TransCmp cmp] {he} :
+    t.getKey? (t.minKey he) = some (t.minKey he) :=
+  Impl.getKey?_minKey t.wf
+
+@[simp]
+theorem getKey_minKey [TransCmp cmp] {he hc} :
+    t.getKey (t.minKey he) hc = t.minKey he :=
+  Impl.getKey_minKey t.wf
+
+@[simp]
+theorem getKey!_minKey [TransCmp cmp] [Inhabited α] {he} :
+    t.getKey! (t.minKey he) = t.minKey he :=
+  Impl.getKey!_minKey t.wf
+
+@[simp]
+theorem getKeyD_minKey [TransCmp cmp] {he fallback} :
+    t.getKeyD (t.minKey he) fallback = t.minKey he :=
+  Impl.getKeyD_minKey t.wf
+
+@[simp]
+theorem minKey_erase_eq_iff_not_compare_eq_minKey [TransCmp cmp] {k he} :
+    (t.erase k |>.minKey he) =
+        t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false he) ↔
+      ¬ cmp k (t.minKey <| isEmpty_eq_false_of_isEmpty_erase_eq_false he) = .eq :=
+  Impl.minKey_erase_eq_iff_not_compare_eq_minKey t.wf
+
+theorem minKey_erase_eq_of_not_compare_eq_minKey [TransCmp cmp] {k he} :
+    (hc : ¬ cmp k (t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false he)) = .eq) →
+    (t.erase k |>.minKey he) =
+      t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false he) :=
+  Impl.minKey_erase_eq_of_not_compare_eq_minKey t.wf
+
+theorem minKey_le_minKey_erase [TransCmp cmp] {k he} :
+    cmp (t.minKey <| isEmpty_eq_false_of_isEmpty_erase_eq_false he)
+      (t.erase k |>.minKey he) |>.isLE :=
+  Impl.minKey_le_minKey_erase t.wf
+
+theorem minKey_insertIfNew [TransCmp cmp] {k v} :
+    (t.insertIfNew k v).minKey isEmpty_insertIfNew =
+      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] {k v he} :
+    cmp (t.insertIfNew k v |>.minKey isEmpty_insertIfNew)
+      (t.minKey he) |>.isLE :=
+  Impl.minKey_insertIfNew_le_minKey t.wf
+
+theorem minKey_insertIfNew_le_self [TransCmp cmp] {k v} :
+    cmp (t.insertIfNew k v |>.minKey <| isEmpty_insertIfNew) k |>.isLE :=
+  Impl.minKey_insertIfNew_le_self t.wf
+
+@[simp]
+theorem minKey_modify [TransCmp cmp] [LawfulEqCmp cmp] {k f he} :
+    (t.modify k f).minKey he = t.minKey (cast (congrArg (· = false) isEmpty_modify) he) :=
+  Impl.minKey_modify t.wf
+
+theorem minKey_alter_eq_self [TransCmp cmp] [LawfulEqCmp cmp] {k f he} :
+    (t.alter k f).minKey he = k ↔
+      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.minKey_alter_eq_self t.wf
+
+namespace Const
+
+variable {β : Type v} {t : DTreeMap α β cmp}
+
+theorem minKey_modify [TransCmp cmp] {k f he} :
+    (modify t k f).minKey he =
+      if cmp (t.minKey <| cast (congrArg (· = false) isEmpty_modify) he) k = .eq then
+        k
+      else
+        (t.minKey <| cast (congrArg (· = false) isEmpty_modify) he) :=
+  Impl.Const.minKey_modify t.wf
+
+@[simp]
+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
+
+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 :=
+  Impl.Const.compare_minKey_modify_eq t.wf
+
+theorem minKey_alter_eq_self [TransCmp cmp] {k f he} :
+    (alter t k f).minKey he = k ↔
+      (f (get? t k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  Impl.Const.minKey_alter_eq_self t.wf
+
+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 5d31c94fd9..1fe9236a72 100644
--- a/src/Std/Data/DTreeMap/Raw/Lemmas.lean
+++ b/src/Std/Data/DTreeMap/Raw/Lemmas.lean
@@ -146,6 +146,15 @@ 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) :
+    t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
+  Impl.isEmpty_eq_isEmpty_erase!_and_not_containsKey h k
+
+theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] (h : t.WF) {k : α}
+    (he : (t.erase k).isEmpty = false) :
+    t.isEmpty = false :=
+  Impl.isEmpty_eq_false_of_isEmpty_erase!_eq_false h he
+
 @[simp]
 theorem contains_erase [TransCmp cmp] (h : t.WF) {k a : α} :
     (t.erase k).contains a = (cmp k a != .eq && t.contains a) :=
@@ -2802,6 +2811,10 @@ theorem isSome_minKey?_eq_not_isEmpty [TransCmp cmp] (h : t.WF) :
     t.minKey?.isSome = !t.isEmpty :=
   Impl.isSome_minKey?_eq_not_isEmpty h
 
+theorem isSome_minKey?_iff_isEmpty_eq_false [TransCmp cmp] (h : t.WF) :
+    t.minKey?.isSome ↔ t.isEmpty = false :=
+  Impl.isSome_minKey?_iff_isEmpty_eq_false h
+
 theorem minKey?_insert [TransCmp cmp] (h : t.WF) {k v} :
     (t.insert k v).minKey? =
       some (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
diff --git a/src/Std/Data/Internal/List/Associative.lean b/src/Std/Data/Internal/List/Associative.lean
index 92991e5d6f..a5bf07591e 100644
--- a/src/Std/Data/Internal/List/Associative.lean
+++ b/src/Std/Data/Internal/List/Associative.lean
@@ -1770,6 +1770,27 @@ theorem containsKey_eraseKey [BEq α] [PartialEquivBEq α] {l : List ((a : α) 
     (hl : DistinctKeys l) : containsKey a (eraseKey k l) = (!(k == a) && containsKey a l) := by
   simp [containsKey_eq_isSome_getEntry?, getEntry?_eraseKey hl, apply_ite]
 
+theorem isEmpty_eq_isEmpty_eraseKey_and_not_containsKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) (k : α) :
+    l.isEmpty = ((eraseKey k l).isEmpty && !(containsKey k l)) := by
+  rw [Bool.eq_iff_iff, Bool.and_eq_true, Bool.not_eq_true']
+  simp only [isEmpty_iff_forall_containsKey, containsKey_eraseKey hd, Bool.and_eq_false_iff,
+    Bool.not_eq_false']
+  constructor
+  · exact fun h => ⟨fun a => Or.inr (h a), h k⟩
+  · rintro ⟨h₁, h₂⟩ a
+    specialize h₁ a
+    cases hbeq : k == a
+    · simp_all
+    · simp only [hbeq, true_or] at h₁
+      exact containsKey_congr hbeq ▸ h₂
+
+theorem isEmpty_eq_false_of_isEmpty_eraseKey_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k : α}
+    (he : (eraseKey k l).isEmpty = false) :
+    l.isEmpty = false := by
+  simp_all [isEmpty_eq_isEmpty_eraseKey_and_not_containsKey hd k]
+
 theorem getValueCast?_eraseKey [BEq α] [LawfulBEq α] {l : List ((a : α) × β a)} {k a : α}
     (hl : DistinctKeys l) :
     getValueCast? a (eraseKey k l) = if k == a then none else getValueCast? a l := by
@@ -4348,6 +4369,10 @@ theorem isSome_minKey?_eq_not_isEmpty [Ord α] {l : List ((a : α) × β a)} :
     (minKey? l).isSome = !l.isEmpty := by
   simpa [minKey?] using isSome_minEntry?_eq_not_isEmpty
 
+theorem isSome_minKey?_iff_isEmpty_eq_false [Ord α] {l : List ((a : α) × β a)} :
+    (minKey? l).isSome ↔ l.isEmpty = false := by
+  simp [isSome_minKey?_eq_not_isEmpty]
+
 theorem min_apply [Ord α] {e₁ e₂ : (a : α) × β a} {f : (a : α) × β a → (a : α) × β a}
     (hf : compare e₁.1 e₂.1 = compare (f e₁).1 (f e₂).1) :
    min (f e₁) (f e₂) = f (min e₁ e₂) := by
@@ -4689,7 +4714,7 @@ theorem minKey?_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k f
       have := TransCmp.lt_of_isLE_of_lt hkm hcmp
       simp [this] at h
 
-theorem minKey?_modifyKey_of_lawfulEqOrd [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+theorem minKey?_modifyKey_eq_minKey? [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
     [LawfulEqOrd α] {k f} {l : List ((_ : α) × β)} (hd : DistinctKeys l) :
     minKey? (modifyKey k f l) = minKey? l := by
   simp only [minKey?_modifyKey hd]
@@ -4741,6 +4766,185 @@ theorem minKey?_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd 
 
 end Const
 
+/-- Given a proof that the list is nonempty, returns the smallest key in an associative list. -/
+def minKey [Ord α] (xs : List ((a : α) × β a)) (h : xs.isEmpty = false) : α :=
+  minKey? xs |>.get (by simp [isSome_minKey?_eq_not_isEmpty, h])
+
+theorem minKey_of_perm [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l l' : List ((a : α) × β a)}
+    {hl} (hd : DistinctKeys l) (hp : l.Perm l') :
+    minKey l hl = minKey l' (hp.isEmpty_eq ▸ hl) := by
+  simp [minKey, minKey?_of_perm hd hp]
+
+theorem minKey_eq_get_minKey? [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} {he} :
+    minKey l he = (minKey? l |>.get (by simp [isSome_minKey?_eq_not_isEmpty, he])) :=
+  rfl
+
+theorem minKey?_eq_some_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} {he} :
+    minKey? l = some (minKey l he) := by
+  simp [minKey_eq_get_minKey?]
+
+private theorem Option.get_eq_iff_eq_some {o : Option α} {h k} :
+    o.get h = k ↔ o = some k := by
+  simp [Option.eq_some_iff_get_eq, exists_prop_of_true h]
+
+private theorem Option.eq_get_iff_some_eq {o : Option α} {h k} :
+    k = o.get h ↔ some k = o := by
+  conv => congr <;> rw [eq_comm]
+  exact get_eq_iff_eq_some
+
+theorem minKey_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 km} :
+    minKey l he = km ↔ getKey? km l = some km ∧ ∀ k, containsKey k l → (compare km k).isLE := by
+  simp [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some,
+    minKey?_eq_some_iff_getKey?_eq_self_and_forall hd]
+
+theorem minKey_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 km} :
+    minKey l he = km ↔ containsKey km l ∧ ∀ k, containsKey k l → (compare km k).isLE := by
+  simp [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some, minKey?_eq_some_iff_mem_and_forall hd]
+
+theorem minKey_insertEntry [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {l : List ((a : α) × β a)}
+    (hd : DistinctKeys l) {k v} :
+    (insertEntry k v l |> minKey <| isEmpty_insertEntry) =
+      ((minKey? l).elim k fun k' => if compare k k'|>.isLE then k else k') := by
+  simp [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some, minKey?_insertEntry hd]
+
+theorem minKey_insertEntry_le_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v he} :
+    compare (insertEntry k v l |> minKey <| isEmpty_insertEntry) (minKey l he) |>.isLE := by
+  simp only [minKey_eq_get_minKey?]
+  exact minKey?_insertEntry_le_minKey? hd (by simp) rfl
+
+theorem minKey_insertEntry_le_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    compare (insertEntry k v l |> minKey <| isEmpty_insertEntry) k |>.isLE := by
+  simp only [minKey_eq_get_minKey?]
+  exact minKey?_insertEntry_le_self hd rfl
+
+theorem containsKey_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he} :
+    containsKey (minKey l he) l :=
+  containsKey_minKey? hd minKey?_eq_some_minKey
+
+theorem minKey_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) :
+    compare (minKey l <| isEmpty_eq_false_iff_exists_containsKey.mpr ⟨k, hc⟩) k |>.isLE :=
+   minKey?_le_of_containsKey hd hc minKey_eq_get_minKey?.symm
+
+theorem le_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k he} :
+    (compare k (minKey l he)).isLE ↔ (∀ k', containsKey k' l → (compare k k').isLE) := by
+  simp only [minKey_eq_get_minKey?, ← le_minKey? hd, Option.eq_some_iff_get_eq]
+  simp only [exists_prop_of_true (isSome_minKey?_iff_isEmpty_eq_false.mpr he), forall_eq']
+
+theorem getKey?_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he} :
+    getKey? (minKey l he) l = some (minKey l he) :=
+  getKey?_minKey? hd minKey?_eq_some_minKey
+
+theorem getKey_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he} :
+    getKey (minKey l he) l (containsKey_minKey hd) = minKey l he := by
+  simpa [getKey?_eq_some_getKey (containsKey_minKey hd)] 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 he) l = minKey l he := by
+  simpa [getKey_eq_getKey!] using getKey_minKey hd
+
+theorem getKeyD_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he fallback} :
+    getKeyD (minKey l he) l fallback = minKey l he := by
+  simpa [getKey_eq_getKeyD (fallback := fallback)] using getKey_minKey hd
+
+theorem minKey_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 he} :
+    (eraseKey k l |> minKey <| he) =
+        minKey l (isEmpty_eq_false_of_isEmpty_eraseKey_eq_false hd he) ↔
+      (k == (minKey l <| isEmpty_eq_false_of_isEmpty_eraseKey_eq_false hd he)) = false := by
+  simp only [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some, Option.some_get]
+  constructor
+  · intro h
+    exact minKey?_eraseKey_eq_iff_beq_minKey?_eq_false hd |>.mp h (by rw [Option.some_get])
+  · intro h
+    apply minKey?_eraseKey_eq_iff_beq_minKey?_eq_false hd |>.mpr
+    intro km hkm
+    simp_all only [Option.get_some]
+
+theorem minKey_eraseKey_eq_of_beq_minKey_eq_false [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k he} :
+    (hc : (k == (minKey l (isEmpty_eq_false_of_isEmpty_eraseKey_eq_false hd he))) = false) →
+    (eraseKey k l |> minKey <| he) =
+      minKey l (isEmpty_eq_false_of_isEmpty_eraseKey_eq_false hd he) :=
+  minKey_eraseKey_eq_iff_beq_minKey_eq_false hd |>.mpr
+
+theorem minKey_le_minKey_erase [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k he} :
+    compare (minKey l <| isEmpty_eq_false_of_isEmpty_eraseKey_eq_false hd he)
+      (eraseKey k l |> minKey <| he) |>.isLE :=
+  minKey?_le_minKey?_eraseKey hd minKey?_eq_some_minKey minKey_eq_get_minKey?.symm
+
+theorem minKey_insertEntryIfNew [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    (insertEntryIfNew k v l |> minKey <| isEmpty_insertEntryIfNew) =
+      (minKey? l).elim k fun k' => if compare k k' = .lt then k else k' := by
+  simp [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some, ← minKey?_insertEntryIfNew hd (v := v)]
+
+theorem minKey_insertEntryIfNew_le_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v he} :
+    compare (insertEntryIfNew k v l |> minKey <| isEmpty_insertEntryIfNew)
+      (minKey l he) |>.isLE :=
+  minKey?_insertEntryIfNew_le_minKey? hd minKey?_eq_some_minKey minKey_eq_get_minKey?.symm
+
+theorem minKey_insertEntryIfNew_le_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k v} :
+    compare (insertEntryIfNew k v l  |> minKey <| isEmpty_insertEntryIfNew) k |>.isLE :=
+  minKey?_insertEntryIfNew_le_self hd minKey_eq_get_minKey?.symm
+
+theorem minKey_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α] {k f}
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {he} :
+    (modifyKey k f l |> minKey <| he) = minKey l (isEmpty_modifyKey k f l ▸ he):= by
+  simp [minKey_eq_get_minKey?, minKey?_modifyKey hd]
+
+theorem minKey_alterKey_eq_self [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    {l : List ((a : α) × β a)} (hd : DistinctKeys l) {k f he} :
+    (alterKey k f l |> minKey <| he) = k ↔
+      (f (getValueCast? k l)).isSome ∧ ∀ k', containsKey k' l → (compare k k').isLE := by
+  simp [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some, minKey?_alterKey_eq_self hd]
+
+namespace Const
+
+variable {β : Type v}
+
+theorem minKey_modifyKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f he} :
+    (modifyKey k f l |> minKey <| he) =
+      if (minKey l <| isEmpty_modifyKey k f l ▸ he) == k then
+        k
+      else
+        (minKey l <| isEmpty_modifyKey k f l ▸ he) := by
+  simp [minKey_eq_get_minKey?, minKey?_modifyKey hd, Option.get_map]
+
+theorem minKey_modifyKey_eq_minKey [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [LawfulEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f he} :
+    (modifyKey k f l |> minKey <| he) = minKey l (isEmpty_modifyKey k f l ▸ he) := by
+  simp [minKey_eq_get_minKey?, minKey?_modifyKey_eq_minKey? hd]
+
+theorem minKey_modifyKey_beq [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α]
+    {l : List ((_ : α) × β)} (hd : DistinctKeys l) {k f he} :
+    (modifyKey k f l |> minKey <| he) == (minKey l <| isEmpty_modifyKey k f l ▸ he) :=
+  minKey?_modifyKey_beq hd minKey?_eq_some_minKey rfl
+
+theorem minKey_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 |> minKey <| he) = k ↔
+      (f (getValue? k l)).isSome ∧ ∀ k', containsKey k' l → (compare k k').isLE := by
+  simp [minKey_eq_get_minKey?, Option.get_eq_iff_eq_some, minKey?_alterKey_eq_self hd]
+
+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 afe6531703..ff9abb064c 100644
--- a/src/Std/Data/TreeMap/Lemmas.lean
+++ b/src/Std/Data/TreeMap/Lemmas.lean
@@ -142,6 +142,15 @@ 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) :
+    t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
+  DTreeMap.isEmpty_eq_isEmpty_erase_and_not_containsKey k
+
+theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] {k : α}
+    (he : (t.erase k).isEmpty = false) :
+    t.isEmpty = false :=
+  DTreeMap.isEmpty_eq_false_of_isEmpty_erase_eq_false he
+
 @[simp]
 theorem contains_erase [TransCmp cmp] {k a : α} :
     (t.erase k).contains a = (cmp k a != .eq && t.contains a) :=
@@ -310,6 +319,11 @@ theorem getElem?_eq_some_getElem! [TransCmp cmp] [Inhabited β] {a : α} :
     a ∈ t → t[a]? = some t[a]! :=
   DTreeMap.Const.get?_eq_some_get!
 
+theorem getElem!_eq_get!_getElem? [TransCmp cmp] [Inhabited β] {a : α} :
+    t[a]! = t[a]?.get! :=
+  DTreeMap.Const.get!_eq_get!_get?
+
+@[deprecated getElem!_eq_get!_getElem? (since := "2025-03-19")]
 theorem getElem!_eq_getElem!_getElem? [TransCmp cmp] [Inhabited β] {a : α} :
     t[a]! = t[a]?.get! :=
   DTreeMap.Const.get!_eq_get!_get?
@@ -1766,6 +1780,10 @@ theorem isSome_minKey?_eq_not_isEmpty [TransCmp cmp] :
     t.minKey?.isSome = !t.isEmpty :=
   DTreeMap.isSome_minKey?_eq_not_isEmpty
 
+theorem isSome_minKey?_iff_isEmpty_eq_false [TransCmp cmp] :
+    t.minKey?.isSome ↔ t.isEmpty = false :=
+  DTreeMap.isSome_minKey?_iff_isEmpty_eq_false
+
 theorem minKey?_insert [TransCmp cmp] {k v} :
     (t.insert k v).minKey? =
       some (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
@@ -1891,9 +1909,133 @@ theorem compare_minKey?_modify_eq [TransCmp cmp] {k f km kmm} :
 
 theorem minKey?_alter_eq_self [TransCmp cmp] {k f} :
     (t.alter k f).minKey? = some k ↔
-      (f (t.get? k)).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+      (f t[k]?).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
   DTreeMap.Const.minKey?_alter_eq_self
 
+theorem minKey_eq_get_minKey? [TransCmp cmp] {he} :
+    t.minKey he = t.minKey?.get (isSome_minKey?_iff_isEmpty_eq_false.mpr he) :=
+  DTreeMap.minKey_eq_get_minKey?
+
+theorem minKey?_eq_some_minKey [TransCmp cmp] {he} :
+    t.minKey? = some (t.minKey he) :=
+  DTreeMap.minKey?_eq_some_minKey
+
+theorem minKey_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] {he km} :
+    t.minKey he = km ↔ t.getKey? km = some km ∧ ∀ k ∈ t, (cmp km k).isLE :=
+  DTreeMap.minKey_eq_iff_getKey?_eq_self_and_forall
+
+theorem minKey_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] {he km} :
+    t.minKey he = km ↔ km ∈ t ∧ ∀ k ∈ t, (cmp km k).isLE :=
+  DTreeMap.minKey_eq_some_iff_mem_and_forall
+
+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' :=
+  DTreeMap.minKey_insert
+
+theorem minKey_insert_le_minKey [TransCmp cmp] {k v he} :
+    cmp (t.insert k v |>.minKey isEmpty_insert) (t.minKey he) |>.isLE :=
+  DTreeMap.minKey_insert_le_minKey
+
+theorem minKey_insert_le_self [TransCmp cmp] {k v} :
+    cmp (t.insert k v |>.minKey isEmpty_insert) k |>.isLE :=
+  DTreeMap.minKey_insert_le_self
+
+theorem contains_minKey [TransCmp cmp] {he} :
+    t.contains (t.minKey he) :=
+  DTreeMap.contains_minKey
+
+theorem minKey_mem [TransCmp cmp] {he} :
+    t.minKey he ∈ t :=
+  DTreeMap.minKey_mem
+
+theorem minKey_le_of_contains [TransCmp cmp] {k} (hc : t.contains k) :
+    cmp (t.minKey <| isEmpty_eq_false_iff_exists_contains_eq_true.mpr ⟨k, hc⟩) k |>.isLE :=
+  DTreeMap.minKey_le_of_contains hc
+
+theorem minKey_le_of_mem [TransCmp cmp] {k} (hc : k ∈ t) :
+    cmp (t.minKey <| isEmpty_eq_false_iff_exists_contains_eq_true.mpr ⟨k, hc⟩) k |>.isLE :=
+  DTreeMap.minKey_le_of_mem hc
+
+theorem le_minKey [TransCmp cmp] {k he} :
+    (cmp k (t.minKey he)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.le_minKey
+
+@[simp]
+theorem getKey?_minKey [TransCmp cmp] {he} :
+    t.getKey? (t.minKey he) = some (t.minKey he) :=
+  DTreeMap.getKey?_minKey
+
+@[simp]
+theorem getKey_minKey [TransCmp cmp] {he hc} :
+    t.getKey (t.minKey he) hc = t.minKey he :=
+  DTreeMap.getKey_minKey
+
+@[simp]
+theorem getKey!_minKey [TransCmp cmp] [Inhabited α] {he} :
+    t.getKey! (t.minKey he) = t.minKey he :=
+  DTreeMap.getKey!_minKey
+
+@[simp]
+theorem getKeyD_minKey [TransCmp cmp] {he fallback} :
+    t.getKeyD (t.minKey he) fallback = t.minKey he :=
+  DTreeMap.getKeyD_minKey
+
+@[simp]
+theorem minKey_erase_eq_iff_not_cmp_eq_minKey [TransCmp cmp] {k he} :
+    (t.erase k |>.minKey he) =
+        t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false he) ↔
+      ¬ cmp k (t.minKey <| isEmpty_eq_false_of_isEmpty_erase_eq_false he) = .eq :=
+  DTreeMap.minKey_erase_eq_iff_not_compare_eq_minKey
+
+theorem minKey_erase_eq_of_not_cmp_eq_minKey [TransCmp cmp] {k he} :
+    (hc : ¬ cmp k (t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false he)) = .eq) →
+    (t.erase k |>.minKey he) =
+      t.minKey (isEmpty_eq_false_of_isEmpty_erase_eq_false he) :=
+  DTreeMap.minKey_erase_eq_of_not_compare_eq_minKey
+
+theorem minKey_le_minKey_erase [TransCmp cmp] {k he} :
+    cmp (t.minKey <| isEmpty_eq_false_of_isEmpty_erase_eq_false he)
+      (t.erase k |>.minKey he) |>.isLE :=
+  DTreeMap.minKey_le_minKey_erase
+
+theorem minKey_insertIfNew [TransCmp cmp] {k v} :
+    (t.insertIfNew k v).minKey isEmpty_insertIfNew =
+      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] {k v he} :
+    cmp (t.insertIfNew k v |>.minKey isEmpty_insertIfNew)
+      (t.minKey he) |>.isLE :=
+  DTreeMap.minKey_insertIfNew_le_minKey (t := t.inner) (he := he)
+
+theorem minKey_insertIfNew_le_self [TransCmp cmp] {k v} :
+    cmp (t.insertIfNew k v |>.minKey <| isEmpty_insertIfNew) k |>.isLE :=
+  DTreeMap.minKey_insertIfNew_le_self
+
+theorem minKey_modify [TransCmp cmp] {k f he} :
+    (modify t k f).minKey he =
+      if cmp (t.minKey <| cast (congrArg (· = false) isEmpty_modify) he) k = .eq then
+        k
+      else
+        (t.minKey <| cast (congrArg (· = false) isEmpty_modify) he) :=
+  DTreeMap.Const.minKey_modify
+
+@[simp]
+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) :=
+  DTreeMap.Const.minKey_modify_eq_minKey
+
+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 :=
+  DTreeMap.Const.compare_minKey_modify_eq
+
+theorem minKey_alter_eq_self [TransCmp cmp] {k f he} :
+    (alter t k f).minKey he = k ↔
+      (f t[k]?).isSome ∧ ∀ k', k' ∈ t → (cmp k k').isLE :=
+  DTreeMap.Const.minKey_alter_eq_self
+
 end Min
 
 end Std.TreeMap
diff --git a/src/Std/Data/TreeMap/Raw/Lemmas.lean b/src/Std/Data/TreeMap/Raw/Lemmas.lean
index e92f7d1eb5..8ca8ee261e 100644
--- a/src/Std/Data/TreeMap/Raw/Lemmas.lean
+++ b/src/Std/Data/TreeMap/Raw/Lemmas.lean
@@ -143,6 +143,15 @@ 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) :
+    t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
+  DTreeMap.Raw.isEmpty_eq_isEmpty_erase_and_not_containsKey h k
+
+theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] (h : t.WF) {k : α}
+    (he : (t.erase k).isEmpty = false) :
+    t.isEmpty = false :=
+  DTreeMap.Raw.isEmpty_eq_false_of_isEmpty_erase_eq_false h he
+
 @[simp]
 theorem contains_erase [TransCmp cmp] (h : t.WF) {k a : α} :
     (t.erase k).contains a = (cmp k a != .eq && t.contains a) :=
@@ -311,6 +320,11 @@ theorem getElem?_eq_some_getElem! [TransCmp cmp] [Inhabited β] (h : t.WF) {a :
     a ∈ t → t[a]? = some t[a]! :=
   DTreeMap.Raw.Const.get?_eq_some_get! h
 
+theorem getElem!_eq_get!_getElem? [TransCmp cmp] [Inhabited β] (h : t.WF) {a : α} :
+    t[a]! = t[a]?.get! :=
+  DTreeMap.Raw.Const.get!_eq_get!_get? h
+
+@[deprecated getElem!_eq_get!_getElem? (since := "2025-03-19")]
 theorem getElem!_eq_getElem!_getElem? [TransCmp cmp] [Inhabited β] (h : t.WF) {a : α} :
     t[a]! = t[a]?.get! :=
   DTreeMap.Raw.Const.get!_eq_get!_get? h
@@ -1775,6 +1789,10 @@ theorem isSome_minKey?_eq_not_isEmpty [TransCmp cmp] (h : t.WF) :
     t.minKey?.isSome = !t.isEmpty :=
   DTreeMap.Raw.isSome_minKey?_eq_not_isEmpty h
 
+theorem isSome_minKey?_iff_isEmpty_eq_false [TransCmp cmp] (h : t.WF) :
+    t.minKey?.isSome ↔ t.isEmpty = false :=
+  DTreeMap.Raw.isSome_minKey?_iff_isEmpty_eq_false h
+
 theorem minKey?_insert [TransCmp cmp] (h : t.WF) {k v} :
     (t.insert k v).minKey? =
       some (t.minKey?.elim k fun k' => if cmp k k'|>.isLE then k else k') :=
diff --git a/src/Std/Data/TreeSet/Lemmas.lean b/src/Std/Data/TreeSet/Lemmas.lean
index ec2361ccf6..6eee9f52b4 100644
--- a/src/Std/Data/TreeSet/Lemmas.lean
+++ b/src/Std/Data/TreeSet/Lemmas.lean
@@ -148,6 +148,15 @@ 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) :
+    t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
+  DTreeMap.isEmpty_eq_isEmpty_erase_and_not_containsKey k
+
+theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] {k : α}
+    (he : (t.erase k).isEmpty = false) :
+    t.isEmpty = false :=
+  TreeMap.isEmpty_eq_false_of_isEmpty_erase_eq_false he
+
 @[simp]
 theorem contains_erase [TransCmp cmp] {k a : α} :
     (t.erase k).contains a = (cmp k a != .eq && t.contains a) :=
@@ -689,6 +698,10 @@ theorem isSome_min?_eq_not_isEmpty [TransCmp cmp] :
     t.min?.isSome = !t.isEmpty :=
   TreeMap.isSome_minKey?_eq_not_isEmpty
 
+theorem isSome_min?_iff_isEmpty_eq_false [TransCmp cmp] :
+    t.min?.isSome ↔ t.isEmpty = false :=
+  DTreeMap.isSome_minKey?_iff_isEmpty_eq_false
+
 theorem min?_insert [TransCmp cmp] {k} :
     (t.insert k).min? =
       t.min?.elim k fun k' => if cmp k k' = .lt then k else k' :=
@@ -767,6 +780,94 @@ theorem min?_le_min?_erase [TransCmp cmp] {k km kme} :
     cmp km kme |>.isLE :=
   TreeMap.minKey?_le_minKey?_erase
 
+theorem min_eq_get_min? [TransCmp cmp] {he} :
+    t.min he = t.min?.get (isSome_min?_iff_isEmpty_eq_false.mpr he) :=
+  TreeMap.minKey_eq_get_minKey?
+
+theorem min?_eq_some_min [TransCmp cmp] {he} :
+    t.min? = some (t.min he) :=
+  TreeMap.minKey?_eq_some_minKey
+
+theorem min_eq_iff_getKey?_eq_self_and_forall [TransCmp cmp] {he km} :
+    t.min he = km ↔ t.get? km = some km ∧ ∀ k ∈ t, (cmp km k).isLE :=
+  TreeMap.minKey_eq_iff_getKey?_eq_self_and_forall
+
+theorem min_eq_some_iff_mem_and_forall [TransCmp cmp] [LawfulEqCmp cmp] {he km} :
+    t.min he = km ↔ km ∈ t ∧ ∀ k ∈ t, (cmp km k).isLE :=
+  TreeMap.minKey_eq_some_iff_mem_and_forall
+
+theorem min_insert [TransCmp cmp] {k} :
+    (t.insert k).min isEmpty_insert =
+      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] {k he} :
+    cmp (t.insert k |>.min isEmpty_insert)
+      (t.min he) |>.isLE :=
+  DTreeMap.minKey_insertIfNew_le_minKey
+
+theorem min_insert_le_self [TransCmp cmp] {k} :
+    cmp (t.insert k |>.min <| isEmpty_insert) k |>.isLE :=
+  DTreeMap.minKey_insertIfNew_le_self
+
+theorem contains_min [TransCmp cmp] {he} :
+    t.contains (t.min he) :=
+  DTreeMap.contains_minKey
+
+theorem min_mem [TransCmp cmp] {he} :
+    t.min he ∈ t :=
+  DTreeMap.minKey_mem
+
+theorem min_le_of_contains [TransCmp cmp] {k} (hc : t.contains k) :
+    cmp (t.min <| isEmpty_eq_false_iff_exists_contains_eq_true.mpr ⟨k, hc⟩) k |>.isLE :=
+  DTreeMap.minKey_le_of_contains hc
+
+theorem min_le_of_mem [TransCmp cmp] {k} (hc : k ∈ t) :
+    cmp (t.min <| isEmpty_eq_false_iff_exists_contains_eq_true.mpr ⟨k, hc⟩) k |>.isLE :=
+  DTreeMap.minKey_le_of_mem hc
+
+theorem le_min [TransCmp cmp] {k he} :
+    (cmp k (t.min he)).isLE ↔ (∀ k', k' ∈ t → (cmp k k').isLE) :=
+  DTreeMap.le_minKey
+
+@[simp]
+theorem get?_min [TransCmp cmp] {he} :
+    t.get? (t.min he) = some (t.min he) :=
+  DTreeMap.getKey?_minKey
+
+@[simp]
+theorem get_min [TransCmp cmp] {he hc} :
+    t.get (t.min he) hc = t.min he :=
+  DTreeMap.getKey_minKey
+
+@[simp]
+theorem get!_min [TransCmp cmp] [Inhabited α] {he} :
+    t.get! (t.min he) = t.min he :=
+  DTreeMap.getKey!_minKey
+
+@[simp]
+theorem getD_min [TransCmp cmp] {he fallback} :
+    t.getD (t.min he) fallback = t.min he :=
+  DTreeMap.getKeyD_minKey
+
+@[simp]
+theorem min_erase_eq_iff_not_cmp_eq_min [TransCmp cmp] {k he} :
+    (t.erase k |>.min he) =
+        t.min (isEmpty_eq_false_of_isEmpty_erase_eq_false he) ↔
+      ¬ cmp k (t.min <| isEmpty_eq_false_of_isEmpty_erase_eq_false he) = .eq :=
+  DTreeMap.minKey_erase_eq_iff_not_compare_eq_minKey
+
+theorem min_erase_eq_of_not_cmp_eq_min [TransCmp cmp] {k he} :
+    (hc : ¬ cmp k (t.min (isEmpty_eq_false_of_isEmpty_erase_eq_false he)) = .eq) →
+    (t.erase k |>.min he) =
+      t.min (isEmpty_eq_false_of_isEmpty_erase_eq_false he) :=
+  DTreeMap.minKey_erase_eq_of_not_compare_eq_minKey
+
+theorem min_le_min_erase [TransCmp cmp] {k he} :
+    cmp (t.min <| isEmpty_eq_false_of_isEmpty_erase_eq_false he)
+      (t.erase k |>.min he) |>.isLE :=
+  DTreeMap.minKey_le_minKey_erase
+
 end Min
 
 end Std.TreeSet
diff --git a/src/Std/Data/TreeSet/Raw/Lemmas.lean b/src/Std/Data/TreeSet/Raw/Lemmas.lean
index 7d3a3cacbe..9a5a3fd5e1 100644
--- a/src/Std/Data/TreeSet/Raw/Lemmas.lean
+++ b/src/Std/Data/TreeSet/Raw/Lemmas.lean
@@ -149,6 +149,15 @@ 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) :
+    t.isEmpty = ((t.erase k).isEmpty && !(t.contains k)) :=
+  TreeMap.Raw.isEmpty_eq_isEmpty_erase_and_not_containsKey h k
+
+theorem isEmpty_eq_false_of_isEmpty_erase_eq_false [TransCmp cmp] (h : t.WF) {k : α}
+    (he : (t.erase k).isEmpty = false) :
+    t.isEmpty = false :=
+  TreeMap.Raw.isEmpty_eq_false_of_isEmpty_erase_eq_false h he
+
 @[simp]
 theorem contains_erase [TransCmp cmp] (h : t.WF) {k a : α} :
     (t.erase k).contains a = (cmp k a != .eq && t.contains a) :=
@@ -687,6 +696,10 @@ theorem isSome_min?_eq_not_isEmpty [TransCmp cmp] (h : t.WF) :
     t.min?.isSome = !t.isEmpty :=
   TreeMap.Raw.isSome_minKey?_eq_not_isEmpty h
 
+theorem isSome_min?_iff_isEmpty_eq_false [TransCmp cmp] (h : t.WF) :
+    t.min?.isSome ↔ t.isEmpty = false :=
+  TreeMap.Raw.isSome_minKey?_iff_isEmpty_eq_false h
+
 theorem min?_insert [TransCmp cmp] (h : t.WF) {k} :
     (t.insert k).min? =
       t.min?.elim k fun k' => if cmp k k' = .lt then k else k' :=