feat: add lemmas relating getMin/getMin?/getMin!/getMinD and insertion to the empty (D)TreeMap/TreeSet (#11231)

This PR adds several lemmas that relate
`getMin`/`getMin?`/`getMin!`/`getMinD` and insertion to the empty
(D)TreeMap/TreeSet and their extensional variants.

---------

Co-authored-by: Markus Himmel <markus@himmel-villmar.de>

src/Std/Data/DTreeMap/Internal/Lemmas.lean

@@ -5948,6 +5948,94 @@ theorem isSome_minKey?_insert! [TransOrd α] (h : t.WF) {k v} :
     (t.insert! k v).minKey?.isSome := by
   simpa only [insert_eq_insert!] using isSome_minKey?_insert h
 
+theorem minKey_insert_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v h.balanced).impl.minKey (isEmpty_insert h) = k := by
+  revert he
+  simp_to_model [isEmpty, insert, minKey] using List.minKey_insertEntry_of_isEmpty
+
+theorem minKey_insert!_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert! k v).minKey (isEmpty_insert! h) = k := by
+  simpa only [insert_eq_insert!] using minKey_insert_of_isEmpty h he
+
+theorem minKey?_insert_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v h.balanced).impl.minKey? = some k := by
+  revert he
+  simp_to_model [isEmpty, insert, minKey?] using List.minKey?_insertEntry_of_isEmpty
+
+theorem minKey?_insert!_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert! k v).minKey? = some k := by
+  simpa only [insert_eq_insert!] using minKey?_insert_of_isEmpty h he
+
+theorem minKey!_insert_of_isEmpty [TransOrd α] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v h.balanced).impl.minKey! = k := by
+  revert he
+  simp_to_model [isEmpty, insert, minKey!]
+  intro he
+  apply List.minKey!_insertEntry_of_isEmpty
+  · wf_trivial
+  · exact he
+
+theorem minKey!_insert!_of_isEmpty [TransOrd α] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert! k v).minKey! = k := by
+  simpa only [insert_eq_insert!] using minKey!_insert_of_isEmpty h he
+
+theorem minKeyD_insert_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v h.balanced).impl.minKeyD fallback = k := by
+  revert he
+  simp_to_model [isEmpty, insert, minKeyD]
+  intro he
+  apply List.minKeyD_insertEntry_of_isEmpty
+  · wf_trivial
+  · exact he
+
+theorem minKeyD_insert!_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert! k v).minKeyD fallback = k := by
+  simpa only [insert_eq_insert!] using minKeyD_insert_of_isEmpty h he
+
+theorem minKey_insertIfNew_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v h.balanced).impl.minKey (isEmpty_insertIfNew h) = k := by
+  revert he
+  simp_to_model [isEmpty, insertIfNew, minKey] using List.minKey_insertEntryIfNew_of_isEmpty
+
+theorem minKey_insertIfNew!_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew! k v).minKey (isEmpty_insertIfNew! h) = k := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKey_insertIfNew_of_isEmpty h he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v h.balanced).impl.minKey? = some k := by
+  revert he
+  simp_to_model [isEmpty, insertIfNew, minKey?] using List.minKey?_insertEntryIfNew_of_isEmpty
+
+theorem minKey?_insertIfNew!_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew! k v).minKey? = some k := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKey?_insertIfNew_of_isEmpty h he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransOrd α] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v h.balanced).impl.minKey! = k := by
+  revert he
+  simp_to_model [isEmpty, insertIfNew, minKey!]
+  intro he
+  apply List.minKey!_insertEntryIfNew_of_isEmpty
+  · wf_trivial
+  · exact he
+
+theorem minKey!_insertIfNew!_of_isEmpty [TransOrd α] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew! k v).minKey! = k := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKey!_insertIfNew_of_isEmpty h he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v h.balanced).impl.minKeyD fallback = k := by
+  revert he
+  simp_to_model [isEmpty, insertIfNew, minKeyD]
+  intro he
+  apply List.minKeyD_insertEntryIfNew_of_isEmpty
+  · wf_trivial
+  · exact he
+
+theorem minKeyD_insertIfNew!_of_isEmpty [TransOrd α] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew! k v).minKeyD fallback = k := by
+  simpa only [insertIfNew_eq_insertIfNew!] using minKeyD_insertIfNew_of_isEmpty h he
+
 theorem minKey?_insert_le_minKey? [TransOrd α] (h : t.WF) {k v km kmi} :
     (hkm : t.minKey? = some km) →
     (hkmi : (t.insert k v h.balanced |>.impl.minKey? |>.get <| isSome_minKey?_insert h) = kmi) →

src/Std/Data/DTreeMap/Lemmas.lean

@@ -3619,6 +3619,38 @@ theorem isSome_minKey?_iff_isEmpty_eq_false [TransCmp cmp] :
     (t.insert k v).minKey?.isSome :=
   Impl.isSome_minKey?_insert t.wf
 
+theorem minKey_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey isEmpty_insert = k :=
+  Impl.minKey_insert_of_isEmpty t.wf he
+
+theorem minKey?_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey? = some k :=
+  Impl.minKey?_insert_of_isEmpty t.wf he
+
+theorem minKey!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey! = k :=
+  Impl.minKey!_insert_of_isEmpty t.wf he
+
+theorem minKeyD_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v).minKeyD fallback = k :=
+  Impl.minKeyD_insert_of_isEmpty t.wf he
+
+theorem minKey_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey isEmpty_insertIfNew = k :=
+  Impl.minKey_insertIfNew_of_isEmpty t.wf he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey? = some k :=
+  Impl.minKey?_insertIfNew_of_isEmpty t.wf he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey! = k :=
+  Impl.minKey!_insertIfNew_of_isEmpty t.wf he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v).minKeyD fallback = k :=
+  Impl.minKeyD_insertIfNew_of_isEmpty t.wf he
+
 theorem minKey?_insert_le_minKey? [TransCmp cmp] {k v km kmi} :
     (hkm : t.minKey? = some km) →
     (hkmi : (t.insert k v |>.minKey? |>.get isSome_minKey?_insert) = kmi) →

src/Std/Data/DTreeMap/Raw/Lemmas.lean

@@ -3780,6 +3780,30 @@ theorem isSome_minKey?_insert [TransCmp cmp] (h : t.WF) {k v} :
     (t.insert k v).minKey?.isSome :=
   Impl.isSome_minKey?_insert! h
 
+theorem minKey?_insert_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey? = some k :=
+  Impl.minKey?_insert!_of_isEmpty h he
+
+theorem minKey!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey! = k :=
+  Impl.minKey!_insert!_of_isEmpty h he
+
+theorem minKeyD_insert_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v).minKeyD fallback = k :=
+  Impl.minKeyD_insert!_of_isEmpty h he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey? = some k :=
+  Impl.minKey?_insertIfNew!_of_isEmpty h he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey! = k :=
+  Impl.minKey!_insertIfNew!_of_isEmpty h he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v).minKeyD fallback = k :=
+  Impl.minKeyD_insertIfNew!_of_isEmpty h he
+
 theorem minKey?_insert_le_minKey? [TransCmp cmp] (h : t.WF) {k v km kmi} :
     (hkm : t.minKey? = some km) →
     (hkmi : (t.insert k v |>.minKey? |>.get <| isSome_minKey?_insert h) = kmi) →

src/Std/Data/ExtDTreeMap/Lemmas.lean

@@ -3407,6 +3407,42 @@ theorem isSome_minKey?_iff_ne_empty [TransCmp cmp] :
     (t.insert k v).minKey?.isSome :=
   t.inductionOn fun _ => DTreeMap.isSome_minKey?_insert
 
+theorem minKey_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey insert_ne_empty = k := by
+  induction t
+  case mk a =>
+    exact DTreeMap.minKey_insert_of_isEmpty he
+
+theorem minKey?_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey? = some k :=
+  t.inductionOn (fun _ he => DTreeMap.minKey?_insert_of_isEmpty he) he
+
+theorem minKey!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey! = k :=
+  t.inductionOn (fun _ he => DTreeMap.minKey!_insert_of_isEmpty he) he
+
+theorem minKeyD_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v).minKeyD fallback = k :=
+  t.inductionOn (fun _ he => DTreeMap.minKeyD_insert_of_isEmpty he) he
+
+theorem minKey_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey insertIfNew_ne_empty = k := by
+  induction t
+  case mk a =>
+    exact DTreeMap.minKey_insertIfNew_of_isEmpty he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey? = some k :=
+  t.inductionOn (fun _ he => DTreeMap.minKey?_insertIfNew_of_isEmpty he) he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey! = k :=
+  t.inductionOn (fun _ he => DTreeMap.minKey!_insertIfNew_of_isEmpty he) he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v).minKeyD fallback = k := t.inductionOn
+    (fun _ he => DTreeMap.minKeyD_insertIfNew_of_isEmpty he) he
+
 theorem minKey?_insert_le_minKey? [TransCmp cmp] {k v km kmi} :
     (hkm : t.minKey? = some km) →
     (hkmi : (t.insert k v |>.minKey? |>.get isSome_minKey?_insert) = kmi) →

src/Std/Data/ExtTreeMap/Lemmas.lean

@@ -2081,6 +2081,38 @@ theorem isSome_minKey?_iff_ne_empty [TransCmp cmp] :
     (t.insert k v).minKey?.isSome :=
   ExtDTreeMap.isSome_minKey?_insert
 
+theorem minKey_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey insert_ne_empty = k :=
+  ExtDTreeMap.minKey_insert_of_isEmpty he
+
+theorem minKey?_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey? = some k :=
+  ExtDTreeMap.minKey?_insert_of_isEmpty he
+
+theorem minKey!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey! = k :=
+  ExtDTreeMap.minKey!_insert_of_isEmpty he
+
+theorem minKeyD_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v).minKeyD fallback = k :=
+  ExtDTreeMap.minKeyD_insert_of_isEmpty he
+
+theorem minKey_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey insertIfNew_ne_empty = k :=
+  ExtDTreeMap.minKey_insertIfNew_of_isEmpty he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey? = some k :=
+  ExtDTreeMap.minKey?_insertIfNew_of_isEmpty he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey! = k :=
+  ExtDTreeMap.minKey!_insertIfNew_of_isEmpty he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v).minKeyD fallback = k :=
+  ExtDTreeMap.minKeyD_insertIfNew_of_isEmpty he
+
 theorem minKey?_insert_le_minKey? [TransCmp cmp] {k v km kmi} :
     (hkm : t.minKey? = some km) →
     (hkmi : (t.insert k v |>.minKey? |>.get isSome_minKey?_insert) = kmi) →

src/Std/Data/ExtTreeSet/Lemmas.lean

@@ -887,6 +887,22 @@ theorem isSome_min?_insert [TransCmp cmp] {k} :
     (t.insert k).min?.isSome :=
   ExtTreeMap.isSome_minKey?_insertIfNew
 
+theorem min_insert_of_isEmpty [TransCmp cmp] {k} (he : t.isEmpty) :
+    (t.insert k).min insert_ne_empty = k :=
+  ExtTreeMap.minKey_insertIfNew_of_isEmpty he
+
+theorem min?_insert_of_isEmpty [TransCmp cmp] {k} (he : t.isEmpty) :
+    (t.insert k).min? = some k :=
+  ExtTreeMap.minKey?_insertIfNew_of_isEmpty he
+
+theorem min!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] {k} (he : t.isEmpty) :
+    (t.insert k).min! = k :=
+  ExtTreeMap.minKey!_insertIfNew_of_isEmpty he
+
+theorem minD_insert_of_isEmpty [TransCmp cmp] {k} (he : t.isEmpty) {fallback : α} :
+    (t.insert k).minD fallback = k :=
+  ExtTreeMap.minKeyD_insertIfNew_of_isEmpty he
+
 theorem min?_insert_le_min? [TransCmp cmp] {k km kmi} :
     (hkm : t.min? = some km) →
     (hkmi : (t.insert k |>.min? |>.get isSome_min?_insert) = kmi) →

src/Std/Data/Internal/List/Associative.lean

@@ -7958,6 +7958,21 @@ theorem minKeyD_eq_getD_minKey? [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α
     minKeyD l fallback = (minKey? l).getD fallback :=
   (rfl)
 
+theorem minKey_insertEntry_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) :
+    List.minKey (insertEntry k v l) isEmpty_insertEntry = k := by
+  simp [minKey, minKey?_insertEntry hl, minKey?_of_isEmpty he]
+
+theorem minKey?_insertEntry_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) :
+    minKey? (insertEntry k v l) = some k := by
+  simp [minKey?_insertEntry hl, minKey?_of_isEmpty he]
+
+theorem minKeyD_insertEntry_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) {fallback : α} :
+    minKeyD (insertEntry k v l) fallback = k := by
+  simp [minKeyD, minKey?_insertEntry hl, minKey?_of_isEmpty he]
+
 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
@@ -7973,6 +7988,33 @@ theorem minKey!_eq_minKeyD_default [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd
     minKey! l = minKeyD l default := by
   simp [minKey!_eq_get!_minKey?, minKeyD_eq_getD_minKey?, Option.get!_eq_getD]
 
+theorem minKey!_insertEntry_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) :
+    minKey! (insertEntry k v l) = k := by
+  simp [minKey!_eq_minKeyD_default]
+  apply minKeyD_insertEntry_of_isEmpty hl he
+
+theorem minKey_insertEntryIfNew_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) :
+    List.minKey (insertEntryIfNew k v l) isEmpty_insertEntryIfNew = k := by
+  simp [minKey, minKey?_insertEntryIfNew hl, minKey?_of_isEmpty he]
+
+theorem minKey?_insertEntryIfNew_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) :
+    minKey? (insertEntryIfNew k v l) = some k := by
+  simp [minKey?_insertEntryIfNew hl, minKey?_of_isEmpty he]
+
+theorem minKeyD_insertEntryIfNew_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) {fallback : α} :
+    minKeyD (insertEntryIfNew k v l) fallback = k := by
+  simp [minKeyD, minKey?_insertEntryIfNew hl, minKey?_of_isEmpty he]
+
+theorem minKey!_insertEntryIfNew_of_isEmpty [Ord α] [TransOrd α] [BEq α] [LawfulBEqOrd α] [Inhabited α] {k : α} {v : β k}
+    {l : List ((a : α) × β a)} (hl : DistinctKeys l) (he : l.isEmpty) :
+    minKey! (insertEntryIfNew k v l) = k := by
+  simp [minKey!_eq_minKeyD_default]
+  apply minKeyD_insertEntryIfNew_of_isEmpty hl he
+
 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

src/Std/Data/TreeMap/Lemmas.lean

@@ -2330,6 +2330,38 @@ theorem isSome_minKey?_iff_isEmpty_eq_false [TransCmp cmp] :
       some (t.minKey?.elim k fun k' => if cmp k k' |>.isLE then k else k') :=
   DTreeMap.minKey?_insert
 
+theorem minKey_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey isEmpty_insert = k :=
+  DTreeMap.minKey_insert_of_isEmpty he
+
+theorem minKey?_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey? = some k :=
+  DTreeMap.minKey?_insert_of_isEmpty he
+
+theorem minKey!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey! = k :=
+  DTreeMap.minKey!_insert_of_isEmpty he
+
+theorem minKeyD_insert_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v).minKeyD fallback = k :=
+  DTreeMap.minKeyD_insert_of_isEmpty he
+
+theorem minKey_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey isEmpty_insertIfNew = k :=
+  DTreeMap.minKey_insertIfNew_of_isEmpty he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey? = some k :=
+  DTreeMap.minKey?_insertIfNew_of_isEmpty he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransCmp cmp] [Inhabited α] {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey! = k :=
+  DTreeMap.minKey!_insertIfNew_of_isEmpty he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransCmp cmp] {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v).minKeyD fallback = k :=
+  DTreeMap.minKeyD_insertIfNew_of_isEmpty he
+
 @[grind =] theorem isSome_minKey?_insert [TransCmp cmp] {k v} :
     (t.insert k v).minKey?.isSome :=
   DTreeMap.isSome_minKey?_insert

src/Std/Data/TreeMap/Raw/Lemmas.lean

@@ -2380,6 +2380,30 @@ theorem minKey?_insert [TransCmp cmp] (h : t.WF) {k v} :
       some (t.minKey?.elim k fun k' => if cmp k k' |>.isLE then k else k') :=
   DTreeMap.Raw.minKey?_insert h
 
+theorem minKey?_insert_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey? = some k :=
+  DTreeMap.Raw.minKey?_insert_of_isEmpty h he
+
+theorem minKey!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insert k v).minKey! = k :=
+  DTreeMap.Raw.minKey!_insert_of_isEmpty h he
+
+theorem minKeyD_insert_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insert k v).minKeyD fallback = k :=
+  DTreeMap.Raw.minKeyD_insert_of_isEmpty h he
+
+theorem minKey?_insertIfNew_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey? = some k :=
+  DTreeMap.Raw.minKey?_insertIfNew_of_isEmpty h he
+
+theorem minKey!_insertIfNew_of_isEmpty [TransCmp cmp] [Inhabited α] (h : t.WF) {k v} (he : t.isEmpty) :
+    (t.insertIfNew k v).minKey! = k :=
+  DTreeMap.Raw.minKey!_insertIfNew_of_isEmpty h he
+
+theorem minKeyD_insertIfNew_of_isEmpty [TransCmp cmp] (h : t.WF) {k v} (he : t.isEmpty) {fallback : α} :
+    (t.insertIfNew k v).minKeyD fallback = k :=
+  DTreeMap.Raw.minKeyD_insertIfNew_of_isEmpty h he
+
 @[grind =]
 theorem isSome_minKey?_insert [TransCmp cmp] (h : t.WF) {k v} :
     (t.insert k v).minKey?.isSome :=

src/Std/Data/TreeSet/Lemmas.lean

@@ -1060,6 +1060,22 @@ theorem isSome_min?_insert [TransCmp cmp] {k} :
     (t.insert k).min?.isSome :=
   TreeMap.isSome_minKey?_insertIfNew
 
+theorem min_insert_of_isEmpty [TransCmp cmp] {k} (he : t.isEmpty) :
+    (t.insert k).min isEmpty_insert = k :=
+  TreeMap.minKey_insertIfNew_of_isEmpty he
+
+theorem min?_insert_of_isEmpty [TransCmp cmp] {k} (he : t.isEmpty) :
+    (t.insert k).min? = some k :=
+  TreeMap.minKey?_insertIfNew_of_isEmpty he
+
+theorem min!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] {k} (he : t.isEmpty) :
+    (t.insert k).min! = k :=
+  TreeMap.minKey!_insertIfNew_of_isEmpty he
+
+theorem minD_insert_of_isEmpty [TransCmp cmp] {k} (he : t.isEmpty) {fallback : α} :
+    (t.insert k).minD fallback = k :=
+  TreeMap.minKeyD_insertIfNew_of_isEmpty he
+
 theorem min?_insert_le_min? [TransCmp cmp] {k km kmi} :
     (hkm : t.min? = some km) →
     (hkmi : (t.insert k |>.min? |>.get isSome_min?_insert) = kmi) →

src/Std/Data/TreeSet/Raw/Lemmas.lean

@@ -1096,6 +1096,18 @@ theorem isSome_min?_insert [TransCmp cmp] (h : t.WF) {k} :
     (t.insert k).min?.isSome :=
   TreeMap.Raw.isSome_minKey?_insertIfNew h
 
+theorem min?_insert_of_isEmpty [TransCmp cmp] (h : t.WF) {k} (he : t.isEmpty) :
+    (t.insert k).min? = some k :=
+  TreeMap.Raw.minKey?_insertIfNew_of_isEmpty h he
+
+theorem min!_insert_of_isEmpty [TransCmp cmp] [Inhabited α] (h : t.WF) {k} (he : t.isEmpty) :
+    (t.insert k).min! = k :=
+  TreeMap.Raw.minKey!_insertIfNew_of_isEmpty h he
+
+theorem minD_insert_of_isEmpty [TransCmp cmp] (h : t.WF) {k} (he : t.isEmpty) {fallback : α} :
+    (t.insert k).minD fallback = k :=
+  TreeMap.Raw.minKeyD_insertIfNew_of_isEmpty h he
+
 theorem min?_insert_le_min? [TransCmp cmp] (h : t.WF) {k km kmi} :
     (hkm : t.min? = some km) →
     (hkmi : (t.insert k |>.min? |>.get <| isSome_min?_insert h) = kmi) →