feat: align {List/Array/Vector}.{attach,attachWith,pmap} lemmas (lean…

…prover#6723)

This PR completes the alignment of
{List/Array/Vector}.{attach,attachWith,pmap} lemmas. I had to fill in a
number of gaps in the List API.

src/Init/Data/Array/Attach.lean

@@ -6,6 +6,7 @@ Authors: Joachim Breitner, Mario Carneiro
 prelude
 import Init.Data.Array.Mem
 import Init.Data.Array.Lemmas
+import Init.Data.Array.Count
 import Init.Data.List.Attach
 
 namespace Array
@@ -142,10 +143,16 @@ theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
   cases l
   simp [List.pmap_eq_map_attach]
 
+@[simp]
+theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l H) :
+    pmap (fun a h => ⟨a, f a h⟩) l H = l.attachWith q (fun x h => f x (H x h)) := by
+  cases l
+  simp [List.pmap_eq_attachWith]
+
 theorem attach_map_coe (l : Array α) (f : α → β) :
     (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
   cases l
-  simp [List.attach_map_coe]
+  simp
 
 theorem attach_map_val (l : Array α) (f : α → β) : (l.attach.map fun i => f i.val) = l.map f :=
   attach_map_coe _ _
@@ -172,6 +179,12 @@ theorem mem_attach (l : Array α) : ∀ x, x ∈ l.attach
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
+@[simp]
+theorem mem_attachWith (l : Array α) {q : α → Prop} (H) (x : {x // q x}) :
+    x ∈ l.attachWith q H ↔ x.1 ∈ l := by
+  cases l
+  simp
+
 @[simp]
 theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
     b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
@@ -223,16 +236,16 @@ theorem attachWith_ne_empty_iff {l : Array α} {P : α → Prop} {H : ∀ a ∈
   cases l; simp
 
 @[simp]
-theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) (n : Nat) :
-    (pmap f l h)[n]? = Option.pmap f l[n]? fun x H => h x (mem_of_getElem? H) := by
+theorem getElem?_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) (i : Nat) :
+    (pmap f l h)[i]? = Option.pmap f l[i]? fun x H => h x (mem_of_getElem? H) := by
   cases l; simp
 
 @[simp]
-theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) {n : Nat}
-    (hn : n < (pmap f l h).size) :
-    (pmap f l h)[n] =
-      f (l[n]'(@size_pmap _ _ p f l h ▸ hn))
-        (h _ (getElem_mem (@size_pmap _ _ p f l h ▸ hn))) := by
+theorem getElem_pmap {p : α → Prop} (f : ∀ a, p a → β) {l : Array α} (h : ∀ a ∈ l, p a) {i : Nat}
+    (hi : i < (pmap f l h).size) :
+    (pmap f l h)[i] =
+      f (l[i]'(@size_pmap _ _ p f l h ▸ hi))
+        (h _ (getElem_mem (@size_pmap _ _ p f l h ▸ hi))) := by
   cases l; simp
 
 @[simp]
@@ -256,6 +269,18 @@ theorem getElem_attach {xs : Array α} {i : Nat} (h : i < xs.attach.size) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
+@[simp] theorem pmap_attach (l : Array α) {p : {x // x ∈ l} → Prop} (f : ∀ a, p a → β) (H) :
+    pmap f l.attach H =
+      l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
+  ext <;> simp
+
+@[simp] theorem pmap_attachWith (l : Array α) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
+    pmap f (l.attachWith q H₁) H₂ =
+      l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
+  ext <;> simp
+
 theorem foldl_pmap (l : Array α) {P : α → Prop} (f : (a : α) → P a → β)
   (H : ∀ (a : α), a ∈ l → P a) (g : γ → β → γ) (x : γ) :
     (l.pmap f H).foldl g x = l.attach.foldl (fun acc a => g acc (f a.1 (H _ a.2))) x := by
@@ -313,11 +338,7 @@ theorem attachWith_map {l : Array α} (f : α → β) {P : β → Prop} {H : ∀
     (l.map f).attachWith P H = (l.attachWith (P ∘ f) (fun _ h => H _ (mem_map_of_mem f h))).map
       fun ⟨x, h⟩ => ⟨f x, h⟩ := by
   cases l
-  ext
-  · simp
-  · simp only [List.map_toArray, List.attachWith_toArray, List.getElem_toArray,
-      List.getElem_attachWith, List.getElem_map, Function.comp_apply]
-    erw [List.getElem_attachWith] -- Why is `erw` needed here?
+  simp [List.attachWith_map]
 
 theorem map_attachWith {l : Array α} {P : α → Prop} {H : ∀ (a : α), a ∈ l → P a}
     (f : { x // P x } → β) :
@@ -347,7 +368,23 @@ theorem attach_filter {l : Array α} (p : α → Bool) :
   simp [List.attach_filter, List.map_filterMap, Function.comp_def]
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
--- Also missing are `filterMap_attach`, `filter_attach`, `filterMap_attachWith` and `filter_attachWith`.
+
+@[simp]
+theorem filterMap_attachWith {q : α → Prop} {l : Array α} {f : {x // q x} → Option β} (H)
+    (w : stop = (l.attachWith q H).size) :
+    (l.attachWith q H).filterMap f 0 stop = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
+  subst w
+  cases l
+  simp [Function.comp_def]
+
+@[simp]
+theorem filter_attachWith {q : α → Prop} {l : Array α} {p : {x // q x} → Bool} (H)
+    (w : stop = (l.attachWith q H).size) :
+    (l.attachWith q H).filter p 0 stop =
+      (l.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
+  subst w
+  cases l
+  simp [Function.comp_def, List.filter_map]
 
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
     pmap f (pmap g l H₁) H₂ =
@@ -427,16 +464,48 @@ theorem reverse_attach (xs : Array α) :
 
 @[simp] theorem back?_attachWith {P : α → Prop} {xs : Array α}
     {H : ∀ (a : α), a ∈ xs → P a} :
-    (xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back?_eq_some h)⟩) := by
+    (xs.attachWith P H).back? = xs.back?.pbind (fun a h => some ⟨a, H _ (mem_of_back? h)⟩) := by
   cases xs
   simp
 
 @[simp]
 theorem back?_attach {xs : Array α} :
-    xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back?_eq_some h⟩ := by
+    xs.attach.back? = xs.back?.pbind fun a h => some ⟨a, mem_of_back? h⟩ := by
   cases xs
   simp
 
+@[simp]
+theorem countP_attach (l : Array α) (p : α → Bool) :
+    l.attach.countP (fun a : {x // x ∈ l} => p a) = l.countP p := by
+  cases l
+  simp [Function.comp_def]
+
+@[simp]
+theorem countP_attachWith {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) (q : α → Bool) :
+    (l.attachWith p H).countP (fun a : {x // p x} => q a) = l.countP q := by
+  cases l
+  simp
+
+@[simp]
+theorem count_attach [DecidableEq α] (l : Array α) (a : {x // x ∈ l}) :
+    l.attach.count a = l.count ↑a := by
+  rcases l with ⟨l⟩
+  simp only [List.attach_toArray, List.attachWith_mem_toArray, List.count_toArray]
+  rw [List.map_attach, List.count_eq_countP]
+  simp only [Subtype.beq_iff]
+  rw [List.countP_pmap, List.countP_attach (p := (fun x => x == a.1)), List.count]
+
+@[simp]
+theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : Array α) (H : ∀ a ∈ l, p a) (a : {x // p x}) :
+    (l.attachWith p H).count a = l.count ↑a := by
+  cases l
+  simp
+
+@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (l : Array α) (H₁) :
+    (l.pmap g H₁).countP f =
+      l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
+  simp [pmap_eq_map_attach, countP_map, Function.comp_def]
+
 /-! ## unattach
 
 `Array.unattach` is the (one-sided) inverse of `Array.attach`. It is a synonym for `Array.map Subtype.val`.
@@ -455,7 +524,7 @@ and is ideally subsequently simplified away by `unattach_attach`.
 
 If not, usually the right approach is `simp [Array.unattach, -Array.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) := l.map (·.val)
+def unattach {α : Type _} {p : α → Prop} (l : Array { x // p x }) : Array α := l.map (·.val)
 
 @[simp] theorem unattach_nil {p : α → Prop} : (#[] : Array { x // p x }).unattach = #[] := rfl
 @[simp] theorem unattach_push {p : α → Prop} {a : { x // p x }} {l : Array { x // p x }} :
@@ -578,4 +647,16 @@ and simplifies these to the function directly taking the value.
   cases l₂
   simp
 
+@[simp] theorem unattach_flatten {p : α → Prop} {l : Array (Array { x // p x })} :
+    l.flatten.unattach = (l.map unattach).flatten := by
+  unfold unattach
+  cases l using array₂_induction
+  simp only [flatten_toArray, List.map_map, Function.comp_def, List.map_id_fun', id_eq,
+    List.map_toArray, List.map_flatten, map_subtype, map_id_fun', List.unattach_toArray, mk.injEq]
+  simp only [List.unattach]
+
+@[simp] theorem unattach_mkArray {p : α → Prop} {n : Nat} {x : { x // p x }} :
+    (Array.mkArray n x).unattach = Array.mkArray n x.1 := by
+  simp [unattach]
+
 end Array

src/Init/Data/Array/Lemmas.lean

@@ -3253,9 +3253,11 @@ theorem back!_eq_back? [Inhabited α] (a : Array α) : a.back! = a.back?.getD de
 @[simp] theorem back!_push [Inhabited α] (a : Array α) : (a.push x).back! = x := by
   simp [back!_eq_back?]
 
-theorem mem_of_back?_eq_some {xs : Array α} {a : α} (h : xs.back? = some a) : a ∈ xs := by
+theorem mem_of_back? {xs : Array α} {a : α} (h : xs.back? = some a) : a ∈ xs := by
   cases xs
-  simpa using List.mem_of_getLast?_eq_some (by simpa using h)
+  simpa using List.mem_of_getLast? (by simpa using h)
+
+@[deprecated mem_of_back? (since := "2024-10-21")] abbrev mem_of_back?_eq_some := @mem_of_back?
 
 theorem getElem?_push_lt (a : Array α) (x : α) (i : Nat) (h : i < a.size) :
     (a.push x)[i]? = some a[i] := by

src/Init/Data/Bool.lean

@@ -620,3 +620,12 @@ but may be used locally.
 -/
 def boolRelToRel : Coe (α → α → Bool) (α → α → Prop) where
   coe r := fun a b => Eq (r a b) true
+
+/-! ### subtypes -/
+
+@[simp] theorem Subtype.beq_iff {α : Type u} [DecidableEq α] {p : α → Prop} {x y : {a : α // p a}} :
+    (x == y) = (x.1 == y.1) := by
+  cases x
+  cases y
+  rw [Bool.eq_iff_iff]
+  simp [beq_iff_eq]

src/Init/Data/List/Attach.lean

@@ -111,6 +111,14 @@ theorem pmap_eq_map_attach {p : α → Prop} (f : ∀ a, p a → β) (l H) :
     pmap f l H = l.attach.map fun x => f x.1 (H _ x.2) := by
   rw [attach, attachWith, map_pmap]; exact pmap_congr_left l fun _ _ _ _ => rfl
 
+@[simp]
+theorem pmap_eq_attachWith {p q : α → Prop} (f : ∀ a, p a → q a) (l H) :
+    pmap (fun a h => ⟨a, f a h⟩) l H = l.attachWith q (fun x h => f x (H x h)) := by
+  induction l with
+  | nil => rfl
+  | cons a l ih =>
+    simp [pmap, attachWith, ih]
+
 theorem attach_map_coe (l : List α) (f : α → β) :
     (l.attach.map fun (i : {i // i ∈ l}) => f i) = l.map f := by
   rw [attach, attachWith, map_pmap]; exact pmap_eq_map _ _ _ _
@@ -136,10 +144,23 @@ theorem attachWith_map_subtype_val {p : α → Prop} (l : List α) (H : ∀ a 
 @[simp]
 theorem mem_attach (l : List α) : ∀ x, x ∈ l.attach
   | ⟨a, h⟩ => by
-    have := mem_map.1 (by rw [attach_map_subtype_val] <;> exact h)
+    have := mem_map.1 (by rw [attach_map_subtype_val]; exact h)
     rcases this with ⟨⟨_, _⟩, m, rfl⟩
     exact m
 
+@[simp]
+theorem mem_attachWith (l : List α) {q : α → Prop} (H) (x : {x // q x}) :
+    x ∈ l.attachWith q H ↔ x.1 ∈ l := by
+  induction l with
+  | nil => simp
+  | cons a l ih =>
+    simp [ih]
+    constructor
+    · rintro (_ | _) <;> simp_all
+    · rintro (h | h)
+      · simp [← h]
+      · simp_all
+
 @[simp]
 theorem mem_pmap {p : α → Prop} {f : ∀ a, p a → β} {l H b} :
     b ∈ pmap f l H ↔ ∃ (a : _) (h : a ∈ l), f a (H a h) = b := by
@@ -266,6 +287,18 @@ theorem getElem_attach {xs : List α} {i : Nat} (h : i < xs.attach.length) :
     xs.attach[i] = ⟨xs[i]'(by simpa using h), getElem_mem (by simpa using h)⟩ :=
   getElem_attachWith h
 
+@[simp] theorem pmap_attach (l : List α) {p : {x // x ∈ l} → Prop} (f : ∀ a, p a → β) (H) :
+    pmap f l.attach H =
+      l.pmap (P := fun a => ∃ h : a ∈ l, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨h, H ⟨a, h⟩ (by simp)⟩) := by
+  apply ext_getElem <;> simp
+
+@[simp] theorem pmap_attachWith (l : List α) {p : {x // q x} → Prop} (f : ∀ a, p a → β) (H₁ H₂) :
+    pmap f (l.attachWith q H₁) H₂ =
+      l.pmap (P := fun a => ∃ h : q a, p ⟨a, h⟩)
+        (fun a h => f ⟨a, h.1⟩ h.2) (fun a h => ⟨H₁ _ h, H₂ ⟨a, H₁ _ h⟩ (by simpa)⟩) := by
+  apply ext_getElem <;> simp
+
 @[simp] theorem head?_pmap {P : α → Prop} (f : (a : α) → P a → β) (xs : List α)
     (H : ∀ (a : α), a ∈ xs → P a) :
     (xs.pmap f H).head? = xs.attach.head?.map fun ⟨a, m⟩ => f a (H a m) := by
@@ -431,7 +464,25 @@ theorem attach_filter {l : List α} (p : α → Bool) :
   split <;> simp
 
 -- We are still missing here `attachWith_filterMap` and `attachWith_filter`.
--- Also missing are `filterMap_attach`, `filter_attach`, `filterMap_attachWith` and `filter_attachWith`.
+
+@[simp]
+theorem filterMap_attachWith {q : α → Prop} {l : List α} {f : {x // q x} → Option β} (H) :
+    (l.attachWith q H).filterMap f = l.attach.filterMap (fun ⟨x, h⟩ => f ⟨x, H _ h⟩) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [attachWith_cons, filterMap_cons]
+    split <;> simp_all [Function.comp_def]
+
+@[simp]
+theorem filter_attachWith {q : α → Prop} {l : List α} {p : {x // q x} → Bool} (H) :
+    (l.attachWith q H).filter p =
+      (l.attach.filter (fun ⟨x, h⟩ => p ⟨x, H _ h⟩)).map (fun ⟨x, h⟩ => ⟨x, H _ h⟩) := by
+  induction l with
+  | nil => rfl
+  | cons x xs ih =>
+    simp only [attachWith_cons, filter_cons]
+    split <;> simp_all [Function.comp_def, filter_map]
 
 theorem pmap_pmap {p : α → Prop} {q : β → Prop} (g : ∀ a, p a → β) (f : ∀ b, q b → γ) (l H₁ H₂) :
     pmap f (pmap g l H₁) H₂ =
@@ -520,7 +571,7 @@ theorem reverse_attach (xs : List α) :
 
 @[simp] theorem getLast?_attachWith {P : α → Prop} {xs : List α}
     {H : ∀ (a : α), a ∈ xs → P a} :
-    (xs.attachWith P H).getLast? = xs.getLast?.pbind (fun a h => some ⟨a, H _ (mem_of_getLast?_eq_some h)⟩) := by
+    (xs.attachWith P H).getLast? = xs.getLast?.pbind (fun a h => some ⟨a, H _ (mem_of_getLast? h)⟩) := by
   rw [getLast?_eq_head?_reverse, reverse_attachWith, head?_attachWith]
   simp
 
@@ -531,7 +582,7 @@ theorem reverse_attach (xs : List α) :
 
 @[simp]
 theorem getLast?_attach {xs : List α} :
-    xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast?_eq_some h⟩ := by
+    xs.attach.getLast? = xs.getLast?.pbind fun a h => some ⟨a, mem_of_getLast? h⟩ := by
   rw [getLast?_eq_head?_reverse, reverse_attach, head?_map, head?_attach]
   simp
 
@@ -560,6 +611,11 @@ theorem count_attachWith [DecidableEq α] {p : α → Prop} (l : List α) (H : 
     (l.attachWith p H).count a = l.count ↑a :=
   Eq.trans (countP_congr fun _ _ => by simp [Subtype.ext_iff]) <| countP_attachWith _ _ _
 
+@[simp] theorem countP_pmap {p : α → Prop} (g : ∀ a, p a → β) (f : β → Bool) (l : List α) (H₁) :
+    (l.pmap g H₁).countP f =
+      l.attach.countP (fun ⟨a, m⟩ => f (g a (H₁ a m))) := by
+  simp [pmap_eq_map_attach, countP_map, Function.comp_def]
+
 /-! ## unattach
 
 `List.unattach` is the (one-sided) inverse of `List.attach`. It is a synonym for `List.map Subtype.val`.
@@ -578,7 +634,7 @@ and is ideally subsequently simplified away by `unattach_attach`.
 
 If not, usually the right approach is `simp [List.unattach, -List.map_subtype]` to unfold.
 -/
-def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) := l.map (·.val)
+def unattach {α : Type _} {p : α → Prop} (l : List { x // p x }) : List α := l.map (·.val)
 
 @[simp] theorem unattach_nil {p : α → Prop} : ([] : List { x // p x }).unattach = [] := rfl
 @[simp] theorem unattach_cons {p : α → Prop} {a : { x // p x }} {l : List { x // p x }} :

src/Init/Data/List/Lemmas.lean

@@ -2750,10 +2750,12 @@ theorem getLast?_eq_some_iff {xs : List α} {a : α} : xs.getLast? = some a ↔
   rw [getLast?_eq_head?_reverse, head?_isSome]
   simp
 
-theorem mem_of_getLast?_eq_some {xs : List α} {a : α} (h : xs.getLast? = some a) : a ∈ xs := by
+theorem mem_of_getLast? {xs : List α} {a : α} (h : xs.getLast? = some a) : a ∈ xs := by
   obtain ⟨ys, rfl⟩ := getLast?_eq_some_iff.1 h
   exact mem_concat_self ys a
 
+@[deprecated mem_of_getLast? (since := "2024-10-21")] abbrev mem_of_getLast?_eq_some := @mem_of_getLast?
+
 @[simp] theorem getLast_reverse {l : List α} (h : l.reverse ≠ []) :
     l.reverse.getLast h = l.head (by simp_all) := by
   simp [getLast_eq_head_reverse]

src/Init/Data/Subtype.lean

@@ -5,6 +5,7 @@ Authors: Johannes Hölzl
 -/
 prelude
 import Init.Ext
+import Init.Core
 
 namespace Subtype