chore: move Array.mapIdx lemmas to new file

src/Init/Data/Array.lean

@@ -16,3 +16,4 @@ import Init.Data.Array.Lemmas
 import Init.Data.Array.TakeDrop
 import Init.Data.Array.Bootstrap
 import Init.Data.Array.GetLit
+import Init.Data.Array.MapIdx

src/Init/Data/Array/Lemmas.lean

@@ -12,9 +12,7 @@ import Init.Data.Array.Mem
 import Init.TacticsExtra
 
 /-!
-## Bootstrapping theorems about arrays
-
-This file contains some theorems about `Array` and `List` needed for `Init.Data.List.Impl`.
+## Theorems about `Array`.
 -/
 
 namespace Array
@@ -811,57 +809,6 @@ theorem map_spec (as : Array α) (f : α → β) (p : Fin as.size → β → Pro
   · simp
   · simp only [getElem_map, getElem_pop, size_map]
 
-/-! ### mapIdx -/
-
--- This could also be proved from `SatisfiesM_mapIdxM` in Batteries.
-theorem mapIdx_induction (as : Array α) (f : Fin as.size → α → β)
-    (motive : Nat → Prop) (h0 : motive 0)
-    (p : Fin as.size → β → Prop)
-    (hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
-    motive as.size ∧ ∃ eq : (Array.mapIdx as f).size = as.size,
-      ∀ i h, p ⟨i, h⟩ ((Array.mapIdx as f)[i]) := by
-  let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p ⟨i, h⟩ bs[i]) (hm : motive j) :
-    let arr : Array β := Array.mapIdxM.map (m := Id) as f i j h bs
-    motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i, h⟩ arr[i] := by
-    induction i generalizing j bs with simp [mapIdxM.map]
-    | zero =>
-      have := (Nat.zero_add _).symm.trans h
-      exact ⟨this ▸ hm, h₁ ▸ this, fun _ _ => h₂ ..⟩
-    | succ i ih =>
-      apply @ih (bs.push (f ⟨j, by omega⟩ as[j])) (j + 1) (by omega) (by simp; omega)
-      · intro i i_lt h'
-        rw [get_push]
-        split
-        · apply h₂
-        · simp only [size_push] at h'
-          obtain rfl : i = j := by omega
-          apply (hs ⟨i, by omega⟩ hm).1
-      · exact (hs ⟨j, by omega⟩ hm).2
-  simp [mapIdx, mapIdxM]; exact go rfl nofun h0
-
-theorem mapIdx_spec (as : Array α) (f : Fin as.size → α → β)
-    (p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
-    ∃ eq : (Array.mapIdx as f).size = as.size,
-      ∀ i h, p ⟨i, h⟩ ((Array.mapIdx as f)[i]) :=
-  (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
-
-@[simp] theorem size_mapIdx (a : Array α) (f : Fin a.size → α → β) : (a.mapIdx f).size = a.size :=
-  (mapIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
-
-@[simp] theorem size_zipWithIndex (as : Array α) : as.zipWithIndex.size = as.size :=
-  Array.size_mapIdx _ _
-
-@[simp] theorem getElem_mapIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat)
-    (h : i < (mapIdx a f).size) :
-    (a.mapIdx f)[i] = f ⟨i, by simp_all⟩ (a[i]'(by simp_all)) :=
-  (mapIdx_spec _ _ (fun i b => b = f i a[i]) fun _ => rfl).2 i _
-
-@[simp] theorem getElem?_mapIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat) :
-    (a.mapIdx f)[i]? =
-      a[i]?.pbind fun b h => f ⟨i, (getElem?_eq_some_iff.1 h).1⟩ b := by
-  simp only [getElem?_def, size_mapIdx, getElem_mapIdx]
-  split <;> simp_all
-
 /-! ### modify -/
 
 @[simp] theorem size_modify (a : Array α) (i : Nat) (f : α → α) : (a.modify i f).size = a.size := by

src/Init/Data/Array/MapIdx.lean

@@ -0,0 +1,64 @@
+/-
+Copyright (c) 2022 Mario Carneiro. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Mario Carneiro, Kim Morrison
+-/
+prelude
+import Init.Data.Array.Lemmas
+import Init.Data.List.MapIdx
+
+namespace Array
+
+
+/-! ### mapIdx -/
+
+-- This could also be proved from `SatisfiesM_mapIdxM` in Batteries.
+theorem mapIdx_induction (as : Array α) (f : Fin as.size → α → β)
+    (motive : Nat → Prop) (h0 : motive 0)
+    (p : Fin as.size → β → Prop)
+    (hs : ∀ i, motive i.1 → p i (f i as[i]) ∧ motive (i + 1)) :
+    motive as.size ∧ ∃ eq : (Array.mapIdx as f).size = as.size,
+      ∀ i h, p ⟨i, h⟩ ((Array.mapIdx as f)[i]) := by
+  let rec go {bs i j h} (h₁ : j = bs.size) (h₂ : ∀ i h h', p ⟨i, h⟩ bs[i]) (hm : motive j) :
+    let arr : Array β := Array.mapIdxM.map (m := Id) as f i j h bs
+    motive as.size ∧ ∃ eq : arr.size = as.size, ∀ i h, p ⟨i, h⟩ arr[i] := by
+    induction i generalizing j bs with simp [mapIdxM.map]
+    | zero =>
+      have := (Nat.zero_add _).symm.trans h
+      exact ⟨this ▸ hm, h₁ ▸ this, fun _ _ => h₂ ..⟩
+    | succ i ih =>
+      apply @ih (bs.push (f ⟨j, by omega⟩ as[j])) (j + 1) (by omega) (by simp; omega)
+      · intro i i_lt h'
+        rw [get_push]
+        split
+        · apply h₂
+        · simp only [size_push] at h'
+          obtain rfl : i = j := by omega
+          apply (hs ⟨i, by omega⟩ hm).1
+      · exact (hs ⟨j, by omega⟩ hm).2
+  simp [mapIdx, mapIdxM]; exact go rfl nofun h0
+
+theorem mapIdx_spec (as : Array α) (f : Fin as.size → α → β)
+    (p : Fin as.size → β → Prop) (hs : ∀ i, p i (f i as[i])) :
+    ∃ eq : (Array.mapIdx as f).size = as.size,
+      ∀ i h, p ⟨i, h⟩ ((Array.mapIdx as f)[i]) :=
+  (mapIdx_induction _ _ (fun _ => True) trivial p fun _ _ => ⟨hs .., trivial⟩).2
+
+@[simp] theorem size_mapIdx (a : Array α) (f : Fin a.size → α → β) : (a.mapIdx f).size = a.size :=
+  (mapIdx_spec (p := fun _ _ => True) (hs := fun _ => trivial)).1
+
+@[simp] theorem size_zipWithIndex (as : Array α) : as.zipWithIndex.size = as.size :=
+  Array.size_mapIdx _ _
+
+@[simp] theorem getElem_mapIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat)
+    (h : i < (mapIdx a f).size) :
+    (a.mapIdx f)[i] = f ⟨i, by simp_all⟩ (a[i]'(by simp_all)) :=
+  (mapIdx_spec _ _ (fun i b => b = f i a[i]) fun _ => rfl).2 i _
+
+@[simp] theorem getElem?_mapIdx (a : Array α) (f : Fin a.size → α → β) (i : Nat) :
+    (a.mapIdx f)[i]? =
+      a[i]?.pbind fun b h => f ⟨i, (getElem?_eq_some_iff.1 h).1⟩ b := by
+  simp only [getElem?_def, size_mapIdx, getElem_mapIdx]
+  split <;> simp_all
+
+end Array