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+/- Arrays/slices -/
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.IList
+import Base.Primitives.Scalar
+import Base.Primitives.Range
+import Base.Arith
+import Base.Progress.Base
+
+namespace Primitives
+
+open Result Error
+
+abbrev Array (α : Type u) (n : Usize) := { l : List α // l.length = n.val }
+
+instance (a : Type u) (n : Usize) : Arith.HasIntProp (Array a n) where
+  prop_ty := λ v => v.val.len = n.val
+  prop := λ ⟨ _, l ⟩ => by simp[Scalar.max, List.len_eq_length, *]
+
+instance {α : Type u} {n : Usize} (p : Array α n → Prop) : Arith.HasIntProp (Subtype p) where
+  prop_ty := λ x => p x
+  prop := λ x => x.property
+
+@[simp]
+abbrev Array.length {α : Type u} {n : Usize} (v : Array α n) : Int := v.val.len
+
+@[simp]
+abbrev Array.v {α : Type u} {n : Usize} (v : Array α n) : List α := v.val
+
+example {α: Type u} {n : Usize} (v : Array α n) : v.length ≤ Scalar.max ScalarTy.Usize := by
+  scalar_tac
+
+def Array.mk (α : Type u) (n : Usize) (init : List α) (hl : init.len = n.val := by decide) :
+  Array α n := ⟨ init, by simp [← List.len_eq_length]; apply hl ⟩
+
+example : Array Int (Usize.ofInt 2) := Array.mk Int (Usize.ofInt 2) [0, 1]
+
+-- Remark: not used yet, but could be used if explicit calls to Len are used in Rust
+-- TODO: very annoying that the α and the n are explicit parameters
+def Array.len (α : Type u) (n : Usize) (v : Array α n) : Usize :=
+  Usize.ofIntCore v.val.len (by scalar_tac) (by scalar_tac)
+
+@[simp]
+theorem Array.len_val {α : Type u} {n : Usize} (v : Array α n) : (Array.len α n v).val = v.length :=
+  by rfl
+
+@[simp]
+abbrev Array.index {α : Type u} {n : Usize} [Inhabited α] (v : Array α n) (i : Int) : α :=
+  v.val.index i
+
+@[simp]
+abbrev Array.slice {α : Type u} {n : Usize} [Inhabited α] (v : Array α n) (i j : Int) : List α :=
+  v.val.slice i j
+
+def Array.index_shared (α : Type u) (n : Usize) (v: Array α n) (i: Usize) : Result α :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some x => ret x
+
+/- In the theorems below: we don't always need the `∃ ..`, but we use one
+   so that `progress` introduces an opaque variable and an equality. This
+   helps control the context.
+ -/
+
+@[pspec]
+theorem Array.index_shared_spec {α : Type u} {n : Usize}[Inhabited α] (v: Array α n) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index_shared α n i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_shared]
+  -- TODO: dependent rewrite
+  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+  simp [*]
+
+-- This shouldn't be used
+def Array.index_shared_back (α : Type u) (n : Usize) (v: Array α n) (i: Usize) (_: α) : Result Unit :=
+  if i.val < List.length v.val then
+    .ret ()
+  else
+    .fail arrayOutOfBounds
+
+def Array.index_mut (α : Type u) (v: Array α n) (i: Usize) : Result α :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some x => ret x
+
+@[pspec]
+theorem Array.index_mut_spec {α : Type u} {n : Usize} [Inhabited α] (v: Array α n) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index_mut α i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_mut]
+  -- TODO: dependent rewrite
+  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+  simp [*]
+
+def Array.index_mut_back (α : Type u) (v: Array α n) (i: Usize) (x: α) : Result (Array α n) :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some _ =>
+    .ret ⟨ v.val.update i.val x, by have := v.property; simp [*] ⟩
+
+@[pspec]
+theorem Array.index_mut_back_spec {α : Type u} {n : Usize} (v: Array α n) (i: Usize) (x : α)
+  (hbound : i.val < v.length) :
+  ∃ nv, v.index_mut_back α i x = ret nv ∧
+  nv.val = v.val.update i.val x
+  := by
+  simp only [index_mut_back]
+  have h := List.indexOpt_bounds v.val i.val
+  split
+  . simp_all [length]; cases h <;> scalar_tac
+  . simp_all
+
+def Slice (α : Type u) := { l : List α // l.length ≤ Usize.max }
+
+instance (a : Type u) : Arith.HasIntProp (Slice a) where
+  prop_ty := λ v => 0 ≤ v.val.len ∧ v.val.len ≤ Scalar.max ScalarTy.Usize
+  prop := λ ⟨ _, l ⟩ => by simp[Scalar.max, List.len_eq_length, *]
+
+instance {α : Type u} (p : Slice α → Prop) : Arith.HasIntProp (Subtype p) where
+  prop_ty := λ x => p x
+  prop := λ x => x.property
+
+@[simp]
+abbrev Slice.length {α : Type u} (v : Slice α) : Int := v.val.len
+
+@[simp]
+abbrev Slice.v {α : Type u} (v : Slice α) : List α := v.val
+
+example {a: Type u} (v : Slice a) : v.length ≤ Scalar.max ScalarTy.Usize := by
+  scalar_tac
+
+def Slice.new (α : Type u): Slice α := ⟨ [], by apply Scalar.cMax_suffices .Usize; simp ⟩
+
+-- TODO: very annoying that the α is an explicit parameter
+def Slice.len (α : Type u) (v : Slice α) : Usize :=
+  Usize.ofIntCore v.val.len (by scalar_tac) (by scalar_tac)
+
+@[simp]
+theorem Slice.len_val {α : Type u} (v : Slice α) : (Slice.len α v).val = v.length :=
+  by rfl
+
+@[simp]
+abbrev Slice.index {α : Type u} [Inhabited α] (v: Slice α) (i: Int) : α :=
+  v.val.index i
+
+@[simp]
+abbrev Slice.slice {α : Type u} [Inhabited α] (s : Slice α) (i j : Int) : List α :=
+  s.val.slice i j
+
+def Slice.index_shared (α : Type u) (v: Slice α) (i: Usize) : Result α :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some x => ret x
+
+/- In the theorems below: we don't always need the `∃ ..`, but we use one
+   so that `progress` introduces an opaque variable and an equality. This
+   helps control the context.
+ -/
+
+@[pspec]
+theorem Slice.index_shared_spec {α : Type u} [Inhabited α] (v: Slice α) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index_shared α i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_shared]
+  -- TODO: dependent rewrite
+  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+  simp [*]
+
+-- This shouldn't be used
+def Slice.index_shared_back (α : Type u) (v: Slice α) (i: Usize) (_: α) : Result Unit :=
+  if i.val < List.length v.val then
+    .ret ()
+  else
+    .fail arrayOutOfBounds
+
+def Slice.index_mut (α : Type u) (v: Slice α) (i: Usize) : Result α :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some x => ret x
+
+@[pspec]
+theorem Slice.index_mut_spec {α : Type u} [Inhabited α] (v: Slice α) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index_mut α i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_mut]
+  -- TODO: dependent rewrite
+  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+  simp [*]
+
+def Slice.index_mut_back (α : Type u) (v: Slice α) (i: Usize) (x: α) : Result (Slice α) :=
+  match v.val.indexOpt i.val with
+  | none => fail .arrayOutOfBounds
+  | some _ =>
+    .ret ⟨ v.val.update i.val x, by have := v.property; simp [*] ⟩
+
+@[pspec]
+theorem Slice.index_mut_back_spec {α : Type u} (v: Slice α) (i: Usize) (x : α)
+  (hbound : i.val < v.length) :
+  ∃ nv, v.index_mut_back α i x = ret nv ∧
+  nv.val = v.val.update i.val x
+  := by
+  simp only [index_mut_back]
+  have h := List.indexOpt_bounds v.val i.val
+  split
+  . simp_all [length]; cases h <;> scalar_tac
+  . simp_all
+
+/- Array to slice/subslices -/
+def Array.to_slice (α : Type u) (n : Usize) (v : Array α n) : Result (Slice α) :=
+  ret ⟨ v.val, by simp [← List.len_eq_length]; scalar_tac ⟩
+
+@[pspec]
+theorem Array.to_slice_spec {α : Type u} {n : Usize} (v : Array α n) :
+  ∃ s, to_slice α n v = ret s ∧ v.val = s.val := by simp [to_slice]
+
+def Array.to_mut_slice (α : Type u) (n : Usize) (v : Array α n) : Result (Slice α) :=
+  to_slice α n v
+
+@[pspec]
+theorem Array.to_mut_slice_spec {α : Type u} {n : Usize} (v : Array α n) :
+  ∃ s, Array.to_slice α n v = ret s ∧ v.val = s.val := to_slice_spec v
+
+def Array.to_mut_slice_back (α : Type u) (n : Usize) (_ : Array α n) (s : Slice α) : Result (Array α n) :=
+  if h: s.val.len = n.val then
+    ret ⟨ s.val, by simp [← List.len_eq_length, *] ⟩
+  else fail panic
+
+@[pspec]
+theorem Array.to_mut_slice_back_spec {α : Type u} {n : Usize} (a : Array α n) (ns : Slice α) (h : ns.val.len = n.val) :
+  ∃ na, to_mut_slice_back α n a ns = ret na ∧ na.val = ns.val
+  := by simp [to_mut_slice_back, *]
+
+def Array.shared_subslice (α : Type u) (n : Usize) (a : Array α n) (r : Range Usize) : Result (Slice α) :=
+  -- TODO: not completely sure here
+  if r.start.val < r.end_.val ∧ r.end_.val ≤ a.val.len then
+    ret ⟨ a.val.slice r.start.val r.end_.val,
+          by
+            simp [← List.len_eq_length]
+            have := a.val.slice_len_le r.start.val r.end_.val
+            scalar_tac ⟩
+  else
+    fail panic
+
+@[pspec]
+theorem Array.shared_subslice_spec {α : Type u} {n : Usize} [Inhabited α] (a : Array α n) (r : Range Usize)
+  (h0 : r.start.val < r.end_.val) (h1 : r.end_.val ≤ a.val.len) :
+  ∃ s, shared_subslice α n a r = ret s ∧
+  s.val = a.val.slice r.start.val r.end_.val ∧
+  (∀ i, 0 ≤ i → i + r.start.val < r.end_.val → s.val.index i = a.val.index (r.start.val + i))
+  := by
+  simp [shared_subslice, *]
+  intro i _ _
+  have := List.index_slice r.start.val r.end_.val i a.val (by scalar_tac) (by scalar_tac) (by trivial) (by scalar_tac)
+  simp [*]
+
+def Array.mut_subslice (α : Type u) (n : Usize) (a : Array α n) (r : Range Usize) : Result (Slice α) :=
+  Array.shared_subslice α n a r
+
+@[pspec]
+theorem Array.mut_subslice_spec {α : Type u} {n : Usize} [Inhabited α] (a : Array α n) (r : Range Usize)
+  (h0 : r.start.val < r.end_.val) (h1 : r.end_.val ≤ a.val.len) :
+  ∃ s, mut_subslice α n a r = ret s ∧
+  s.val = a.slice r.start.val r.end_.val ∧
+  (∀ i, 0 ≤ i → i + r.start.val < r.end_.val → s.val.index i = a.val.index (r.start.val + i))
+  := shared_subslice_spec a r h0 h1
+
+def Array.mut_subslice_back (α : Type u) (n : Usize) (a : Array α n) (r : Range Usize) (s : Slice α) : Result (Array α n) :=
+  -- TODO: not completely sure here
+  if h: r.start.val < r.end_.val ∧ r.end_.val ≤ a.length ∧ s.val.len = r.end_.val - r.start.val then
+    let s_beg := a.val.itake r.start.val
+    let s_end := a.val.idrop r.end_.val
+    have : s_beg.len = r.start.val := by
+      apply List.itake_len
+      . simp_all; scalar_tac
+      . scalar_tac
+    have : s_end.len = a.val.len - r.end_.val := by
+      apply List.idrop_len
+      . scalar_tac
+      . scalar_tac
+    let na := s_beg.append (s.val.append s_end)
+    have : na.len = a.val.len := by simp [*]
+    ret ⟨ na, by simp_all [← List.len_eq_length]; scalar_tac ⟩
+  else
+    fail panic
+
+-- TODO: it is annoying to write `.val` everywhere. We could leverage coercions,
+-- but: some symbols like `+` are already overloaded to be notations for monadic
+-- operations/
+-- We should introduce special symbols for the monadic arithmetic operations
+-- (the use will never write those symbols directly).
+@[pspec]
+theorem Array.mut_subslice_back_spec {α : Type u} {n : Usize} [Inhabited α] (a : Array α n) (r : Range Usize) (s : Slice α)
+  (_ : r.start.val < r.end_.val) (_ : r.end_.val ≤ a.length) (_ : s.length = r.end_.val - r.start.val) :
+  ∃ na, mut_subslice_back α n a r s = ret na ∧
+  (∀ i, 0 ≤ i → i < r.start.val → na.index i = a.index i) ∧
+  (∀ i, r.start.val ≤ i → i < r.end_.val → na.index i = s.index (i - r.start.val)) ∧
+  (∀ i, r.end_.val ≤ i → i < n.val → na.index i = a.index i) := by
+  simp [mut_subslice_back, *]
+  have h := List.replace_slice_index r.start.val r.end_.val a.val s.val
+    (by scalar_tac) (by scalar_tac) (by scalar_tac) (by scalar_tac)
+  simp [List.replace_slice] at h
+  have ⟨ h0, h1, h2 ⟩ := h
+  clear h
+  split_conjs
+  . intro i _ _
+    have := h0 i (by int_tac) (by int_tac)
+    simp [*]
+  . intro i _ _
+    have := h1 i (by int_tac) (by int_tac)
+    simp [*]
+  . intro i _ _
+    have := h2 i (by int_tac) (by int_tac)
+    simp [*]
+
+def Slice.shared_subslice (α : Type u) (s : Slice α) (r : Range Usize) : Result (Slice α) :=
+  -- TODO: not completely sure here
+  if r.start.val < r.end_.val ∧ r.end_.val ≤ s.length then
+    ret ⟨ s.val.slice r.start.val r.end_.val,
+          by
+            simp [← List.len_eq_length]
+            have := s.val.slice_len_le r.start.val r.end_.val
+            scalar_tac ⟩
+  else
+    fail panic
+
+@[pspec]
+theorem Slice.shared_subslice_spec {α : Type u} [Inhabited α] (s : Slice α) (r : Range Usize)
+  (h0 : r.start.val < r.end_.val) (h1 : r.end_.val ≤ s.val.len) :
+  ∃ ns, shared_subslice α s r = ret ns ∧
+  ns.val = s.slice r.start.val r.end_.val ∧
+  (∀ i, 0 ≤ i → i + r.start.val < r.end_.val → ns.index i = s.index (r.start.val + i))
+  := by
+  simp [shared_subslice, *]
+  intro i _ _
+  have := List.index_slice r.start.val r.end_.val i s.val (by scalar_tac) (by scalar_tac) (by trivial) (by scalar_tac)
+  simp [*]
+
+def Slice.mut_subslice (α : Type u) (s : Slice α) (r : Range Usize) : Result (Slice α) :=
+  Slice.shared_subslice α s r
+
+@[pspec]
+theorem Slice.mut_subslice_spec {α : Type u} [Inhabited α] (s : Slice α) (r : Range Usize)
+  (h0 : r.start.val < r.end_.val) (h1 : r.end_.val ≤ s.val.len) :
+  ∃ ns, mut_subslice α s r = ret ns ∧
+  ns.val = s.slice r.start.val r.end_.val ∧
+  (∀ i, 0 ≤ i → i + r.start.val < r.end_.val → ns.index i = s.index (r.start.val + i))
+  := shared_subslice_spec s r h0 h1
+
+attribute [pp_dot] List.len List.length List.index -- use the dot notation when printing
+set_option pp.coercions false -- do not print coercions with ↑ (this doesn't parse)
+
+def Slice.mut_subslice_back (α : Type u) (s : Slice α) (r : Range Usize) (ss : Slice α) : Result (Slice α) :=
+  -- TODO: not completely sure here
+  if h: r.start.val < r.end_.val ∧ r.end_.val ≤ s.length ∧ ss.val.len = r.end_.val - r.start.val then
+    let s_beg := s.val.itake r.start.val
+    let s_end := s.val.idrop r.end_.val
+    have : s_beg.len = r.start.val := by
+      apply List.itake_len
+      . simp_all; scalar_tac
+      . scalar_tac
+    have : s_end.len = s.val.len - r.end_.val := by
+      apply List.idrop_len
+      . scalar_tac
+      . scalar_tac
+    let ns := s_beg.append (ss.val.append s_end)
+    have : ns.len = s.val.len := by simp [*]
+    ret ⟨ ns, by simp_all [← List.len_eq_length]; scalar_tac ⟩
+  else
+    fail panic
+
+@[pspec]
+theorem Slice.mut_subslice_back_spec {α : Type u} [Inhabited α] (a : Slice α) (r : Range Usize) (ss : Slice α)
+  (_ : r.start.val < r.end_.val) (_ : r.end_.val ≤ a.length) (_ : ss.length = r.end_.val - r.start.val) :
+  ∃ na, mut_subslice_back α a r ss = ret na ∧
+  (∀ i, 0 ≤ i → i < r.start.val → na.index i = a.index i) ∧
+  (∀ i, r.start.val ≤ i → i < r.end_.val → na.index i = ss.index (i - r.start.val)) ∧
+  (∀ i, r.end_.val ≤ i → i < a.length → na.index i = a.index i) := by
+  simp [mut_subslice_back, *]
+  have h := List.replace_slice_index r.start.val r.end_.val a.val ss.val
+    (by scalar_tac) (by scalar_tac) (by scalar_tac) (by scalar_tac)
+  simp [List.replace_slice, *] at h
+  have ⟨ h0, h1, h2 ⟩ := h
+  clear h
+  split_conjs
+  . intro i _ _
+    have := h0 i (by int_tac) (by int_tac)
+    simp [*]
+  . intro i _ _
+    have := h1 i (by int_tac) (by int_tac)
+    simp [*]
+  . intro i _ _
+    have := h2 i (by int_tac) (by int_tac)
+    simp [*]
+
+end Primitives