Merge branch 'main' into son_fixes2

8 files changed, 843 insertions, 453 deletions

diff --git a/backends/lean/Base/Primitives/Alloc.lean b/backends/lean/Base/Primitives/Alloc.lean
new file mode 100644
index 00000000..6c89c6bb
--- /dev/null
+++ b/backends/lean/Base/Primitives/Alloc.lean
@@ -0,0 +1,37 @@
+import Lean
+import Base.Primitives.Base
+import Base.Primitives.CoreOps
+
+open Primitives
+open Result
+
+namespace alloc
+
+namespace boxed -- alloc.boxed
+
+namespace Box -- alloc.boxed.Box
+
+def deref (T : Type) (x : T) : Result T := ret x
+def deref_mut (T : Type) (x : T) : Result T := ret x
+def deref_mut_back (T : Type) (_ : T) (x : T) : Result T := ret x
+
+/-- Trait instance -/
+def coreopsDerefInst (Self : Type) :
+  core.ops.deref.Deref Self := {
+  Target := Self
+  deref := deref Self
+}
+
+/-- Trait instance -/
+def coreopsDerefMutInst (Self : Type) :
+  core.ops.deref.DerefMut Self := {
+  derefInst := coreopsDerefInst Self
+  deref_mut := deref_mut Self
+  deref_mut_back := deref_mut_back Self
+}
+
+end Box -- alloc.boxed.Box
+
+end boxed -- alloc.boxed
+
+end alloc
diff --git a/backends/lean/Base/Primitives/Array.lean b/backends/lean/Base/Primitives/Array.lean
deleted file mode 100644
index 6c95fd78..00000000
--- a/backends/lean/Base/Primitives/Array.lean
+++ /dev/null
@@ -1,394 +0,0 @@
-/- 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
-
-def 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.make (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.make Int (Usize.ofInt 2) [0, 1]
-
-@[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) (n : Usize) (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 n 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) (n : Usize) (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 n 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 -/
-
-/- We could make this function not use the `Result` type. By making it monadic, we
-   push the user to use the `Array.to_slice_shared_spec` spec theorem below (through the
-   `progress` tactic), meaning `Array.to_slice_shared` should be considered as opaque.
-   All what the spec theorem reveals is that the "representative" lists are the same. -/
-def Array.to_slice_shared (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_shared_spec {α : Type u} {n : Usize} (v : Array α n) :
-  ∃ s, to_slice_shared α n v = ret s ∧ v.val = s.val := by simp [to_slice_shared]
-
-def Array.to_slice_mut (α : Type u) (n : Usize) (v : Array α n) : Result (Slice α) :=
-  to_slice_shared α n v
-
-@[pspec]
-theorem Array.to_slice_mut_spec {α : Type u} {n : Usize} (v : Array α n) :
-  ∃ s, Array.to_slice_shared α n v = ret s ∧ v.val = s.val := to_slice_shared_spec v
-
-def Array.to_slice_mut_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_slice_mut_back_spec {α : Type u} {n : Usize} (a : Array α n) (ns : Slice α) (h : ns.val.len = n.val) :
-  ∃ na, to_slice_mut_back α n a ns = ret na ∧ na.val = ns.val
-  := by simp [to_slice_mut_back, *]
-
-def Array.subslice_shared (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.subslice_shared_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, subslice_shared 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 [subslice_shared, *]
-  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.subslice_mut (α : Type u) (n : Usize) (a : Array α n) (r : Range Usize) : Result (Slice α) :=
-  Array.subslice_shared α n a r
-
-@[pspec]
-theorem Array.subslice_mut_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, subslice_mut 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))
-  := subslice_shared_spec a r h0 h1
-
-def Array.subslice_mut_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.subslice_mut_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, subslice_mut_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 [subslice_mut_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.subslice_shared (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.subslice_shared_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, subslice_shared 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 [subslice_shared, *]
-  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.subslice_mut (α : Type u) (s : Slice α) (r : Range Usize) : Result (Slice α) :=
-  Slice.subslice_shared α s r
-
-@[pspec]
-theorem Slice.subslice_mut_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, subslice_mut 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))
-  := subslice_shared_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.subslice_mut_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.subslice_mut_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, subslice_mut_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 [subslice_mut_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
diff --git a/backends/lean/Base/Primitives/ArraySlice.lean b/backends/lean/Base/Primitives/ArraySlice.lean
new file mode 100644
index 00000000..f68c0846
--- /dev/null
+++ b/backends/lean/Base/Primitives/ArraySlice.lean
@@ -0,0 +1,552 @@
+/- 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.Primitives.CoreOps
+import Base.Arith
+import Base.Progress.Base
+
+namespace Primitives
+
+open Result Error core.ops.range
+
+def 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.make (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.make Int (Usize.ofInt 2) [0, 1]
+
+@[simp]
+abbrev Array.index_s {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_usize (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
+
+-- For initialization
+def Array.repeat (α : Type u) (n : Usize) (x : α) : Array α n :=
+  ⟨ List.ireplicate n.val x, by have h := n.hmin; simp_all [Scalar.min] ⟩
+
+@[pspec]
+theorem Array.repeat_spec {α : Type u} (n : Usize) (x : α) :
+  ∃ a, Array.repeat α n x = a ∧ a.val = List.ireplicate n.val x := by
+  simp [Array.repeat]
+
+/- 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_usize_spec {α : Type u} {n : Usize} [Inhabited α] (v: Array α n) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index_usize α n i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_usize]
+  -- TODO: dependent rewrite
+  have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
+  simp [*]
+
+def Array.update_usize (α : Type u) (n : Usize) (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.update_usize_spec {α : Type u} {n : Usize} (v: Array α n) (i: Usize) (x : α)
+  (hbound : i.val < v.length) :
+  ∃ nv, v.update_usize α n i x = ret nv ∧
+  nv.val = v.val.update i.val x
+  := by
+  simp only [update_usize]
+  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_s {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_usize (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_usize_spec {α : Type u} [Inhabited α] (v: Slice α) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index_usize α i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_usize]
+  -- 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.update_usize (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.update_usize_spec {α : Type u} (v: Slice α) (i: Usize) (x : α)
+  (hbound : i.val < v.length) :
+  ∃ nv, v.update_usize α i x = ret nv ∧
+  nv.val = v.val.update i.val x
+  := by
+  simp only [update_usize]
+  have h := List.indexOpt_bounds v.val i.val
+  split
+  . simp_all [length]; cases h <;> scalar_tac
+  . simp_all
+
+/- Array to slice/subslices -/
+
+/- We could make this function not use the `Result` type. By making it monadic, we
+   push the user to use the `Array.to_slice_spec` spec theorem below (through the
+   `progress` tactic), meaning `Array.to_slice` should be considered as opaque.
+   All what the spec theorem reveals is that the "representative" lists are the same. -/
+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.from_slice (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.from_slice_spec {α : Type u} {n : Usize} (a : Array α n) (ns : Slice α) (h : ns.val.len = n.val) :
+  ∃ na, from_slice α n a ns = ret na ∧ na.val = ns.val
+  := by simp [from_slice, *]
+
+def Array.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.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, 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 [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.update_subslice (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.update_subslice_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, update_subslice α n a r s = ret na ∧
+  (∀ i, 0 ≤ i → i < r.start.val → na.index_s i = a.index_s i) ∧
+  (∀ i, r.start.val ≤ i → i < r.end_.val → na.index_s i = s.index_s (i - r.start.val)) ∧
+  (∀ i, r.end_.val ≤ i → i < n.val → na.index_s i = a.index_s i) := by
+  simp [update_subslice, *]
+  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.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.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, 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_s i = s.index_s (r.start.val + i))
+  := by
+  simp [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 [*]
+
+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.update_subslice (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.update_subslice_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, update_subslice α a r ss = ret na ∧
+  (∀ i, 0 ≤ i → i < r.start.val → na.index_s i = a.index_s i) ∧
+  (∀ i, r.start.val ≤ i → i < r.end_.val → na.index_s i = ss.index_s (i - r.start.val)) ∧
+  (∀ i, r.end_.val ≤ i → i < a.length → na.index_s i = a.index_s i) := by
+  simp [update_subslice, *]
+  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 [*]
+
+/- Trait declaration: [core::slice::index::private_slice_index::Sealed] -/
+structure core.slice.index.private_slice_index.Sealed (Self : Type) where
+
+/- Trait declaration: [core::slice::index::SliceIndex] -/
+structure core.slice.index.SliceIndex (Self T : Type) where
+  sealedInst : core.slice.index.private_slice_index.Sealed Self
+  Output : Type
+  get : Self → T → Result (Option Output)
+  get_mut : Self → T → Result (Option Output)
+  get_mut_back : Self → T → Option Output → Result T
+  get_unchecked : Self → ConstRawPtr T → Result (ConstRawPtr Output)
+  get_unchecked_mut : Self → MutRawPtr T → Result (MutRawPtr Output)
+  index : Self → T → Result Output
+  index_mut : Self → T → Result Output
+  index_mut_back : Self → T → Output → Result T
+
+/- [core::slice::index::[T]::index]: forward function -/
+def core.slice.index.Slice.index
+  (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T))
+  (slice : Slice T) (i : I) : Result inst.Output := do
+  let x ← inst.get i slice
+  match x with
+  | none => fail panic
+  | some x => ret x
+
+/- [core::slice::index::Range:::get]: forward function -/
+def core.slice.index.RangeUsize.get (T : Type) (i : Range Usize) (slice : Slice T) :
+  Result (Option (Slice T)) :=
+  sorry -- TODO
+
+/- [core::slice::index::Range::get_mut]: forward function -/
+def core.slice.index.RangeUsize.get_mut
+  (T : Type) : Range Usize → Slice T → Result (Option (Slice T)) :=
+  sorry -- TODO
+
+/- [core::slice::index::Range::get_mut]: backward function 0 -/
+def core.slice.index.RangeUsize.get_mut_back
+  (T : Type) :
+  Range Usize → Slice T → Option (Slice T) → Result (Slice T) :=
+  sorry -- TODO
+
+/- [core::slice::index::Range::get_unchecked]: forward function -/
+def core.slice.index.RangeUsize.get_unchecked
+  (T : Type) :
+  Range Usize → ConstRawPtr (Slice T) → Result (ConstRawPtr (Slice T)) :=
+  -- Don't know what the model should be - for now we always fail to make
+  -- sure code which uses it fails
+  fun _ _ => fail panic
+
+/- [core::slice::index::Range::get_unchecked_mut]: forward function -/
+def core.slice.index.RangeUsize.get_unchecked_mut
+  (T : Type) :
+  Range Usize → MutRawPtr (Slice T) → Result (MutRawPtr (Slice T)) :=
+  -- Don't know what the model should be - for now we always fail to make
+  -- sure code which uses it fails
+  fun _ _ => fail panic
+
+/- [core::slice::index::Range::index]: forward function -/
+def core.slice.index.RangeUsize.index
+  (T : Type) : Range Usize → Slice T → Result (Slice T) :=
+  sorry -- TODO
+
+/- [core::slice::index::Range::index_mut]: forward function -/
+def core.slice.index.RangeUsize.index_mut
+  (T : Type) : Range Usize → Slice T → Result (Slice T) :=
+  sorry -- TODO
+
+/- [core::slice::index::Range::index_mut]: backward function 0 -/
+def core.slice.index.RangeUsize.index_mut_back
+  (T : Type) : Range Usize → Slice T → Slice T → Result (Slice T) :=
+  sorry -- TODO
+
+/- [core::slice::index::[T]::index_mut]: forward function -/
+def core.slice.index.Slice.index_mut
+  (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T)) :
+  Slice T → I → Result inst.Output :=
+  sorry -- TODO
+
+/- [core::slice::index::[T]::index_mut]: backward function 0 -/
+def core.slice.index.Slice.index_mut_back
+  (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T)) :
+  Slice T → I → inst.Output → Result (Slice T) :=
+  sorry -- TODO
+
+/- [core::array::[T; N]::index]: forward function -/
+def core.array.Array.index
+  (T I : Type) (N : Usize) (inst : core.ops.index.Index (Slice T) I)
+  (a : Array T N) (i : I) : Result inst.Output :=
+  sorry -- TODO
+
+/- [core::array::[T; N]::index_mut]: forward function -/
+def core.array.Array.index_mut
+  (T I : Type) (N : Usize) (inst : core.ops.index.IndexMut (Slice T) I)
+  (a : Array T N) (i : I) : Result inst.indexInst.Output :=
+  sorry -- TODO
+
+/- [core::array::[T; N]::index_mut]: backward function 0 -/
+def core.array.Array.index_mut_back
+  (T I : Type) (N : Usize) (inst : core.ops.index.IndexMut (Slice T) I)
+  (a : Array T N) (i : I) (x : inst.indexInst.Output) : Result (Array T N) :=
+  sorry -- TODO
+
+/- Trait implementation: [core::slice::index::private_slice_index::Range] -/
+def core.slice.index.private_slice_index.SealedRangeUsizeInst
+  : core.slice.index.private_slice_index.Sealed (Range Usize) := {}
+
+/- Trait implementation: [core::slice::index::Range] -/
+def core.slice.index.SliceIndexRangeUsizeSliceTInst (T : Type) :
+  core.slice.index.SliceIndex (Range Usize) (Slice T) := {
+  sealedInst := core.slice.index.private_slice_index.SealedRangeUsizeInst
+  Output := Slice T
+  get := core.slice.index.RangeUsize.get T
+  get_mut := core.slice.index.RangeUsize.get_mut T
+  get_mut_back := core.slice.index.RangeUsize.get_mut_back T
+  get_unchecked := core.slice.index.RangeUsize.get_unchecked T
+  get_unchecked_mut := core.slice.index.RangeUsize.get_unchecked_mut T
+  index := core.slice.index.RangeUsize.index T
+  index_mut := core.slice.index.RangeUsize.index_mut T
+  index_mut_back := core.slice.index.RangeUsize.index_mut_back T
+}
+
+/- Trait implementation: [core::slice::index::[T]] -/
+def core.ops.index.IndexSliceTIInst (T I : Type)
+  (inst : core.slice.index.SliceIndex I (Slice T)) :
+  core.ops.index.Index (Slice T) I := {
+  Output := inst.Output
+  index := core.slice.index.Slice.index T I inst
+}
+
+/- Trait implementation: [core::slice::index::[T]] -/
+def core.ops.index.IndexMutSliceTIInst (T I : Type)
+  (inst : core.slice.index.SliceIndex I (Slice T)) :
+  core.ops.index.IndexMut (Slice T) I := {
+  indexInst := core.ops.index.IndexSliceTIInst T I inst
+  index_mut := core.slice.index.Slice.index_mut T I inst
+  index_mut_back := core.slice.index.Slice.index_mut_back T I inst
+}
+
+/- Trait implementation: [core::array::[T; N]] -/
+def core.ops.index.IndexArrayIInst (T I : Type) (N : Usize)
+  (inst : core.ops.index.Index (Slice T) I) :
+  core.ops.index.Index (Array T N) I := {
+  Output := inst.Output
+  index := core.array.Array.index T I N inst
+}
+
+/- Trait implementation: [core::array::[T; N]] -/
+def core.ops.index.IndexMutArrayIInst (T I : Type) (N : Usize)
+  (inst : core.ops.index.IndexMut (Slice T) I) :
+  core.ops.index.IndexMut (Array T N) I := {
+  indexInst := core.ops.index.IndexArrayIInst T I N inst.indexInst
+  index_mut := core.array.Array.index_mut T I N inst
+  index_mut_back := core.array.Array.index_mut_back T I N inst
+}
+
+/- [core::slice::index::usize::get]: forward function -/
+def core.slice.index.Usize.get
+  (T : Type) : Usize → Slice T → Result (Option T) :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::get_mut]: forward function -/
+def core.slice.index.Usize.get_mut
+  (T : Type) : Usize → Slice T → Result (Option T) :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::get_mut]: backward function 0 -/
+def core.slice.index.Usize.get_mut_back
+  (T : Type) : Usize → Slice T → Option T → Result (Slice T) :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::get_unchecked]: forward function -/
+def core.slice.index.Usize.get_unchecked
+  (T : Type) : Usize → ConstRawPtr (Slice T) → Result (ConstRawPtr T) :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::get_unchecked_mut]: forward function -/
+def core.slice.index.Usize.get_unchecked_mut
+  (T : Type) : Usize → MutRawPtr (Slice T) → Result (MutRawPtr T) :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::index]: forward function -/
+def core.slice.index.Usize.index (T : Type) : Usize → Slice T → Result T :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::index_mut]: forward function -/
+def core.slice.index.Usize.index_mut (T : Type) : Usize → Slice T → Result T :=
+  sorry -- TODO
+
+/- [core::slice::index::usize::index_mut]: backward function 0 -/
+def core.slice.index.Usize.index_mut_back
+  (T : Type) : Usize → Slice T → T → Result (Slice T) :=
+  sorry -- TODO
+
+/- Trait implementation: [core::slice::index::private_slice_index::usize] -/
+def core.slice.index.private_slice_index.SealedUsizeInst
+  : core.slice.index.private_slice_index.Sealed Usize := {}
+
+/- Trait implementation: [core::slice::index::usize] -/
+def core.slice.index.SliceIndexUsizeSliceTInst (T : Type) :
+  core.slice.index.SliceIndex Usize (Slice T) := {
+  sealedInst := core.slice.index.private_slice_index.SealedUsizeInst
+  Output := T
+  get := core.slice.index.Usize.get T
+  get_mut := core.slice.index.Usize.get_mut T
+  get_mut_back := core.slice.index.Usize.get_mut_back T
+  get_unchecked := core.slice.index.Usize.get_unchecked T
+  get_unchecked_mut := core.slice.index.Usize.get_unchecked_mut T
+  index := core.slice.index.Usize.index T
+  index_mut := core.slice.index.Usize.index_mut T
+  index_mut_back := core.slice.index.Usize.index_mut_back T
+}
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Base.lean b/backends/lean/Base/Primitives/Base.lean
index 7c0fa3bb..7fc33251 100644
--- a/backends/lean/Base/Primitives/Base.lean
+++ b/backends/lean/Base/Primitives/Base.lean
@@ -120,11 +120,18 @@ def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
 -- MISC --
 ----------
 
-@[simp] def mem.replace (a : Type) (x : a) (_ : a) : a := x
-@[simp] def mem.replace_back (a : Type) (_ : a) (y : a) : a := y
+@[simp] def core.mem.replace (a : Type) (x : a) (_ : a) : a := x
+@[simp] def core.mem.replace_back (a : Type) (_ : a) (y : a) : a := y
 
 /-- Aeneas-translated function -- useful to reduce non-recursive definitions.
  Use with `simp [ aeneas ]` -/
 register_simp_attr aeneas
 
+-- We don't really use raw pointers for now
+structure MutRawPtr (T : Type) where
+  v : T
+
+structure ConstRawPtr (T : Type) where
+  v : T
+
 end Primitives
diff --git a/backends/lean/Base/Primitives/CoreOps.lean b/backends/lean/Base/Primitives/CoreOps.lean
new file mode 100644
index 00000000..da458f66
--- /dev/null
+++ b/backends/lean/Base/Primitives/CoreOps.lean
@@ -0,0 +1,37 @@
+import Lean
+import Base.Primitives.Base
+
+open Primitives
+open Result
+
+namespace core.ops
+
+namespace index -- core.ops.index
+
+/- Trait declaration: [core::ops::index::Index] -/
+structure Index (Self Idx : Type) where
+  Output : Type
+  index : Self → Idx → Result Output
+
+/- Trait declaration: [core::ops::index::IndexMut] -/
+structure IndexMut (Self Idx : Type) where
+  indexInst : Index Self Idx
+  index_mut : Self → Idx → Result indexInst.Output
+  index_mut_back : Self → Idx → indexInst.Output → Result Self
+
+end index -- core.ops.index
+
+namespace deref -- core.ops.deref
+
+structure Deref (Self : Type) where
+  Target : Type
+  deref : Self → Result Target
+
+structure DerefMut (Self : Type) where
+  derefInst : Deref Self
+  deref_mut : Self → Result derefInst.Target
+  deref_mut_back : Self → derefInst.Target → Result Self
+
+end deref -- core.ops.deref
+
+end core.ops
diff --git a/backends/lean/Base/Primitives/Range.lean b/backends/lean/Base/Primitives/Range.lean
index 26cbee42..a268bcba 100644
--- a/backends/lean/Base/Primitives/Range.lean
+++ b/backends/lean/Base/Primitives/Range.lean
@@ -11,7 +11,7 @@ import Base.Progress.Base
 
 namespace Primitives
 
-structure Range (α : Type u) where
+structure core.ops.range.Range (α : Type u) where
   mk ::
   start: α
   end_: α
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
index fecb0d1d..f74fecd4 100644
--- a/backends/lean/Base/Primitives/Scalar.lean
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -230,6 +230,20 @@ def Scalar.cMax (ty : ScalarTy) : Int :=
   | .Usize => Scalar.max .U32
   | _ => Scalar.max ty
 
+theorem Scalar.min_lt_max (ty : ScalarTy) : Scalar.min ty < Scalar.max ty := by
+  cases ty <;> simp [Scalar.min, Scalar.max]
+  . simp [Isize.min, Isize.max]
+    have h1 := Isize.refined_min.property
+    have h2 := Isize.refined_max.property
+    cases h1 <;> cases h2 <;> simp [*]
+  . simp [Usize.max]
+    have h := Usize.refined_max.property
+    cases h <;> simp [*]
+
+theorem Scalar.min_le_max (ty : ScalarTy) : Scalar.min ty ≤ Scalar.max ty := by
+  have := Scalar.min_lt_max ty
+  int_tac
+
 theorem Scalar.cMin_bound ty : Scalar.min ty ≤ Scalar.cMin ty := by
   cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
   have h := Isize.refined_min.property
@@ -372,10 +386,28 @@ def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar
 def Scalar.mul {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
   Scalar.tryMk ty (x.val * y.val)
 
--- TODO: instances of +, -, * etc. for scalars
+-- TODO: shift left
+def Scalar.shiftl {ty0 ty1 : ScalarTy} (x : Scalar ty0) (y : Scalar ty1) : Result (Scalar ty0) :=
+  sorry
+
+-- TODO: shift right
+def Scalar.shiftr {ty0 ty1 : ScalarTy} (x : Scalar ty0) (y : Scalar ty1) : Result (Scalar ty0) :=
+  sorry
+
+-- TODO: xor
+def Scalar.xor {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Scalar ty :=
+  sorry
+
+-- TODO: and
+def Scalar.and {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Scalar ty :=
+  sorry
+
+-- TODO: or
+def Scalar.or {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Scalar ty :=
+  sorry
 
 -- Cast an integer from a [src_ty] to a [tgt_ty]
--- TODO: check the semantics of casts in Rust
+-- TODO: double-check the semantics of casts in Rust
 def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Result (Scalar tgt_ty) :=
   Scalar.tryMk tgt_ty x.val
 
@@ -398,6 +430,34 @@ def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Re
 instance (ty : ScalarTy) : Inhabited (Scalar ty) := by
   constructor; cases ty <;> apply (Scalar.ofInt 0)
 
+-- TODO: reducible?
+@[reducible] def core_isize_min : Isize := Scalar.ofInt Isize.min (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Isize))
+@[reducible] def core_isize_max : Isize := Scalar.ofInt Isize.max (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Isize))
+@[reducible] def core_i8_min : I8 := Scalar.ofInt I8.min
+@[reducible] def core_i8_max : I8 := Scalar.ofInt I8.max
+@[reducible] def core_i16_min : I16 := Scalar.ofInt I16.min
+@[reducible] def core_i16_max : I16 := Scalar.ofInt I16.max
+@[reducible] def core_i32_min : I32 := Scalar.ofInt I32.min
+@[reducible] def core_i32_max : I32 := Scalar.ofInt I32.max
+@[reducible] def core_i64_min : I64 := Scalar.ofInt I64.min
+@[reducible] def core_i64_max : I64 := Scalar.ofInt I64.max
+@[reducible] def core_i128_min : I128 := Scalar.ofInt I128.min
+@[reducible] def core_i128_max : I128 := Scalar.ofInt I128.max
+
+-- TODO: reducible?
+@[reducible] def core_usize_min : Usize := Scalar.ofInt Usize.min
+@[reducible] def core_usize_max : Usize := Scalar.ofInt Usize.max (by simp [Scalar.min, Scalar.max]; apply (Scalar.min_le_max .Usize))
+@[reducible] def core_u8_min : U8 := Scalar.ofInt U8.min
+@[reducible] def core_u8_max : U8 := Scalar.ofInt U8.max
+@[reducible] def core_u16_min : U16 := Scalar.ofInt U16.min
+@[reducible] def core_u16_max : U16 := Scalar.ofInt U16.max
+@[reducible] def core_u32_min : U32 := Scalar.ofInt U32.min
+@[reducible] def core_u32_max : U32 := Scalar.ofInt U32.max
+@[reducible] def core_u64_min : U64 := Scalar.ofInt U64.min
+@[reducible] def core_u64_max : U64 := Scalar.ofInt U64.max
+@[reducible] def core_u128_min : U128 := Scalar.ofInt U128.min
+@[reducible] def core_u128_max : U128 := Scalar.ofInt U128.max
+
 -- TODO: below: not sure this is the best way.
 -- Should we rather overload operations like +, -, etc.?
 -- Also, it is possible to automate the generation of those definitions
@@ -447,6 +507,26 @@ instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
 instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
   hMod x y := Scalar.rem x y
 
+-- Shift left
+instance {ty0 ty1} : HShiftLeft (Scalar ty0) (Scalar ty1) (Result (Scalar ty0)) where
+  hShiftLeft x y := Scalar.shiftl x y
+
+-- Shift right
+instance {ty0 ty1} : HShiftRight (Scalar ty0) (Scalar ty1) (Result (Scalar ty0)) where
+  hShiftRight x y := Scalar.shiftr x y
+
+-- Xor
+instance {ty} : HXor (Scalar ty) (Scalar ty) (Scalar ty) where
+  hXor x y := Scalar.xor x y
+
+-- Or
+instance {ty} : HOr (Scalar ty) (Scalar ty) (Scalar ty) where
+  hOr x y := Scalar.or x y
+
+-- And
+instance {ty} : HAnd (Scalar ty) (Scalar ty) (Scalar ty) where
+  hAnd x y := Scalar.and x y
+
 -- Generic theorem - shouldn't be used much
 @[cpspec]
 theorem Scalar.add_spec {ty} {x y : Scalar ty}
@@ -864,33 +944,33 @@ theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : S
 
 -- ofIntCore
 -- TODO: typeclass?
-@[reducible] def Isize.ofIntCore := @Scalar.ofIntCore .Isize
-@[reducible] def I8.ofIntCore    := @Scalar.ofIntCore .I8
-@[reducible] def I16.ofIntCore   := @Scalar.ofIntCore .I16
-@[reducible] def I32.ofIntCore   := @Scalar.ofIntCore .I32
-@[reducible] def I64.ofIntCore   := @Scalar.ofIntCore .I64
-@[reducible] def I128.ofIntCore  := @Scalar.ofIntCore .I128
-@[reducible] def Usize.ofIntCore := @Scalar.ofIntCore .Usize
-@[reducible] def U8.ofIntCore    := @Scalar.ofIntCore .U8
-@[reducible] def U16.ofIntCore   := @Scalar.ofIntCore .U16
-@[reducible] def U32.ofIntCore   := @Scalar.ofIntCore .U32
-@[reducible] def U64.ofIntCore   := @Scalar.ofIntCore .U64
-@[reducible] def U128.ofIntCore  := @Scalar.ofIntCore .U128
+def Isize.ofIntCore := @Scalar.ofIntCore .Isize
+def I8.ofIntCore    := @Scalar.ofIntCore .I8
+def I16.ofIntCore   := @Scalar.ofIntCore .I16
+def I32.ofIntCore   := @Scalar.ofIntCore .I32
+def I64.ofIntCore   := @Scalar.ofIntCore .I64
+def I128.ofIntCore  := @Scalar.ofIntCore .I128
+def Usize.ofIntCore := @Scalar.ofIntCore .Usize
+def U8.ofIntCore    := @Scalar.ofIntCore .U8
+def U16.ofIntCore   := @Scalar.ofIntCore .U16
+def U32.ofIntCore   := @Scalar.ofIntCore .U32
+def U64.ofIntCore   := @Scalar.ofIntCore .U64
+def U128.ofIntCore  := @Scalar.ofIntCore .U128
 
 --  ofInt
 -- TODO: typeclass?
-@[reducible] def Isize.ofInt := @Scalar.ofInt .Isize
-@[reducible] def I8.ofInt    := @Scalar.ofInt .I8
-@[reducible] def I16.ofInt   := @Scalar.ofInt .I16
-@[reducible] def I32.ofInt   := @Scalar.ofInt .I32
-@[reducible] def I64.ofInt   := @Scalar.ofInt .I64
-@[reducible] def I128.ofInt  := @Scalar.ofInt .I128
-@[reducible] def Usize.ofInt := @Scalar.ofInt .Usize
-@[reducible] def U8.ofInt    := @Scalar.ofInt .U8
-@[reducible] def U16.ofInt   := @Scalar.ofInt .U16
-@[reducible] def U32.ofInt   := @Scalar.ofInt .U32
-@[reducible] def U64.ofInt   := @Scalar.ofInt .U64
-@[reducible] def U128.ofInt  := @Scalar.ofInt .U128
+abbrev Isize.ofInt := @Scalar.ofInt .Isize
+abbrev I8.ofInt    := @Scalar.ofInt .I8
+abbrev I16.ofInt   := @Scalar.ofInt .I16
+abbrev I32.ofInt   := @Scalar.ofInt .I32
+abbrev I64.ofInt   := @Scalar.ofInt .I64
+abbrev I128.ofInt  := @Scalar.ofInt .I128
+abbrev Usize.ofInt := @Scalar.ofInt .Usize
+abbrev U8.ofInt    := @Scalar.ofInt .U8
+abbrev U16.ofInt   := @Scalar.ofInt .U16
+abbrev U32.ofInt   := @Scalar.ofInt .U32
+abbrev U64.ofInt   := @Scalar.ofInt .U64
+abbrev U128.ofInt  := @Scalar.ofInt .U128
 
 postfix:max "#isize" => Isize.ofInt
 postfix:max "#i8"    => I8.ofInt
@@ -908,9 +988,46 @@ postfix:max "#u128"  => U128.ofInt
 -- Testing the notations
 example : Result Usize := 0#usize + 1#usize
 
+-- TODO: factor those lemmas out
 @[simp] theorem Scalar.ofInt_val_eq {ty} (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty) : (Scalar.ofInt x h).val = x := by
   simp [Scalar.ofInt, Scalar.ofIntCore]
 
+@[simp] theorem Isize.ofInt_val_eq (h : Scalar.min ScalarTy.Isize ≤ x ∧ x ≤ Scalar.max ScalarTy.Isize) : (Isize.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I8.ofInt_val_eq (h : Scalar.min ScalarTy.I8 ≤ x ∧ x ≤ Scalar.max ScalarTy.I8) : (I8.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I16.ofInt_val_eq (h : Scalar.min ScalarTy.I16 ≤ x ∧ x ≤ Scalar.max ScalarTy.I16) : (I16.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I32.ofInt_val_eq (h : Scalar.min ScalarTy.I32 ≤ x ∧ x ≤ Scalar.max ScalarTy.I32) : (I32.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I64.ofInt_val_eq (h : Scalar.min ScalarTy.I64 ≤ x ∧ x ≤ Scalar.max ScalarTy.I64) : (I64.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem I128.ofInt_val_eq (h : Scalar.min ScalarTy.I128 ≤ x ∧ x ≤ Scalar.max ScalarTy.I128) : (I128.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem Usize.ofInt_val_eq (h : Scalar.min ScalarTy.Usize ≤ x ∧ x ≤ Scalar.max ScalarTy.Usize) : (Usize.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U8.ofInt_val_eq (h : Scalar.min ScalarTy.U8 ≤ x ∧ x ≤ Scalar.max ScalarTy.U8) : (U8.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U16.ofInt_val_eq (h : Scalar.min ScalarTy.U16 ≤ x ∧ x ≤ Scalar.max ScalarTy.U16) : (U16.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U32.ofInt_val_eq (h : Scalar.min ScalarTy.U32 ≤ x ∧ x ≤ Scalar.max ScalarTy.U32) : (U32.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U64.ofInt_val_eq (h : Scalar.min ScalarTy.U64 ≤ x ∧ x ≤ Scalar.max ScalarTy.U64) : (U64.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
+@[simp] theorem U128.ofInt_val_eq (h : Scalar.min ScalarTy.U128 ≤ x ∧ x ≤ Scalar.max ScalarTy.U128) : (U128.ofInt x h).val = x := by
+  apply Scalar.ofInt_val_eq h
+
 -- Comparisons
 instance {ty} : LT (Scalar ty) where
   lt a b := LT.lt a.val b.val
diff --git a/backends/lean/Base/Primitives/Vec.lean b/backends/lean/Base/Primitives/Vec.lean
index 2d48a641..2c3fce91 100644
--- a/backends/lean/Base/Primitives/Vec.lean
+++ b/backends/lean/Base/Primitives/Vec.lean
@@ -6,7 +6,7 @@ import Mathlib.Tactic.RunCmd
 import Mathlib.Tactic.Linarith
 import Base.IList
 import Base.Primitives.Scalar
-import Base.Primitives.Array
+import Base.Primitives.ArraySlice
 import Base.Arith
 import Base.Progress.Base
 
@@ -14,6 +14,8 @@ namespace Primitives
 
 open Result Error
 
+namespace alloc.vec
+
 def Vec (α : Type u) := { l : List α // l.length ≤ Usize.max }
 
 instance (a : Type u) : Arith.HasIntProp (Vec a) where
@@ -83,7 +85,7 @@ theorem Vec.insert_spec {α : Type u} (v: Vec α) (i: Usize) (x: α)
   ∃ nv, v.insert α i x = ret nv ∧ nv.val = v.val.update i.val x := by
   simp [insert, *]
 
-def Vec.index_shared (α : Type u) (v: Vec α) (i: Usize) : Result α :=
+def Vec.index_usize {α : Type u} (v: Vec α) (i: Usize) : Result α :=
   match v.val.indexOpt i.val with
   | none => fail .arrayOutOfBounds
   | some x => ret x
@@ -94,51 +96,83 @@ def Vec.index_shared (α : Type u) (v: Vec α) (i: Usize) : Result α :=
  -/
 
 @[pspec]
-theorem Vec.index_shared_spec {α : Type u} [Inhabited α] (v: Vec 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 Vec.index_back (α : Type u) (v: Vec α) (i: Usize) (_: α) : Result Unit :=
-  if i.val < List.length v.val then
-    .ret ()
-  else
-    .fail arrayOutOfBounds
-
-def Vec.index_mut (α : Type u) (v: Vec α) (i: Usize) : Result α :=
-  match v.val.indexOpt i.val with
-  | none => fail .arrayOutOfBounds
-  | some x => ret x
-
-@[pspec]
-theorem Vec.index_mut_spec {α : Type u} [Inhabited α] (v: Vec α) (i: Usize)
+theorem Vec.index_usize_spec {α : Type u} [Inhabited α] (v: Vec 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]
+  ∃ x, v.index_usize i = ret x ∧ x = v.val.index i.val := by
+  simp only [index_usize]
   -- TODO: dependent rewrite
   have h := List.indexOpt_eq_index v.val i.val (by scalar_tac) (by simp [*])
   simp [*]
 
-def Vec.index_mut_back (α : Type u) (v: Vec α) (i: Usize) (x: α) : Result (Vec α) :=
+def Vec.update_usize {α : Type u} (v: Vec α) (i: Usize) (x: α) : Result (Vec 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 Vec.index_mut_back_spec {α : Type u} (v: Vec α) (i: Usize) (x : α)
+theorem Vec.update_usize_spec {α : Type u} (v: Vec α) (i: Usize) (x : α)
   (hbound : i.val < v.length) :
-  ∃ nv, v.index_mut_back α i x = ret nv ∧
+  ∃ nv, v.update_usize i x = ret nv ∧
   nv.val = v.val.update i.val x
   := by
-  simp only [index_mut_back]
+  simp only [update_usize]
   have h := List.indexOpt_bounds v.val i.val
   split
   . simp_all [length]; cases h <;> scalar_tac
   . simp_all
 
+/- [alloc::vec::Vec::index]: forward function -/
+def Vec.index (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T))
+  (self : Vec T) (i : I) : Result inst.Output :=
+  sorry -- TODO
+
+/- [alloc::vec::Vec::index_mut]: forward function -/
+def Vec.index_mut (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T))
+  (self : Vec T) (i : I) : Result inst.Output :=
+  sorry -- TODO
+
+/- [alloc::vec::Vec::index_mut]: backward function 0 -/
+def Vec.index_mut_back
+  (T I : Type) (inst : core.slice.index.SliceIndex I (Slice T))
+  (self : Vec T) (i : I) (x : inst.Output) : Result (alloc.vec.Vec T) :=
+  sorry -- TODO
+
+/- Trait implementation: [alloc::vec::Vec] -/
+def Vec.coreopsindexIndexInst (T I : Type)
+  (inst : core.slice.index.SliceIndex I (Slice T)) :
+  core.ops.index.Index (alloc.vec.Vec T) I := {
+  Output := inst.Output
+  index := Vec.index T I inst
+}
+
+/- Trait implementation: [alloc::vec::Vec] -/
+def Vec.coreopsindexIndexMutInst (T I : Type)
+  (inst : core.slice.index.SliceIndex I (Slice T)) :
+  core.ops.index.IndexMut (alloc.vec.Vec T) I := {
+  indexInst := Vec.coreopsindexIndexInst T I inst
+  index_mut := Vec.index_mut T I inst
+  index_mut_back := Vec.index_mut_back T I inst
+}
+
+@[simp]
+theorem Vec.index_slice_index {α : Type} (v : Vec α) (i : Usize) :
+  Vec.index α Usize (core.slice.index.SliceIndexUsizeSliceTInst α) v i =
+  Vec.index_usize v i :=
+  sorry
+
+@[simp]
+theorem Vec.index_mut_slice_index {α : Type} (v : Vec α) (i : Usize) :
+  Vec.index_mut α Usize (core.slice.index.SliceIndexUsizeSliceTInst α) v i =
+  Vec.index_usize v i :=
+  sorry
+
+@[simp]
+theorem Vec.index_mut_back_slice_index {α : Type} (v : Vec α) (i : Usize) (x : α) :
+  Vec.index_mut_back α Usize (core.slice.index.SliceIndexUsizeSliceTInst α) v i x =
+  Vec.update_usize v i x :=
+  sorry
+
+end alloc.vec
+
 end Primitives