Merge branch 'main' into afromher_shifts

15 files changed, 971 insertions, 470 deletions

diff --git a/backends/lean/Base/Arith/Base.lean b/backends/lean/Base/Arith/Base.lean
index 9c11ed45..8ada4171 100644
--- a/backends/lean/Base/Arith/Base.lean
+++ b/backends/lean/Base/Arith/Base.lean
@@ -57,4 +57,16 @@ theorem int_pos_ind (p : Int → Prop) :
 -- TODO: there is probably something more general to do
 theorem nat_zero_eq_int_zero : (0 : Nat) = (0 : Int) := by simp
 
+-- This is mostly used in termination proofs
+theorem to_int_to_nat_lt (x y : ℤ) (h0 : 0 ≤ x) (h1 : x < y) :
+  ↑(x.toNat) < y := by
+  simp [*]
+
+-- This is mostly used in termination proofs
+theorem to_int_sub_to_nat_lt (x y : ℤ) (x' : ℕ)
+  (h0 : ↑x' ≤ x) (h1 : x - ↑x' < y) :
+  ↑(x.toNat - x') < y := by
+  have : 0 ≤ x := by linarith
+  simp [Int.toNat_sub_of_le, *]
+
 end Arith
diff --git a/backends/lean/Base/Arith/Int.lean b/backends/lean/Base/Arith/Int.lean
index 3359ecdb..a57f8bb1 100644
--- a/backends/lean/Base/Arith/Int.lean
+++ b/backends/lean/Base/Arith/Int.lean
@@ -162,7 +162,7 @@ def introInstances (declToUnfold : Name) (lookup : Expr → MetaM (Option Expr))
     -- Add a declaration
     let nval ← Utils.addDeclTac name e type (asLet := false)
     -- Simplify to unfold the declaration to unfold (i.e., the projector)
-    Utils.simpAt [declToUnfold] [] [] (Tactic.Location.targets #[mkIdent name] false)
+    Utils.simpAt true [declToUnfold] [] [] (Location.targets #[mkIdent name] false)
     -- Return the new value
     pure nval
 
@@ -240,7 +240,7 @@ def intTac (splitGoalConjs : Bool) (extraPreprocess :  Tactic.TacticM Unit) : Ta
   -- the goal. I think before leads to a smaller proof term?
   Tactic.allGoals (intTacPreprocess extraPreprocess)
   -- More preprocessing
-  Tactic.allGoals (Utils.tryTac (Utils.simpAt [] [``nat_zero_eq_int_zero] [] .wildcard))
+  Tactic.allGoals (Utils.tryTac (Utils.simpAt true [] [``nat_zero_eq_int_zero] [] .wildcard))
   -- Split the conjunctions in the goal
   if splitGoalConjs then Tactic.allGoals (Utils.repeatTac Utils.splitConjTarget)
   -- Call linarith
@@ -270,6 +270,17 @@ elab "int_tac" args:(" split_goal"?): tactic =>
   let split := args.raw.getArgs.size > 0
   intTac split (do pure ())
 
+-- For termination proofs
+syntax "int_decr_tac" : tactic
+macro_rules
+  | `(tactic| int_decr_tac) =>
+    `(tactic|
+      simp_wf;
+      -- TODO: don't use a macro (namespace problems)
+      (first | apply Arith.to_int_to_nat_lt
+             | apply Arith.to_int_sub_to_nat_lt) <;>
+      simp_all <;> int_tac)
+
 example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
   int_tac_preprocess
   linarith
diff --git a/backends/lean/Base/Arith/Scalar.lean b/backends/lean/Base/Arith/Scalar.lean
index 47751c8a..2342cce6 100644
--- a/backends/lean/Base/Arith/Scalar.lean
+++ b/backends/lean/Base/Arith/Scalar.lean
@@ -17,7 +17,7 @@ def scalarTacExtraPreprocess : Tactic.TacticM Unit := do
    add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Usize []])
    add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Isize []])
    -- Reveal the concrete bounds, simplify calls to [ofInt]
-   Utils.simpAt [``Scalar.min, ``Scalar.max, ``Scalar.cMin, ``Scalar.cMax,
+   Utils.simpAt true [``Scalar.min, ``Scalar.max, ``Scalar.cMin, ``Scalar.cMax,
                  ``I8.min, ``I16.min, ``I32.min, ``I64.min, ``I128.min,
                  ``I8.max, ``I16.max, ``I32.max, ``I64.max, ``I128.max,
                  ``U8.min, ``U16.min, ``U32.min, ``U64.min, ``U128.min,
@@ -36,6 +36,17 @@ def scalarTac (splitGoalConjs : Bool) : Tactic.TacticM Unit := do
 elab "scalar_tac" : tactic =>
   scalarTac false
 
+-- For termination proofs
+syntax "scalar_decr_tac" : tactic
+macro_rules
+  | `(tactic| scalar_decr_tac) =>
+    `(tactic|
+      simp_wf;
+      -- TODO: don't use a macro (namespace problems)
+      (first | apply Arith.to_int_to_nat_lt
+             | apply Arith.to_int_sub_to_nat_lt) <;>
+      simp_all <;> scalar_tac)
+
 instance (ty : ScalarTy) : HasIntProp (Scalar ty) where
   -- prop_ty is inferred
   prop := λ x => And.intro x.hmin x.hmax
diff --git a/backends/lean/Base/IList/IList.lean b/backends/lean/Base/IList/IList.lean
index a940da25..f71f2de2 100644
--- a/backends/lean/Base/IList/IList.lean
+++ b/backends/lean/Base/IList/IList.lean
@@ -112,7 +112,13 @@ def pairwise_rel
 
 section Lemmas
 
-variable {α : Type u} 
+variable {α : Type u}
+
+def ireplicate {α : Type u} (i : ℤ) (x : α) : List α :=
+  if i ≤ 0 then []
+  else x :: ireplicate (i - 1) x
+termination_by ireplicate i x => i.toNat
+decreasing_by int_decr_tac
 
 @[simp] theorem update_nil : update ([] : List α) i y = [] := by simp [update]
 @[simp] theorem update_zero_cons : update ((x :: tl) : List α) 0 y = y :: tl := by simp [update]
@@ -129,6 +135,10 @@ variable {α : Type u}
 @[simp] theorem slice_nil : slice i j ([] : List α) = [] := by simp [slice]
 @[simp] theorem slice_zero : slice 0 0 (ls : List α) = [] := by cases ls <;> simp [slice]
 
+@[simp] theorem ireplicate_zero : ireplicate 0 x = [] := by rw [ireplicate]; simp
+@[simp] theorem ireplicate_nzero_cons (hne : 0 < i) : ireplicate i x = x :: ireplicate (i - 1) x := by
+  rw [ireplicate]; simp [*]; intro; linarith
+
 @[simp]
 theorem slice_nzero_cons (i j : Int) (x : α) (tl : List α) (hne : i ≠ 0) : slice i j ((x :: tl) : List α) = slice (i - 1) (j - 1) tl :=
   match tl with
@@ -144,6 +154,33 @@ theorem slice_nzero_cons (i j : Int) (x : α) (tl : List α) (hne : i ≠ 0) : s
         conv at this => lhs; simp [slice, *]
         simp [*, slice]
 
+@[simp]
+theorem ireplicate_replicate {α : Type u} (l : ℤ) (x : α) (h : 0 ≤ l) :
+  ireplicate l x = replicate l.toNat x :=
+  if hz: l = 0 then by
+    simp [*]
+  else by
+    have : 0 < l := by int_tac
+    have hr := ireplicate_replicate (l - 1) x (by int_tac)
+    simp [*]
+    have hl : l.toNat = .succ (l.toNat - 1) := by
+      cases hl: l.toNat <;> simp_all
+    conv => rhs; rw[hl]
+termination_by ireplicate_replicate l x h => l.toNat
+decreasing_by int_decr_tac
+
+@[simp]
+theorem ireplicate_len {α : Type u} (l : ℤ) (x : α) (h : 0 ≤ l) :
+  (ireplicate l x).len = l :=
+  if hz: l = 0 then by
+    simp [*]
+  else by
+    have : 0 < l := by int_tac
+    have hr := ireplicate_len (l - 1) x (by int_tac)
+    simp [*]
+termination_by ireplicate_len l x h => l.toNat
+decreasing_by int_decr_tac
+
 theorem len_eq_length (ls : List α) : ls.len = ls.length := by
   induction ls
   . rfl
diff --git a/backends/lean/Base/Primitives.lean b/backends/lean/Base/Primitives.lean
index 6b7b0792..613b6076 100644
--- a/backends/lean/Base/Primitives.lean
+++ b/backends/lean/Base/Primitives.lean
@@ -1,4 +1,6 @@
 import Base.Primitives.Base
 import Base.Primitives.Scalar
-import Base.Primitives.Array
+import Base.Primitives.ArraySlice
 import Base.Primitives.Vec
+import Base.Primitives.Alloc
+import Base.Primitives.CoreOps
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 55227a9f..ec9665a5 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
@@ -395,6 +409,34 @@ def Scalar.cast {src_ty : ScalarTy} (tgt_ty : ScalarTy) (x : Scalar src_ty) : Re
 @[reducible] def U64   := Scalar .U64
 @[reducible] def U128  := Scalar .U128
 
+-- 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
@@ -861,33 +903,33 @@ 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
@@ -905,9 +947,46 @@ 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 c4c4d9f2..e600a151 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
@@ -79,7 +81,7 @@ 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
@@ -90,51 +92,83 @@ 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
diff --git a/backends/lean/Base/Progress/Progress.lean b/backends/lean/Base/Progress/Progress.lean
index 8b0759c5..ba63f09d 100644
--- a/backends/lean/Base/Progress/Progress.lean
+++ b/backends/lean/Base/Progress/Progress.lean
@@ -8,6 +8,27 @@ namespace Progress
 open Lean Elab Term Meta Tactic
 open Utils
 
+-- TODO: the scalar types annoyingly often get reduced when we use the progress
+-- tactic. We should find a way of controling reduction. For now we use rewriting
+-- lemmas to make sure the goal remains clean, but this complexifies proof terms.
+-- It seems there used to be a `fold` tactic.
+theorem scalar_isize_eq : Primitives.Scalar .Isize = Primitives.Isize := by rfl
+theorem scalar_i8_eq    : Primitives.Scalar .I8 = Primitives.I8 := by rfl
+theorem scalar_i16_eq   : Primitives.Scalar .I16 = Primitives.I16 := by rfl
+theorem scalar_i32_eq   : Primitives.Scalar .I32 = Primitives.I32 := by rfl
+theorem scalar_i64_eq   : Primitives.Scalar .I64 = Primitives.I64 := by rfl
+theorem scalar_i128_eq  : Primitives.Scalar .I128 = Primitives.I128 := by rfl
+theorem scalar_usize_eq : Primitives.Scalar .Usize = Primitives.Usize := by rfl
+theorem scalar_u8_eq    : Primitives.Scalar .U8 = Primitives.U8 := by rfl
+theorem scalar_u16_eq   : Primitives.Scalar .U16 = Primitives.U16 := by rfl
+theorem scalar_u32_eq   : Primitives.Scalar .U32 = Primitives.U32 := by rfl
+theorem scalar_u64_eq   : Primitives.Scalar .U64 = Primitives.U64 := by rfl
+theorem scalar_u128_eq  : Primitives.Scalar .U128 = Primitives.U128 := by rfl
+def scalar_eqs := [
+  ``scalar_isize_eq, ``scalar_i8_eq, ``scalar_i16_eq, ``scalar_i32_eq, ``scalar_i64_eq, ``scalar_i128_eq,
+  ``scalar_usize_eq, ``scalar_u8_eq, ``scalar_u16_eq, ``scalar_u32_eq, ``scalar_u64_eq, ``scalar_u128_eq
+]
+
 inductive TheoremOrLocal where
 | Theorem (thName : Name)
 | Local (asm : LocalDecl)
@@ -111,8 +132,11 @@ def progressWith (fExpr : Expr) (th : TheoremOrLocal)
     splitEqAndPost fun hEq hPost ids => do
     trace[Progress] "eq and post:\n{hEq} : {← inferType hEq}\n{hPost}"
     tryTac (
-      simpAt [] [``Primitives.bind_tc_ret, ``Primitives.bind_tc_fail, ``Primitives.bind_tc_div]
+      simpAt true []
+             [``Primitives.bind_tc_ret, ``Primitives.bind_tc_fail, ``Primitives.bind_tc_div]
              [hEq.fvarId!] (.targets #[] true))
+    -- TODO: remove this (some types get unfolded too much: we "fold" them back)
+    tryTac (simpAt true [] scalar_eqs [] .wildcard_dep)
     -- Clear the equality, unless the user requests not to do so
     let mgoal ← do
       if keep.isSome then getMainGoal
@@ -359,6 +383,7 @@ namespace Test
   -- #eval showStoredPSpec
   -- #eval showStoredPSpecClass
   -- #eval showStoredPSpecExprClass
+  open alloc.vec
 
   example {ty} {x y : Scalar ty}
     (hmin : Scalar.min ty ≤ x.val + y.val)
@@ -384,7 +409,7 @@ namespace Test
      `α : Type u` where u is quantified, while here we use `α : Type 0` -/
   example {α : Type} (v: Vec α) (i: Usize) (x : α)
     (hbounds : 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
     progress
     simp [*]
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
index 5224e1c3..b917a789 100644
--- a/backends/lean/Base/Utils.lean
+++ b/backends/lean/Base/Utils.lean
@@ -604,16 +604,12 @@ example (h : ∃ x y z, x + y + z ≥ 0) : ∃ x, x ≥ 0 := by
   rename_i x y z
   exists x + y + z
 
-/- Call the simp tactic.
-   The initialization of the context is adapted from Tactic.elabSimpArgs.
-   Something very annoying is that there is no function which allows to
-   initialize a simp context without doing an elaboration - as a consequence
-   we write our own here. -/
-def simpAt (declsToUnfold : List Name) (thms : List Name) (hypsToUse : List FVarId)
-  (loc : Tactic.Location) :
-  Tactic.TacticM Unit := do
-  -- Initialize with the builtin simp theorems
-  let simpThms ← Tactic.simpOnlyBuiltins.foldlM (·.addConst ·) ({} : SimpTheorems)
+def mkSimpCtx (simpOnly : Bool) (declsToUnfold : List Name) (thms : List Name) (hypsToUse : List FVarId) :
+  Tactic.TacticM Simp.Context := do
+  -- Initialize either with the builtin simp theorems or with all the simp theorems
+  let simpThms ←
+    if simpOnly then Tactic.simpOnlyBuiltins.foldlM (·.addConst ·) ({} : SimpTheorems)
+    else getSimpTheorems
   -- Add the equational theorem for the declarations to unfold
   let simpThms ←
     declsToUnfold.foldlM (fun thms decl => thms.addDeclToUnfold decl) simpThms
@@ -637,8 +633,63 @@ def simpAt (declsToUnfold : List Name) (thms : List Name) (hypsToUse : List FVar
         throwError "Not a proposition: {thmName}"
       ) simpThms
   let congrTheorems ← getSimpCongrTheorems
-  let ctx : Simp.Context := { simpTheorems := #[simpThms], congrTheorems }
+  pure { simpTheorems := #[simpThms], congrTheorems }
+
+
+inductive Location where
+  /-- Apply the tactic everywhere. Same as `Tactic.Location.wildcard` -/
+  | wildcard
+  /-- Apply the tactic everywhere, including in the variable types (i.e., in
+      assumptions which are not propositions).  --/
+  | wildcard_dep
+  /-- Same as Tactic.Location -/
+  | targets (hypotheses : Array Syntax) (type : Bool)
+
+-- Comes from Tactic.simpLocation
+def customSimpLocation (ctx : Simp.Context) (discharge? : Option Simp.Discharge := none)
+  (loc : Location) : TacticM Simp.UsedSimps := do
+  match loc with
+  | Location.targets hyps simplifyTarget =>
+    withMainContext do
+      let fvarIds ← Lean.Elab.Tactic.getFVarIds hyps
+      go fvarIds simplifyTarget
+  | Location.wildcard =>
+    withMainContext do
+      go (← (← getMainGoal).getNondepPropHyps) (simplifyTarget := true)
+  | Location.wildcard_dep =>
+    withMainContext do
+      let ctx ← Lean.MonadLCtx.getLCtx
+      let decls ← ctx.getDecls
+      let tgts := (decls.map (fun d => d.fvarId)).toArray
+      go tgts (simplifyTarget := true)
+where
+  go (fvarIdsToSimp : Array FVarId) (simplifyTarget : Bool) : TacticM Simp.UsedSimps := do
+    let mvarId ← getMainGoal
+    let (result?, usedSimps) ← simpGoal mvarId ctx (simplifyTarget := simplifyTarget) (discharge? := discharge?) (fvarIdsToSimp := fvarIdsToSimp)
+    match result? with
+    | none => replaceMainGoal []
+    | some (_, mvarId) => replaceMainGoal [mvarId]
+    return usedSimps
+
+/- Call the simp tactic.
+   The initialization of the context is adapted from Tactic.elabSimpArgs.
+   Something very annoying is that there is no function which allows to
+   initialize a simp context without doing an elaboration - as a consequence
+   we write our own here. -/
+def simpAt (simpOnly : Bool) (declsToUnfold : List Name) (thms : List Name) (hypsToUse : List FVarId)
+  (loc : Location) :
+  Tactic.TacticM Unit := do
+  -- Initialize the simp context
+  let ctx ← mkSimpCtx simpOnly declsToUnfold thms hypsToUse
+  -- Apply the simplifier
+  let _ ← customSimpLocation ctx (discharge? := .none) loc
+
+-- Call the simpAll tactic
+def simpAll (declsToUnfold : List Name) (thms : List Name) (hypsToUse : List FVarId) :
+  Tactic.TacticM Unit := do
+  -- Initialize the simp context
+  let ctx ← mkSimpCtx false declsToUnfold thms hypsToUse
   -- Apply the simplifier
-  let _ ← Tactic.simpLocation ctx (discharge? := .none) loc
+  let _ ← Lean.Meta.simpAll (← getMainGoal) ctx
 
 end Utils