19 files changed, 5441 insertions, 0 deletions

diff --git a/backends/lean/Base/Arith.lean b/backends/lean/Base/Arith.lean
new file mode 100644
index 00000000..c0d09fd2
--- /dev/null
+++ b/backends/lean/Base/Arith.lean
@@ -0,0 +1,2 @@
+import Base.Arith.Int
+import Base.Arith.Scalar
diff --git a/backends/lean/Base/Arith/Arith.lean b/backends/lean/Base/Arith/Arith.lean
new file mode 100644
index 00000000..e69de29b
--- /dev/null
+++ b/backends/lean/Base/Arith/Arith.lean
diff --git a/backends/lean/Base/Arith/Base.lean b/backends/lean/Base/Arith/Base.lean
new file mode 100644
index 00000000..9c11ed45
--- /dev/null
+++ b/backends/lean/Base/Arith/Base.lean
@@ -0,0 +1,60 @@
+import Lean
+import Std.Data.Int.Lemmas
+import Mathlib.Tactic.Linarith
+
+namespace Arith
+
+open Lean Elab Term Meta
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Arith
+
+-- TODO: move?
+theorem ne_zero_is_lt_or_gt {x : Int} (hne : x ≠ 0) : x < 0 ∨ x > 0 := by
+  cases h: x <;> simp_all
+  . rename_i n;
+    cases n <;> simp_all
+  . apply Int.negSucc_lt_zero
+
+-- TODO: move?
+theorem ne_is_lt_or_gt {x y : Int} (hne : x ≠ y) : x < y ∨ x > y := by
+  have hne : x - y ≠ 0 := by
+    simp
+    intro h
+    have: x = y := by linarith
+    simp_all
+  have h := ne_zero_is_lt_or_gt hne
+  match h with
+  | .inl _ => left; linarith
+  | .inr _ => right; linarith
+
+-- TODO: move?
+theorem add_one_le_iff_le_ne (n m : Nat) (h1 : m ≤ n) (h2 : m ≠ n) : m + 1 ≤ n := by
+  -- Damn, those proofs on natural numbers are hard - I wish Omega was in mathlib4...
+  simp [Nat.add_one_le_iff]
+  simp [Nat.lt_iff_le_and_ne]
+  simp_all
+
+/- Induction over positive integers -/
+-- TODO: move
+theorem int_pos_ind (p : Int → Prop) :
+  (zero:p 0) → (pos:∀ i, 0 ≤ i → p i → p (i + 1)) → ∀ i, 0 ≤ i → p i := by
+  intro h0 hr i hpos
+--  have heq : Int.toNat i = i := by
+--    cases i <;> simp_all
+  have ⟨ n, heq ⟩  : {n:Nat // n = i } := ⟨ Int.toNat i, by cases i <;> simp_all ⟩
+  revert i
+  induction n
+  . intro i hpos heq
+    cases i <;> simp_all
+  . rename_i n hi
+    intro i hpos heq
+    cases i <;> simp_all
+    rename_i m
+    cases m <;> simp_all
+
+-- We sometimes need this to make sure no natural numbers appear in the goals
+-- TODO: there is probably something more general to do
+theorem nat_zero_eq_int_zero : (0 : Nat) = (0 : Int) := by simp
+
+end Arith
diff --git a/backends/lean/Base/Arith/Int.lean b/backends/lean/Base/Arith/Int.lean
new file mode 100644
index 00000000..7a5bbe98
--- /dev/null
+++ b/backends/lean/Base/Arith/Int.lean
@@ -0,0 +1,280 @@
+/- This file contains tactics to solve arithmetic goals -/
+
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+-- TODO: there is no Omega tactic for now - it seems it hasn't been ported yet
+--import Mathlib.Tactic.Omega
+import Base.Utils
+import Base.Arith.Base
+
+namespace Arith
+
+open Utils
+
+-- Remark: I tried a version of the shape `HasScalarProp {a : Type} (x : a)`
+-- but the lookup didn't work
+class HasIntProp (a : Sort u) where
+  prop_ty : a → Prop
+  prop : ∀ x:a, prop_ty x
+
+class PropHasImp (x : Prop) where
+  concl : Prop
+  prop : x → concl
+
+instance (p : Int → Prop) : HasIntProp (Subtype p) where
+  prop_ty := λ x => p x
+  prop := λ x => x.property
+
+-- This also works for `x ≠ y` because this expression reduces to `¬ x = y`
+-- and `Ne` is marked as `reducible`
+instance (x y : Int) : PropHasImp (¬ x = y) where
+  concl := x < y ∨ x > y
+  prop := λ (h:x ≠ y) => ne_is_lt_or_gt h
+
+-- Check if a proposition is a linear integer proposition.
+-- We notably use this to check the goals.
+class IsLinearIntProp (x : Prop) where
+
+instance (x y : Int) : IsLinearIntProp (x < y) where
+instance (x y : Int) : IsLinearIntProp (x > y) where
+instance (x y : Int) : IsLinearIntProp (x ≤ y) where
+instance (x y : Int) : IsLinearIntProp (x ≥ y) where
+instance (x y : Int) : IsLinearIntProp (x ≥ y) where
+instance (x y : Int) : IsLinearIntProp (x = y) where
+/- It seems we don't need to do any special preprocessing when the *goal*
+   has the following shape - I guess `linarith` automatically calls `intro` -/
+instance (x y : Int) : IsLinearIntProp (¬ x = y) where
+
+open Lean Lean.Elab Lean.Meta
+
+-- Explore a term by decomposing the applications (we explore the applied
+-- functions and their arguments, but ignore lambdas, forall, etc. -
+-- should we go inside?).
+partial def foldTermApps (k : α → Expr → MetaM α) (s : α) (e : Expr) : MetaM α := do
+  -- We do it in a very simpler manner: we deconstruct applications,
+  -- and recursively explore the sub-expressions. Note that we do
+  -- not go inside foralls and abstractions (should we?).
+  e.withApp fun f args => do
+    let s ← k s f
+    args.foldlM (foldTermApps k) s
+
+-- Provided a function `k` which lookups type class instances on an expression,
+-- collect all the instances lookuped by applying `k` on the sub-expressions of `e`.
+def collectInstances
+  (k : Expr → MetaM (Option Expr)) (s : HashSet Expr) (e : Expr) : MetaM (HashSet Expr) := do
+  let k s e := do
+    match ← k e with
+    | none => pure s
+    | some i => pure (s.insert i)
+  foldTermApps k s e
+
+-- Similar to `collectInstances`, but explores all the local declarations in the
+-- main context.
+def collectInstancesFromMainCtx (k : Expr → MetaM (Option Expr)) : Tactic.TacticM (HashSet Expr) := do
+  Tactic.withMainContext do
+  -- Get the local context
+  let ctx ← Lean.MonadLCtx.getLCtx
+  -- Just a matter of precaution
+  let ctx ← instantiateLCtxMVars ctx
+  -- Initialize the hashset
+  let hs := HashSet.empty
+  -- Explore the declarations
+  let decls ← ctx.getDecls
+  decls.foldlM (fun hs d => collectInstances k hs d.toExpr) hs
+
+-- Helper
+def lookupProp (fName : String) (className : Name) (e : Expr) : MetaM (Option Expr) := do
+  trace[Arith] fName
+  -- TODO: do we need Lean.observing?
+  -- This actually eliminates the error messages
+  Lean.observing? do
+    trace[Arith] m!"{fName}: observing"
+    let ty ← Lean.Meta.inferType e
+    let hasProp ← mkAppM className #[ty]
+    let hasPropInst ← trySynthInstance hasProp
+    match hasPropInst with
+    | LOption.some i =>
+      trace[Arith] "Found {fName} instance"
+      let i_prop ← mkProjection i (Name.mkSimple "prop")
+      some (← mkAppM' i_prop #[e])
+    | _ => none
+
+-- Return an instance of `HasIntProp` for `e` if it has some
+def lookupHasIntProp (e : Expr) : MetaM (Option Expr) :=
+  lookupProp "lookupHasIntProp" ``HasIntProp e
+
+-- Collect the instances of `HasIntProp` for the subexpressions in the context
+def collectHasIntPropInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
+  collectInstancesFromMainCtx lookupHasIntProp
+
+-- Return an instance of `PropHasImp` for `e` if it has some
+def lookupPropHasImp (e : Expr) : MetaM (Option Expr) := do
+  trace[Arith] "lookupPropHasImp"
+  -- TODO: do we need Lean.observing?
+  -- This actually eliminates the error messages
+  Lean.observing? do
+    trace[Arith] "lookupPropHasImp: observing"
+    let ty ← Lean.Meta.inferType e
+    trace[Arith] "lookupPropHasImp: ty: {ty}"
+    let cl ← mkAppM ``PropHasImp #[ty]
+    let inst ← trySynthInstance cl
+    match inst with
+    | LOption.some i =>
+      trace[Arith] "Found PropHasImp instance"
+      let i_prop ←  mkProjection i (Name.mkSimple "prop")
+      some (← mkAppM' i_prop #[e])
+    | _ => none
+
+-- Collect the instances of `PropHasImp` for the subexpressions in the context
+def collectPropHasImpInstancesFromMainCtx : Tactic.TacticM (HashSet Expr) := do
+  collectInstancesFromMainCtx lookupPropHasImp
+
+elab "display_prop_has_imp_instances" : tactic => do
+  trace[Arith] "Displaying the PropHasImp instances"
+  let hs ← collectPropHasImpInstancesFromMainCtx
+  hs.forM fun e => do
+    trace[Arith] "+ PropHasImp instance: {e}"
+
+example (x y : Int) (_ : x ≠ y) (_ : ¬ x = y) : True := by
+  display_prop_has_imp_instances
+  simp
+
+-- Lookup instances in a context and introduce them with additional declarations.
+def introInstances (declToUnfold : Name) (lookup : Expr → MetaM (Option Expr)) : Tactic.TacticM (Array Expr) := do
+  let hs ← collectInstancesFromMainCtx lookup
+  hs.toArray.mapM fun e => do
+    let type ← inferType e
+    let name ← mkFreshAnonPropUserName
+    -- 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)
+    -- Return the new value
+    pure nval
+
+def introHasIntPropInstances : Tactic.TacticM (Array Expr) := do
+  trace[Arith] "Introducing the HasIntProp instances"
+  introInstances ``HasIntProp.prop_ty lookupHasIntProp
+
+-- Lookup the instances of `HasIntProp for all the sub-expressions in the context,
+-- and introduce the corresponding assumptions
+elab "intro_has_int_prop_instances" : tactic => do
+  let _ ← introHasIntPropInstances
+
+-- Lookup the instances of `PropHasImp for all the sub-expressions in the context,
+-- and introduce the corresponding assumptions
+elab "intro_prop_has_imp_instances" : tactic => do
+  trace[Arith] "Introducing the PropHasImp instances"
+  let _ ← introInstances ``PropHasImp.concl lookupPropHasImp
+
+example (x y : Int) (h0 : x ≤ y) (h1 : x ≠ y) : x < y := by
+  intro_prop_has_imp_instances
+  rename_i h
+  split_disj h
+  . linarith
+  . linarith
+
+/- Boosting a bit the linarith tac.
+
+   We do the following:
+   - for all the assumptions of the shape `(x : Int) ≠ y` or `¬ (x = y), we
+     introduce two goals with the assumptions `x < y` and `x > y`
+     TODO: we could create a PR for mathlib.
+ -/
+def intTacPreprocess (extraPreprocess :  Tactic.TacticM Unit) : Tactic.TacticM Unit := do
+  Tactic.withMainContext do
+  -- Lookup the instances of PropHasImp (this is how we detect assumptions
+  -- of the proper shape), introduce assumptions in the context and split
+  -- on those
+  -- TODO: get rid of the assumptions that we split
+  let rec splitOnAsms (asms : List Expr) : Tactic.TacticM Unit :=
+    match asms with
+    | [] => pure ()
+    | asm :: asms =>
+      let k := splitOnAsms asms
+      Utils.splitDisjTac asm k k
+  -- Introduce the scalar bounds
+  let _ ← introHasIntPropInstances
+  -- Extra preprocessing, before we split on the disjunctions
+  extraPreprocess
+  -- Split
+  let asms ← introInstances ``PropHasImp.concl lookupPropHasImp
+  splitOnAsms asms.toList
+
+elab "int_tac_preprocess" : tactic =>
+  intTacPreprocess (do pure ())
+
+-- Check if the goal is a linear arithmetic goal
+def goalIsLinearInt : Tactic.TacticM Bool := do
+  Tactic.withMainContext do
+  let gty ← Tactic.getMainTarget
+  match ← trySynthInstance (← mkAppM ``IsLinearIntProp #[gty]) with
+  | .some _ => pure true
+  | _ => pure false
+
+def intTac (splitGoalConjs : Bool) (extraPreprocess :  Tactic.TacticM Unit) : Tactic.TacticM Unit := do
+  Tactic.withMainContext do
+  Tactic.focus do
+  let g ← Tactic.getMainGoal
+  trace[Arith] "Original goal: {g}"
+  -- Introduce all the universally quantified variables (includes the assumptions)
+  let (_, g) ← g.intros
+  Tactic.setGoals [g]
+  -- Preprocess - wondering if we should do this before or after splitting
+  -- the goal. I think before leads to a smaller proof term?
+  Tactic.allGoals (intTacPreprocess extraPreprocess)
+  -- More preprocessing
+  Tactic.allGoals (Utils.simpAt [] [``nat_zero_eq_int_zero] [] .wildcard)
+  -- Split the conjunctions in the goal
+  if splitGoalConjs then Tactic.allGoals (Utils.repeatTac Utils.splitConjTarget)
+  -- Call linarith
+  let linarith := do
+    let cfg : Linarith.LinarithConfig := {
+      -- We do this with our custom preprocessing
+      splitNe := false
+    }
+    Tactic.liftMetaFinishingTactic <| Linarith.linarith false [] cfg
+  Tactic.allGoals do
+    -- We check if the goal is a linear arithmetic goal: if yes, we directly
+    -- call linarith, otherwise we first apply exfalso (we do this because
+    -- linarith is too general and sometimes fails to do this correctly).
+    if ← goalIsLinearInt then do
+      trace[Arith] "linarith goal: {← Tactic.getMainGoal}"
+      linarith
+    else do
+      let g ← Tactic.getMainGoal
+      let gs ← g.apply (Expr.const ``False.elim [.zero])
+      let goals ← Tactic.getGoals
+      Tactic.setGoals (gs ++ goals)
+      Tactic.allGoals do
+        trace[Arith] "linarith goal: {← Tactic.getMainGoal}"
+        linarith
+
+elab "int_tac" args:(" split_goal"?): tactic =>
+  let split := args.raw.getArgs.size > 0
+  intTac split (do pure ())
+
+example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
+  int_tac_preprocess
+  linarith
+  linarith
+
+example (x : Int) (h0: 0 ≤ x) (h1: x ≠ 0) : 0 < x := by
+  int_tac
+
+-- Checking that things append correctly when there are several disjunctions
+example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y := by
+  int_tac split_goal
+
+-- Checking that things append correctly when there are several disjunctions
+example (x y : Int) (h0: 0 ≤ x) (h1: x ≠ 0) (h2 : 0 ≤ y) (h3 : y ≠ 0) : 0 < x ∧ 0 < y ∧ x + y ≥ 2 := by
+  int_tac split_goal
+
+-- Checking that we can prove exfalso
+example (a : Prop) (x : Int) (h0: 0 < x) (h1: x < 0) : a := by
+  int_tac
+
+end Arith
diff --git a/backends/lean/Base/Arith/Scalar.lean b/backends/lean/Base/Arith/Scalar.lean
new file mode 100644
index 00000000..b792ff21
--- /dev/null
+++ b/backends/lean/Base/Arith/Scalar.lean
@@ -0,0 +1,49 @@
+import Base.Arith.Int
+import Base.Primitives.Scalar
+
+/- Automation for scalars - TODO: not sure it is worth having two files (Int.lean and Scalar.lean) -/
+namespace Arith
+
+open Lean Lean.Elab Lean.Meta
+open Primitives
+
+def scalarTacExtraPreprocess : Tactic.TacticM Unit := do
+   Tactic.withMainContext do
+   -- Inroduce the bounds for the isize/usize types
+   let add (e : Expr) : Tactic.TacticM Unit := do
+     let ty ← inferType e
+     let _ ← Utils.addDeclTac (← Utils.mkFreshAnonPropUserName) e ty (asLet := false)
+   add (← mkAppM ``Scalar.cMin_bound #[.const ``ScalarTy.Isize []])
+   add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Usize []])
+   add (← mkAppM ``Scalar.cMax_bound #[.const ``ScalarTy.Isize []])
+   -- Reveal the concrete bounds
+   Utils.simpAt [``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,
+                 ``U8.max, ``U16.max, ``U32.max, ``U64.max, ``U128.max,
+                 ``Usize.min
+                 ] [] [] .wildcard
+
+elab "scalar_tac_preprocess" : tactic =>
+  intTacPreprocess scalarTacExtraPreprocess
+
+-- A tactic to solve linear arithmetic goals in the presence of scalars
+def scalarTac (splitGoalConjs : Bool) : Tactic.TacticM Unit := do
+  intTac splitGoalConjs scalarTacExtraPreprocess
+
+elab "scalar_tac" : tactic =>
+  scalarTac false
+
+instance (ty : ScalarTy) : HasIntProp (Scalar ty) where
+  -- prop_ty is inferred
+  prop := λ x => And.intro x.hmin x.hmax
+
+example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
+  intro_has_int_prop_instances
+  simp [*]
+
+example (x y : U32) : x.val ≤ Scalar.max ScalarTy.U32 := by
+  scalar_tac
+
+end Arith
diff --git a/backends/lean/Base/Diverge.lean b/backends/lean/Base/Diverge.lean
new file mode 100644
index 00000000..c9a2eec2
--- /dev/null
+++ b/backends/lean/Base/Diverge.lean
@@ -0,0 +1,7 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.Diverge.Base
+import Base.Diverge.Elab
diff --git a/backends/lean/Base/Diverge/Base.lean b/backends/lean/Base/Diverge/Base.lean
new file mode 100644
index 00000000..1d548389
--- /dev/null
+++ b/backends/lean/Base/Diverge/Base.lean
@@ -0,0 +1,1138 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Mathlib.Tactic.Linarith
+import Base.Primitives.Base
+import Base.Arith.Base
+
+/- TODO: this is very useful, but is there more? -/
+set_option profiler true
+set_option profiler.threshold 100
+
+namespace Diverge
+
+namespace Fix
+
+  open Primitives
+  open Result
+
+  variable {a : Type u} {b : a → Type v}
+  variable {c d : Type w} -- TODO: why do we have to make them both : Type w?
+
+  /-! # The least fixed point definition and its properties -/
+
+  def least_p (p : Nat → Prop) (n : Nat) : Prop := p n ∧ (∀ m, m < n → ¬ p m)
+  noncomputable def least (p : Nat → Prop) : Nat :=
+    Classical.epsilon (least_p p)
+
+  -- Auxiliary theorem for [least_spec]: if there exists an `n` satisfying `p`,
+  -- there there exists a least `m` satisfying `p`.
+  theorem least_spec_aux (p : Nat → Prop) : ∀ (n : Nat), (hn : p n) → ∃ m, least_p p m := by
+    apply Nat.strongRec'
+    intros n hi hn
+    -- Case disjunction on: is n the smallest n satisfying p?
+    match Classical.em (∀ m, m < n → ¬ p m) with
+    | .inl hlt =>
+      -- Yes: trivial
+      exists n
+    | .inr hlt =>
+      simp at *
+      let ⟨ m, ⟨ hmlt, hm ⟩ ⟩ := hlt
+      have hi := hi m hmlt hm
+      apply hi
+
+  -- The specification of [least]: either `p` is never satisfied, or it is satisfied
+  -- by `least p` and no `n < least p` satisfies `p`.
+  theorem least_spec (p : Nat → Prop) : (∀ n, ¬ p n) ∨ (p (least p) ∧ ∀ n, n < least p → ¬ p n) := by
+    -- Case disjunction on the existence of an `n` which satisfies `p`
+    match Classical.em (∀ n, ¬ p n) with
+    | .inl h =>
+      -- There doesn't exist: trivial
+      apply (Or.inl h)
+    | .inr h =>
+      -- There exists: we simply use `least_spec_aux` in combination with the property
+      -- of the epsilon operator
+      simp at *
+      let ⟨ n, hn ⟩ := h
+      apply Or.inr
+      have hl := least_spec_aux p n hn
+      have he := Classical.epsilon_spec hl
+      apply he
+
+  /-! # The fixed point definitions -/
+
+  def fix_fuel (n : Nat) (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
+    Result (b x) :=
+    match n with
+    | 0 => .div
+    | n + 1 =>
+      f (fix_fuel n f) x
+
+  @[simp] def fix_fuel_pred (f : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (x : a) (n : Nat) :=
+    not (div? (fix_fuel n f x))
+
+  def fix_fuel_P (f : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (x : a) (n : Nat) : Prop :=
+    fix_fuel_pred f x n
+
+  partial
+  def fixImpl (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) : Result (b x) :=
+    f (fixImpl f) x
+
+  -- The fact that `fix` is implemented by `fixImpl` allows us to not mark the
+  -- functions defined with the fixed-point as noncomputable. One big advantage
+  -- is that it allows us to evaluate those functions, for instance with #eval.
+  @[implemented_by fixImpl]
+  def fix (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) : Result (b x) :=
+    fix_fuel (least (fix_fuel_P f x)) f x
+
+  /-! # The validity property -/
+
+  -- Monotonicity relation over results
+  -- TODO: generalize (we should parameterize the definition by a relation over `a`)
+  def result_rel {a : Type u} (x1 x2 : Result a) : Prop :=
+    match x1 with
+    | div => True
+    | fail _ => x2 = x1
+    | ret _ => x2 = x1 -- TODO: generalize
+
+  -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
+  def karrow_rel (k1 k2 : (x:a) → Result (b x)) : Prop :=
+    ∀ x, result_rel (k1 x) (k2 x)
+
+  -- Monotonicity property for function bodies
+  def is_mono (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
+    ∀ {{k1 k2}}, karrow_rel k1 k2 → karrow_rel (f k1) (f k2)
+
+  -- "Continuity" property.
+  -- We need this, and this looks a lot like continuity. Also see this paper:
+  -- https://inria.hal.science/file/index/docid/216187/filename/tarski.pdf
+  -- We define our "continuity" criteria so that it gives us what we need to
+  -- prove the fixed-point equation, and we can also easily manipulate it.
+  def is_cont (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
+    ∀ x, (Hdiv : ∀ n, fix_fuel (.succ n) f x = div) → f (fix f) x = div
+
+  /-! # The proof of the fixed-point equation -/
+  theorem fix_fuel_mono {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}
+    (Hmono : is_mono f) :
+    ∀ {{n m}}, n ≤ m → karrow_rel (fix_fuel n f) (fix_fuel m f) := by
+    intros n
+    induction n
+    case zero => simp [karrow_rel, fix_fuel, result_rel]
+    case succ n1 Hi =>
+      intros m Hle x
+      simp [result_rel]
+      match m with
+      | 0 =>
+        exfalso
+        zify at *
+        linarith
+      | Nat.succ m1 =>
+        simp_arith at Hle
+        simp [fix_fuel]
+        have Hi := Hi Hle
+        have Hmono := Hmono Hi x
+        simp [result_rel] at Hmono
+        apply Hmono
+
+  @[simp] theorem neg_fix_fuel_P
+    {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} {x : a} {n : Nat} :
+    ¬ fix_fuel_P f x n ↔ (fix_fuel n f x = div) := by
+    simp [fix_fuel_P, div?]
+    cases fix_fuel n f x <;> simp  
+
+  theorem fix_fuel_fix_mono {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} (Hmono : is_mono f) :
+    ∀ n, karrow_rel (fix_fuel n f) (fix f) := by
+    intros n x
+    simp [result_rel]
+    have Hl := least_spec (fix_fuel_P f x)
+    simp at Hl
+    match Hl with
+    | .inl Hl => simp [*]
+    | .inr ⟨ Hl, Hn ⟩ =>
+      match Classical.em (fix_fuel n f x = div) with
+      | .inl Hd =>
+        simp [*]
+      | .inr Hd =>
+        have Hineq : least (fix_fuel_P f x) ≤ n := by
+          -- Proof by contradiction
+          cases Classical.em (least (fix_fuel_P f x) ≤ n) <;> simp [*]
+          simp at *
+          rename_i Hineq
+          have Hn := Hn n Hineq
+          contradiction
+        have Hfix : ¬ (fix f x = div) := by
+          simp [fix]
+          -- By property of the least upper bound
+          revert Hd Hl
+          -- TODO: there is no conversion to select the head of a function!
+          conv => lhs; apply congr_fun; apply congr_fun; apply congr_fun; simp [fix_fuel_P, div?]
+          cases fix_fuel (least (fix_fuel_P f x)) f x <;> simp
+        have Hmono := fix_fuel_mono Hmono Hineq x
+        simp [result_rel] at Hmono
+        simp [fix] at *
+        cases Heq: fix_fuel (least (fix_fuel_P f x)) f x <;>
+        cases Heq':fix_fuel n f x <;>
+        simp_all
+
+  theorem fix_fuel_P_least {f : ((x:a) → Result (b x)) → (x:a) → Result (b x)} (Hmono : is_mono f) :
+    ∀ {{x n}}, fix_fuel_P f x n → fix_fuel_P f x (least (fix_fuel_P f x)) := by
+    intros x n Hf
+    have Hfmono := fix_fuel_fix_mono Hmono n x
+    -- TODO: there is no conversion to select the head of a function!
+    conv => apply congr_fun; simp [fix_fuel_P]
+    simp [fix_fuel_P] at Hf
+    revert Hf Hfmono
+    simp [div?, result_rel, fix]
+    cases fix_fuel n f x <;> simp_all
+
+  -- Prove the fixed point equation in the case there exists some fuel for which
+  -- the execution terminates
+  theorem fix_fixed_eq_terminates (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (Hmono : is_mono f)
+    (x : a) (n : Nat) (He : fix_fuel_P f x n) :
+    fix f x = f (fix f) x := by
+    have Hl := fix_fuel_P_least Hmono He
+    -- TODO: better control of simplification
+    conv at Hl =>
+      apply congr_fun
+      simp [fix_fuel_P]
+    -- The least upper bound is > 0
+    have ⟨ n, Hsucc ⟩ : ∃ n, least (fix_fuel_P f x) = Nat.succ n := by
+      revert Hl
+      simp [div?]
+      cases least (fix_fuel_P f x) <;> simp [fix_fuel]
+    simp [Hsucc] at Hl
+    revert Hl
+    simp [*, div?, fix, fix_fuel]
+    -- Use the monotonicity
+    have Hfixmono := fix_fuel_fix_mono Hmono n
+    have Hvm := Hmono Hfixmono x
+    -- Use functional extensionality
+    simp [result_rel, fix] at Hvm
+    revert Hvm
+    split <;> simp [*] <;> intros <;> simp [*]
+
+  theorem fix_fixed_eq_forall {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+    (Hmono : is_mono f) (Hcont : is_cont f) :
+    ∀ x, fix f x = f (fix f) x := by
+    intros x
+    -- Case disjunction: is there a fuel such that the execution successfully execute?
+    match Classical.em (∃ n, fix_fuel_P f x n) with
+    | .inr He =>
+      -- No fuel: the fixed point evaluates to `div`
+      --simp [fix] at *
+      simp at *
+      conv => lhs; simp [fix]
+      have Hel := He (Nat.succ (least (fix_fuel_P f x))); simp [*, fix_fuel] at *; clear Hel
+      -- Use the "continuity" of `f`
+      have He : ∀ n, fix_fuel (.succ n) f x = div := by intros; simp [*]
+      have Hcont := Hcont x He
+      simp [Hcont]
+    | .inl ⟨ n, He ⟩ => apply fix_fixed_eq_terminates f Hmono x n He
+
+  -- The final fixed point equation
+  theorem fix_fixed_eq {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+    (Hmono : is_mono f) (Hcont : is_cont f) :
+    fix f = f (fix f) := by
+    have Heq := fix_fixed_eq_forall Hmono Hcont
+    have Heq1 : fix f = (λ x => fix f x) := by simp
+    rw [Heq1]
+    conv => lhs; ext; simp [Heq]
+
+  /-! # Making the proofs of validity manageable (and automatable) -/
+
+  -- Monotonicity property for expressions
+  def is_mono_p (e : ((x:a) → Result (b x)) → Result c) : Prop :=
+    ∀ {{k1 k2}}, karrow_rel k1 k2 → result_rel (e k1) (e k2)
+
+  theorem is_mono_p_same (x : Result c) :
+    @is_mono_p a b c (λ _ => x) := by
+    simp [is_mono_p, karrow_rel, result_rel]
+    split <;> simp
+
+  theorem is_mono_p_rec (x : a) :
+    @is_mono_p a b (b x) (λ f => f x) := by
+    simp_all [is_mono_p, karrow_rel, result_rel]
+
+  -- The important lemma about `is_mono_p`
+  theorem is_mono_p_bind
+    (g : ((x:a) → Result (b x)) → Result c)
+    (h : c → ((x:a) → Result (b x)) → Result d) :
+    is_mono_p g →
+    (∀ y, is_mono_p (h y)) →
+    @is_mono_p a b d (λ k => @Bind.bind Result _ c d (g k) (fun y => h y k)) := by
+--    @is_mono_p a b d (λ k => do let (y : c) ← g k; h y k) := by
+    intro hg hh
+    simp [is_mono_p]
+    intro fg fh Hrgh
+    simp [karrow_rel, result_rel]
+    have hg := hg Hrgh; simp [result_rel] at hg
+    cases heq0: g fg <;> simp_all
+    rename_i y _
+    have hh := hh y Hrgh; simp [result_rel] at hh
+    simp_all
+
+  -- Continuity property for expressions - note that we take the continuation
+  -- as parameter
+  def is_cont_p (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (e : ((x:a) → Result (b x)) → Result c) : Prop :=
+    (Hc : ∀ n, e (fix_fuel n k) = .div) →
+    e (fix k) = .div
+
+  theorem is_cont_p_same (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (x : Result c) :
+    is_cont_p k (λ _ => x) := by
+    simp [is_cont_p]
+
+  theorem is_cont_p_rec (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
+    is_cont_p f (λ f => f x) := by
+    simp_all [is_cont_p, fix]
+
+  -- The important lemma about `is_cont_p`
+  theorem is_cont_p_bind
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (Hkmono : is_mono k)
+    (g : ((x:a) → Result (b x)) → Result c)
+    (h : c → ((x:a) → Result (b x)) → Result d) :
+    is_mono_p g →
+    is_cont_p k g →
+    (∀ y, is_mono_p (h y)) →
+    (∀ y, is_cont_p k (h y)) →
+    is_cont_p k (λ k => do let y ← g k; h y k) := by
+    intro Hgmono Hgcont Hhmono Hhcont
+    simp [is_cont_p]
+    intro Hdiv
+    -- Case on `g (fix... k)`: is there an n s.t. it terminates?
+    cases Classical.em (∀ n, g (fix_fuel n k) = .div) <;> rename_i Hn
+    . -- Case 1: g diverges
+      have Hgcont := Hgcont Hn
+      simp_all
+    . -- Case 2: g doesn't diverge
+      simp at Hn
+      let ⟨ n, Hn ⟩ := Hn
+      have Hdivn := Hdiv n
+      have Hffmono := fix_fuel_fix_mono Hkmono n
+      have Hgeq := Hgmono Hffmono
+      simp [result_rel] at Hgeq
+      cases Heq: g (fix_fuel n k) <;> rename_i y <;> simp_all
+      -- Remains the .ret case
+      -- Use Hdiv to prove that: ∀ n, h y (fix_fuel n f) = div
+      -- We do this in two steps: first we prove it for m ≥ n
+      have Hhdiv: ∀ m, h y (fix_fuel m k) = .div := by
+        have Hhdiv : ∀ m, n ≤ m → h y (fix_fuel m k) = .div := by
+          -- We use the fact that `g (fix_fuel n f) = .div`, combined with Hdiv
+          intro m Hle
+          have Hdivm := Hdiv m
+          -- Monotonicity of g
+          have Hffmono := fix_fuel_mono Hkmono Hle
+          have Hgmono := Hgmono Hffmono
+          -- We need to clear Hdiv because otherwise simp_all rewrites Hdivm with Hdiv
+          clear Hdiv
+          simp_all [result_rel]
+        intro m
+        -- TODO: we shouldn't need the excluded middle here because it is decidable
+        cases Classical.em (n ≤ m) <;> rename_i Hl
+        . apply Hhdiv; assumption
+        . simp at Hl
+          -- Make a case disjunction on `h y (fix_fuel m k)`: if it is not equal
+          -- to div, use the monotonicity of `h y`
+          have Hle : m ≤ n := by linarith
+          have Hffmono := fix_fuel_mono Hkmono Hle
+          have Hmono := Hhmono y Hffmono
+          simp [result_rel] at Hmono
+          cases Heq: h y (fix_fuel m k) <;> simp_all
+      -- We can now use the continuity hypothesis for h
+      apply Hhcont; assumption
+
+  -- The validity property for an expression
+  def is_valid_p (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (e : ((x:a) → Result (b x)) → Result c) : Prop :=
+    is_mono_p e ∧
+    (is_mono k → is_cont_p k e)
+
+  @[simp] theorem is_valid_p_same
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : Result c) :
+    is_valid_p k (λ _ => x) := by
+    simp [is_valid_p, is_mono_p_same, is_cont_p_same]
+
+  @[simp] theorem is_valid_p_rec
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x)) (x : a) :
+    is_valid_p k (λ k => k x) := by
+    simp_all [is_valid_p, is_mono_p_rec, is_cont_p_rec]
+
+  theorem is_valid_p_ite
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 e2 : ((x:a) → Result (b x)) → Result c}
+    (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
+    is_valid_p k (ite cond e1 e2) := by
+    split <;> assumption
+
+  theorem is_valid_p_dite
+    (k : ((x:a) → Result (b x)) → (x:a) → Result (b x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 : cond → ((x:a) → Result (b x)) → Result c}
+    {e2 : Not cond → ((x:a) → Result (b x)) → Result c}
+    (he1: ∀ x, is_valid_p k (e1 x)) (he2 : ∀ x, is_valid_p k (e2 x)) :
+    is_valid_p k (dite cond e1 e2) := by
+    split <;> simp [*]
+
+  -- Lean is good at unification: we can write a very general version
+  -- (in particular, it will manage to figure out `g` and `h` when we
+  -- apply the lemma)
+  theorem is_valid_p_bind
+    {{k : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+    {{g : ((x:a) → Result (b x)) → Result c}}
+    {{h : c → ((x:a) → Result (b x)) → Result d}}
+    (Hgvalid : is_valid_p k g)
+    (Hhvalid : ∀ y, is_valid_p k (h y)) :
+    is_valid_p k (λ k => do let y ← g k; h y k) := by
+    let ⟨ Hgmono, Hgcont ⟩ := Hgvalid
+    simp [is_valid_p, forall_and] at Hhvalid
+    have ⟨ Hhmono, Hhcont ⟩ := Hhvalid
+    simp [← imp_forall_iff] at Hhcont
+    simp [is_valid_p]; constructor
+    . -- Monotonicity
+      apply is_mono_p_bind <;> assumption
+    . -- Continuity
+      intro Hkmono
+      have Hgcont := Hgcont Hkmono
+      have Hhcont := Hhcont Hkmono
+      apply is_cont_p_bind <;> assumption
+
+  def is_valid (f : ((x:a) → Result (b x)) → (x:a) → Result (b x)) : Prop :=
+    ∀ k x, is_valid_p k (λ k => f k x)
+
+  theorem is_valid_p_imp_is_valid {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+    (Hvalid : is_valid f) :
+    is_mono f ∧ is_cont f := by
+    have Hmono : is_mono f := by
+      intro f h Hr x
+      have Hmono := Hvalid (λ _ _ => .div) x
+      have Hmono := Hmono.left
+      apply Hmono; assumption
+    have Hcont : is_cont f := by
+      intro x Hdiv
+      have Hcont := (Hvalid f x).right Hmono
+      simp [is_cont_p] at Hcont
+      apply Hcont
+      intro n
+      have Hdiv := Hdiv n
+      simp [fix_fuel] at Hdiv
+      simp [*]
+    simp [*]
+
+  theorem is_valid_fix_fixed_eq {{f : ((x:a) → Result (b x)) → (x:a) → Result (b x)}}
+    (Hvalid : is_valid f) :
+    fix f = f (fix f) := by
+    have ⟨ Hmono, Hcont ⟩ := is_valid_p_imp_is_valid Hvalid
+    exact fix_fixed_eq Hmono Hcont
+
+end Fix
+
+namespace FixI
+  /- Indexed fixed-point: definitions with indexed types, convenient to use for mutually
+     recursive definitions. We simply port the definitions and proofs from Fix to a more
+     specific case.
+   -/
+  open Primitives Fix
+
+  -- The index type
+  variable {id : Type u}
+
+  -- The input/output types
+  variable {a : id → Type v} {b : (i:id) → a i → Type w}
+
+  -- Monotonicity relation over monadic arrows (i.e., Kleisli arrows)
+  def karrow_rel (k1 k2 : (i:id) → (x:a i) → Result (b i x)) : Prop :=
+    ∀ i x, result_rel (k1 i x) (k2 i x)
+
+  def kk_to_gen (k : (i:id) → (x:a i) → Result (b i x)) :
+    (x: (i:id) × a i) → Result (b x.fst x.snd) :=
+    λ ⟨ i, x ⟩ => k i x
+
+  def kk_of_gen (k : (x: (i:id) × a i) → Result (b x.fst x.snd)) :
+    (i:id) → (x:a i) → Result (b i x) :=
+    λ i x => k ⟨ i, x ⟩
+
+  def k_to_gen (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) :
+    ((x: (i:id) × a i) → Result (b x.fst x.snd)) → (x: (i:id) × a i) → Result (b x.fst x.snd) :=
+    λ kk => kk_to_gen (k (kk_of_gen kk))
+
+  def k_of_gen (k : ((x: (i:id) × a i) → Result (b x.fst x.snd)) → (x: (i:id) × a i) → Result (b x.fst x.snd)) :
+    ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x) :=
+    λ kk => kk_of_gen (k (kk_to_gen kk))
+
+  def e_to_gen (e : ((i:id) → (x:a i) → Result (b i x)) → Result c) :
+    ((x: (i:id) × a i) → Result (b x.fst x.snd)) → Result c :=
+    λ k => e (kk_of_gen k)
+
+  def is_valid_p (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
+    (e : ((i:id) → (x:a i) → Result (b i x)) → Result c) : Prop :=
+    Fix.is_valid_p (k_to_gen k) (e_to_gen e)
+
+  def is_valid (f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) : Prop :=
+    ∀ k i x, is_valid_p k (λ k => f k i x)
+
+  def fix
+    (f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) :
+    (i:id) → (x:a i) → Result (b i x) :=
+    kk_of_gen (Fix.fix (k_to_gen f))
+
+  theorem is_valid_fix_fixed_eq
+    {{f : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)}}
+    (Hvalid : is_valid f) :
+    fix f = f (fix f) := by
+    have Hvalid' : Fix.is_valid (k_to_gen f) := by
+      intro k x
+      simp only [is_valid, is_valid_p] at Hvalid
+      let ⟨ i, x ⟩ := x
+      have Hvalid := Hvalid (k_of_gen k) i x
+      simp only [k_to_gen, k_of_gen, kk_to_gen, kk_of_gen] at Hvalid
+      refine Hvalid
+    have Heq := Fix.is_valid_fix_fixed_eq Hvalid'
+    simp [fix]
+    conv => lhs; rw [Heq]
+
+  /- Some utilities to define the mutually recursive functions -/
+
+  -- TODO: use more
+  abbrev kk_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
+    (i:id) → (x:a i) → Result (b i x)
+  abbrev k_ty (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :=
+    kk_ty id a b → kk_ty id a b
+
+  abbrev in_out_ty : Type (imax (u + 1) (v + 1)) := (in_ty : Type u) × ((x:in_ty) → Type v)
+  -- TODO: remove?
+  abbrev mk_in_out_ty (in_ty : Type u) (out_ty : in_ty → Type v) :
+    in_out_ty :=
+    Sigma.mk in_ty out_ty
+
+  -- Initially, we had left out the parameters id, a and b.
+  -- However, by parameterizing Funs with those parameters, we can state
+  -- and prove lemmas like Funs.is_valid_p_is_valid_p
+  inductive Funs (id : Type u) (a : id → Type v) (b : (i:id) → (x:a i) → Type w) :
+    List in_out_ty.{v, w} → Type (max (u + 1) (max (v + 1) (w + 1))) :=
+    | Nil : Funs id a b []
+    | Cons {ity : Type v} {oty : ity → Type w} {tys : List in_out_ty}
+           (f : kk_ty id a b → (x:ity) → Result (oty x)) (tl : Funs id a b tys) :
+           Funs id a b (⟨ ity, oty ⟩ :: tys)
+
+  def get_fun {tys : List in_out_ty} (fl : Funs id a b tys) :
+    (i : Fin tys.length) → kk_ty id a b → (x : (tys.get i).fst) →
+         Result ((tys.get i).snd x) :=
+    match fl with
+    | .Nil => λ i => by have h:= i.isLt; simp at h
+    | @Funs.Cons id a b ity oty tys1 f tl =>
+      λ ⟨ i, iLt ⟩ =>
+      match i with
+      | 0 =>
+        Eq.mp (by simp [List.get]) f
+      | .succ j =>
+        have jLt: j < tys1.length := by
+          simp at iLt
+          revert iLt
+          simp_arith
+        let j: Fin tys1.length := ⟨ j, jLt ⟩
+        Eq.mp (by simp) (get_fun tl j)
+
+  def for_all_fin_aux {n : Nat} (f : Fin n → Prop) (m : Nat) (h : m ≤ n) : Prop :=
+    if heq: m = n then True
+    else
+      f ⟨ m, by simp_all [Nat.lt_iff_le_and_ne] ⟩ ∧
+      for_all_fin_aux f (m + 1) (by simp_all [Arith.add_one_le_iff_le_ne])
+  termination_by for_all_fin_aux n _ m h => n - m
+  decreasing_by
+    simp_wf
+    apply Nat.sub_add_lt_sub <;> simp
+    simp_all [Arith.add_one_le_iff_le_ne]
+
+  def for_all_fin {n : Nat} (f : Fin n → Prop) := for_all_fin_aux f 0 (by simp)
+
+  theorem for_all_fin_aux_imp_forall {n : Nat} (f : Fin n → Prop) (m : Nat) :
+    (h : m ≤ n) →
+    for_all_fin_aux f m h → ∀ i, m ≤ i.val → f i
+    := by
+    generalize h: (n - m) = k
+    revert m
+    induction k -- TODO: induction h rather?
+    case zero =>
+      simp_all
+      intro m h1 h2
+      have h: n = m := by
+        linarith
+      unfold for_all_fin_aux; simp_all
+      simp_all
+      -- There is no i s.t. m ≤ i
+      intro i h3; cases i; simp_all
+      linarith
+    case succ k hi =>
+      simp_all
+      intro m hk hmn
+      intro hf i hmi
+      have hne: m ≠ n := by
+        have hineq := Nat.lt_of_sub_eq_succ hk
+        linarith
+      -- m = i?
+      if heq: m = i then
+        -- Yes: simply use the `for_all_fin_aux` hyp
+        unfold for_all_fin_aux at hf
+        simp_all
+        tauto
+      else
+        -- No: use the induction hypothesis
+        have hlt: m < i := by simp_all [Nat.lt_iff_le_and_ne]
+        have hineq: m + 1 ≤ n := by
+          have hineq := Nat.lt_of_sub_eq_succ hk
+          simp [*, Nat.add_one_le_iff]
+        have heq1: n - (m + 1) = k := by
+          -- TODO: very annoying arithmetic proof
+          simp [Nat.sub_eq_iff_eq_add hineq]
+          have hineq1: m ≤ n := by linarith
+          simp [Nat.sub_eq_iff_eq_add hineq1] at hk
+          simp_arith [hk]
+        have hi := hi (m + 1) heq1 hineq
+        apply hi <;> simp_all
+        . unfold for_all_fin_aux at hf
+          simp_all
+        . simp_all [Arith.add_one_le_iff_le_ne]
+
+  -- TODO: this is not necessary anymore
+  theorem for_all_fin_imp_forall (n : Nat) (f : Fin n → Prop) :
+    for_all_fin f → ∀ i, f i
+    := by
+    intro Hf i
+    apply for_all_fin_aux_imp_forall <;> try assumption
+    simp
+
+  /- Automating the proofs -/
+  @[simp] theorem is_valid_p_same
+    (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (x : Result c) :
+    is_valid_p k (λ _ => x) := by
+    simp [is_valid_p, k_to_gen, e_to_gen]
+
+  @[simp] theorem is_valid_p_rec
+     (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)) (i : id) (x : a i) :
+    is_valid_p k (λ k => k i x) := by
+    simp [is_valid_p, k_to_gen, e_to_gen, kk_to_gen, kk_of_gen]
+
+  theorem is_valid_p_ite
+    (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 e2 : ((i:id) → (x:a i) → Result (b i x)) → Result c}
+    (he1: is_valid_p k e1) (he2 : is_valid_p k e2) :
+    is_valid_p k (λ k => ite cond (e1 k) (e2 k)) := by
+    split <;> assumption
+
+  theorem is_valid_p_dite
+    (k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x))
+    (cond : Prop) [h : Decidable cond]
+    {e1 : ((i:id) → (x:a i) → Result (b i x)) → cond → Result c}
+    {e2 : ((i:id) → (x:a i) → Result (b i x)) → Not cond → Result c}
+    (he1: ∀ x, is_valid_p k (λ k => e1 k x))
+    (he2 : ∀ x, is_valid_p k (λ k => e2 k x)) :
+    is_valid_p k (λ k => dite cond (e1 k) (e2 k)) := by
+    split <;> simp [*]
+
+  theorem is_valid_p_bind
+    {{k : ((i:id) → (x:a i) → Result (b i x)) → (i:id) → (x:a i) → Result (b i x)}}
+    {{g : ((i:id) → (x:a i) → Result (b i x)) → Result c}}
+    {{h : c → ((i:id) → (x:a i) → Result (b i x)) → Result d}}
+    (Hgvalid : is_valid_p k g)
+    (Hhvalid : ∀ y, is_valid_p k (h y)) :
+    is_valid_p k (λ k => do let y ← g k; h y k) := by
+    apply Fix.is_valid_p_bind
+    . apply Hgvalid
+    . apply Hhvalid
+
+  def Funs.is_valid_p
+    (k : k_ty id a b)
+    (fl : Funs id a b tys) :
+    Prop :=
+    match fl with
+    | .Nil => True
+    | .Cons f fl => (∀ x, FixI.is_valid_p k (λ k => f k x)) ∧ fl.is_valid_p k
+
+  theorem Funs.is_valid_p_Nil (k : k_ty id a b) :
+    Funs.is_valid_p k Funs.Nil := by simp [Funs.is_valid_p]
+
+  def Funs.is_valid_p_is_valid_p_aux
+    {k : k_ty id a b}
+    {tys : List in_out_ty}
+    (fl : Funs id a b tys) (Hvalid : is_valid_p k fl) :
+    ∀ (i : Fin tys.length) (x : (tys.get i).fst), FixI.is_valid_p k (fun k => get_fun fl i k x) := by
+    -- Prepare the induction
+    have ⟨ n, Hn ⟩ : { n : Nat // tys.length = n } := ⟨ tys.length, by rfl ⟩
+    revert tys fl Hvalid
+    induction n
+    --
+    case zero =>
+      intro tys fl Hvalid Hlen;
+      have Heq: tys = [] := by cases tys <;> simp_all
+      intro i x
+      simp_all
+      have Hi := i.isLt
+      simp_all
+    case succ n Hn =>
+      intro tys fl Hvalid Hlen i x;
+      cases fl <;> simp at Hlen i x Hvalid
+      rename_i ity oty tys f fl
+      have ⟨ Hvf, Hvalid ⟩ := Hvalid
+      have Hvf1: is_valid_p k fl := by
+        simp [Hvalid, Funs.is_valid_p]
+      have Hn := @Hn tys fl Hvf1 (by simp [*])
+      -- Case disjunction on i
+      match i with
+      | ⟨ 0, _ ⟩ =>
+        simp at x
+        simp [get_fun]
+        apply (Hvf x)
+      | ⟨ .succ j, HiLt ⟩ =>
+        simp_arith at HiLt
+        simp at x
+        let j : Fin (List.length tys) := ⟨ j, by simp_arith [HiLt] ⟩
+        have Hn := Hn j x
+        apply Hn
+
+  def Funs.is_valid_p_is_valid_p
+    (tys : List in_out_ty)
+    (k : k_ty (Fin (List.length tys)) (λ i => (tys.get i).fst) (fun i x => (List.get tys i).snd x))
+    (fl : Funs (Fin tys.length) (λ i => (tys.get i).fst) (λ i x => (tys.get i).snd x) tys) :
+    fl.is_valid_p k  →
+    ∀ (i : Fin tys.length) (x : (tys.get i).fst),
+      FixI.is_valid_p k (fun k => get_fun fl i k x)
+    := by
+    intro Hvalid
+    apply is_valid_p_is_valid_p_aux; simp [*]
+
+end FixI
+
+namespace Ex1
+  /- An example of use of the fixed-point -/
+  open Primitives Fix
+
+  variable {a : Type} (k : (List a × Int) → Result a)
+
+  def list_nth_body (x : (List a × Int)) : Result a :=
+    let (ls, i) := x
+    match ls with
+    | [] => .fail .panic
+    | hd :: tl =>
+      if i = 0 then .ret hd
+      else k (tl, i - 1)
+
+  theorem list_nth_body_is_valid: ∀ k x, is_valid_p k (λ k => @list_nth_body a k x) := by
+    intro k x
+    simp [list_nth_body]
+    split <;> simp
+    split <;> simp
+
+  def list_nth (ls : List a) (i : Int) : Result a := fix list_nth_body (ls, i)
+
+  -- The unfolding equation - diverges if `i < 0`
+  theorem list_nth_eq (ls : List a) (i : Int) :
+    list_nth ls i =
+      match ls with
+      | [] => .fail .panic
+      | hd :: tl =>
+        if i = 0 then .ret hd
+        else list_nth tl (i - 1)
+    := by
+    have Heq := is_valid_fix_fixed_eq (@list_nth_body_is_valid a)
+    simp [list_nth]
+    conv => lhs; rw [Heq]
+
+end Ex1
+
+namespace Ex2
+  /- Same as Ex1, but we make the body of nth non tail-rec (this is mostly
+     to see what happens when there are let-bindings) -/
+  open Primitives Fix
+
+  variable {a : Type} (k : (List a × Int) → Result a)
+
+  def list_nth_body (x : (List a × Int)) : Result a :=
+    let (ls, i) := x
+    match ls with
+    | [] => .fail .panic
+    | hd :: tl =>
+      if i = 0 then .ret hd
+      else
+        do
+          let y ← k (tl, i - 1)
+          .ret y
+
+  theorem list_nth_body_is_valid: ∀ k x, is_valid_p k (λ k => @list_nth_body a k x) := by
+    intro k x
+    simp [list_nth_body]
+    split <;> simp
+    split <;> simp
+    apply is_valid_p_bind <;> intros <;> simp_all
+
+  def list_nth (ls : List a) (i : Int) : Result a := fix list_nth_body (ls, i)
+
+  -- The unfolding equation - diverges if `i < 0`
+  theorem list_nth_eq (ls : List a) (i : Int) :
+    (list_nth ls i =
+       match ls with
+       | [] => .fail .panic
+       | hd :: tl =>
+         if i = 0 then .ret hd
+         else
+           do
+             let y ← list_nth tl (i - 1)
+             .ret y)
+    := by
+    have Heq := is_valid_fix_fixed_eq (@list_nth_body_is_valid a)
+    simp [list_nth]
+    conv => lhs; rw [Heq]
+
+end Ex2
+
+namespace Ex3
+  /- Mutually recursive functions - first encoding (see Ex4 for a better encoding) -/
+  open Primitives Fix
+
+  /- Because we have mutually recursive functions, we use a sum for the inputs
+     and the output types:
+     - inputs: the sum allows to select the function to call in the recursive
+       calls (and the functions may not have the same input types)
+     - outputs: this case is degenerate because `even` and `odd` have the same
+       return type `Bool`, but generally speaking we need a sum type because
+       the functions in the mutually recursive group may have different
+       return types.
+   -/
+  variable (k : (Int ⊕ Int) → Result (Bool ⊕ Bool))
+
+  def is_even_is_odd_body (x : (Int ⊕ Int)) : Result (Bool ⊕ Bool) :=
+    match x with
+    | .inl i =>
+      -- Body of `is_even`
+      if i = 0
+      then .ret (.inl true) -- We use .inl because this is `is_even`
+      else
+        do
+          let b ←
+            do
+              -- Call `odd`: we need to wrap the input value in `.inr`, then
+              -- extract the output value
+              let r ← k (.inr (i- 1))
+              match r with
+              | .inl _ => .fail .panic -- Invalid output
+              | .inr b => .ret b
+          -- Wrap the return value
+          .ret (.inl b)
+    | .inr i =>
+      -- Body of `is_odd`
+      if i = 0
+      then .ret (.inr false) -- We use .inr because this is `is_odd`
+      else
+        do
+          let b ←
+            do
+              -- Call `is_even`: we need to wrap the input value in .inr, then
+              -- extract the output value
+              let r ← k (.inl (i- 1))
+              match r with
+              | .inl b => .ret b
+              | .inr _ => .fail .panic -- Invalid output
+          -- Wrap the return value
+          .ret (.inr b)
+
+  theorem is_even_is_odd_body_is_valid:
+    ∀ k x, is_valid_p k (λ k => is_even_is_odd_body k x) := by
+    intro k x
+    simp [is_even_is_odd_body]
+    split <;> simp <;> split <;> simp
+    apply is_valid_p_bind; simp
+    intros; split <;> simp
+    apply is_valid_p_bind; simp
+    intros; split <;> simp
+
+  def is_even (i : Int): Result Bool :=
+    do
+      let r ← fix is_even_is_odd_body (.inl i)
+      match r with
+      | .inl b => .ret b
+      | .inr _ => .fail .panic
+
+  def is_odd (i : Int): Result Bool :=
+    do
+      let r ← fix is_even_is_odd_body (.inr i)
+      match r with
+      | .inl _ => .fail .panic
+      | .inr b => .ret b
+
+  -- The unfolding equation for `is_even` - diverges if `i < 0`
+  theorem is_even_eq (i : Int) :
+    is_even i = (if i = 0 then .ret true else is_odd (i - 1))
+    := by
+    have Heq := is_valid_fix_fixed_eq is_even_is_odd_body_is_valid
+    simp [is_even, is_odd]
+    conv => lhs; rw [Heq]; simp; rw [is_even_is_odd_body]; simp
+    -- Very annoying: we need to swap the matches
+    -- Doing this with rewriting lemmas is hard generally speaking
+    -- (especially as we may have to generate lemmas for user-defined
+    -- inductives on the fly).
+    -- The simplest is to repeatedly split then simplify (we identify
+    -- the outer match or monadic let-binding, and split on its scrutinee)
+    split <;> simp
+    cases H0 : fix is_even_is_odd_body (Sum.inr (i - 1)) <;> simp
+    rename_i v
+    split <;> simp
+
+  -- The unfolding equation for `is_odd` - diverges if `i < 0`
+  theorem is_odd_eq (i : Int) :
+    is_odd i = (if i = 0 then .ret false else is_even (i - 1))
+    := by
+    have Heq := is_valid_fix_fixed_eq is_even_is_odd_body_is_valid
+    simp [is_even, is_odd]
+    conv => lhs; rw [Heq]; simp; rw [is_even_is_odd_body]; simp
+    -- Same remark as for `even`
+    split <;> simp
+    cases H0 : fix is_even_is_odd_body (Sum.inl (i - 1)) <;> simp
+    rename_i v
+    split <;> simp
+
+end Ex3
+
+namespace Ex4
+  /- Mutually recursive functions - 2nd encoding -/
+  open Primitives FixI
+
+  /- We make the input type and output types dependent on a parameter -/
+  @[simp] def tys : List in_out_ty := [mk_in_out_ty Int (λ _ => Bool), mk_in_out_ty Int (λ _ => Bool)]
+  @[simp] def input_ty (i : Fin 2) : Type := (tys.get i).fst
+  @[simp] def output_ty (i : Fin 2) (x : input_ty i) : Type :=
+    (tys.get i).snd x
+
+  /- The bodies are more natural -/
+  def is_even_body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i : Int) : Result Bool :=
+    if i = 0
+      then .ret true
+    else do
+      let b ← k 1 (i - 1)
+      .ret b
+
+  def is_odd_body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i : Int) : Result Bool :=
+    if i = 0
+      then .ret false
+    else do
+      let b ← k 0 (i - 1)
+      .ret b
+
+  @[simp] def bodies :
+    Funs (Fin 2) input_ty output_ty
+      [mk_in_out_ty Int (λ _ => Bool), mk_in_out_ty Int (λ _ => Bool)] :=
+    Funs.Cons (is_even_body) (Funs.Cons (is_odd_body) Funs.Nil)
+
+  def body (k : (i : Fin 2) → (x : input_ty i) → Result (output_ty i x)) (i: Fin 2) :
+    (x : input_ty i) → Result (output_ty i x) := get_fun bodies i k
+
+  theorem body_is_valid : is_valid body := by
+    -- Split the proof into proofs of validity of the individual bodies
+    rw [is_valid]
+    simp only [body]
+    intro k
+    apply (Funs.is_valid_p_is_valid_p tys)
+    simp [Funs.is_valid_p]
+    (repeat (apply And.intro)) <;> intro x <;> simp at x <;>
+    simp only [is_even_body, is_odd_body]
+    -- Prove the validity of the individual bodies
+    . split <;> simp
+      apply is_valid_p_bind <;> simp
+    . split <;> simp
+      apply is_valid_p_bind <;> simp
+
+  theorem body_fix_eq : fix body = body (fix body) :=
+    is_valid_fix_fixed_eq body_is_valid
+
+  def is_even (i : Int) : Result Bool := fix body 0 i
+  def is_odd (i : Int) : Result Bool := fix body 1 i
+
+  theorem is_even_eq (i : Int) : is_even i =
+    (if i = 0
+       then .ret true
+     else do
+       let b ← is_odd (i - 1)
+       .ret b) := by
+    simp [is_even, is_odd];
+    conv => lhs; rw [body_fix_eq]
+
+  theorem is_odd_eq (i : Int) : is_odd i =
+    (if i = 0
+       then .ret false
+     else do
+       let b ← is_even (i - 1)
+       .ret b) := by
+    simp [is_even, is_odd];
+    conv => lhs; rw [body_fix_eq]
+end Ex4
+
+namespace Ex5
+  /- Higher-order example -/
+  open Primitives Fix
+
+  variable {a b : Type}
+
+  /- An auxiliary function, which doesn't require the fixed-point -/
+  def map (f : a → Result b) (ls : List a) : Result (List b) :=
+    match ls with
+    | [] => .ret []
+    | hd :: tl =>
+      do
+        let hd ← f hd
+        let tl ← map f tl
+        .ret (hd :: tl)
+
+  /- The validity theorem for `map`, generic in `f` -/
+  theorem map_is_valid
+    {{f : (a → Result b) → a → Result c}}
+    (Hfvalid : ∀ k x, is_valid_p k (λ k => f k x))
+    (k : (a → Result b) → a → Result b)
+    (ls : List a) :
+    is_valid_p k (λ k => map (f k) ls) := by
+    induction ls <;> simp [map]
+    apply is_valid_p_bind <;> simp_all
+    intros
+    apply is_valid_p_bind <;> simp_all
+
+  /- An example which uses map -/
+  inductive Tree (a : Type) :=
+  | leaf (x : a)
+  | node (tl : List (Tree a))
+
+  def id_body (k : Tree a → Result (Tree a)) (t : Tree a) : Result (Tree a) :=
+    match t with
+    | .leaf x => .ret (.leaf x)
+    | .node tl =>
+      do
+        let tl ← map k tl
+        .ret (.node tl)
+
+  theorem id_body_is_valid :
+    ∀ k x, is_valid_p k (λ k => @id_body a k x) := by
+    intro k x
+    simp only [id_body]
+    split <;> simp
+    apply is_valid_p_bind <;> simp [*]
+    -- We have to show that `map k tl` is valid
+    apply map_is_valid;
+    -- Remark: if we don't do the intro, then the last step is expensive:
+    -- "typeclass inference of Nonempty took 119ms"
+    intro k x
+    simp only [is_valid_p_same, is_valid_p_rec]
+
+  def id (t : Tree a) := fix id_body t
+
+  -- The unfolding equation
+  theorem id_eq (t : Tree a) :
+    (id t =
+       match t with
+       | .leaf x => .ret (.leaf x)
+       | .node tl =>
+       do
+         let tl ← map id tl
+         .ret (.node tl))
+    := by
+    have Heq := is_valid_fix_fixed_eq (@id_body_is_valid a)
+    simp [id]
+    conv => lhs; rw [Heq]; simp; rw [id_body]
+
+end Ex5
+
+namespace Ex6
+  /- `list_nth` again, but this time we use FixI -/
+  open Primitives FixI
+
+  @[simp] def tys.{u} : List in_out_ty :=
+    [mk_in_out_ty ((a:Type u) × (List a × Int)) (λ ⟨ a, _ ⟩ => a)]
+
+  @[simp] def input_ty (i : Fin 1) := (tys.get i).fst
+  @[simp] def output_ty (i : Fin 1) (x : input_ty i) :=
+    (tys.get i).snd x
+
+  def list_nth_body.{u} (k : (i:Fin 1) → (x:input_ty i) → Result (output_ty i x))
+    (x : (a : Type u) × List a × Int) : Result x.fst :=
+    let ⟨ a, ls, i ⟩ := x
+    match ls with
+    | [] => .fail .panic
+    | hd :: tl =>
+      if i = 0 then .ret hd
+      else k 0 ⟨ a, tl, i - 1 ⟩
+
+  @[simp] def bodies :
+    Funs (Fin 1) input_ty output_ty tys :=
+    Funs.Cons list_nth_body Funs.Nil
+
+  def body (k : (i : Fin 1) → (x : input_ty i) → Result (output_ty i x)) (i: Fin 1) :
+    (x : input_ty i) → Result (output_ty i x) := get_fun bodies i k
+
+  theorem body_is_valid: is_valid body := by
+    -- Split the proof into proofs of validity of the individual bodies
+    rw [is_valid]
+    simp only [body]
+    intro k
+    apply (Funs.is_valid_p_is_valid_p tys)
+    simp [Funs.is_valid_p]
+    (repeat (apply And.intro)); intro x; simp at x
+    simp only [list_nth_body]
+    -- Prove the validity of the individual bodies
+    intro k x
+    simp [list_nth_body]
+    split <;> simp
+    split <;> simp
+
+  -- Writing the proof terms explicitly
+  theorem list_nth_body_is_valid' (k : k_ty (Fin 1) input_ty output_ty)
+    (x : (a : Type u) × List a × Int) : is_valid_p k (fun k => list_nth_body k x) :=
+    let ⟨ a, ls, i ⟩ := x
+    match ls with
+    | [] => is_valid_p_same k (.fail .panic)
+    | hd :: tl =>
+      is_valid_p_ite k (Eq i 0) (is_valid_p_same k (.ret hd)) (is_valid_p_rec k 0 ⟨a, tl, i-1⟩)
+
+  theorem body_is_valid' : is_valid body :=
+    fun k =>
+    Funs.is_valid_p_is_valid_p tys k bodies
+      (And.intro (list_nth_body_is_valid' k) (Funs.is_valid_p_Nil k))
+
+  def list_nth {a: Type u} (ls : List a) (i : Int) : Result a :=
+    fix body 0 ⟨ a, ls , i ⟩
+
+  -- The unfolding equation - diverges if `i < 0`
+  theorem list_nth_eq (ls : List a) (i : Int) :
+    list_nth ls i =
+      match ls with
+      | [] => .fail .panic
+      | hd :: tl =>
+        if i = 0 then .ret hd
+        else list_nth tl (i - 1)
+    := by
+    have Heq := is_valid_fix_fixed_eq body_is_valid
+    simp [list_nth]
+    conv => lhs; rw [Heq]
+
+  -- Write the proof term explicitly: the generation of the proof term (without tactics)
+  -- is automatable, and the proof term is actually a lot simpler and smaller when we
+  -- don't use tactics.
+  theorem list_nth_eq'.{u} {a : Type u} (ls : List a) (i : Int) :
+    list_nth ls i =
+      match ls with
+      | [] => .fail .panic
+      | hd :: tl =>
+        if i = 0 then .ret hd
+        else list_nth tl (i - 1)
+    :=
+    -- Use the fixed-point equation
+    have Heq := is_valid_fix_fixed_eq body_is_valid.{u}
+    -- Add the index
+    have Heqi := congr_fun Heq 0
+    -- Add the input
+    have Heqix := congr_fun Heqi { fst := a, snd := (ls, i) }
+    -- Done
+    Heqix
+
+end Ex6
diff --git a/backends/lean/Base/Diverge/Elab.lean b/backends/lean/Base/Diverge/Elab.lean
new file mode 100644
index 00000000..f109e847
--- /dev/null
+++ b/backends/lean/Base/Diverge/Elab.lean
@@ -0,0 +1,1162 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Init.Data.List.Basic
+import Mathlib.Tactic.RunCmd
+import Base.Utils
+import Base.Diverge.Base
+import Base.Diverge.ElabBase
+
+namespace Diverge
+
+/- Automating the generation of the encoding and the proofs so as to use nice
+   syntactic sugar. -/
+
+syntax (name := divergentDef)
+  declModifiers "divergent" "def" declId ppIndent(optDeclSig) declVal : command
+
+open Lean Elab Term Meta Primitives Lean.Meta
+open Utils
+
+/- The following was copied from the `wfRecursion` function. -/
+
+open WF in
+
+def mkProd (x y : Expr) : MetaM Expr :=
+  mkAppM ``Prod.mk #[x, y]
+
+def mkInOutTy (x y : Expr) : MetaM Expr :=
+  mkAppM ``FixI.mk_in_out_ty #[x, y]
+
+-- Return the `a` in `Return a`
+def getResultTy (ty : Expr) : MetaM Expr :=
+  ty.withApp fun f args => do
+  if ¬ f.isConstOf ``Result ∨ args.size ≠ 1 then
+    throwError "Invalid argument to getResultTy: {ty}"
+  else
+    pure (args.get! 0)
+
+/- Deconstruct a sigma type.
+
+   For instance, deconstructs `(a : Type) × List a` into
+   `Type` and `λ a => List a`.
+ -/
+def getSigmaTypes (ty : Expr) : MetaM (Expr × Expr) := do
+  ty.withApp fun f args => do
+  if ¬ f.isConstOf ``Sigma ∨ args.size ≠ 2 then
+    throwError "Invalid argument to getSigmaTypes: {ty}"
+  else
+    pure (args.get! 0, args.get! 1)
+
+/- Generate a Sigma type from a list of *variables* (all the expressions
+   must be variables).
+
+   Example:
+   - xl = [(a:Type), (ls:List a), (i:Int)]
+
+   Generates:
+   `(a:Type) × (ls:List a) × (i:Int)`
+
+ -/
+def mkSigmasType (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] => do
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: []"
+    pure (Expr.const ``PUnit.unit [])
+  | [x] => do
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}]"
+    let ty ← Lean.Meta.inferType x
+    pure ty
+  | x :: xl => do
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]"
+    let alpha ← Lean.Meta.inferType x
+    let sty ← mkSigmasType xl
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: [{x}::{xl}]: alpha={alpha}, sty={sty}"
+    let beta ← mkLambdaFVars #[x] sty
+    trace[Diverge.def.sigmas] "mkSigmasOfTypes: ({alpha}) ({beta})"
+    mkAppOptM ``Sigma #[some alpha, some beta]
+
+/- Apply a lambda expression to some arguments, simplifying the lambdas -/
+def applyLambdaToArgs (e : Expr) (xs : Array Expr) : MetaM Expr := do
+  lambdaTelescopeN e xs.size fun vars body =>
+  -- Create the substitution
+  let s : HashMap FVarId Expr := HashMap.ofList (List.zip (vars.toList.map Expr.fvarId!) xs.toList)
+  -- Substitute in the body
+  pure (body.replace fun e =>
+    match e with
+    | Expr.fvar fvarId => match s.find? fvarId with
+      | none   => e
+      | some v => v
+    | _ => none)
+
+/- Group a list of expressions into a dependent tuple.
+
+   Example:
+   xl = [`a : Type`, `ls : List a`]
+   returns:
+   `⟨ (a:Type), (ls: List a) ⟩`
+
+   We need the type argument because as the elements in the tuple are
+   "concrete", we can't in all generality figure out the type of the tuple.
+
+   Example:
+   `⟨ True, 3 ⟩ : (x : Bool) × (if x then Int else Unit)`
+ -/
+def mkSigmasVal (ty : Expr) (xl : List Expr) : MetaM Expr :=
+  match xl with
+  | [] => do
+    trace[Diverge.def.sigmas] "mkSigmasVal: []"
+    pure (Expr.const ``PUnit.unit [])
+  | [x] => do
+    trace[Diverge.def.sigmas] "mkSigmasVal: [{x}]"
+    pure x
+  | fst :: xl => do
+    trace[Diverge.def.sigmas] "mkSigmasVal: [{fst}::{xl}]"
+    -- Deconstruct the type
+    let (alpha, beta) ← getSigmaTypes ty
+    -- Compute the "second" field
+    -- Specialize beta for fst
+    let nty ← applyLambdaToArgs beta #[fst]
+    -- Recursive call
+    let snd ← mkSigmasVal nty xl
+    -- Put everything together
+    trace[Diverge.def.sigmas] "mkSigmasVal:\n{alpha}\n{beta}\n{fst}\n{snd}"
+    mkAppOptM ``Sigma.mk #[some alpha, some beta, some fst, some snd]
+
+def mkAnonymous (s : String) (i : Nat) : Name :=
+  .num (.str .anonymous s) i
+
+/- Given a list of values `[x0:ty0, ..., xn:ty1]`, where every `xi` might use the previous
+   `xj` (j < i) and a value `out` which uses `x0`, ..., `xn`, generate the following
+   expression:
+   ```
+   fun x:((x0:ty0) × ... × (xn:tyn) => -- **Dependent** tuple
+   match x with
+   | (x0, ..., xn) => out
+   ```
+
+   The `index` parameter is used for naming purposes: we use it to numerotate the
+   bound variables that we introduce.
+
+   We use this function to currify functions (the function bodies given to the
+   fixed-point operator must be unary functions).
+
+   Example:
+   ========
+   - xl = `[a:Type, ls:List a, i:Int]`
+   - out = `a`
+   - index = 0
+
+   generates (getting rid of most of the syntactic sugar):
+   ```
+   λ scrut0 => match scrut0 with
+   | Sigma.mk x scrut1 =>
+     match scrut1 with
+     | Sigma.mk ls i =>
+       a
+   ```
+-/
+partial def mkSigmasMatch (xl : List Expr) (out : Expr) (index : Nat := 0) : MetaM Expr :=
+  match xl with
+  | [] => do
+    -- This would be unexpected
+    throwError "mkSigmasMatch: empyt list of input parameters"
+  | [x] => do
+    -- In the example given for the explanations: this is the inner match case
+    trace[Diverge.def.sigmas] "mkSigmasMatch: [{x}]"
+    mkLambdaFVars #[x] out
+  | fst :: xl => do
+    -- In the example given for the explanations: this is the outer match case
+    -- Remark: for the naming purposes, we use the same convention as for the
+    -- fields and parameters in `Sigma.casesOn` and `Sigma.mk` (looking at
+    -- those definitions might help)
+    --
+    -- We want to build the match expression:
+    -- ```
+    -- λ scrut =>
+    -- match scrut with
+    -- | Sigma.mk x ...  -- the hole is given by a recursive call on the tail
+    -- ```
+    trace[Diverge.def.sigmas] "mkSigmasMatch: [{fst}::{xl}]"
+    let alpha ← Lean.Meta.inferType fst
+    let snd_ty ← mkSigmasType xl
+    let beta ← mkLambdaFVars #[fst] snd_ty
+    let snd ← mkSigmasMatch xl out (index + 1)
+    let mk ← mkLambdaFVars #[fst] snd
+    -- Introduce the "scrut" variable
+    let scrut_ty ← mkSigmasType (fst :: xl)
+    withLocalDeclD (mkAnonymous "scrut" index) scrut_ty fun scrut => do
+    trace[Diverge.def.sigmas] "mkSigmasMatch: scrut: ({scrut}) : ({← inferType scrut})"
+    -- TODO: make the computation of the motive more efficient
+    let motive ← do
+      let out_ty ← inferType out
+      match out_ty  with
+      | .sort _ | .lit _ | .const .. =>
+        -- The type of the motive doesn't depend on the scrutinee
+        mkLambdaFVars #[scrut] out_ty
+      | _ =>
+        -- The type of the motive *may* depend on the scrutinee
+        -- TODO: make this more efficient (we could change the output type of
+        -- mkSigmasMatch
+        mkSigmasMatch (fst :: xl) out_ty
+    -- The final expression: putting everything together
+    trace[Diverge.def.sigmas] "mkSigmasMatch:\n  ({alpha})\n  ({beta})\n  ({motive})\n  ({scrut})\n  ({mk})"
+    let sm ← mkAppOptM ``Sigma.casesOn #[some alpha, some beta, some motive, some scrut, some mk]
+    -- Abstracting the "scrut" variable
+    let sm ← mkLambdaFVars #[scrut] sm
+    trace[Diverge.def.sigmas] "mkSigmasMatch: sm: {sm}"
+    pure sm
+
+/- Small tests for list_nth: give a model of what `mkSigmasMatch` should generate -/
+private def list_nth_out_ty_inner (a :Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) :=
+  @Sigma.casesOn (List a)
+                 (fun (_ls : List a) => Int)
+                 (fun (_scrut1:@Sigma (List a) (fun (_ls : List a) => Int)) => Type)
+                 scrut1
+                 (fun (_ls : List a) (_i : Int) => Primitives.Result a)
+
+private def list_nth_out_ty_outer (scrut0 : @Sigma (Type) (fun (a:Type) =>
+                      @Sigma (List a) (fun (_ls : List a) => Int))) :=
+  @Sigma.casesOn (Type)
+                 (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))
+                 (fun (_scrut0:@Sigma (Type) (fun (a:Type) => @Sigma (List a) (fun (_ls : List a) => Int))) => Type)
+                 scrut0
+                 (fun (a : Type) (scrut1: @Sigma (List a) (fun (_ls : List a) => Int)) =>
+                  list_nth_out_ty_inner a scrut1)
+/- -/
+
+-- Return the expression: `Fin n`
+-- TODO: use more
+def mkFin (n : Nat) : Expr :=
+  mkAppN (.const ``Fin []) #[.lit (.natVal n)]
+
+-- Return the expression: `i : Fin n`
+def mkFinVal (n i : Nat) : MetaM Expr := do
+  let n_lit : Expr := .lit (.natVal (n - 1))
+  let i_lit : Expr := .lit (.natVal i)
+  -- We could use `trySynthInstance`, but as we know the instance that we are
+  -- going to use, we can save the lookup
+  let ofNat ← mkAppOptM ``Fin.instOfNatFinHAddNatInstHAddInstAddNatOfNat #[n_lit, i_lit]
+  mkAppOptM ``OfNat.ofNat #[none, none, ofNat]
+
+/- Generate and declare as individual definitions the bodies for the individual funcions:
+   - replace the recursive calls with calls to the continutation `k`
+   - make those bodies take one single dependent tuple as input
+
+   We name the declarations: "[original_name].body".
+   We return the new declarations.
+ -/
+def mkDeclareUnaryBodies (grLvlParams : List Name) (kk_var : Expr)
+  (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+  MetaM (Array Expr) := do
+  let grSize := preDefs.size
+
+  -- Compute the map from name to (index × input type).
+  -- Remark: the continuation has an indexed type; we use the index (a finite number of
+  -- type `Fin`) to control which function we call at the recursive call site.
+  let nameToInfo : HashMap Name (Nat × Expr) :=
+    let bl := preDefs.mapIdx fun i d => (d.declName, (i.val, (inOutTys.get! i.val).fst))
+    HashMap.ofList bl.toList
+
+  trace[Diverge.def.genBody] "nameToId: {nameToInfo.toList}"
+
+  -- Auxiliary function to explore the function bodies and replace the
+  -- recursive calls
+  let visit_e (i : Nat) (e : Expr) : MetaM Expr := do
+    trace[Diverge.def.genBody] "visiting expression (dept: {i}): {e}"
+    let ne ← do
+      match e with
+      | .app .. => do
+        e.withApp fun f args => do
+          trace[Diverge.def.genBody] "this is an app: {f} {args}"
+          -- Check if this is a recursive call
+          if f.isConst then
+            let name := f.constName!
+            match nameToInfo.find? name with
+            | none => pure e
+            | some (id, in_ty) =>
+              trace[Diverge.def.genBody] "this is a recursive call"
+              -- This is a recursive call: replace it
+              -- Compute the index
+              let i ← mkFinVal grSize id
+              -- Put the arguments in one big dependent tuple
+              let args ← mkSigmasVal in_ty args.toList
+              mkAppM' kk_var #[i, args]
+          else
+            -- Not a recursive call: do nothing
+            pure e
+       | .const name _ =>
+         -- Sanity check: we eliminated all the recursive calls
+         if (nameToInfo.find? name).isSome then
+           throwError "mkUnaryBodies: a recursive call was not eliminated"
+         else pure e
+       | _ => pure e
+    trace[Diverge.def.genBody] "done with expression (depth: {i}): {e}"
+    pure ne
+
+  -- Explore the bodies
+  preDefs.mapM fun preDef => do
+    -- Replace the recursive calls
+    trace[Diverge.def.genBody] "About to replace recursive calls in {preDef.declName}"
+    let body ← mapVisit visit_e preDef.value
+    trace[Diverge.def.genBody] "Body after replacement of the recursive calls: {body}"
+
+    -- Currify the function by grouping the arguments into a dependent tuple
+    -- (over which we match to retrieve the individual arguments).
+    lambdaTelescope body fun args body => do
+    let body ← mkSigmasMatch args.toList body 0
+
+    -- Add the declaration
+    let value ← mkLambdaFVars #[kk_var] body
+    let name := preDef.declName.append "body"
+    let levelParams := grLvlParams
+    let decl := Declaration.defnDecl {
+      name := name
+      levelParams := levelParams
+      type := ← inferType value -- TODO: change the type
+      value := value
+      hints := ReducibilityHints.regular (getMaxHeight (← getEnv) value + 1)
+      safety := .safe
+      all := [name]
+    }
+    addDecl decl
+    trace[Diverge.def] "individual body of {preDef.declName}: {body}"
+    -- Return the constant
+    let body := Lean.mkConst name (levelParams.map .param)
+    -- let body ← mkAppM' body #[kk_var]
+    trace[Diverge.def] "individual body (after decl): {body}"
+    pure body
+
+-- Generate a unique function body from the bodies of the mutually recursive group,
+-- and add it as a declaration in the context.
+-- We return the list of bodies (of type `FixI.Funs ...`) and the mutually recursive body.
+def mkDeclareMutRecBody (grName : Name) (grLvlParams : List Name)
+  (kk_var i_var : Expr)
+  (in_ty out_ty : Expr) (inOutTys : List (Expr × Expr))
+  (bodies : Array Expr) : MetaM (Expr × Expr) := do
+  -- Generate the body
+  let grSize := bodies.size
+  let finTypeExpr := mkFin grSize
+  -- TODO: not very clean
+  let inOutTyType ← do
+    let (x, y) := inOutTys.get! 0
+    inferType (← mkInOutTy x y)
+  let rec mkFuns (inOutTys : List (Expr × Expr)) (bl : List Expr) : MetaM Expr :=
+    match inOutTys, bl with
+    | [], [] =>
+      mkAppOptM ``FixI.Funs.Nil #[finTypeExpr, in_ty, out_ty]
+    | (ity, oty) :: inOutTys, b :: bl => do
+      -- Retrieving ity and oty - this is not very clean
+      let inOutTysExpr ← mkListLit inOutTyType (← inOutTys.mapM (λ (x, y) => mkInOutTy x y))
+      let fl ← mkFuns inOutTys bl
+      mkAppOptM ``FixI.Funs.Cons #[finTypeExpr, in_ty, out_ty, ity, oty, inOutTysExpr, b, fl]
+    | _, _ => throwError "mkDeclareMutRecBody: `tys` and `bodies` don't have the same length"
+  let bodyFuns ← mkFuns inOutTys bodies.toList
+  -- Wrap in `get_fun`
+  let body ← mkAppM ``FixI.get_fun #[bodyFuns, i_var, kk_var]
+  -- Add the index `i` and the continuation `k` as a variables
+  let body ← mkLambdaFVars #[kk_var, i_var] body
+  trace[Diverge.def] "mkDeclareMutRecBody: body: {body}"
+  -- Add the declaration
+  let name := grName.append "mut_rec_body"
+  let levelParams := grLvlParams
+  let decl := Declaration.defnDecl {
+    name := name
+    levelParams := levelParams
+    type := ← inferType body
+    value := body
+    hints := ReducibilityHints.regular (getMaxHeight (← getEnv) body + 1)
+    safety := .safe
+    all := [name]
+  }
+  addDecl decl
+  -- Return the bodies and the constant
+  pure (bodyFuns, Lean.mkConst name (levelParams.map .param))
+
+def isCasesExpr (e : Expr) : MetaM Bool := do
+  let e := e.getAppFn
+  if e.isConst then
+    return isCasesOnRecursor (← getEnv) e.constName
+  else return false
+
+structure MatchInfo where
+  matcherName       : Name
+  matcherLevels     : Array Level
+  params            : Array Expr
+  motive            : Expr
+  scruts            : Array Expr
+  branchesNumParams : Array Nat
+  branches          : Array Expr
+
+instance : ToMessageData MatchInfo where
+  -- This is not a very clean formatting, but we don't need more
+  toMessageData := fun me => m!"\n- matcherName: {me.matcherName}\n- params: {me.params}\n- motive: {me.motive}\n- scruts: {me.scruts}\n- branchesNumParams: {me.branchesNumParams}\n- branches: {me.branches}"
+
+-- Small helper: prove that an expression which doesn't use the continuation `kk`
+-- is valid, and return the proof.
+def proveNoKExprIsValid (k_var : Expr) (e : Expr) : MetaM Expr := do
+  trace[Diverge.def.valid] "proveNoKExprIsValid: {e}"
+  let eIsValid ← mkAppM ``FixI.is_valid_p_same #[k_var, e]
+  trace[Diverge.def.valid] "proveNoKExprIsValid: result:\n{eIsValid}:\n{← inferType eIsValid}"
+  pure eIsValid
+
+mutual
+
+/- Prove that an expression is valid, and return the proof.
+
+   More precisely, if `e` is an expression which potentially uses the continution
+   `kk`, return an expression of type:
+   ```
+   is_valid_p k (λ kk => e)
+   ```
+ -/
+partial def proveExprIsValid (k_var kk_var : Expr) (e : Expr) : MetaM Expr := do
+  trace[Diverge.def.valid] "proveValid: {e}"
+  match e with
+  | .const _ _ => throwError "Unimplemented" -- Shouldn't get there?
+  | .bvar _
+  | .fvar _
+  | .lit _
+  | .mvar _
+  | .sort _ => throwError "Unreachable"
+  | .lam .. => throwError "Unimplemented"
+  | .forallE .. => throwError "Unreachable" -- Shouldn't get there
+  | .letE .. => do
+    -- Telescope all the let-bindings (remark: this also telescopes the lambdas)
+    lambdaLetTelescope e fun xs body => do
+    -- Note that we don't visit the bound values: there shouldn't be
+    -- recursive calls, lambda expressions, etc. inside
+    -- Prove that the body is valid
+    let isValid ← proveExprIsValid k_var kk_var body
+    -- Add the let-bindings around.
+    -- Rem.: the let-binding should be *inside* the `is_valid_p`, not outside,
+    -- but because it reduces in the end it doesn't matter. More precisely:
+    -- `P (let x := v in y)` and `let x := v in P y` reduce to the same expression.
+    mkLambdaFVars xs isValid (usedLetOnly := false)
+  | .mdata _ b => proveExprIsValid k_var kk_var b
+  | .proj _ _ _ =>
+    -- The projection shouldn't use the continuation
+    proveNoKExprIsValid k_var e
+  | .app .. =>
+    e.withApp fun f args => do
+      -- There are several cases: first, check if this is a match/if
+      -- Check if the expression is a (dependent) if then else.
+      -- We treat the if then else expressions differently from the other matches,
+      -- and have dedicated theorems for them.
+      let isIte := e.isIte
+      if isIte || e.isDIte then do
+        e.withApp fun f args => do
+        trace[Diverge.def.valid] "ite/dite: {f}:\n{args}"
+        if args.size ≠ 5 then
+           throwError "Wrong number of parameters for {f}: {args}"
+        let cond := args.get! 1
+        let dec := args.get! 2
+        -- Prove that the branches are valid
+        let br0 := args.get! 3
+        let br1 := args.get! 4
+        let proveBranchValid (br : Expr) : MetaM Expr :=
+          if isIte then proveExprIsValid k_var kk_var br
+          else do
+            -- There is a lambda
+            lambdaOne br fun x br => do
+            let brValid ← proveExprIsValid k_var kk_var br
+            mkLambdaFVars #[x] brValid
+        let br0Valid ← proveBranchValid br0
+        let br1Valid ← proveBranchValid br1
+        let const := if isIte then ``FixI.is_valid_p_ite else ``FixI.is_valid_p_dite
+        let eIsValid ← mkAppOptM const #[none, none, none, none, some k_var, some cond, some dec, none, none, some br0Valid, some br1Valid]
+        trace[Diverge.def.valid] "ite/dite: result:\n{eIsValid}:\n{← inferType eIsValid}"
+        pure eIsValid
+      -- Check if the expression is a match (this case is for when the elaborator
+      -- introduces auxiliary definitions to hide the match behind syntactic
+      -- sugar):
+      else if let some me := ← matchMatcherApp? e then do
+        trace[Diverge.def.valid]
+          "matcherApp:
+           - params: {me.params}
+           - motive: {me.motive}
+           - discrs: {me.discrs}
+           - altNumParams: {me.altNumParams}
+           - alts: {me.alts}
+           - remaining: {me.remaining}"
+        -- matchMatcherApp does all the work for us: we simply need to gather
+        -- the information and call the auxiliary helper `proveMatchIsValid`
+        if me.remaining.size ≠ 0 then
+          throwError "MatcherApp: non empty remaining array: {me.remaining}"
+        let me : MatchInfo := {
+          matcherName := me.matcherName
+          matcherLevels := me.matcherLevels
+          params := me.params
+          motive := me.motive
+          scruts := me.discrs
+          branchesNumParams := me.altNumParams
+          branches := me.alts
+        }
+        proveMatchIsValid k_var kk_var me
+      -- Check if the expression is a raw match (this case is for when the expression
+      -- is a direct call to the primitive `casesOn` function, without syntactic sugar).
+      -- We have to check this case because functions like `mkSigmasMatch`, which we
+      -- use to currify function bodies, introduce such raw matches.
+      else if ← isCasesExpr f then do
+        trace[Diverge.def.valid] "rawMatch: {e}"
+        -- Deconstruct the match, and call the auxiliary helper `proveMatchIsValid`.
+        --
+        -- The casesOn definition is always of the following shape:
+        -- - input parameters (implicit parameters)
+        -- - motive (implicit), -- the motive gives the return type of the match
+        -- - scrutinee (explicit)
+        -- - branches (explicit).
+        -- In particular, we notice that the scrutinee is the first *explicit*
+        -- parameter - this is how we spot it.
+        let matcherName := f.constName!
+        let matcherLevels := f.constLevels!.toArray
+        -- Find the first explicit parameter: this is the scrutinee
+        forallTelescope (← inferType f) fun xs _ => do
+        let rec findFirstExplicit (i : Nat) : MetaM Nat := do
+          if i ≥ xs.size then throwError "Unexpected: could not find an explicit parameter"
+          else
+            let x := xs.get! i
+            let xFVarId := x.fvarId!
+            let localDecl ← xFVarId.getDecl
+            match localDecl.binderInfo with
+            | .default => pure i
+            | _ => findFirstExplicit (i + 1)
+        let scrutIdx ← findFirstExplicit 0
+        -- Split the arguments
+        let params := args.extract 0 (scrutIdx - 1)
+        let motive := args.get! (scrutIdx - 1)
+        let scrut := args.get! scrutIdx
+        let branches := args.extract (scrutIdx + 1) args.size
+        -- Compute the number of parameters for the branches: for this we use
+        -- the type of the uninstantiated casesOn constant (we can't just
+        -- destruct the lambdas in the branch expressions because the result
+        -- of a match might be a lambda expression).
+        let branchesNumParams : Array Nat ← do
+          let env ← getEnv
+          let decl := env.constants.find! matcherName
+          let ty := decl.type
+          forallTelescope ty fun xs _ => do
+          let xs := xs.extract (scrutIdx + 1) xs.size
+          xs.mapM fun x => do
+          let xty ← inferType x
+          forallTelescope xty fun ys _ => do
+          pure ys.size
+        let me : MatchInfo := {
+          matcherName,
+          matcherLevels,
+          params,
+          motive,
+          scruts := #[scrut],
+          branchesNumParams,
+          branches,
+        }
+        proveMatchIsValid k_var kk_var me
+      -- Check if this is a monadic let-binding
+      else if f.isConstOf ``Bind.bind then do
+        trace[Diverge.def.valid] "bind:\n{args}"
+        -- We simply need to prove that the subexpressions are valid, and call
+        -- the appropriate lemma.
+        let x := args.get! 4
+        let y := args.get! 5
+        -- Prove that the subexpressions are valid
+        let xValid ← proveExprIsValid k_var kk_var x
+        trace[Diverge.def.valid] "bind: xValid:\n{xValid}:\n{← inferType xValid}"
+        let yValid ← do
+          -- This is a lambda expression
+          lambdaOne y fun x y => do
+          trace[Diverge.def.valid] "bind: y: {y}"
+          let yValid ← proveExprIsValid k_var kk_var y
+          trace[Diverge.def.valid] "bind: yValid (no forall): {yValid}"
+          trace[Diverge.def.valid] "bind: yValid: x: {x}"
+          let yValid ← mkLambdaFVars #[x] yValid
+          trace[Diverge.def.valid] "bind: yValid (forall): {yValid}: {← inferType yValid}"
+          pure yValid
+        -- Put everything together
+        trace[Diverge.def.valid] "bind:\n- xValid: {xValid}: {← inferType xValid}\n- yValid: {yValid}: {← inferType yValid}"
+        mkAppM ``FixI.is_valid_p_bind #[xValid, yValid]
+      -- Check if this is a recursive call, i.e., a call to the continuation `kk`
+      else if f.isFVarOf kk_var.fvarId! then do
+        trace[Diverge.def.valid] "rec: args: \n{args}"
+        if args.size ≠ 2 then throwError "Recursive call with invalid number of parameters: {args}"
+        let i_arg := args.get! 0
+        let x_arg := args.get! 1
+        let eIsValid ← mkAppM ``FixI.is_valid_p_rec #[k_var, i_arg, x_arg]
+        trace[Diverge.def.valid] "rec: result: \n{eIsValid}"
+        pure eIsValid
+      else do
+        -- Remaining case: normal application.
+        -- It shouldn't use the continuation.
+        proveNoKExprIsValid k_var e
+
+-- Prove that a match expression is valid.
+partial def proveMatchIsValid (k_var kk_var : Expr) (me : MatchInfo) : MetaM Expr := do
+  trace[Diverge.def.valid] "proveMatchIsValid: {me}"
+  -- Prove the validity of the branch expressions
+  let branchesValid:Array Expr ← me.branches.mapIdxM fun idx br => do
+    -- Go inside the lambdas - note that we have to be careful: some of the
+    -- binders might come from the match, and some of the binders might come
+    -- from the fact that the expression in the match is a lambda expression:
+    -- we use the branchesNumParams field for this reason
+    let numParams := me.branchesNumParams.get! idx
+    lambdaTelescopeN br numParams fun xs br => do
+    -- Prove that the branch expression is valid
+    let brValid ← proveExprIsValid k_var kk_var br
+    -- Reconstruct the lambda expression
+    mkLambdaFVars xs brValid
+  trace[Diverge.def.valid] "branchesValid:\n{branchesValid}"
+  -- Compute the motive, which has the following shape:
+  -- ```
+  -- λ scrut => is_valid_p k (λ k => match scrut with ...)
+  --                                 ^^^^^^^^^^^^^^^^^^^^
+  --                         this is the original match expression, with the
+  --                        the difference that the scrutinee(s) is a variable
+  -- ```
+  let validMotive : Expr ← do
+    -- The motive is a function of the scrutinees (i.e., a lambda expression):
+    -- introduce binders for the scrutinees
+    let declInfos := me.scruts.mapIdx fun idx scrut =>
+      let name : Name := mkAnonymous "scrut" idx
+      let ty := λ (_ : Array Expr) => inferType scrut
+      (name, ty)
+    withLocalDeclsD declInfos fun scrutVars => do
+    -- Create a match expression but where the scrutinees have been replaced
+    -- by variables
+    let params : Array (Option Expr) := me.params.map some
+    let motive : Option Expr := some me.motive
+    let scruts : Array (Option Expr) := scrutVars.map some
+    let branches : Array (Option Expr) := me.branches.map some
+    let args := params ++ [motive] ++ scruts ++ branches
+    let matchE ← mkAppOptM me.matcherName args
+    -- Wrap in the `is_valid_p` predicate
+    let matchE ← mkLambdaFVars #[kk_var] matchE
+    let validMotive ← mkAppM ``FixI.is_valid_p #[k_var, matchE]
+    -- Abstract away the scrutinee variables
+    mkLambdaFVars scrutVars validMotive
+  trace[Diverge.def.valid] "valid motive: {validMotive}"
+  -- Put together
+  let valid ← do
+    -- We let Lean infer the parameters
+    let params : Array (Option Expr) := me.params.map (λ _ => none)
+    let motive := some validMotive
+    let scruts := me.scruts.map some
+    let branches := branchesValid.map some
+    let args := params ++ [motive] ++ scruts ++ branches
+    mkAppOptM me.matcherName args
+  trace[Diverge.def.valid] "proveMatchIsValid:\n{valid}:\n{← inferType valid}"
+  pure valid
+
+end
+
+-- Prove that a single body (in the mutually recursive group) is valid.
+--
+-- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+-- we prove that `is_even.body` and `is_odd.body` are valid.
+partial def proveSingleBodyIsValid
+  (k_var : Expr) (preDef : PreDefinition) (bodyConst : Expr) :
+  MetaM Expr := do
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: bodyConst: {bodyConst}"
+  -- Lookup the definition (`bodyConst` is a const, we want to retrieve its
+  -- definition to dive inside)
+  let name := bodyConst.constName!
+  let env ← getEnv
+  let body := (env.constants.find! name).value!
+  trace[Diverge.def.valid] "body: {body}"
+  lambdaTelescope body fun xs body => do
+  assert! xs.size = 2
+  let kk_var := xs.get! 0
+  let x_var := xs.get! 1
+  -- State the type of the theorem to prove
+  let thmTy ← mkAppM ``FixI.is_valid_p
+    #[k_var, ← mkLambdaFVars #[kk_var] (← mkAppM' bodyConst #[kk_var, x_var])]
+  trace[Diverge.def.valid] "thmTy: {thmTy}"
+  -- Prove that the body is valid
+  let proof ← proveExprIsValid k_var kk_var body
+  let proof ← mkLambdaFVars #[k_var, x_var] proof
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: proof:\n{proof}:\n{← inferType proof}"
+  -- The target type (we don't have to do this: this is simply a sanity check,
+  -- and this allows a nicer debugging output)
+  let thmTy ← do
+    let body ← mkAppM' bodyConst #[kk_var, x_var]
+    let body ← mkLambdaFVars #[kk_var] body
+    let ty ← mkAppM ``FixI.is_valid_p #[k_var, body]
+    mkForallFVars #[k_var, x_var] ty
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: thmTy\n{thmTy}:\n{← inferType thmTy}"
+  -- Save the theorem
+  let name := preDef.declName ++ "body_is_valid"
+  let decl := Declaration.thmDecl {
+    name
+    levelParams := preDef.levelParams
+    type := thmTy
+    value := proof
+    all := [name]
+  }
+  addDecl decl
+  trace[Diverge.def.valid] "proveSingleBodyIsValid: added thm: {name}"
+  -- Return the theorem
+  pure (Expr.const name (preDef.levelParams.map .param))
+
+-- Prove that the list of bodies are valid.
+--
+-- For instance, if we define the mutually recursive group [`is_even`, `is_odd`],
+-- we prove that `Funs.Cons is_even.body (Funs.Cons is_odd.body Funs.Nil)` is
+-- valid.
+partial def proveFunsBodyIsValid (inOutTys: Expr) (bodyFuns : Expr)
+  (k_var : Expr) (bodiesValid : Array Expr) : MetaM Expr := do
+  -- Create the big "and" expression, which groups the validity proof of the individual bodies
+  let rec mkValidConj (i : Nat) : MetaM Expr := do
+    if i = bodiesValid.size then
+      -- We reached the end
+      mkAppM ``FixI.Funs.is_valid_p_Nil #[k_var]
+    else do
+      -- We haven't reached the end: introduce a conjunction
+      let valid := bodiesValid.get! i
+      let valid ← mkAppM' valid #[k_var]
+      mkAppM ``And.intro #[valid, ← mkValidConj (i + 1)]
+  let andExpr ← mkValidConj 0
+  -- Wrap in the `is_valid_p_is_valid_p` theorem, and abstract the continuation
+  let isValid ← mkAppM ``FixI.Funs.is_valid_p_is_valid_p #[inOutTys, k_var, bodyFuns, andExpr]
+  mkLambdaFVars #[k_var] isValid
+
+-- Prove that the mut rec body (i.e., the unary body which groups the bodies
+-- of all the functions in the mutually recursive group and on which we will
+-- apply the fixed-point operator) is valid.
+--
+-- We save the proof in the theorem "[GROUP_NAME]."mut_rec_body_is_valid",
+-- which we return.
+--
+-- TODO: maybe this function should introduce k_var itself
+def proveMutRecIsValid
+  (grName : Name) (grLvlParams : List Name)
+  (inOutTys : Expr) (bodyFuns mutRecBodyConst : Expr)
+  (k_var : Expr) (preDefs : Array PreDefinition)
+  (bodies : Array Expr) : MetaM Expr := do
+  -- First prove that the individual bodies are valid
+  let bodiesValid ←
+    bodies.mapIdxM fun idx body => do
+      let preDef := preDefs.get! idx
+      trace[Diverge.def.valid] "## Proving that the body {body} is valid"
+      proveSingleBodyIsValid k_var preDef body
+  -- Then prove that the mut rec body is valid
+  trace[Diverge.def.valid] "## Proving that the 'Funs' body is valid"
+  let isValid ← proveFunsBodyIsValid inOutTys bodyFuns k_var bodiesValid
+  -- Save the theorem
+  let thmTy ← mkAppM ``FixI.is_valid #[mutRecBodyConst]
+  let name := grName ++ "mut_rec_body_is_valid"
+  let decl := Declaration.thmDecl {
+    name
+    levelParams := grLvlParams
+    type := thmTy
+    value := isValid
+    all := [name]
+  }
+  addDecl decl
+  trace[Diverge.def.valid] "proveFunsBodyIsValid: added thm: {name}:\n{thmTy}"
+  -- Return the theorem
+  pure (Expr.const name (grLvlParams.map .param))
+
+-- Generate the final definions by using the mutual body and the fixed point operator.
+--
+-- For instance:
+-- ```
+-- def is_even (i : Int) : Result Bool := mut_rec_body 0 i
+-- def is_odd (i : Int) : Result Bool := mut_rec_body 1 i
+-- ```
+def mkDeclareFixDefs (mutRecBody : Expr) (inOutTys : Array (Expr × Expr)) (preDefs : Array PreDefinition) :
+  TermElabM (Array Name) := do
+  let grSize := preDefs.size
+  let defs ← preDefs.mapIdxM fun idx preDef => do
+    lambdaTelescope preDef.value fun xs _ => do
+    -- Retrieve the input type
+    let in_ty := (inOutTys.get! idx.val).fst
+    -- Create the index
+    let idx ← mkFinVal grSize idx.val
+    -- Group the inputs into a dependent tuple
+    let input ← mkSigmasVal in_ty xs.toList
+    -- Apply the fixed point
+    let fixedBody ← mkAppM ``FixI.fix #[mutRecBody, idx, input]
+    let fixedBody ← mkLambdaFVars xs fixedBody
+    -- Create the declaration
+    let name := preDef.declName
+    let decl := Declaration.defnDecl {
+      name := name
+      levelParams := preDef.levelParams
+      type := preDef.type
+      value := fixedBody
+      hints := ReducibilityHints.regular (getMaxHeight (← getEnv) fixedBody + 1)
+      safety := .safe
+      all := [name]
+    }
+    addDecl decl
+    pure name
+  pure defs
+
+-- Prove the equations that we will use as unfolding theorems
+partial def proveUnfoldingThms (isValidThm : Expr) (inOutTys : Array (Expr × Expr))
+  (preDefs : Array PreDefinition) (decls : Array Name) : MetaM Unit := do
+  let grSize := preDefs.size
+  let proveIdx (i : Nat) : MetaM Unit := do
+    let preDef := preDefs.get! i
+    let defName := decls.get! i
+    -- Retrieve the arguments
+    lambdaTelescope preDef.value fun xs body => do
+    trace[Diverge.def.unfold] "proveUnfoldingThms: xs: {xs}"
+    trace[Diverge.def.unfold] "proveUnfoldingThms: body: {body}"
+    -- The theorem statement
+    let thmTy ← do
+      -- The equation: the declaration gives the lhs, the pre-def gives the rhs
+      let lhs ← mkAppOptM defName (xs.map some)
+      let rhs := body
+      let eq ← mkAppM ``Eq #[lhs, rhs]
+      mkForallFVars xs eq
+    trace[Diverge.def.unfold] "proveUnfoldingThms: thm statement: {thmTy}"
+    -- The proof
+    -- Use the fixed-point equation
+    let proof ← mkAppM ``FixI.is_valid_fix_fixed_eq #[isValidThm]
+    -- Add the index
+    let idx ← mkFinVal grSize i
+    let proof ← mkAppM ``congr_fun #[proof, idx]
+    -- Add the input argument
+    let arg ← mkSigmasVal (inOutTys.get! i).fst xs.toList
+    let proof ← mkAppM ``congr_fun #[proof, arg]
+    -- Abstract the arguments away
+    let proof ← mkLambdaFVars xs proof
+    trace[Diverge.def.unfold] "proveUnfoldingThms: proof: {proof}:\n{← inferType proof}"
+    -- Declare the theorem
+    let name := preDef.declName ++ "unfold"
+    let decl := Declaration.thmDecl {
+      name
+      levelParams := preDef.levelParams
+      type := thmTy
+      value := proof
+      all := [name]
+    }
+    addDecl decl
+    -- Add the unfolding theorem to the equation compiler
+    eqnsAttribute.add preDef.declName #[name]
+    trace[Diverge.def.unfold] "proveUnfoldingThms: added thm: {name}:\n{thmTy}"
+  let rec prove (i : Nat) : MetaM Unit := do
+    if i = preDefs.size then pure ()
+    else do
+      proveIdx i
+      prove (i + 1)
+  --
+  prove 0
+
+def divRecursion (preDefs : Array PreDefinition) : TermElabM Unit := do
+  let msg := toMessageData <| preDefs.map fun pd => (pd.declName, pd.levelParams, pd.type, pd.value)
+  trace[Diverge.def] ("divRecursion: defs:\n" ++ msg)
+
+  -- TODO: what is this?
+  for preDef in preDefs do
+    applyAttributesOf #[preDef] AttributeApplicationTime.afterCompilation
+
+  -- Retrieve the name of the first definition, that we will use as the namespace
+  -- for the definitions common to the group
+  let def0 := preDefs[0]!
+  let grName := def0.declName
+  trace[Diverge.def] "group name: {grName}"
+
+  /- # Compute the input/output types of the continuation `k`. -/
+  let grLvlParams := def0.levelParams
+  trace[Diverge.def] "def0 universe levels: {def0.levelParams}"
+
+  -- We first compute the list of pairs: (input type × output type)
+  let inOutTys : Array (Expr × Expr) ←
+      preDefs.mapM (fun preDef => do
+        withRef preDef.ref do -- is the withRef useful?
+        -- Check the universe parameters - TODO: I'm not sure what the best thing
+        -- to do is. In practice, all the type parameters should be in Type 0, so
+        -- we shouldn't have universe issues.
+        if preDef.levelParams ≠ grLvlParams then
+          throwError "Non-uniform polymorphism in the universes"
+        forallTelescope preDef.type (fun in_tys out_ty => do
+          let in_ty ← liftM (mkSigmasType in_tys.toList)
+          -- Retrieve the type in the "Result"
+          let out_ty ← getResultTy out_ty
+          let out_ty ← liftM (mkSigmasMatch in_tys.toList out_ty)
+          pure (in_ty, out_ty)
+        )
+      )
+  trace[Diverge.def] "inOutTys: {inOutTys}"
+  -- Turn the list of input/output type pairs into an expresion
+  let inOutTysExpr ← inOutTys.mapM (λ (x, y) => mkInOutTy x y)
+  let inOutTysExpr ← mkListLit (← inferType (inOutTysExpr.get! 0)) inOutTysExpr.toList
+
+  -- From the list of pairs of input/output types, actually compute the
+  -- type of the continuation `k`.
+  -- We first introduce the index `i : Fin n` where `n` is the number of
+  -- functions in the group.
+  let i_var_ty := mkFin preDefs.size
+  withLocalDeclD (mkAnonymous "i" 0) i_var_ty fun i_var => do
+  let in_out_ty ← mkAppM ``List.get #[inOutTysExpr, i_var]
+  trace[Diverge.def] "in_out_ty := {in_out_ty} : {← inferType in_out_ty}"
+  -- Add an auxiliary definition for `in_out_ty`
+  let in_out_ty ← do
+    let value ← mkLambdaFVars #[i_var] in_out_ty
+    let name := grName.append "in_out_ty"
+    let levelParams := grLvlParams
+    let decl := Declaration.defnDecl {
+      name := name
+      levelParams := levelParams
+      type := ← inferType value
+      value := value
+      hints := .abbrev
+      safety := .safe
+      all := [name]
+    }
+    addDecl decl
+    -- Return the constant
+    let in_out_ty := Lean.mkConst name (levelParams.map .param)
+    mkAppM' in_out_ty #[i_var]
+  trace[Diverge.def] "in_out_ty (after decl) := {in_out_ty} : {← inferType in_out_ty}"
+  let in_ty ← mkAppM ``Sigma.fst #[in_out_ty]
+  trace[Diverge.def] "in_ty: {in_ty}"
+  withLocalDeclD (mkAnonymous "x" 1) in_ty fun input => do
+  let out_ty ← mkAppM' (← mkAppM ``Sigma.snd #[in_out_ty]) #[input]
+  trace[Diverge.def] "out_ty: {out_ty}"
+
+  -- Introduce the continuation `k`
+  let in_ty ← mkLambdaFVars #[i_var] in_ty
+  let out_ty ← mkLambdaFVars #[i_var, input] out_ty
+  let kk_var_ty ← mkAppM ``FixI.kk_ty #[i_var_ty, in_ty, out_ty]
+  trace[Diverge.def] "kk_var_ty: {kk_var_ty}"
+  withLocalDeclD (mkAnonymous "kk" 2) kk_var_ty fun kk_var => do
+  trace[Diverge.def] "kk_var: {kk_var}"
+
+  -- Replace the recursive calls in all the function bodies by calls to the
+  -- continuation `k` and and generate for those bodies declarations
+  trace[Diverge.def] "# Generating the unary bodies"
+  let bodies ← mkDeclareUnaryBodies grLvlParams kk_var inOutTys preDefs
+  trace[Diverge.def] "Unary bodies (after decl): {bodies}"
+  -- Generate the mutually recursive body
+  trace[Diverge.def] "# Generating  the mut rec body"
+  let (bodyFuns, mutRecBody) ← mkDeclareMutRecBody grName grLvlParams kk_var i_var in_ty out_ty inOutTys.toList bodies
+  trace[Diverge.def] "mut rec body (after decl): {mutRecBody}"
+
+  -- Prove that the mut rec body satisfies the validity criteria required by
+  -- our fixed-point
+  let k_var_ty ← mkAppM ``FixI.k_ty #[i_var_ty, in_ty, out_ty]
+  withLocalDeclD (mkAnonymous "k" 3) k_var_ty fun k_var => do
+  trace[Diverge.def] "# Proving that the mut rec body is valid"
+  let isValidThm ← proveMutRecIsValid grName grLvlParams inOutTysExpr bodyFuns mutRecBody k_var preDefs bodies
+
+  -- Generate the final definitions
+  trace[Diverge.def] "# Generating the final definitions"
+  let decls ← mkDeclareFixDefs mutRecBody inOutTys preDefs
+
+  -- Prove the unfolding theorems
+  trace[Diverge.def] "# Proving the unfolding theorems"
+  proveUnfoldingThms isValidThm inOutTys preDefs decls
+
+  -- Generating code -- TODO
+  addAndCompilePartialRec preDefs
+
+-- The following function is copy&pasted from Lean.Elab.PreDefinition.Main
+-- This is the only part where we make actual changes and hook into the equation compiler.
+-- (I've removed all the well-founded stuff to make it easier to read though.)
+
+open private ensureNoUnassignedMVarsAtPreDef betaReduceLetRecApps partitionPreDefs
+  addAndCompilePartial addAsAxioms from Lean.Elab.PreDefinition.Main
+
+def addPreDefinitions (preDefs : Array PreDefinition) : TermElabM Unit := withLCtx {} {} do
+  for preDef in preDefs do
+    trace[Diverge.elab] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
+  let preDefs ← preDefs.mapM ensureNoUnassignedMVarsAtPreDef
+  let preDefs ← betaReduceLetRecApps preDefs
+  let cliques := partitionPreDefs preDefs
+  let mut hasErrors := false
+  for preDefs in cliques do
+    trace[Diverge.elab] "{preDefs.map (·.declName)}"
+    try
+      withRef (preDefs[0]!.ref) do
+        divRecursion preDefs
+    catch ex =>
+      -- If it failed, we add the functions as partial functions
+      hasErrors := true
+      logException ex
+      let s ← saveState
+      try
+        if preDefs.all fun preDef => preDef.kind == DefKind.def ||
+           preDefs.all fun preDef => preDef.kind == DefKind.abbrev then
+          -- try to add as partial definition
+          try
+            addAndCompilePartial preDefs (useSorry := true)
+          catch _ =>
+            -- Compilation failed try again just as axiom
+            s.restore
+            addAsAxioms preDefs
+        else return ()
+      catch _ => s.restore
+
+-- The following two functions are copy-pasted from Lean.Elab.MutualDef
+
+open private elabHeaders levelMVarToParamHeaders getAllUserLevelNames withFunLocalDecls elabFunValues
+  instantiateMVarsAtHeader instantiateMVarsAtLetRecToLift checkLetRecsToLiftTypes withUsed from Lean.Elab.MutualDef
+
+def Term.elabMutualDef (vars : Array Expr) (views : Array DefView) : TermElabM Unit := do
+    let scopeLevelNames ← getLevelNames
+    let headers ← elabHeaders views
+    let headers ← levelMVarToParamHeaders views headers
+    let allUserLevelNames := getAllUserLevelNames headers
+    withFunLocalDecls headers fun funFVars => do
+      for view in views, funFVar in funFVars do
+        addLocalVarInfo view.declId funFVar
+        -- Add fake use site to prevent "unused variable" warning (if the
+        -- function is actually not recursive, Lean would print this warning).
+        -- Remark: we could detect this case and encode the function without
+        -- using the fixed-point. In practice it shouldn't happen however:
+        -- we define non-recursive functions with the `divergent` keyword
+        -- only for testing purposes.
+        addTermInfo' view.declId funFVar
+      let values ←
+        try
+          let values ← elabFunValues headers
+          Term.synthesizeSyntheticMVarsNoPostponing
+          values.mapM (instantiateMVars ·)
+        catch ex =>
+          logException ex
+          headers.mapM fun header => mkSorry header.type (synthetic := true)
+      let headers ← headers.mapM instantiateMVarsAtHeader
+      let letRecsToLift ← getLetRecsToLift
+      let letRecsToLift ← letRecsToLift.mapM instantiateMVarsAtLetRecToLift
+      checkLetRecsToLiftTypes funFVars letRecsToLift
+      withUsed vars headers values letRecsToLift fun vars => do
+        let preDefs ← MutualClosure.main vars headers funFVars values letRecsToLift
+        for preDef in preDefs do
+          trace[Diverge.elab] "{preDef.declName} : {preDef.type} :=\n{preDef.value}"
+        let preDefs ← withLevelNames allUserLevelNames <| levelMVarToParamPreDecls preDefs
+        let preDefs ← instantiateMVarsAtPreDecls preDefs
+        let preDefs ← fixLevelParams preDefs scopeLevelNames allUserLevelNames
+        for preDef in preDefs do
+          trace[Diverge.elab] "after eraseAuxDiscr, {preDef.declName} : {preDef.type} :=\n{preDef.value}"
+        checkForHiddenUnivLevels allUserLevelNames preDefs
+        addPreDefinitions preDefs
+
+open Command in
+def Command.elabMutualDef (ds : Array Syntax) : CommandElabM Unit := do
+  let views ← ds.mapM fun d => do
+    let `($mods:declModifiers divergent def $id:declId $sig:optDeclSig $val:declVal) := d
+      | throwUnsupportedSyntax
+    let modifiers ← elabModifiers mods
+    let (binders, type) := expandOptDeclSig sig
+    let deriving? := none
+    pure { ref := d, kind := DefKind.def, modifiers,
+           declId := id, binders, type? := type, value := val, deriving? }
+  runTermElabM fun vars => Term.elabMutualDef vars views
+
+-- Special command so that we don't fall back to the built-in mutual when we produce an error.
+local syntax "_divergent" Parser.Command.mutual : command
+elab_rules : command | `(_divergent mutual $decls* end) => Command.elabMutualDef decls
+
+macro_rules
+  | `(mutual $decls* end) => do
+    unless !decls.isEmpty && decls.all (·.1.getKind == ``divergentDef) do
+      Macro.throwUnsupported
+    `(command| _divergent mutual $decls* end)
+
+open private setDeclIdName from Lean.Elab.Declaration
+elab_rules : command
+  | `($mods:declModifiers divergent%$tk def $id:declId $sig:optDeclSig $val:declVal) => do
+    let (name, _) := expandDeclIdCore id
+    if (`_root_).isPrefixOf name then throwUnsupportedSyntax
+    let view := extractMacroScopes name
+    let .str ns shortName := view.name | throwUnsupportedSyntax
+    let shortName' := { view with name := shortName }.review
+    let cmd ← `(mutual $mods:declModifiers divergent%$tk def $(⟨setDeclIdName id shortName'⟩):declId $sig:optDeclSig $val:declVal end)
+    if ns matches .anonymous then
+      Command.elabCommand cmd
+    else
+      Command.elabCommand <| ← `(namespace $(mkIdentFrom id ns) $cmd end $(mkIdentFrom id ns))
+
+namespace Tests
+  /- Some examples of partial functions -/
+
+  divergent def list_nth {a: Type} (ls : List a) (i : Int) : Result a :=
+    match ls with
+    | [] => .fail .panic
+    | x :: ls =>
+      if i = 0 then return x
+      else return (← list_nth ls (i - 1))
+
+  #check list_nth.unfold
+
+  example {a: Type} (ls : List a) :
+    ∀ (i : Int),
+    0 ≤ i → i < ls.length →
+    ∃ x, list_nth ls i = .ret x := by
+    induction ls
+    . intro i hpos h; simp at h; linarith
+    . rename_i hd tl ih
+      intro i hpos h
+      -- We can directly use `rw [list_nth]`!
+      rw [list_nth]; simp
+      split <;> simp [*]
+      . tauto
+      . -- TODO: we shouldn't have to do that
+        have hneq : 0 < i := by cases i <;> rename_i a _ <;> simp_all; cases a <;> simp_all
+        simp at h
+        have ⟨ x, ih ⟩ := ih (i - 1) (by linarith) (by linarith)
+        simp [ih]
+        tauto
+
+  mutual
+    divergent def is_even (i : Int) : Result Bool :=
+      if i = 0 then return true else return (← is_odd (i - 1))
+
+    divergent def is_odd (i : Int) : Result Bool :=
+      if i = 0 then return false else return (← is_even (i - 1))
+  end
+
+  #check is_even.unfold
+  #check is_odd.unfold
+
+  mutual
+    divergent def foo (i : Int) : Result Nat :=
+      if i > 10 then return (← foo (i / 10)) + (← bar i) else bar 10
+
+    divergent def bar (i : Int) : Result Nat :=
+      if i > 20 then foo (i / 20) else .ret 42
+  end
+
+  #check foo.unfold
+  #check bar.unfold
+
+  -- Testing dependent branching and let-bindings
+  -- TODO: why the linter warning?
+  divergent def isNonZero (i : Int) : Result Bool :=
+    if _h:i = 0 then return false
+    else
+      let b := true
+      return b
+
+  #check isNonZero.unfold
+
+  -- Testing let-bindings
+  divergent def iInBounds {a : Type} (ls : List a) (i : Int) : Result Bool :=
+    let i0 := ls.length
+    if i < i0
+    then Result.ret True
+    else Result.ret False
+
+  #check iInBounds.unfold
+
+  divergent def isCons
+    {a : Type} (ls : List a) : Result Bool :=
+    let ls1 := ls
+    match ls1 with
+    | [] => Result.ret False
+    | _ :: _ => Result.ret True
+
+  #check isCons.unfold
+
+  -- Testing what happens when we use concrete arguments in dependent tuples
+  divergent def test1
+    (_ : Option Bool) (_ : Unit) :
+    Result Unit
+    :=
+    test1 Option.none ()
+
+  #check test1.unfold
+
+end Tests
+
+end Diverge
diff --git a/backends/lean/Base/Diverge/ElabBase.lean b/backends/lean/Base/Diverge/ElabBase.lean
new file mode 100644
index 00000000..fedb1c74
--- /dev/null
+++ b/backends/lean/Base/Diverge/ElabBase.lean
@@ -0,0 +1,15 @@
+import Lean
+
+namespace Diverge
+
+open Lean Elab Term Meta
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Diverge.elab
+initialize registerTraceClass `Diverge.def
+initialize registerTraceClass `Diverge.def.sigmas
+initialize registerTraceClass `Diverge.def.genBody
+initialize registerTraceClass `Diverge.def.valid
+initialize registerTraceClass `Diverge.def.unfold
+
+end Diverge
diff --git a/backends/lean/Base/IList.lean b/backends/lean/Base/IList.lean
new file mode 100644
index 00000000..31b66ffa
--- /dev/null
+++ b/backends/lean/Base/IList.lean
@@ -0,0 +1 @@
+import Base.IList.IList
diff --git a/backends/lean/Base/IList/IList.lean b/backends/lean/Base/IList/IList.lean
new file mode 100644
index 00000000..93047a1b
--- /dev/null
+++ b/backends/lean/Base/IList/IList.lean
@@ -0,0 +1,284 @@
+/- Complementary list functions and lemmas which operate on integers rather
+   than natural numbers. -/
+
+import Std.Data.Int.Lemmas
+import Base.Arith
+
+namespace List
+
+def len (ls : List α) : Int :=
+  match ls with
+  | [] => 0
+  | _ :: tl => 1 + len tl
+
+@[simp] theorem len_nil : len ([] : List α) = 0 := by simp [len]
+@[simp] theorem len_cons : len ((x :: tl) : List α) = 1 + len tl := by simp [len]
+
+theorem len_pos : 0 ≤ (ls : List α).len := by
+  induction ls <;> simp [*]
+  linarith
+
+instance (a : Type u) : Arith.HasIntProp (List a) where
+  prop_ty := λ ls => 0 ≤ ls.len
+  prop := λ ls => ls.len_pos
+
+-- Remark: if i < 0, then the result is none
+def indexOpt (ls : List α) (i : Int) : Option α :=
+  match ls with
+  | [] => none
+  | hd :: tl => if i = 0 then some hd else indexOpt tl (i - 1)
+
+@[simp] theorem indexOpt_nil : indexOpt ([] : List α) i = none := by simp [indexOpt]
+@[simp] theorem indexOpt_zero_cons : indexOpt ((x :: tl) : List α) 0 = some x := by simp [indexOpt]
+@[simp] theorem indexOpt_nzero_cons (hne : i ≠ 0) : indexOpt ((x :: tl) : List α) i = indexOpt tl (i - 1) := by simp [*, indexOpt]
+
+-- Remark: if i < 0, then the result is the defaul element
+def index [Inhabited α] (ls : List α) (i : Int) : α :=
+  match ls with
+  | [] => Inhabited.default
+  | x :: tl =>
+    if i = 0 then x else index tl (i - 1)
+
+@[simp] theorem index_zero_cons [Inhabited α] : index ((x :: tl) : List α) 0 = x := by simp [index]
+@[simp] theorem index_nzero_cons [Inhabited α] (hne : i ≠ 0) : index ((x :: tl) : List α) i = index tl (i - 1) := by simp [*, index]
+
+theorem indexOpt_bounds (ls : List α) (i : Int) :
+  ls.indexOpt i = none ↔ i < 0 ∨ ls.len ≤ i :=
+  match ls with
+  | [] =>
+    have : ¬ (i < 0) → 0 ≤ i := by int_tac
+    by simp; tauto
+  | _ :: tl =>
+    have := indexOpt_bounds tl (i - 1)
+    if h: i = 0 then
+      by
+        simp [*];
+        int_tac
+    else by
+      simp [*]
+      constructor <;> intros <;>
+      casesm* _ ∨ _ <;> -- splits all the disjunctions
+      first | left; int_tac | right; int_tac
+
+theorem indexOpt_eq_index [Inhabited α] (ls : List α) (i : Int) :
+  0 ≤ i →
+  i < ls.len →
+  ls.indexOpt i = some (ls.index i) :=
+  match ls with
+  | [] => by simp; intros; linarith
+  | hd :: tl =>
+    if h: i = 0 then
+      by simp [*]
+    else
+      have hi := indexOpt_eq_index tl (i - 1)
+      by simp [*]; intros;  apply hi <;> int_tac
+
+-- Remark: the list is unchanged if the index is not in bounds (in particular
+-- if it is < 0)
+def update (ls : List α) (i : Int) (y : α) : List α :=
+  match ls with
+  | [] => []
+  | x :: tl => if i = 0 then y :: tl else x :: update tl (i - 1) y
+
+-- Remark: the whole list is dropped if the index is not in bounds (in particular
+-- if it is < 0)
+def idrop (i : Int) (ls : List α) : List α :=
+  match ls with
+  | [] => []
+  | x :: tl => if i = 0 then x :: tl else idrop (i - 1) tl
+
+section Lemmas
+
+variable {α : Type u} 
+
+@[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]
+@[simp] theorem update_nzero_cons (hne : i ≠ 0) : update ((x :: tl) : List α) i y = x :: update tl (i - 1) y := by simp [*, update]
+
+@[simp] theorem idrop_nil : idrop i ([] : List α) = [] := by simp [idrop]
+@[simp] theorem idrop_zero : idrop 0 (ls : List α) = ls := by cases ls <;> simp [idrop]
+@[simp] theorem idrop_nzero_cons (hne : i ≠ 0) : idrop i ((x :: tl) : List α) = idrop (i - 1) tl := by simp [*, idrop]
+
+theorem len_eq_length (ls : List α) : ls.len = ls.length := by
+  induction ls
+  . rfl
+  . simp [*, Int.ofNat_succ, Int.add_comm]
+
+@[simp] theorem len_append (l1 l2 : List α) : (l1 ++ l2).len = l1.len + l2.len := by
+  -- Remark: simp loops here because of the following rewritings:
+  -- @Nat.cast_add: ↑(List.length l1 + List.length l2) ==> ↑(List.length l1) + ↑(List.length l2)
+  -- Int.ofNat_add_ofNat: ↑(List.length l1) + ↑(List.length l2) ==> ↑(List.length l1 + List.length l2)
+  -- TODO: post an issue?
+  simp only [len_eq_length]
+  simp only [length_append]
+  simp only [Int.ofNat_add]
+
+@[simp]
+theorem length_update (ls : List α) (i : Int) (x : α) : (ls.update i x).length = ls.length := by
+  revert i
+  induction ls <;> simp_all [length, update]
+  intro; split <;> simp [*]
+
+@[simp]
+theorem len_update (ls : List α) (i : Int) (x : α) : (ls.update i x).len = ls.len := by
+  simp [len_eq_length]
+
+@[simp]
+theorem len_map (ls : List α) (f : α → β) : (ls.map f).len = ls.len := by
+  simp [len_eq_length]
+
+theorem left_length_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l1.length = l1'.length) :
+  l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+  revert l1'
+  induction l1
+  . intro l1'; cases l1' <;> simp [*]
+  . intro l1'; cases l1' <;> simp_all; tauto
+
+theorem right_length_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l2.length = l2'.length) :
+  l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+  have := left_length_eq_append_eq l1 l2 l1' l2'
+  constructor <;> intro heq2 <;>
+  have : l1.length + l2.length = l1'.length + l2'.length := by
+    have : (l1 ++ l2).length = (l1' ++ l2').length := by simp [*]
+    simp only [length_append] at this
+    apply this
+  . simp [heq] at this
+    tauto
+  . tauto
+
+theorem left_len_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l1.len = l1'.len) :
+  l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+  simp [len_eq_length] at heq
+  apply left_length_eq_append_eq
+  assumption
+
+theorem right_len_eq_append_eq (l1 l2 l1' l2' : List α) (heq : l2.len = l2'.len) :
+  l1 ++ l2 = l1' ++ l2' ↔ l1 = l1' ∧ l2 = l2' := by
+  simp [len_eq_length] at heq
+  apply right_length_eq_append_eq
+  assumption
+
+open Arith in
+theorem idrop_eq_nil_of_le (hineq : ls.len ≤ i) : idrop i ls = [] := by
+  revert i
+  induction ls <;> simp [*]
+  rename_i hd tl hi
+  intro i hineq
+  if heq: i = 0 then
+    simp [*] at *
+    have := tl.len_pos
+    linarith
+  else
+    simp at hineq
+    have : 0 < i := by int_tac
+    simp [*]
+    apply hi
+    linarith
+
+@[simp]
+theorem index_ne
+  {α : Type u} [Inhabited α] (l: List α) (i: ℤ) (j: ℤ) (x: α) :
+   0 ≤ i → i < l.len → 0 ≤ j → j < l.len → j ≠ i →
+    (l.update i x).index j = l.index j
+  :=
+  λ _ _ _ _ _ => match l with
+  | [] => by simp at *
+  | hd :: tl =>
+    if h: i = 0 then
+      have : j ≠ 0 := by scalar_tac
+      by simp [*]
+    else if h : j = 0 then
+      have : i ≠ 0 := by scalar_tac
+      by simp [*]
+    else
+      by
+        simp [*]
+        simp at *
+        apply index_ne <;> scalar_tac
+
+@[simp]
+theorem index_eq
+  {α : Type u} [Inhabited α] (l: List α) (i: ℤ) (x: α) :
+   0 ≤ i → i < l.len →
+    (l.update i x).index i = x
+  :=
+  fun _ _ => match l with
+  | [] => by simp at *; scalar_tac
+  | hd :: tl =>
+    if h: i = 0 then
+      by
+        simp [*]
+    else
+      by
+        simp [*]
+        simp at *
+        apply index_eq <;> scalar_tac
+
+theorem update_map_eq {α : Type u} {β : Type v} (ls : List α) (i : Int) (x : α) (f : α → β) :
+  (ls.update i x).map f = (ls.map f).update i (f x) :=
+  match ls with
+  | [] => by simp
+  | hd :: tl =>
+    if h : i = 0 then by simp [*]
+    else
+      have hi := update_map_eq tl (i - 1) x f
+      by simp [*]
+
+theorem len_flatten_update_eq {α : Type u} (ls : List (List α)) (i : Int) (x : List α)
+  (h0 : 0 ≤ i) (h1 : i < ls.len) :
+  (ls.update i x).flatten.len = ls.flatten.len + x.len - (ls.index i).len :=
+  match ls with
+  | [] => by simp at h1; int_tac
+  | hd :: tl => by
+    simp at h1
+    if h : i = 0 then simp [*]; int_tac
+    else
+      have hi := len_flatten_update_eq tl (i - 1) x (by int_tac) (by int_tac)
+      simp [*]
+      int_tac
+
+@[simp]
+theorem index_map_eq {α : Type u} {β : Type v} [Inhabited α] [Inhabited β] (ls : List α) (i : Int) (f : α → β)
+  (h0 : 0 ≤ i) (h1 : i < ls.len) :
+  (ls.map f).index i = f (ls.index i) :=
+  match ls with
+  | [] => by simp at h1; int_tac
+  | hd :: tl =>
+    if h : i = 0 then by
+      simp [*]
+    else
+      have hi := index_map_eq tl (i - 1) f (by int_tac) (by simp at h1; int_tac)
+      by
+        simp [*]
+
+def allP {α : Type u} (l : List α) (p: α → Prop) : Prop :=
+  foldr (fun a r => p a ∧ r) True l
+
+@[simp]
+theorem allP_nil {α : Type u} (p: α → Prop) : allP [] p :=
+  by simp [allP, foldr]
+
+@[simp]
+theorem allP_cons {α : Type u} (hd: α) (tl : List α) (p: α → Prop) :
+  allP (hd :: tl) p ↔ p hd ∧ allP tl p
+  := by simp [allP, foldr]
+
+def pairwise_rel
+  {α : Type u} (rel : α → α → Prop) (l: List α) : Prop
+  := match l with
+  | [] => True
+  | hd :: tl => allP tl (rel hd) ∧ pairwise_rel rel tl
+
+@[simp]
+theorem pairwise_rel_nil {α : Type u} (rel : α → α → Prop) :
+  pairwise_rel rel []
+  := by simp [pairwise_rel]
+
+@[simp]
+theorem pairwise_rel_cons {α : Type u} (rel : α → α → Prop) (hd: α) (tl: List α) :
+  pairwise_rel rel (hd :: tl) ↔ allP tl (rel hd) ∧ pairwise_rel rel tl
+  := by simp [pairwise_rel]
+
+end Lemmas
+
+end List
diff --git a/backends/lean/Base/Primitives.lean b/backends/lean/Base/Primitives.lean
new file mode 100644
index 00000000..91823cb6
--- /dev/null
+++ b/backends/lean/Base/Primitives.lean
@@ -0,0 +1,3 @@
+import Base.Primitives.Base
+import Base.Primitives.Scalar
+import Base.Primitives.Vec
diff --git a/backends/lean/Base/Primitives/Base.lean b/backends/lean/Base/Primitives/Base.lean
new file mode 100644
index 00000000..7c0fa3bb
--- /dev/null
+++ b/backends/lean/Base/Primitives/Base.lean
@@ -0,0 +1,130 @@
+import Lean
+
+namespace Primitives
+
+--------------------
+-- ASSERT COMMAND --Std.
+--------------------
+
+open Lean Elab Command Term Meta
+
+syntax (name := assert) "#assert" term: command
+
+@[command_elab assert]
+unsafe
+def assertImpl : CommandElab := fun (_stx: Syntax) => do
+  runTermElabM (fun _ => do
+    let r ← evalTerm Bool (mkConst ``Bool) _stx[1]
+    if not r then
+      logInfo ("Assertion failed for:\n" ++ _stx[1])
+      throwError ("Expression reduced to false:\n"  ++ _stx[1])
+    pure ())
+
+#eval 2 == 2
+#assert (2 == 2)
+
+-------------
+-- PRELUDE --
+-------------
+
+-- Results & monadic combinators
+
+inductive Error where
+   | assertionFailure: Error
+   | integerOverflow: Error
+   | divisionByZero: Error
+   | arrayOutOfBounds: Error
+   | maximumSizeExceeded: Error
+   | panic: Error
+deriving Repr, BEq
+
+open Error
+
+inductive Result (α : Type u) where
+  | ret (v: α): Result α
+  | fail (e: Error): Result α
+  | div
+deriving Repr, BEq
+
+open Result
+
+instance Result_Inhabited (α : Type u) : Inhabited (Result α) :=
+  Inhabited.mk (fail panic)
+
+instance Result_Nonempty (α : Type u) : Nonempty (Result α) :=
+  Nonempty.intro div
+
+/- HELPERS -/
+
+def ret? {α: Type u} (r: Result α): Bool :=
+  match r with
+  | ret _ => true
+  | fail _ | div => false
+
+def div? {α: Type u} (r: Result α): Bool :=
+  match r with
+  | div => true
+  | ret _ | fail _ => false
+
+def massert (b:Bool) : Result Unit :=
+  if b then ret () else fail assertionFailure
+
+def eval_global {α: Type u} (x: Result α) (_: ret? x): α :=
+  match x with
+  | fail _ | div => by contradiction
+  | ret x => x
+
+/- DO-DSL SUPPORT -/
+
+def bind {α : Type u} {β : Type v} (x: Result α) (f: α → Result β) : Result β :=
+  match x with
+  | ret v  => f v 
+  | fail v => fail v
+  | div => div
+
+-- Allows using Result in do-blocks
+instance : Bind Result where
+  bind := bind
+
+-- Allows using return x in do-blocks
+instance : Pure Result where
+  pure := fun x => ret x
+
+@[simp] theorem bind_ret (x : α) (f : α → Result β) : bind (.ret x) f = f x := by simp [bind]
+@[simp] theorem bind_fail (x : Error) (f : α → Result β) : bind (.fail x) f = .fail x := by simp [bind]
+@[simp] theorem bind_div (f : α → Result β) : bind .div f = .div := by simp [bind]
+
+/- CUSTOM-DSL SUPPORT -/
+
+-- Let-binding the Result of a monadic operation is oftentimes not sufficient,
+-- because we may need a hypothesis for equational reasoning in the scope. We
+-- rely on subtype, and a custom let-binding operator, in effect recreating our
+-- own variant of the do-dsl
+
+def Result.attach {α: Type} (o : Result α): Result { x : α // o = ret x } :=
+  match o with
+  | ret x => ret ⟨x, rfl⟩
+  | fail e => fail e
+  | div => div
+
+@[simp] theorem bind_tc_ret (x : α) (f : α → Result β) :
+  (do let y ← .ret x; f y) = f x := by simp [Bind.bind, bind]
+
+@[simp] theorem bind_tc_fail (x : Error) (f : α → Result β) :
+  (do let y ← fail x; f y) = fail x := by simp [Bind.bind, bind]
+
+@[simp] theorem bind_tc_div (f : α → Result β) :
+  (do let y ← div; f y) = div := by simp [Bind.bind, bind]
+
+----------
+-- MISC --
+----------
+
+@[simp] def mem.replace (a : Type) (x : a) (_ : a) : a := x
+@[simp] def 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
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Scalar.lean b/backends/lean/Base/Primitives/Scalar.lean
new file mode 100644
index 00000000..2e5be8bf
--- /dev/null
+++ b/backends/lean/Base/Primitives/Scalar.lean
@@ -0,0 +1,831 @@
+import Lean
+import Lean.Meta.Tactic.Simp
+import Mathlib.Tactic.Linarith
+import Base.Primitives.Base
+import Base.Diverge.Base
+import Base.Progress.Base
+import Base.Arith.Int
+
+namespace Primitives
+
+----------------------
+-- MACHINE INTEGERS --
+----------------------
+
+-- We redefine our machine integers types.
+
+-- For Isize/Usize, we reuse `getNumBits` from `USize`. You cannot reduce `getNumBits`
+-- using the simplifier, meaning that proofs do not depend on the compile-time value of
+-- USize.size. (Lean assumes 32 or 64-bit platforms, and Rust doesn't really support, at
+-- least officially, 16-bit microcontrollers, so this seems like a fine design decision
+-- for now.)
+
+-- Note from Chris Bailey: "If there's more than one salient property of your
+-- definition then the subtyping strategy might get messy, and the property part
+-- of a subtype is less discoverable by the simplifier or tactics like
+-- library_search." So, we will not add refinements on the return values of the
+-- operations defined on Primitives, but will rather rely on custom lemmas to
+-- invert on possible return values of the primitive operations.
+
+-- Machine integer constants, done via `ofNatCore`, which requires a proof that
+-- the `Nat` fits within the desired integer type. We provide a custom tactic.
+
+open Result Error
+open System.Platform.getNumBits
+
+-- TODO: is there a way of only importing System.Platform.getNumBits?
+--
+@[simp] def size_num_bits : Nat := (System.Platform.getNumBits ()).val
+
+-- Remark: Lean seems to use < for the comparisons with the upper bounds by convention.
+
+-- The "structured" bounds
+def Isize.smin : Int := - (HPow.hPow 2 (size_num_bits - 1))
+def Isize.smax : Int := (HPow.hPow 2 (size_num_bits - 1)) - 1
+def I8.smin    : Int := - (HPow.hPow 2 7)
+def I8.smax    : Int := HPow.hPow 2 7 - 1
+def I16.smin   : Int := - (HPow.hPow 2 15)
+def I16.smax   : Int := HPow.hPow 2 15 - 1
+def I32.smin   : Int := -(HPow.hPow 2 31)
+def I32.smax   : Int := HPow.hPow 2 31 - 1
+def I64.smin   : Int := -(HPow.hPow 2 63)
+def I64.smax   : Int := HPow.hPow 2 63 - 1
+def I128.smin  : Int := -(HPow.hPow 2 127)
+def I128.smax  : Int := HPow.hPow 2 127 - 1
+def Usize.smin : Int := 0
+def Usize.smax : Int := HPow.hPow 2 size_num_bits - 1
+def U8.smin    : Int := 0
+def U8.smax    : Int := HPow.hPow 2 8 - 1
+def U16.smin   : Int := 0
+def U16.smax   : Int := HPow.hPow 2 16 - 1
+def U32.smin   : Int := 0
+def U32.smax   : Int := HPow.hPow 2 32 - 1
+def U64.smin   : Int := 0
+def U64.smax   : Int := HPow.hPow 2 64 - 1
+def U128.smin  : Int := 0
+def U128.smax  : Int := HPow.hPow 2 128 - 1
+
+-- The "normalized" bounds, that we use in practice
+def I8.min    : Int   := -128
+def I8.max    : Int   := 127
+def I16.min   : Int  := -32768
+def I16.max   : Int  := 32767
+def I32.min   : Int  := -2147483648
+def I32.max   : Int  := 2147483647
+def I64.min   : Int  := -9223372036854775808
+def I64.max   : Int  := 9223372036854775807
+def I128.min  : Int := -170141183460469231731687303715884105728
+def I128.max  : Int := 170141183460469231731687303715884105727
+@[simp]
+def U8.min    : Int   := 0
+def U8.max    : Int   := 255
+@[simp]
+def U16.min   : Int  := 0
+def U16.max   : Int  := 65535
+@[simp]
+def U32.min   : Int  := 0
+def U32.max   : Int  := 4294967295
+@[simp]
+def U64.min   : Int  := 0
+def U64.max   : Int  := 18446744073709551615
+@[simp]
+def U128.min  : Int := 0
+def U128.max  : Int := 340282366920938463463374607431768211455
+@[simp]
+def Usize.min : Int := 0
+
+def Isize.refined_min : { n:Int // n = I32.min ∨ n = I64.min } :=
+  ⟨ Isize.smin, by
+    simp [Isize.smin]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.refined_max : { n:Int // n = I32.max ∨ n = I64.max } :=
+  ⟨ Isize.smax, by
+    simp [Isize.smax]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Usize.refined_max : { n:Int // n = U32.max ∨ n = U64.max } :=
+  ⟨ Usize.smax, by
+    simp [Usize.smax]
+    cases System.Platform.numBits_eq <;>
+    unfold System.Platform.numBits at * <;> simp [*] ⟩
+
+def Isize.min := Isize.refined_min.val
+def Isize.max := Isize.refined_max.val
+def Usize.max := Usize.refined_max.val
+
+inductive ScalarTy :=
+| Isize
+| I8
+| I16
+| I32
+| I64
+| I128
+| Usize
+| U8
+| U16
+| U32
+| U64
+| U128
+
+def ScalarTy.isSigned (ty : ScalarTy) : Bool :=
+  match ty with
+  | Isize
+  | I8
+  | I16
+  | I32
+  | I64
+  | I128 => true
+  | Usize
+  | U8
+  | U16
+  | U32
+  | U64
+  | U128 => false
+
+
+def Scalar.smin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.smin
+  | .I8    => I8.smin
+  | .I16   => I16.smin
+  | .I32   => I32.smin
+  | .I64   => I64.smin
+  | .I128  => I128.smin
+  | .Usize => Usize.smin
+  | .U8    => U8.smin
+  | .U16   => U16.smin
+  | .U32   => U32.smin
+  | .U64   => U64.smin
+  | .U128  => U128.smin
+
+def Scalar.smax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.smax
+  | .I8    => I8.smax
+  | .I16   => I16.smax
+  | .I32   => I32.smax
+  | .I64   => I64.smax
+  | .I128  => I128.smax
+  | .Usize => Usize.smax
+  | .U8    => U8.smax
+  | .U16   => U16.smax
+  | .U32   => U32.smax
+  | .U64   => U64.smax
+  | .U128  => U128.smax
+
+def Scalar.min (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.min
+  | .I8    => I8.min
+  | .I16   => I16.min
+  | .I32   => I32.min
+  | .I64   => I64.min
+  | .I128  => I128.min
+  | .Usize => Usize.min
+  | .U8    => U8.min
+  | .U16   => U16.min
+  | .U32   => U32.min
+  | .U64   => U64.min
+  | .U128  => U128.min
+
+def Scalar.max (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Isize.max
+  | .I8    => I8.max
+  | .I16   => I16.max
+  | .I32   => I32.max
+  | .I64   => I64.max
+  | .I128  => I128.max
+  | .Usize => Usize.max
+  | .U8    => U8.max
+  | .U16   => U16.max
+  | .U32   => U32.max
+  | .U64   => U64.max
+  | .U128  => U128.max
+
+def Scalar.smin_eq (ty : ScalarTy) : Scalar.min ty = Scalar.smin ty := by
+  cases ty <;> rfl
+
+def Scalar.smax_eq (ty : ScalarTy) : Scalar.max ty = Scalar.smax ty := by
+  cases ty <;> rfl
+
+-- "Conservative" bounds
+-- We use those because we can't compare to the isize bounds (which can't
+-- reduce at compile-time). Whenever we perform an arithmetic operation like
+-- addition we need to check that the result is in bounds: we first compare
+-- to the conservative bounds, which reduce, then compare to the real bounds.
+-- This is useful for the various #asserts that we want to reduce at
+-- type-checking time.
+def Scalar.cMin (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Scalar.min .I32
+  | _ => Scalar.min ty
+
+def Scalar.cMax (ty : ScalarTy) : Int :=
+  match ty with
+  | .Isize => Scalar.max .I32
+  | .Usize => Scalar.max .U32
+  | _ => Scalar.max ty
+
+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
+  cases h <;> simp [*, Isize.min]
+
+theorem Scalar.cMax_bound ty : Scalar.cMax ty ≤ Scalar.max ty := by
+  cases ty <;> simp [Scalar.min, Scalar.max, Scalar.cMin, Scalar.cMax] at *
+  . have h := Isize.refined_max.property
+    cases h <;> simp [*, Isize.max]
+  . have h := Usize.refined_max.property
+    cases h <;> simp [*, Usize.max]
+
+theorem Scalar.cMin_suffices ty (h : Scalar.cMin ty ≤ x) : Scalar.min ty ≤ x := by
+  have := Scalar.cMin_bound ty
+  linarith
+
+theorem Scalar.cMax_suffices ty (h : x ≤ Scalar.cMax ty) : x ≤ Scalar.max ty := by
+  have := Scalar.cMax_bound ty
+  linarith
+
+structure Scalar (ty : ScalarTy) where
+  val : Int
+  hmin : Scalar.min ty ≤ val
+  hmax : val ≤ Scalar.max ty
+deriving Repr
+
+theorem Scalar.bound_suffices (ty : ScalarTy) (x : Int) :
+  Scalar.cMin ty ≤ x ∧ x ≤ Scalar.cMax ty ->
+  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty
+  :=
+  λ h => by
+  apply And.intro <;> have hmin := Scalar.cMin_bound ty <;> have hmax := Scalar.cMax_bound ty <;> linarith
+
+def Scalar.ofIntCore {ty : ScalarTy} (x : Int)
+  (hmin : Scalar.min ty ≤ x) (hmax : x ≤ Scalar.max ty) : Scalar ty :=
+  { val := x, hmin := hmin, hmax := hmax }
+
+-- Tactic to prove that integers are in bounds
+-- TODO: use this: https://leanprover.zulipchat.com/#narrow/stream/270676-lean4/topic/instance.20with.20tactic.20autoparam
+syntax "intlit" : tactic
+macro_rules
+  | `(tactic| intlit) => `(tactic| apply Scalar.bound_suffices; decide)
+
+def Scalar.ofInt {ty : ScalarTy} (x : Int)
+  (h : Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by intlit) : Scalar ty :=
+  -- Remark: we initially wrote:
+  --  let ⟨ hmin, hmax ⟩ := h
+  --  Scalar.ofIntCore x hmin hmax
+  -- We updated to the line below because a similar pattern in `Scalar.tryMk`
+  -- made reduction block. Both versions seem to work for `Scalar.ofInt`, though.
+  -- TODO: investigate
+  Scalar.ofIntCore x h.left h.right
+
+@[simp] def Scalar.check_bounds (ty : ScalarTy) (x : Int) : Bool :=
+  (Scalar.cMin ty ≤ x || Scalar.min ty ≤ x) ∧ (x ≤ Scalar.cMax ty || x ≤ Scalar.max ty)
+
+theorem Scalar.check_bounds_prop {ty : ScalarTy} {x : Int} (h: Scalar.check_bounds ty x) :
+  Scalar.min ty ≤ x ∧ x ≤ Scalar.max ty := by
+  simp at *
+  have ⟨ hmin, hmax ⟩ := h
+  have hbmin := Scalar.cMin_bound ty
+  have hbmax := Scalar.cMax_bound ty
+  cases hmin <;> cases hmax <;> apply And.intro <;> linarith
+
+-- Further thoughts: look at what has been done here:
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/Fin/Basic.lean
+-- and
+-- https://github.com/leanprover-community/mathlib4/blob/master/Mathlib/Data/UInt.lean
+-- which both contain a fair amount of reasoning already!
+def Scalar.tryMk (ty : ScalarTy) (x : Int) : Result (Scalar ty) :=
+  if h:Scalar.check_bounds ty x then
+    -- If we do:
+    -- ```
+    -- let ⟨ hmin, hmax ⟩ := (Scalar.check_bounds_prop h)
+    -- Scalar.ofIntCore x hmin hmax
+    -- ```
+    -- then normalization blocks (for instance, some proofs which use reflexivity fail).
+    -- However, the version below doesn't block reduction (TODO: investigate):
+    return Scalar.ofInt x (Scalar.check_bounds_prop h)
+  else fail integerOverflow
+
+def Scalar.neg {ty : ScalarTy} (x : Scalar ty) : Result (Scalar ty) := Scalar.tryMk ty (- x.val)
+
+-- Our custom remainder operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_rem (x y : Int) : Int :=
+  if 0 ≤ x then x % y
+  else - (|x| % |y|)
+
+@[simp]
+def scalar_rem_nonneg {x y : Int} (hx : 0 ≤ x) : scalar_rem x y = x % y := by
+  intros
+  simp [*, scalar_rem]
+
+-- Our custom division operation, which satisfies the semantics of Rust
+-- TODO: is there a better way?
+def scalar_div (x y : Int) : Int :=
+  if 0 ≤ x && 0 ≤ y then x / y
+  else if 0 ≤ x && y < 0 then - (|x| / |y|)
+  else if x < 0 && 0 ≤ y then - (|x| / |y|)
+  else |x| / |y|
+
+@[simp]
+def scalar_div_nonneg {x y : Int} (hx : 0 ≤ x) (hy : 0 ≤ y) : scalar_div x y = x / y := by
+  intros
+  simp [*, scalar_div]
+
+-- Checking that the remainder operation is correct
+#assert scalar_rem 1 2 = 1
+#assert scalar_rem (-1) 2 = -1
+#assert scalar_rem 1 (-2) = 1
+#assert scalar_rem (-1) (-2) = -1
+#assert scalar_rem 7 3 = (1:Int)
+#assert scalar_rem (-7) 3 = -1
+#assert scalar_rem 7 (-3) = 1
+#assert scalar_rem (-7) (-3) = -1
+
+-- Checking that the division operation is correct
+#assert scalar_div 3 2 = 1
+#assert scalar_div (-3) 2 = -1
+#assert scalar_div 3 (-2) = -1
+#assert scalar_div (-3) (-2) = 1
+#assert scalar_div 7 3 = 2
+#assert scalar_div (-7) 3 = -2
+#assert scalar_div 7 (-3) = -2
+#assert scalar_div (-7) (-3) = 2
+
+def Scalar.div {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (scalar_div x.val y.val) else fail divisionByZero
+
+def Scalar.rem {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  if y.val != 0 then Scalar.tryMk ty (scalar_rem x.val y.val) else fail divisionByZero
+
+def Scalar.add {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val + y.val)
+
+def Scalar.sub {ty : ScalarTy} (x : Scalar ty) (y : Scalar ty) : Result (Scalar ty) :=
+  Scalar.tryMk ty (x.val - y.val)
+
+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
+
+-- Cast an integer from a [src_ty] to a [tgt_ty]
+-- TODO: 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
+
+-- The scalar types
+-- We declare the definitions as reducible so that Lean can unfold them (useful
+-- for type class resolution for instance).
+@[reducible] def Isize := Scalar .Isize
+@[reducible] def I8    := Scalar .I8
+@[reducible] def I16   := Scalar .I16
+@[reducible] def I32   := Scalar .I32
+@[reducible] def I64   := Scalar .I64
+@[reducible] def I128  := Scalar .I128
+@[reducible] def Usize := Scalar .Usize
+@[reducible] def U8    := Scalar .U8
+@[reducible] def U16   := Scalar .U16
+@[reducible] def U32   := Scalar .U32
+@[reducible] def U64   := Scalar .U64
+@[reducible] def U128  := Scalar .U128
+
+-- 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
+-- with macros (but would it be a good idea? It would be less easy to
+-- read the file, which is not supposed to change a lot)
+
+-- Negation
+
+/--
+Remark: there is no heterogeneous negation in the Lean prelude: we thus introduce
+one here.
+
+The notation typeclass for heterogeneous addition.
+This enables the notation `- a : β` where `a : α`.
+-/
+class HNeg (α : Type u) (β : outParam (Type v)) where
+  /-- `- a` computes the negation of `a`.
+  The meaning of this notation is type-dependent. -/
+  hNeg : α → β
+
+prefix:75  "-"   => HNeg.hNeg
+
+instance : HNeg Isize (Result Isize) where hNeg x := Scalar.neg x
+instance : HNeg I8 (Result I8) where hNeg x := Scalar.neg x
+instance : HNeg I16 (Result I16) where hNeg x := Scalar.neg x
+instance : HNeg I32 (Result I32) where hNeg x := Scalar.neg x
+instance : HNeg I64 (Result I64) where hNeg x := Scalar.neg x
+instance : HNeg I128 (Result I128) where hNeg x := Scalar.neg x
+
+-- Addition
+instance {ty} : HAdd (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hAdd x y := Scalar.add x y
+
+-- Substraction
+instance {ty} : HSub (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hSub x y := Scalar.sub x y
+
+-- Multiplication
+instance {ty} : HMul (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMul x y := Scalar.mul x y
+
+-- Division
+instance {ty} : HDiv (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hDiv x y := Scalar.div x y
+
+-- Remainder
+instance {ty} : HMod (Scalar ty) (Scalar ty) (Result (Scalar ty)) where
+  hMod x y := Scalar.rem x y
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.add_spec {ty} {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val + y.val)
+  (hmax : x.val + y.val ≤ Scalar.max ty) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  simp [HAdd.hAdd, add, Add.add]
+  simp [tryMk]
+  split
+  . simp [pure]
+    rfl
+  . tauto
+
+theorem Scalar.add_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+  (hmax : x.val + y.val ≤ Scalar.max ty) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  have hmin : Scalar.min ty ≤ x.val + y.val := by
+    have hx := x.hmin
+    have hy := y.hmin
+    cases ty <;> simp [min] at * <;> linarith
+  apply add_spec <;> assumption
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.add_spec {x y : Usize} (hmax : x.val + y.val ≤ Usize.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U8.add_spec {x y : U8} (hmax : x.val + y.val ≤ U8.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U16.add_spec {x y : U16} (hmax : x.val + y.val ≤ U16.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U32.add_spec {x y : U32} (hmax : x.val + y.val ≤ U32.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U64.add_spec {x y : U64} (hmax : x.val + y.val ≤ U64.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U128.add_spec {x y : U128} (hmax : x.val + y.val ≤ U128.max) :
+  ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+  apply Scalar.add_unsigned_spec <;> simp only [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.sub_spec {ty} {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val - y.val)
+  (hmax : x.val - y.val ≤ Scalar.max ty) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  simp [HSub.hSub, sub, Sub.sub]
+  simp [tryMk]
+  split
+  . simp [pure]
+    rfl
+  . tauto
+
+theorem Scalar.sub_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  have : x.val - y.val ≤ Scalar.max ty := by
+    have hx := x.hmin
+    have hxm := x.hmax
+    have hy := y.hmin
+    cases ty <;> simp [min, max] at * <;> linarith
+  intros
+  apply sub_spec <;> assumption
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.sub_spec {x y : Usize} (hmin : Usize.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U8.sub_spec {x y : U8} (hmin : U8.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U16.sub_spec {x y : U16} (hmin : U16.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U32.sub_spec {x y : U32} (hmin : U32.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U64.sub_spec {x y : U64} (hmin : U64.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+@[cepspec] theorem U128.sub_spec {x y : U128} (hmin : U128.min ≤ x.val - y.val) :
+  ∃ z, x - y = ret z ∧ z.val = x.val - y.val := by
+  apply Scalar.sub_unsigned_spec <;> simp only [Scalar.min, *]
+
+-- Generic theorem - shouldn't be used much
+theorem Scalar.mul_spec {ty} {x y : Scalar ty}
+  (hmin : Scalar.min ty ≤ x.val * y.val)
+  (hmax : x.val * y.val ≤ Scalar.max ty) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  simp [HMul.hMul, mul, Mul.mul]
+  simp [tryMk]
+  split
+  . simp [pure]
+    rfl
+  . tauto
+
+theorem Scalar.mul_unsigned_spec {ty} (s: ¬ ty.isSigned) {x y : Scalar ty}
+  (hmax : x.val * y.val ≤ Scalar.max ty) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  have : Scalar.min ty ≤ x.val * y.val := by
+    have hx := x.hmin
+    have hy := y.hmin
+    cases ty <;> simp at * <;> apply mul_nonneg hx hy
+  apply mul_spec <;> assumption
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.mul_spec {x y : Usize} (hmax : x.val * y.val ≤ Usize.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U8.mul_spec {x y : U8} (hmax : x.val * y.val ≤ U8.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U16.mul_spec {x y : U16} (hmax : x.val * y.val ≤ U16.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U32.mul_spec {x y : U32} (hmax : x.val * y.val ≤ U32.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U64.mul_spec {x y : U64} (hmax : x.val * y.val ≤ U64.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+@[cepspec] theorem U128.mul_spec {x y : U128} (hmax : x.val * y.val ≤ U128.max) :
+  ∃ z, x * y = ret z ∧ z.val = x.val * y.val := by
+  apply Scalar.mul_unsigned_spec <;> simp only [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.div_spec {ty} {x y : Scalar ty}
+  (hnz : y.val ≠ 0)
+  (hmin : Scalar.min ty ≤ scalar_div x.val y.val)
+  (hmax : scalar_div x.val y.val ≤ Scalar.max ty) :
+  ∃ z, x / y = ret z ∧ z.val = scalar_div x.val y.val := by
+  simp [HDiv.hDiv, div, Div.div]
+  simp [tryMk, *]
+  simp [pure]
+  rfl
+
+theorem Scalar.div_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
+  (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  have h : Scalar.min ty = 0 := by cases ty <;> simp at *
+  have hx := x.hmin
+  have hy := y.hmin
+  simp [h] at hx hy
+  have hmin : 0 ≤ x.val / y.val := Int.ediv_nonneg hx hy
+  have hmax : x.val / y.val ≤ Scalar.max ty := by
+    have := Int.ediv_le_self y.val hx
+    have := x.hmax
+    linarith
+  have hs := @div_spec ty x y hnz
+  simp [*] at hs
+  apply hs
+
+/- Fine-grained theorems -/
+@[cepspec] theorem Usize.div_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [*]
+
+@[cepspec] theorem U8.div_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U16.div_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U32.div_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U64.div_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U128.div_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+  ∃ z, x / y = ret z ∧ z.val = x.val / y.val := by
+  apply Scalar.div_unsigned_spec <;> simp [Scalar.max, *]
+
+-- Generic theorem - shouldn't be used much
+@[cpspec]
+theorem Scalar.rem_spec {ty} {x y : Scalar ty}
+  (hnz : y.val ≠ 0)
+  (hmin : Scalar.min ty ≤ scalar_rem x.val y.val)
+  (hmax : scalar_rem x.val y.val ≤ Scalar.max ty) :
+  ∃ z, x % y = ret z ∧ z.val = scalar_rem x.val y.val := by
+  simp [HMod.hMod, rem]
+  simp [tryMk, *]
+  simp [pure]
+  rfl
+
+theorem Scalar.rem_unsigned_spec {ty} (s: ¬ ty.isSigned) (x : Scalar ty) {y : Scalar ty}
+  (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  have h : Scalar.min ty = 0 := by cases ty <;> simp at *
+  have hx := x.hmin
+  have hy := y.hmin
+  simp [h] at hx hy
+  have hmin : 0 ≤ x.val % y.val := Int.emod_nonneg x.val hnz
+  have hmax : x.val % y.val ≤ Scalar.max ty := by
+    have h : 0 < y.val := by int_tac
+    have h := Int.emod_lt_of_pos x.val h
+    have := y.hmax
+    linarith
+  have hs := @rem_spec ty x y hnz
+  simp [*] at hs
+  simp [*]
+
+@[cepspec] theorem Usize.rem_spec (x : Usize) {y : Usize} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [*]
+
+@[cepspec] theorem U8.rem_spec (x : U8) {y : U8} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U16.rem_spec (x : U16) {y : U16} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U32.rem_spec (x : U32) {y : U32} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U64.rem_spec (x : U64) {y : U64} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+@[cepspec] theorem U128.rem_spec (x : U128) {y : U128} (hnz : y.val ≠ 0) :
+  ∃ z, x % y = ret z ∧ z.val = x.val % y.val := by
+  apply Scalar.rem_unsigned_spec <;> simp [Scalar.max, *]
+
+-- ofIntCore
+-- TODO: typeclass?
+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?
+def Isize.ofInt := @Scalar.ofInt .Isize
+def I8.ofInt    := @Scalar.ofInt .I8
+def I16.ofInt   := @Scalar.ofInt .I16
+def I32.ofInt   := @Scalar.ofInt .I32
+def I64.ofInt   := @Scalar.ofInt .I64
+def I128.ofInt  := @Scalar.ofInt .I128
+def Usize.ofInt := @Scalar.ofInt .Usize
+def U8.ofInt    := @Scalar.ofInt .U8
+def U16.ofInt   := @Scalar.ofInt .U16
+def U32.ofInt   := @Scalar.ofInt .U32
+def U64.ofInt   := @Scalar.ofInt .U64
+def U128.ofInt  := @Scalar.ofInt .U128
+
+-- 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
+
+instance {ty} : LE (Scalar ty) where le a b := LE.le a.val b.val
+
+instance Scalar.decLt {ty} (a b : Scalar ty) : Decidable (LT.lt a b) := Int.decLt ..
+instance Scalar.decLe {ty} (a b : Scalar ty) : Decidable (LE.le a b) := Int.decLe ..
+
+theorem Scalar.eq_of_val_eq {ty} : ∀ {i j : Scalar ty}, Eq i.val j.val → Eq i j
+  | ⟨_, _, _⟩, ⟨_, _, _⟩, rfl => rfl
+
+theorem Scalar.val_eq_of_eq {ty} {i j : Scalar ty} (h : Eq i j) : Eq i.val j.val :=
+  h ▸ rfl
+
+theorem Scalar.ne_of_val_ne {ty} {i j : Scalar ty} (h : Not (Eq i.val j.val)) : Not (Eq i j) :=
+  fun h' => absurd (val_eq_of_eq h') h
+
+instance (ty : ScalarTy) : DecidableEq (Scalar ty) :=
+  fun i j =>
+    match decEq i.val j.val with
+    | isTrue h  => isTrue (Scalar.eq_of_val_eq h)
+    | isFalse h => isFalse (Scalar.ne_of_val_ne h)
+
+/- Remark: we can't write the following instance because of restrictions about
+   the type class parameters (`ty` doesn't appear in the return type, which is
+   forbidden):
+
+   ```
+   instance Scalar.cast (ty : ScalarTy) : Coe (Scalar ty) Int where coe := λ v => v.val
+   ```
+ -/
+def Scalar.toInt {ty} (n : Scalar ty) : Int := n.val
+
+-- -- We now define a type class that subsumes the various machine integer types, so
+-- -- as to write a concise definition for scalar_cast, rather than exhaustively
+-- -- enumerating all of the possible pairs. We remark that Rust has sane semantics
+-- -- and fails if a cast operation would involve a truncation or modulo.
+
+-- class MachineInteger (t: Type) where
+--   size: Nat
+--   val: t -> Fin size
+--   ofNatCore: (n:Nat) -> LT.lt n size -> t
+
+-- set_option hygiene false in
+-- run_cmd
+--   for typeName in [`UInt8, `UInt16, `UInt32, `UInt64, `USize].map Lean.mkIdent do
+--   Lean.Elab.Command.elabCommand (← `(
+--     namespace $typeName
+--     instance: MachineInteger $typeName where
+--       size := size
+--       val := val
+--       ofNatCore := ofNatCore
+--     end $typeName
+--   ))
+
+-- -- Aeneas only instantiates the destination type (`src` is implicit). We rely on
+-- -- Lean to infer `src`.
+
+-- def scalar_cast { src: Type } (dst: Type) [ MachineInteger src ] [ MachineInteger dst ] (x: src): Result dst :=
+--   if h: MachineInteger.val x < MachineInteger.size dst then
+--     .ret (MachineInteger.ofNatCore (MachineInteger.val x).val h)
+--   else
+--     .fail integerOverflow
+
+end Primitives
diff --git a/backends/lean/Base/Primitives/Vec.lean b/backends/lean/Base/Primitives/Vec.lean
new file mode 100644
index 00000000..a09d6ac2
--- /dev/null
+++ b/backends/lean/Base/Primitives/Vec.lean
@@ -0,0 +1,145 @@
+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.Arith
+import Base.Progress.Base
+
+namespace Primitives
+
+open Result Error
+
+-------------
+-- VECTORS --
+-------------
+
+def Vec (α : Type u) := { l : List α // l.length ≤ Usize.max }
+
+-- TODO: do we really need it? It should be with Subtype by default
+instance Vec.cast (a : Type u): Coe (Vec a) (List a)  where coe := λ v => v.val
+
+instance (a : Type u) : Arith.HasIntProp (Vec 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, *]
+
+@[simp]
+abbrev Vec.length {α : Type u} (v : Vec α) : Int := v.val.len
+
+@[simp]
+abbrev Vec.v {α : Type u} (v : Vec α) : List α := v.val
+
+example {a: Type u} (v : Vec a) : v.length ≤ Scalar.max ScalarTy.Usize := by
+  scalar_tac
+
+def Vec.new (α : Type u): Vec α := ⟨ [], by apply Scalar.cMax_suffices .Usize; simp ⟩
+
+-- TODO: very annoying that the α is an explicit parameter
+def Vec.len (α : Type u) (v : Vec α) : Usize :=
+  Usize.ofIntCore v.val.len (by scalar_tac) (by scalar_tac)
+
+@[simp]
+theorem Vec.len_val {α : Type u} (v : Vec α) : (Vec.len α v).val = v.length :=
+  by rfl
+
+-- This shouldn't be used
+def Vec.push_fwd (α : Type u) (_ : Vec α) (_ : α) : Unit := ()
+
+-- This is actually the backward function
+def Vec.push (α : Type u) (v : Vec α) (x : α) : Result (Vec α)
+  :=
+  let nlen := List.length v.val + 1
+  if h : nlen ≤ U32.max || nlen ≤ Usize.max then
+    have h : nlen ≤ Usize.max := by
+      simp [Usize.max] at *
+      have hm := Usize.refined_max.property
+      cases h <;> cases hm <;> simp [U32.max, U64.max] at * <;> try linarith
+    return ⟨ List.concat v.val x, by simp at *; assumption ⟩
+  else
+    fail maximumSizeExceeded
+
+-- This shouldn't be used
+def Vec.insert_fwd (α : Type u) (v: Vec α) (i: Usize) (_: α) : Result Unit :=
+  if i.val < v.length then
+    .ret ()
+  else
+    .fail arrayOutOfBounds
+
+-- This is actually the backward function
+def Vec.insert (α : Type u) (v: Vec α) (i: Usize) (x: α) : Result (Vec α) :=
+  if i.val < v.length then
+    .ret ⟨ v.val.update i.val x, by have := v.property; simp [*] ⟩
+  else
+    .fail arrayOutOfBounds
+
+@[pspec]
+theorem Vec.insert_spec {α : Type u} (v: Vec α) (i: Usize) (x: α)
+  (hbound : i.val < v.length) :
+  ∃ nv, v.insert α i x = ret nv ∧ nv.val = v.val.update i.val x := by
+  simp [insert, *]
+
+def Vec.index (α : Type u) (v: Vec 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 Vec.index_spec {α : Type u} [Inhabited α] (v: Vec α) (i: Usize)
+  (hbound : i.val < v.length) :
+  ∃ x, v.index α i = ret x ∧ x = v.val.index i.val := by
+  simp only [index]
+  -- 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)
+  (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 [*]
+
+instance {α : Type u} (p : Vec α → Prop) : Arith.HasIntProp (Subtype p) where
+  prop_ty := λ x => p x
+  prop := λ x => x.property
+
+def Vec.index_mut_back (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)
+  (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
+
+end Primitives
diff --git a/backends/lean/Base/Progress.lean b/backends/lean/Base/Progress.lean
new file mode 100644
index 00000000..d812b896
--- /dev/null
+++ b/backends/lean/Base/Progress.lean
@@ -0,0 +1 @@
+import Base.Progress.Progress
diff --git a/backends/lean/Base/Progress/Base.lean b/backends/lean/Base/Progress/Base.lean
new file mode 100644
index 00000000..6f820a84
--- /dev/null
+++ b/backends/lean/Base/Progress/Base.lean
@@ -0,0 +1,316 @@
+import Lean
+import Std.Lean.HashSet
+import Base.Utils
+import Base.Primitives.Base
+
+namespace Progress
+
+open Lean Elab Term Meta
+open Utils
+
+-- We can't define and use trace classes in the same file
+initialize registerTraceClass `Progress
+
+/- # Progress tactic -/
+
+structure PSpecDesc where
+  -- The universally quantified variables
+  fvars : Array Expr
+  -- The existentially quantified variables
+  evars : Array Expr
+  -- The function
+  fExpr : Expr
+  fName : Name
+  -- The function arguments
+  fLevels : List Level
+  args : Array Expr
+  -- The universally quantified variables which appear in the function arguments
+  argsFVars : Array FVarId
+  -- The returned value
+  ret : Expr
+  -- The postcondition (if there is)
+  post : Option Expr
+
+section Methods
+  variable [MonadLiftT MetaM m] [MonadControlT MetaM m] [Monad m] [MonadOptions m]
+  variable [MonadTrace m] [MonadLiftT IO m] [MonadRef m] [AddMessageContext m]
+  variable [MonadError m]
+  variable {a : Type}
+
+  /- Analyze a pspec theorem to decompose its arguments.
+
+     PSpec theorems should be of the following shape:
+     ```
+     ∀ x1 ... xn, H1 → ... Hn → ∃ y1 ... ym. f x1 ... xn = .ret ... ∧ Post1 ∧ ... ∧ Postk
+     ```
+
+     The continuation `k` receives the following inputs:
+     - universally quantified variables
+     - assumptions
+     - existentially quantified variables
+     - function name
+     - function arguments
+     - return
+     - postconditions
+
+     TODO: generalize for when we do inductive proofs
+  -/
+  partial
+  def withPSpec [Inhabited (m a)] [Nonempty (m a)] (th : Expr) (k : PSpecDesc → m a)
+    (sanityChecks : Bool := false) :
+    m a := do
+    trace[Progress] "Proposition: {th}"
+    -- Dive into the quantified variables and the assumptions
+    forallTelescope th.consumeMData fun fvars th => do
+    trace[Progress] "Universally quantified arguments and assumptions: {fvars}"
+    -- Dive into the existentials
+    existsTelescope th.consumeMData fun evars th => do
+    trace[Progress] "Existentials: {evars}"
+    trace[Progress] "Proposition after stripping the quantifiers: {th}"
+    -- Take the first conjunct
+    let (th, post) ← optSplitConj th.consumeMData
+    trace[Progress] "After splitting the conjunction:\n- eq: {th}\n- post: {post}"
+    -- Destruct the equality
+    let (mExpr, ret) ← destEq th.consumeMData
+    trace[Progress] "After splitting the equality:\n- lhs: {th}\n- rhs: {ret}"
+    -- Destruct the monadic application to dive into the bind, if necessary (this
+    -- is for when we use `withPSpec` inside of the `progress` tactic), and
+    -- destruct the application to get the function name
+    mExpr.consumeMData.withApp fun mf margs => do
+    trace[Progress] "After stripping the arguments of the monad expression:\n- mf: {mf}\n- margs: {margs}"
+    let (fExpr, f, args) ← do
+      if mf.isConst ∧ mf.constName = ``Bind.bind then do
+        -- Dive into the bind
+        let fExpr := (margs.get! 4).consumeMData
+        fExpr.withApp fun f args => pure (fExpr, f, args)
+      else pure (mExpr, mf, margs)
+    trace[Progress] "After stripping the arguments of the function call:\n- f: {f}\n- args: {args}"
+    if ¬ f.isConst then throwError "Not a constant: {f}"
+    -- Compute the set of universally quantified variables which appear in the function arguments
+    let allArgsFVars ← args.foldlM (fun hs arg => getFVarIds arg hs) HashSet.empty
+    -- Sanity check
+    if sanityChecks then
+      -- All the variables which appear in the inputs given to the function are
+      -- universally quantified (in particular, they are not *existentially* quantified)
+      let fvarsSet : HashSet FVarId := HashSet.ofArray (fvars.map (fun x => x.fvarId!))
+      let filtArgsFVars := allArgsFVars.toArray.filter (fun fvar => ¬ fvarsSet.contains fvar)
+      if ¬ filtArgsFVars.isEmpty then
+        let filtArgsFVars := filtArgsFVars.map (fun fvarId => Expr.fvar fvarId)
+        throwError "Some of the function inputs are not universally quantified: {filtArgsFVars}"
+    let argsFVars := fvars.map (fun x => x.fvarId!)
+    let argsFVars := argsFVars.filter (fun fvar => allArgsFVars.contains fvar)
+    -- Return
+    trace[Progress] "Function: {f.constName!}";
+    let thDesc := {
+      fvars := fvars
+      evars := evars
+      fExpr
+      fName := f.constName!
+      fLevels := f.constLevels!
+      args := args
+      argsFVars
+      ret := ret
+      post := post
+    }
+    k thDesc
+
+end Methods
+
+def getPSpecFunName (th : Expr) : MetaM Name :=
+  withPSpec th (fun d => do pure d.fName) true
+
+def getPSpecClassFunNames (th : Expr) : MetaM (Name × Name) :=
+  withPSpec th (fun d => do
+    let arg0 := d.args.get! 0
+    arg0.withApp fun f _ => do
+    if ¬ f.isConst then throwError "Not a constant: {f}"
+    pure (d.fName, f.constName)
+    ) true
+
+def getPSpecClassFunNameArg (th : Expr) : MetaM (Name × Expr) :=
+  withPSpec th (fun d => do
+    let arg0 := d.args.get! 0
+    pure (d.fName, arg0)
+    ) true
+
+-- "Regular" pspec attribute
+structure PSpecAttr where
+  attr : AttributeImpl
+  ext  : MapDeclarationExtension Name
+  deriving Inhabited
+
+/- pspec attribute for type classes: we use the name of the type class to
+   lookup another map. We use the *first* argument of the type class to lookup
+   into this second map.
+
+   Example:
+   ========
+   We use type classes for addition. For instance, the addition between two
+   U32 is written (without syntactic sugar) as `HAdd.add (Scalar ty) x y`. As a consequence,
+   we store the theorem through the bindings: HAdd.add → Scalar → ...
+
+   SH: TODO: this (and `PSpecClassExprAttr`) is a bit ad-hoc. For now it works for the
+   specs of the scalar operations, which is what I really need, but I'm not sure it
+   applies well to other situations. A better way would probably to use type classes, but
+   I couldn't get them to work on those cases. It is worth retrying.
+-/
+structure PSpecClassAttr where
+  attr : AttributeImpl
+  ext  : MapDeclarationExtension (NameMap Name)
+  deriving Inhabited
+
+/- Same as `PSpecClassAttr` but we use the full first argument (it works when it
+   is a constant). -/
+structure PSpecClassExprAttr where
+  attr : AttributeImpl
+  ext  : MapDeclarationExtension (HashMap Expr Name)
+  deriving Inhabited
+
+-- TODO: the original function doesn't define correctly the `addImportedFn`. Do a PR?
+def mkMapDeclarationExtension [Inhabited α] (name : Name := by exact decl_name%) : IO (MapDeclarationExtension α) :=
+  registerSimplePersistentEnvExtension {
+    name          := name,
+    addImportedFn := fun a => a.foldl (fun s a => a.foldl (fun s (k, v) => s.insert k v) s) RBMap.empty,
+    addEntryFn    := fun s n => s.insert n.1 n.2 ,
+    toArrayFn     := fun es => es.toArray.qsort (fun a b => Name.quickLt a.1 b.1)
+  }
+
+/- The persistent map from function to pspec theorems. -/
+initialize pspecAttr : PSpecAttr ← do
+  let ext ← mkMapDeclarationExtension `pspecMap
+  let attrImpl : AttributeImpl := {
+    name := `pspec
+    descr := "Marks theorems to use with the `progress` tactic"
+    add := fun thName stx attrKind => do
+      Attribute.Builtin.ensureNoArgs stx
+      -- TODO: use the attribute kind
+      unless attrKind == AttributeKind.global do
+        throwError "invalid attribute 'pspec', must be global"
+      -- Lookup the theorem
+      let env ← getEnv
+      let thDecl := env.constants.find! thName
+      let fName ← MetaM.run' (getPSpecFunName thDecl.type)
+      trace[Progress] "Registering spec theorem for {fName}"
+      let env := ext.addEntry env (fName, thName)
+      setEnv env
+      pure ()
+  }
+  registerBuiltinAttribute attrImpl
+  pure { attr := attrImpl, ext := ext }
+
+/- The persistent map from type classes to pspec theorems -/
+initialize pspecClassAttr : PSpecClassAttr ← do
+  let ext : MapDeclarationExtension (NameMap Name) ← mkMapDeclarationExtension `pspecClassMap
+  let attrImpl : AttributeImpl  := {
+    name := `cpspec
+    descr := "Marks theorems to use for type classes with the `progress` tactic"
+    add := fun thName stx attrKind => do
+      Attribute.Builtin.ensureNoArgs stx
+      -- TODO: use the attribute kind
+      unless attrKind == AttributeKind.global do
+        throwError "invalid attribute 'cpspec', must be global"
+      -- Lookup the theorem
+      let env ← getEnv
+      let thDecl := env.constants.find! thName
+      let (fName, argName) ← MetaM.run' (getPSpecClassFunNames thDecl.type)
+      trace[Progress] "Registering class spec theorem for ({fName}, {argName})"
+      -- Update the entry if there is one, add an entry if there is none
+      let env :=
+        match (ext.getState (← getEnv)).find? fName with
+        | none =>
+          let m := RBMap.ofList [(argName, thName)]
+          ext.addEntry env (fName, m)
+        | some m =>
+          let m := m.insert argName thName
+          ext.addEntry env (fName, m)
+      setEnv env
+      pure ()
+  }
+  registerBuiltinAttribute attrImpl
+  pure { attr := attrImpl, ext := ext }
+
+/- The 2nd persistent map from type classes to pspec theorems -/
+initialize pspecClassExprAttr : PSpecClassExprAttr ← do
+  let ext : MapDeclarationExtension (HashMap Expr Name) ← mkMapDeclarationExtension `pspecClassExprMap
+  let attrImpl : AttributeImpl  := {
+    name := `cepspec
+    descr := "Marks theorems to use for type classes with the `progress` tactic"
+    add := fun thName stx attrKind => do
+      Attribute.Builtin.ensureNoArgs stx
+      -- TODO: use the attribute kind
+      unless attrKind == AttributeKind.global do
+        throwError "invalid attribute 'cpspec', must be global"
+      -- Lookup the theorem
+      let env ← getEnv
+      let thDecl := env.constants.find! thName
+      let (fName, arg) ← MetaM.run' (getPSpecClassFunNameArg thDecl.type)
+      -- Sanity check: no variables appear in the argument
+      MetaM.run' do
+        let fvars ← getFVarIds arg
+        if ¬ fvars.isEmpty then throwError "The first argument ({arg}) contains variables"
+      -- We store two bindings:
+      -- - arg to theorem name
+      -- - reduced arg to theorem name
+      let rarg ← MetaM.run' (reduceAll arg)
+      trace[Progress] "Registering class spec theorem for ({fName}, {arg}) and ({fName}, {rarg})"
+      -- Update the entry if there is one, add an entry if there is none
+      let env :=
+        match (ext.getState (← getEnv)).find? fName with
+        | none =>
+          let m := HashMap.ofList [(arg, thName), (rarg, thName)]
+          ext.addEntry env (fName, m)
+        | some m =>
+          let m := m.insert arg thName
+          let m := m.insert rarg thName
+          ext.addEntry env (fName, m)
+      setEnv env
+      pure ()
+  }
+  registerBuiltinAttribute attrImpl
+  pure { attr := attrImpl, ext := ext }
+
+
+def PSpecAttr.find? (s : PSpecAttr) (name : Name) : MetaM (Option Name) := do
+  return (s.ext.getState (← getEnv)).find? name
+
+def PSpecClassAttr.find? (s : PSpecClassAttr) (className argName : Name) : MetaM (Option Name) := do
+  match (s.ext.getState (← getEnv)).find? className with
+  | none => return none
+  | some map => return map.find? argName
+
+def PSpecClassExprAttr.find? (s : PSpecClassExprAttr) (className : Name) (arg : Expr) : MetaM (Option Name) := do
+  match (s.ext.getState (← getEnv)).find? className with
+  | none => return none
+  | some map => return map.find? arg
+
+def PSpecAttr.getState (s : PSpecAttr) : MetaM (NameMap Name) := do
+  pure (s.ext.getState (← getEnv))
+
+def PSpecClassAttr.getState (s : PSpecClassAttr) : MetaM (NameMap (NameMap Name)) := do
+  pure (s.ext.getState (← getEnv))
+
+def PSpecClassExprAttr.getState (s : PSpecClassExprAttr) : MetaM (NameMap (HashMap Expr Name)) := do
+  pure (s.ext.getState (← getEnv))
+
+def showStoredPSpec : MetaM Unit := do
+  let st ← pspecAttr.getState
+  let s := st.toList.foldl (fun s (f, th) => f!"{s}\n{f} → {th}") f!""
+  IO.println s
+
+def showStoredPSpecClass : MetaM Unit := do
+  let st ← pspecClassAttr.getState
+  let s := st.toList.foldl (fun s (f, m) =>
+    let ms := m.toList.foldl (fun s (f, th) =>
+      f!"{s}\n  {f} → {th}") f!""
+    f!"{s}\n{f} → [{ms}]") f!""
+  IO.println s
+
+def showStoredPSpecExprClass : MetaM Unit := do
+  let st ← pspecClassExprAttr.getState
+  let s := st.toList.foldl (fun s (f, m) =>
+    let ms := m.toList.foldl (fun s (f, th) =>
+      f!"{s}\n  {f} → {th}") f!""
+    f!"{s}\n{f} → [{ms}]") f!""
+  IO.println s
+
+end Progress
diff --git a/backends/lean/Base/Progress/Progress.lean b/backends/lean/Base/Progress/Progress.lean
new file mode 100644
index 00000000..6a4729dc
--- /dev/null
+++ b/backends/lean/Base/Progress/Progress.lean
@@ -0,0 +1,377 @@
+import Lean
+import Base.Arith
+import Base.Progress.Base
+import Base.Primitives -- TODO: remove?
+
+namespace Progress
+
+open Lean Elab Term Meta Tactic
+open Utils
+
+inductive TheoremOrLocal where
+| Theorem (thName : Name)
+| Local (asm : LocalDecl)
+
+instance : ToMessageData TheoremOrLocal where
+  toMessageData := λ x => match x with | .Theorem thName => m!"{thName}" | .Local asm => m!"{asm.userName}"
+
+/- Type to propagate the errors of `progressWith`.
+   We need this because we use the exceptions to backtrack, when trying to
+   use the assumptions for instance. When there is actually an error we want
+   to propagate to the user, we return it. -/
+inductive ProgressError
+| Ok
+| Error (msg : MessageData)
+deriving Inhabited
+
+def progressWith (fExpr : Expr) (th : TheoremOrLocal)
+  (keep : Option Name) (ids : Array (Option Name)) (splitPost : Bool)
+  (asmTac : TacticM Unit) : TacticM ProgressError := do
+  /- Apply the theorem
+     We try to match the theorem with the goal
+     In order to do so, we introduce meta-variables for all the parameters
+     (i.e., quantified variables and assumpions), and unify those with the goal.
+     Remark: we do not introduce meta-variables for the quantified variables
+     which don't appear in the function arguments (we want to let them
+     quantified).
+     We also make sure that all the meta variables which appear in the
+     function arguments have been instantiated
+   -/
+  let env ← getEnv
+  let thTy ← do
+    match th with
+    | .Theorem thName =>
+      let thDecl := env.constants.find! thName
+      -- We have to introduce fresh meta-variables for the universes already
+      let ul : List (Name × Level) ←
+        thDecl.levelParams.mapM (λ x => do pure (x, ← mkFreshLevelMVar))
+      let ulMap : HashMap Name Level := HashMap.ofList ul
+      let thTy := thDecl.type.instantiateLevelParamsCore (λ x => ulMap.find! x)
+      pure thTy
+    | .Local asmDecl => pure asmDecl.type
+  trace[Progress] "Looked up theorem/assumption type: {thTy}"
+  -- TODO: the tactic fails if we uncomment withNewMCtxDepth
+  -- withNewMCtxDepth do
+  let (mvars, binders, thExBody) ← forallMetaTelescope thTy
+  trace[Progress] "After stripping foralls: {thExBody}"
+  -- Introduce the existentially quantified variables and the post-condition
+  -- in the context
+  let thBody ←
+    existsTelescope thExBody.consumeMData fun _evars thBody => do
+    trace[Progress] "After stripping existentials: {thBody}"
+    let (thBody, _) ← optSplitConj thBody
+    trace[Progress] "After splitting the conjunction: {thBody}"
+    let (thBody, _) ← destEq thBody
+    trace[Progress] "After splitting equality: {thBody}"
+    -- There shouldn't be any existential variables in thBody
+    pure thBody.consumeMData
+  -- Match the body with the target
+  trace[Progress] "Matching:\n- body:\n{thBody}\n- target:\n{fExpr}"
+  let ok ← isDefEq thBody fExpr
+  if ¬ ok then throwError "Could not unify the theorem with the target:\n- theorem: {thBody}\n- target: {fExpr}"
+  let mgoal ← Tactic.getMainGoal
+  postprocessAppMVars `progress mgoal mvars binders true true
+  Term.synthesizeSyntheticMVarsNoPostponing
+  let thBody ← instantiateMVars thBody
+  trace[Progress] "thBody (after instantiation): {thBody}"
+  -- Add the instantiated theorem to the assumptions (we apply it on the metavariables).
+  let th ← do
+    match th with
+    | .Theorem thName => mkAppOptM thName (mvars.map some)
+    | .Local decl => mkAppOptM' (mkFVar decl.fvarId) (mvars.map some)
+  let asmName ← do match keep with | none => mkFreshAnonPropUserName | some n => do pure n
+  let thTy ← inferType th
+  let thAsm ← Utils.addDeclTac asmName th thTy (asLet := false)
+  withMainContext do -- The context changed - TODO: remove once addDeclTac is updated
+  let ngoal ← getMainGoal
+  trace[Progress] "current goal: {ngoal}"
+  trace[Progress] "current goal: {← ngoal.isAssigned}"
+  -- The assumption should be of the shape:
+  -- `∃ x1 ... xn, f args = ... ∧ ...`
+  -- We introduce the existentially quantified variables and split the top-most
+  -- conjunction if there is one. We use the provided `ids` list to name the
+  -- introduced variables.
+  let res ← splitAllExistsTac thAsm ids.toList fun h ids => do
+    -- Split the conjunctions.
+    -- For the conjunctions, we split according once to separate the equality `f ... = .ret ...`
+    -- from the postcondition, if there is, then continue to split the postcondition if there
+    -- are remaining ids.
+    let splitEqAndPost (k : Expr → Option Expr → List (Option Name) → TacticM ProgressError) : TacticM ProgressError := do
+      if ← isConj (← inferType h) then do
+        let hName := (← h.fvarId!.getDecl).userName
+        let (optIds, ids) ← do
+          match ids with
+          | [] => do pure (some (hName, ← mkFreshAnonPropUserName), [])
+          | none :: ids => do pure (some (hName, ← mkFreshAnonPropUserName), ids)
+          | some id :: ids => do pure (some (hName, id), ids)
+        splitConjTac h optIds (fun hEq hPost => k hEq (some hPost) ids)
+      else k h none ids
+    -- Simplify the target by using the equality and some monad simplifications,
+    -- then continue splitting the post-condition
+    splitEqAndPost fun hEq hPost ids => do
+    trace[Progress] "eq and post:\n{hEq} : {← inferType hEq}\n{hPost}"
+    simpAt [] [``Primitives.bind_tc_ret, ``Primitives.bind_tc_fail, ``Primitives.bind_tc_div]
+           [hEq.fvarId!] (.targets #[] true)
+    -- Clear the equality, unless the user requests not to do so
+    let mgoal ← do
+      if keep.isSome then getMainGoal
+      else do
+        let mgoal ← getMainGoal
+        mgoal.tryClearMany #[hEq.fvarId!]
+    setGoals (mgoal :: (← getUnsolvedGoals))
+    trace[Progress] "Goal after splitting eq and post and simplifying the target: {mgoal}"
+    -- Continue splitting following the post following the user's instructions
+    match hPost with
+    | none =>
+      -- Sanity check
+      if ¬ ids.isEmpty then
+        return (.Error m!"Too many ids provided ({ids}): there is no postcondition to split")
+      else return .Ok
+    | some hPost => do
+      let rec splitPostWithIds (prevId : Name) (hPost : Expr) (ids0 : List (Option Name)) : TacticM ProgressError := do
+        match ids0 with
+        | [] =>
+          /- We used all the user provided ids.
+             Split the remaining conjunctions by using fresh ids if the user
+             instructed to fully split the post-condition, otherwise stop -/
+          if splitPost then
+            splitFullConjTac true hPost (λ _ => pure .Ok)
+          else pure .Ok
+        | nid :: ids => do
+          trace[Progress] "Splitting post: {← inferType hPost}"
+          -- Split
+          let nid ← do
+            match nid with
+            | none => mkFreshAnonPropUserName
+            | some nid => pure nid
+          trace[Progress] "\n- prevId: {prevId}\n- nid: {nid}\n- remaining ids: {ids}"
+          if ← isConj (← inferType hPost) then
+            splitConjTac hPost (some (prevId, nid)) (λ _ nhPost => splitPostWithIds nid nhPost ids)
+          else return (.Error m!"Too many ids provided ({ids0}) not enough conjuncts to split in the postcondition")
+      let curPostId := (← hPost.fvarId!.getDecl).userName
+      splitPostWithIds curPostId hPost ids
+  match res with
+  | .Error _ => return res -- Can we get there? We're using "return"
+  | .Ok =>
+    -- Update the set of goals
+    let curGoals ← getUnsolvedGoals
+    let newGoals := mvars.map Expr.mvarId!
+    let newGoals ← newGoals.filterM fun mvar => not <$> mvar.isAssigned
+    trace[Progress] "new goals: {newGoals}"
+    setGoals newGoals.toList
+    allGoals asmTac
+    let newGoals ← getUnsolvedGoals
+    setGoals (newGoals ++ curGoals)
+    trace[Progress] "progress: replaced the goals"
+    --
+    pure .Ok
+
+-- Small utility: if `args` is not empty, return the name of the app in the first
+-- arg, if it is a const.
+def getFirstArgAppName (args : Array Expr) : MetaM (Option Name) := do
+  if args.size = 0 then pure none
+  else
+    (args.get! 0).withApp fun f _ => do
+    if f.isConst then pure (some f.constName)
+    else pure none
+
+def getFirstArg (args : Array Expr) : Option Expr := do
+  if args.size = 0 then none
+  else some (args.get! 0)
+
+/- Helper: try to lookup a theorem and apply it, or continue with another tactic
+   if it fails -/
+def tryLookupApply (keep : Option Name) (ids : Array (Option Name)) (splitPost : Bool)
+  (asmTac : TacticM Unit) (fExpr : Expr)
+  (kind : String) (th : Option TheoremOrLocal) (x : TacticM Unit) : TacticM Unit := do
+  let res ← do
+    match th with
+    | none =>
+      trace[Progress] "Could not find a {kind}"
+      pure none
+    | some th => do
+      trace[Progress] "Lookuped up {kind}: {th}"
+      -- Apply the theorem
+      let res ← do
+        try
+          let res ← progressWith fExpr th keep ids splitPost asmTac
+          pure (some res)
+        catch _ => none
+  match res with
+  | some .Ok => return ()
+  | some (.Error msg) => throwError msg
+  | none => x
+
+-- The array of ids are identifiers to use when introducing fresh variables
+def progressAsmsOrLookupTheorem (keep : Option Name) (withTh : Option TheoremOrLocal)
+  (ids : Array (Option Name)) (splitPost : Bool) (asmTac : TacticM Unit) : TacticM Unit := do
+  withMainContext do
+  -- Retrieve the goal
+  let mgoal ← Tactic.getMainGoal
+  let goalTy ← mgoal.getType
+  trace[Progress] "goal: {goalTy}"
+  -- Dive into the goal to lookup the theorem
+  let (fExpr, fName, args) ← do
+    withPSpec goalTy fun desc =>
+    -- TODO: check that no quantified variables in the arguments
+    pure (desc.fExpr, desc.fName, desc.args)
+  trace[Progress] "Function: {fName}"
+  -- If the user provided a theorem/assumption: use it.
+  -- Otherwise, lookup one.
+  match withTh with
+  | some th => do
+    match ← progressWith fExpr th keep ids splitPost asmTac with
+    | .Ok => return ()
+    | .Error msg => throwError msg
+  | none =>
+    -- Try all the assumptions one by one and if it fails try to lookup a theorem.
+    let ctx ← Lean.MonadLCtx.getLCtx
+    let decls ← ctx.getDecls
+    for decl in decls.reverse do
+      trace[Progress] "Trying assumption: {decl.userName} : {decl.type}"
+      let res ← do try progressWith fExpr (.Local decl) keep ids splitPost asmTac catch _ => continue
+      match res with
+      | .Ok => return ()
+      | .Error msg => throwError msg
+    -- It failed: try to lookup a theorem
+    -- TODO: use a list of theorems, and try them one by one?
+    trace[Progress] "No assumption succeeded: trying to lookup a theorem"
+    let pspec ← do
+      let thName ← pspecAttr.find? fName
+      pure (thName.map fun th => .Theorem th)
+    tryLookupApply keep ids splitPost asmTac fExpr "pspec theorem" pspec do
+    -- It failed: try to lookup a *class* expr spec theorem (those are more
+    -- specific than class spec theorems)
+    let pspecClassExpr ← do
+      match getFirstArg args with
+      | none => pure none
+      | some arg => do
+        let thName ← pspecClassExprAttr.find? fName arg
+        pure (thName.map fun th => .Theorem th)
+    tryLookupApply keep ids splitPost asmTac fExpr "pspec class expr theorem" pspecClassExpr do
+    -- It failed: try to lookup a *class* spec theorem
+    let pspecClass ← do
+      match ← getFirstArgAppName args with
+      | none => pure none
+      | some argName => do
+        let thName ← pspecClassAttr.find? fName argName
+        pure (thName.map fun th => .Theorem th)
+    tryLookupApply keep ids splitPost asmTac fExpr "pspec class theorem" pspecClass do
+    -- Try a recursive call - we try the assumptions of kind "auxDecl"
+    let ctx ← Lean.MonadLCtx.getLCtx
+    let decls ← ctx.getAllDecls
+    let decls := decls.filter (λ decl => match decl.kind with
+      | .default | .implDetail => false | .auxDecl => true)
+    for decl in decls.reverse do
+      trace[Progress] "Trying recursive assumption: {decl.userName} : {decl.type}"
+      let res ← do try progressWith fExpr (.Local decl) keep ids splitPost asmTac catch _ => continue
+      match res with
+      | .Ok => return ()
+      | .Error msg => throwError msg
+    -- Nothing worked: failed
+    throwError "Progress failed"
+
+syntax progressArgs := ("keep" (ident <|> "_"))? ("with" ident)? ("as" " ⟨ " (ident <|> "_"),* " .."? " ⟩")?
+
+def evalProgress (args : TSyntax `Progress.progressArgs) : TacticM Unit := do
+  let args := args.raw
+  -- Process the arguments to retrieve the identifiers to use
+  trace[Progress] "Progress arguments: {args}"
+  let (keepArg, withArg, asArgs) ←
+    match args.getArgs.toList with
+    | [keepArg, withArg, asArgs] => do pure (keepArg, withArg, asArgs)
+    | _ => throwError "Unexpected: invalid arguments"
+  let keep : Option Name ← do
+    trace[Progress] "Keep arg: {keepArg}"
+    let args := keepArg.getArgs
+    if args.size > 0 then do
+      trace[Progress] "Keep args: {args}"
+      let arg := args.get! 1
+      trace[Progress] "Keep arg: {arg}"
+      if arg.isIdent then pure (some arg.getId)
+      else do pure (some (← mkFreshAnonPropUserName))
+    else do pure none
+  trace[Progress] "Keep: {keep}"
+  let withArg ← do
+    let withArg := withArg.getArgs
+    if withArg.size > 0 then
+      let id := withArg.get! 1
+      trace[Progress] "With arg: {id}"
+      -- Attempt to lookup a local declaration
+      match (← getLCtx).findFromUserName? id.getId with
+      | some decl => do
+        trace[Progress] "With arg: local decl"
+        pure (some (.Local decl))
+      | none => do
+        -- Not a local declaration: should be a theorem
+        trace[Progress] "With arg: theorem"
+        addCompletionInfo <| CompletionInfo.id id id.getId (danglingDot := false) {} none
+        let cs ← resolveGlobalConstWithInfos id
+        match cs with
+        | [] => throwError "Could not find theorem {id}"
+        | id :: _ =>
+          pure (some (.Theorem id))
+    else pure none
+  let ids :=
+    let args := asArgs.getArgs
+    let args := (args.get! 2).getSepArgs
+    args.map (λ s => if s.isIdent then some s.getId else none)
+  trace[Progress] "User-provided ids: {ids}"
+  let splitPost : Bool :=
+    let args := asArgs.getArgs
+    (args.get! 3).getArgs.size > 0
+  trace[Progress] "Split post: {splitPost}"
+  /- For scalarTac we have a fast track: if the goal is not a linear
+     arithmetic goal, we skip (note that otherwise, scalarTac would try
+     to prove a contradiction) -/
+  let scalarTac : TacticM Unit := do
+    if ← Arith.goalIsLinearInt then
+      -- Also: we don't try to split the goal if it is a conjunction
+      -- (it shouldn't be)
+      Arith.scalarTac false
+    else
+      throwError "Not a linear arithmetic goal"
+  progressAsmsOrLookupTheorem keep withArg ids splitPost (
+    withMainContext do
+    trace[Progress] "trying to solve assumption: {← getMainGoal}"
+    firstTac [assumptionTac, scalarTac])
+  trace[Diverge] "Progress done"
+
+elab "progress" args:progressArgs : tactic =>
+  evalProgress args
+
+namespace Test
+  open Primitives Result
+
+  set_option trace.Progress true
+  set_option pp.rawOnError true
+
+  #eval showStoredPSpec
+  #eval showStoredPSpecClass
+
+  example {ty} {x y : Scalar ty}
+    (hmin : Scalar.min ty ≤ x.val + y.val)
+    (hmax : x.val + y.val ≤ Scalar.max ty) :
+    ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+    progress keep _ as ⟨ z, h1 .. ⟩
+    simp [*, h1]
+
+  example {ty} {x y : Scalar ty}
+    (hmin : Scalar.min ty ≤ x.val + y.val)
+    (hmax : x.val + y.val ≤ Scalar.max ty) :
+    ∃ z, x + y = ret z ∧ z.val = x.val + y.val := by
+    progress keep h with Scalar.add_spec as ⟨ z ⟩
+    simp [*, h]
+
+  /- Checking that universe instantiation works: the original spec uses
+     `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.val = v.val.update i.val x := by
+    progress
+    simp [*]
+
+end Test
+
+end Progress
diff --git a/backends/lean/Base/Utils.lean b/backends/lean/Base/Utils.lean
new file mode 100644
index 00000000..1f8f1455
--- /dev/null
+++ b/backends/lean/Base/Utils.lean
@@ -0,0 +1,640 @@
+import Lean
+import Mathlib.Tactic.Core
+import Mathlib.Tactic.LeftRight
+import Base.UtilsBase
+
+/-
+Mathlib tactics:
+- rcases: https://leanprover-community.github.io/mathlib_docs/tactics.html#rcases
+- split_ifs: https://leanprover-community.github.io/mathlib_docs/tactics.html#split_ifs
+- norm_num: https://leanprover-community.github.io/mathlib_docs/tactics.html#norm_num
+- should we use linarith or omega?
+- hint: https://leanprover-community.github.io/mathlib_docs/tactics.html#hint
+- classical: https://leanprover-community.github.io/mathlib_docs/tactics.html#classical
+-/
+
+/-
+TODO:
+- we want an easier to use cases:
+  - keeps in the goal an equation of the shape: `t = case`
+  - if called on Prop terms, uses Classical.em
+  Actually, the cases from mathlib seems already quite powerful
+  (https://leanprover-community.github.io/mathlib_docs/tactics.html#cases)
+  For instance: cases h : e
+  Also: **casesm**
+- better split tactic
+- we need conversions to operate on the head of applications.
+  Actually, something like this works:
+  ```
+  conv at Hl =>
+    apply congr_fun
+    simp [fix_fuel_P]
+  ```
+  Maybe we need a rpt ... ; focus?
+- simplifier/rewriter have a strange behavior sometimes
+-/
+
+
+namespace List
+
+  -- TODO: I could not find this function??
+  @[simp] def flatten {a : Type u} : List (List a) → List a
+  | [] => []
+  | x :: ls => x ++ flatten ls
+
+end List
+
+-- TODO: move?
+@[simp]
+theorem neq_imp {α : Type u} {x y : α} (h : ¬ x = y) : ¬ y = x := by intro; simp_all
+
+namespace Lean
+
+namespace LocalContext
+
+  open Lean Lean.Elab Command Term Lean.Meta
+
+  -- Small utility: return the list of declarations in the context, from
+  -- the last to the first.
+  def getAllDecls (lctx : Lean.LocalContext) : MetaM (List Lean.LocalDecl) :=
+    lctx.foldrM (fun d ls => do let d ← instantiateLocalDeclMVars d; pure (d :: ls)) []
+
+  -- Return the list of declarations in the context, but filter the
+  -- declarations which are considered as implementation details
+  def getDecls (lctx : Lean.LocalContext) : MetaM (List Lean.LocalDecl) := do
+    let ls ← lctx.getAllDecls
+    pure (ls.filter (fun d => not d.isImplementationDetail))
+
+end LocalContext
+
+end Lean
+
+namespace Utils
+
+open Lean Elab Term Meta Tactic
+
+-- Useful helper to explore definitions and figure out the variant
+-- of their sub-expressions.
+def explore_term (incr : String) (e : Expr) : MetaM Unit :=
+  match e with
+  | .bvar _ => do logInfo m!"{incr}bvar: {e}"; return ()
+  | .fvar _ => do logInfo m!"{incr}fvar: {e}"; return ()
+  | .mvar _ => do logInfo m!"{incr}mvar: {e}"; return ()
+  | .sort _ => do logInfo m!"{incr}sort: {e}"; return ()
+  | .const _ _ => do logInfo m!"{incr}const: {e}"; return ()
+  | .app fn arg => do
+    logInfo m!"{incr}app: {e}"
+    explore_term (incr ++ "  ") fn
+    explore_term (incr ++ "  ") arg
+  | .lam _bName bTy body _binfo => do
+    logInfo m!"{incr}lam: {e}"
+    explore_term (incr ++ "  ") bTy
+    explore_term (incr ++ "  ") body
+  | .forallE _bName bTy body _bInfo => do
+    logInfo m!"{incr}forallE: {e}"
+    explore_term (incr ++ "  ") bTy
+    explore_term (incr ++ "  ") body
+  | .letE _dName ty val body _nonDep => do
+    logInfo m!"{incr}letE: {e}"
+    explore_term (incr ++ "  ") ty
+    explore_term (incr ++ "  ") val
+    explore_term (incr ++ "  ") body
+  | .lit _ => do logInfo m!"{incr}lit: {e}"; return ()
+  | .mdata _ e => do
+    logInfo m!"{incr}mdata: {e}"
+    explore_term (incr ++ "  ") e
+  | .proj _ _ struct => do
+    logInfo m!"{incr}proj: {e}"
+    explore_term (incr ++ "  ") struct
+
+def explore_decl (n : Name) : TermElabM Unit := do
+  logInfo m!"Name: {n}"
+  let env ← getEnv
+  let decl := env.constants.find! n
+  match decl with
+  | .defnInfo val =>
+     logInfo m!"About to explore defn: {decl.name}"
+     logInfo m!"# Type:"
+     explore_term "" val.type
+     logInfo m!"# Value:"
+     explore_term "" val.value
+  | .axiomInfo _  => throwError m!"axiom: {n}"
+  | .thmInfo _    => throwError m!"thm: {n}"
+  | .opaqueInfo _ => throwError m!"opaque: {n}"
+  | .quotInfo _   => throwError m!"quot: {n}"
+  | .inductInfo _ => throwError m!"induct: {n}"
+  | .ctorInfo _   => throwError m!"ctor: {n}"
+  | .recInfo _    => throwError m!"rec: {n}"
+
+syntax (name := printDecl) "print_decl " ident : command
+
+open Lean.Elab.Command
+
+@[command_elab printDecl] def elabPrintDecl : CommandElab := fun stx => do
+  liftTermElabM do
+    let id := stx[1]
+    addCompletionInfo <| CompletionInfo.id id id.getId (danglingDot := false) {} none
+    let cs ← resolveGlobalConstWithInfos id
+    explore_decl cs[0]!
+
+private def test1 : Nat := 0
+private def test2 (x : Nat) : Nat := x
+print_decl test1
+print_decl test2
+
+def printDecls (decls : List LocalDecl) : MetaM Unit := do
+  let decls ← decls.foldrM (λ decl msg => do
+    pure (m!"\n{decl.toExpr} : {← inferType decl.toExpr}" ++ msg)) m!""
+  logInfo m!"# Ctx decls:{decls}"
+
+-- Small utility: print all the declarations in the context (including the "implementation details")
+elab "print_all_ctx_decls" : tactic => do
+  let ctx ← Lean.MonadLCtx.getLCtx
+  let decls ← ctx.getAllDecls
+  printDecls decls
+
+-- Small utility: print all declarations in the context
+elab "print_ctx_decls" : tactic => do
+  let ctx ← Lean.MonadLCtx.getLCtx
+  let decls ← ctx.getDecls
+  printDecls decls
+
+-- A map visitor function for expressions (adapted from `AbstractNestedProofs.visit`)
+-- The continuation takes as parameters:
+-- - the current depth of the expression (useful for printing/debugging)
+-- - the expression to explore
+partial def mapVisit (k : Nat → Expr → MetaM Expr) (e : Expr) : MetaM Expr := do
+  let mapVisitBinders (xs : Array Expr) (k2 : MetaM Expr) : MetaM Expr := do
+    let localInstances ← getLocalInstances
+    let mut lctx ← getLCtx
+    for x in xs do
+      let xFVarId := x.fvarId!
+      let localDecl ← xFVarId.getDecl
+      let type      ← mapVisit k localDecl.type
+      let localDecl := localDecl.setType type
+      let localDecl ← match localDecl.value? with
+         | some value => let value ← mapVisit k value; pure <| localDecl.setValue value
+         | none       => pure localDecl
+      lctx :=lctx.modifyLocalDecl xFVarId fun _ => localDecl
+    withLCtx lctx localInstances k2
+  -- TODO: use a cache? (Lean.checkCache)
+  let rec visit (i : Nat) (e : Expr) : MetaM Expr := do
+    -- Explore
+    let e ← k i e
+    match e with
+    | .bvar _
+    | .fvar _
+    | .mvar _
+    | .sort _
+    | .lit _
+    | .const _ _ => pure e
+    | .app .. => do e.withApp fun f args => return mkAppN f (← args.mapM (visit (i + 1)))
+    | .lam .. =>
+      lambdaLetTelescope e fun xs b =>
+        mapVisitBinders xs do mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+    | .forallE .. => do
+      forallTelescope e fun xs b => mapVisitBinders xs do mkForallFVars xs (← visit (i + 1) b)
+    | .letE .. => do
+      lambdaLetTelescope e fun xs b => mapVisitBinders xs do
+        mkLambdaFVars xs (← visit (i + 1) b) (usedLetOnly := false)
+    | .mdata _ b => return e.updateMData! (← visit (i + 1) b)
+    | .proj _ _ b => return e.updateProj! (← visit (i + 1) b)
+  visit 0 e
+
+-- Generate a fresh user name for an anonymous proposition to introduce in the
+-- assumptions
+def mkFreshAnonPropUserName := mkFreshUserName `_
+
+section Methods
+  variable [MonadLiftT MetaM m] [MonadControlT MetaM m] [Monad m] [MonadError m]
+  variable {a : Type}
+
+  /- Like `lambdaTelescopeN` but only destructs a fixed number of lambdas -/
+  def lambdaTelescopeN (e : Expr) (n : Nat) (k : Array Expr → Expr → m a) : m a :=
+    lambdaTelescope e fun xs body => do
+    if xs.size < n then throwError "lambdaTelescopeN: not enough lambdas"
+    let xs := xs.extract 0 n
+    let ys := xs.extract n xs.size
+    let body ← liftMetaM (mkLambdaFVars ys body)
+    k xs body
+
+  /- Like `lambdaTelescope`, but only destructs one lambda
+     TODO: is there an equivalent of this function somewhere in the
+     standard library? -/
+  def lambdaOne (e : Expr) (k : Expr → Expr → m a) : m a :=
+    lambdaTelescopeN e 1 λ xs b => k (xs.get! 0) b
+
+  def isExists (e : Expr) : Bool := e.getAppFn.isConstOf ``Exists ∧ e.getAppNumArgs = 2
+
+  -- Remark: Lean doesn't find the inhabited and nonempty instances if we don'
+  -- put them explicitely in the signature
+  partial def existsTelescopeProcess [Inhabited (m a)] [Nonempty (m a)]
+    (fvars : Array Expr) (e : Expr) (k : Array Expr → Expr → m a) : m a := do
+    -- Attempt to deconstruct an existential
+    if isExists e then do
+      let p := e.appArg!
+      lambdaOne p fun x ne =>
+      existsTelescopeProcess (fvars.push x) ne k
+    else
+      -- No existential: call the continuation
+      k fvars e
+
+  def existsTelescope [Inhabited (m a)] [Nonempty (m a)] (e : Expr) (k : Array Expr → Expr → m a) : m a := do
+    existsTelescopeProcess #[] e k
+
+end Methods
+
+-- TODO: this should take a continuation
+def addDeclTac (name : Name) (val : Expr) (type : Expr) (asLet : Bool) : TacticM Expr :=
+  -- I don't think we need that
+  withMainContext do
+  -- Insert the new declaration
+  let withDecl := if asLet then withLetDecl name type val else withLocalDeclD name type
+  withDecl fun nval => do
+    -- For debugging
+    let lctx ← Lean.MonadLCtx.getLCtx
+    let fid := nval.fvarId!
+    let decl := lctx.get! fid
+    trace[Arith] "  new decl: \"{decl.userName}\" ({nval}) : {decl.type} := {decl.value}"
+    --
+    -- Tranform the main goal `?m0` to `let x = nval in ?m1`
+    let mvarId ← getMainGoal
+    let newMVar ← mkFreshExprSyntheticOpaqueMVar (← mvarId.getType)
+    let newVal ← mkLetFVars #[nval] newMVar
+    -- There are two cases:
+    -- - asLet is true: newVal is `let $name := $val in $newMVar`
+    -- - asLet is false: ewVal is `λ $name => $newMVar`
+    --   We need to apply it to `val`
+    let newVal := if asLet then newVal else mkAppN newVal #[val]
+    -- Assign the main goal and update the current goal
+    mvarId.assign newVal
+    let goals ← getUnsolvedGoals
+    setGoals (newMVar.mvarId! :: goals)
+    -- Return the new value - note: we are in the *new* context, created
+    -- after the declaration was added, so it will persist
+    pure nval
+
+def addDeclTacSyntax (name : Name) (val : Syntax) (asLet : Bool) : TacticM Unit :=
+  -- I don't think we need that
+  withMainContext do
+  --
+  let val ← Term.elabTerm val .none
+  let type ← inferType val
+  -- In some situations, the type will be left as a metavariable (for instance,
+  -- if the term is `3`, Lean has the choice between `Nat` and `Int` and will
+  -- not choose): we force the instantiation of the meta-variable
+  synthesizeSyntheticMVarsUsingDefault
+  --
+  let _ ← addDeclTac name val type asLet
+
+elab "custom_let " n:ident " := " v:term : tactic => do
+  addDeclTacSyntax n.getId v (asLet := true)
+
+elab "custom_have " n:ident " := " v:term : tactic =>
+  addDeclTacSyntax n.getId v (asLet := false)
+
+example : Nat := by
+  custom_let x := 4
+  custom_have y := 4
+  apply y
+
+example (x : Bool) : Nat := by
+  cases x <;> custom_let x := 3 <;> apply x
+
+-- Repeatedly apply a tactic
+partial def repeatTac (tac : TacticM Unit) : TacticM Unit := do
+  try
+    tac
+    allGoals (focus (repeatTac tac))
+  -- TODO: does this restore the state?
+  catch _ => pure ()
+
+def firstTac (tacl : List (TacticM Unit)) : TacticM Unit := do
+  match tacl with
+  | [] => pure ()
+  | tac :: tacl =>
+    -- Should use try ... catch or Lean.observing?
+    -- Generally speaking we should use Lean.observing? to restore the state,
+    -- but with tactics the try ... catch variant seems to work
+    try do
+      tac
+      -- Check that there are no remaining goals
+      let gl ← Tactic.getUnsolvedGoals
+      if ¬ gl.isEmpty then throwError "tactic failed"
+    catch _ => firstTac tacl
+/-    let res ← Lean.observing? do
+      tac
+      -- Check that there are no remaining goals
+      let gl ← Tactic.getUnsolvedGoals
+      if ¬ gl.isEmpty then throwError "tactic failed"
+    match res with
+    | some _ => pure ()
+    | none => firstTac tacl -/
+
+-- Taken from Lean.Elab.evalAssumption
+def assumptionTac : TacticM Unit :=
+  liftMetaTactic fun mvarId => do mvarId.assumption; pure []
+
+def isConj (e : Expr) : MetaM Bool :=
+  e.consumeMData.withApp fun f args => pure (f.isConstOf ``And ∧ args.size = 2)
+
+-- Return the first conjunct if the expression is a conjunction, or the
+-- expression itself otherwise. Also return the second conjunct if it is a
+-- conjunction.
+def optSplitConj (e : Expr) : MetaM (Expr × Option Expr) := do
+  e.consumeMData.withApp fun f args =>
+  if f.isConstOf ``And ∧ args.size = 2 then pure (args.get! 0, some (args.get! 1))
+  else pure (e, none)
+
+-- Split the goal if it is a conjunction
+def splitConjTarget : TacticM Unit := do
+  withMainContext do
+  let g ← getMainTarget
+  trace[Utils] "splitConjTarget: goal: {g}"
+  -- The tactic was initially implemened with `_root_.Lean.MVarId.apply`
+  -- but it tended to mess the goal by unfolding terms, even when it failed
+  let (l, r) ← optSplitConj g
+  match r with
+  | none => do throwError "The goal is not a conjunction"
+  | some r => do
+    let lmvar ← mkFreshExprSyntheticOpaqueMVar l
+    let rmvar ← mkFreshExprSyntheticOpaqueMVar r
+    let and_intro ← mkAppM ``And.intro #[lmvar, rmvar]
+    let g ← getMainGoal
+    g.assign and_intro
+    let goals ← getUnsolvedGoals
+    setGoals (lmvar.mvarId! :: rmvar.mvarId! :: goals)
+
+-- Destruct an equaliy and return the two sides
+def destEq (e : Expr) : MetaM (Expr × Expr) := do
+  e.withApp fun f args =>
+  if f.isConstOf ``Eq ∧ args.size = 3 then pure (args.get! 1, args.get! 2)
+  else throwError "Not an equality: {e}"
+
+-- Return the set of FVarIds in the expression
+partial def getFVarIds (e : Expr) (hs : HashSet FVarId := HashSet.empty) : MetaM (HashSet FVarId) := do
+  e.withApp fun body args => do
+  let hs := if body.isFVar then hs.insert body.fvarId! else hs
+  args.foldlM (fun hs arg => getFVarIds arg hs) hs
+
+-- Tactic to split on a disjunction.
+-- The expression `h` should be an fvar.
+-- TODO: there must be simpler. Use use _root_.Lean.MVarId.cases for instance
+def splitDisjTac (h : Expr) (kleft kright : TacticM Unit) : TacticM Unit := do
+  trace[Arith] "assumption on which to split: {h}"
+  -- Retrieve the main goal
+  withMainContext do
+  let goalType ← getMainTarget
+  let hDecl := (← getLCtx).get! h.fvarId!
+  let hName := hDecl.userName
+  -- Case disjunction
+  let hTy ← inferType h
+  hTy.withApp fun f xs => do
+  trace[Arith] "as app: {f} {xs}"
+  -- Sanity check
+  if ¬ (f.isConstOf ``Or ∧ xs.size = 2) then throwError "Invalid argument to splitDisjTac"
+  let a := xs.get! 0
+  let b := xs.get! 1
+  -- Introduce the new goals
+  -- Returns:
+  -- - the match branch
+  -- - a fresh new mvar id
+  let mkGoal (hTy : Expr) (nGoalName : String) : MetaM (Expr × MVarId) := do
+    -- Introduce a variable for the assumption (`a` or `b`). Note that we reuse
+    -- the name of the assumption we split.
+    withLocalDeclD hName hTy fun var => do
+    -- The new goal
+    let mgoal ← mkFreshExprSyntheticOpaqueMVar goalType (tag := Name.mkSimple nGoalName)
+    -- Clear the assumption that we split
+    let mgoal ← mgoal.mvarId!.tryClearMany #[h.fvarId!]
+    -- The branch expression
+    let branch ← mkLambdaFVars #[var] (mkMVar mgoal)
+    pure (branch, mgoal)
+  let (inl, mleft) ← mkGoal a "left"
+  let (inr, mright) ← mkGoal b "right"
+  trace[Arith] "left: {inl}: {mleft}"
+  trace[Arith] "right: {inr}: {mright}"
+  -- Create the match expression
+  withLocalDeclD (← mkFreshAnonPropUserName) hTy fun hVar => do
+  let motive ← mkLambdaFVars #[hVar] goalType
+  let casesExpr ← mkAppOptM ``Or.casesOn #[a, b, motive, h, inl, inr]
+  let mgoal ← getMainGoal
+  trace[Arith] "goals: {← getUnsolvedGoals}"
+  trace[Arith] "main goal: {mgoal}"
+  mgoal.assign casesExpr
+  let goals ← getUnsolvedGoals
+  -- Focus on the left
+  setGoals [mleft]
+  withMainContext kleft
+  let leftGoals ← getUnsolvedGoals
+  -- Focus on the right
+  setGoals [mright]
+  withMainContext kright
+  let rightGoals ← getUnsolvedGoals
+  -- Put all the goals back
+  setGoals (leftGoals ++ rightGoals ++ goals)
+  trace[Arith] "new goals: {← getUnsolvedGoals}"
+
+elab "split_disj " n:ident : tactic => do
+  withMainContext do
+  let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
+  let fvar := mkFVar decl.fvarId
+  splitDisjTac fvar (fun _ => pure ()) (fun _ => pure ())
+
+example (x y : Int) (h0 : x ≤ y ∨ x ≥ y) : x ≤ y ∨ x ≥ y := by
+  split_disj h0
+  . left; assumption
+  . right; assumption
+
+
+-- Tactic to split on an exists.
+-- `h` must be an FVar
+def splitExistsTac (h : Expr) (optId : Option Name) (k : Expr → Expr → TacticM α) : TacticM α := do
+  withMainContext do
+  let goal ←  getMainGoal
+  let hTy ← inferType h
+  if isExists hTy then do
+    -- Try to use the user-provided names
+    let altVarNames ← do
+      let hDecl ← h.fvarId!.getDecl
+      let id ← do
+        match optId with
+        | none => mkFreshUserName `x
+        | some id => pure id
+      pure #[{ varNames := [id, hDecl.userName] }]
+    let newGoals ← goal.cases h.fvarId! altVarNames
+    -- There should be exactly one goal
+    match newGoals.toList with
+    | [ newGoal ] =>
+      -- Set the new goal
+      let goals ← getUnsolvedGoals
+      setGoals (newGoal.mvarId :: goals)
+      -- There should be exactly two fields
+      let fields := newGoal.fields
+      withMainContext do
+      k (fields.get! 0) (fields.get! 1)
+    | _ =>
+      throwError "Unreachable"
+  else
+    throwError "Not a conjunction"
+
+-- TODO: move
+def listTryPopHead (ls : List α) : Option α × List α :=
+  match ls with
+  | [] => (none, ls)
+  | hd :: tl => (some hd, tl)
+
+/- Destruct all the existentials appearing in `h`, and introduce them as variables
+   in the context.
+
+   If `ids` is not empty, we use it to name the introduced variables. We
+   transmit the stripped expression and the remaining ids to the continuation.
+ -/
+partial def splitAllExistsTac [Inhabited α] (h : Expr) (ids : List (Option Name)) (k : Expr → List (Option Name) → TacticM α) : TacticM α := do
+  try
+    let (optId, ids) :=
+      match ids with
+      | [] => (none, [])
+      | x :: ids => (x, ids)
+    splitExistsTac h optId (fun _ body => splitAllExistsTac body ids k)
+  catch _ => k h ids
+
+-- Tactic to split on a conjunction.
+def splitConjTac (h : Expr) (optIds : Option (Name × Name)) (k : Expr → Expr → TacticM α)  : TacticM α := do
+  withMainContext do
+  let goal ←  getMainGoal
+  let hTy ← inferType h
+  if ← isConj hTy then do
+    -- Try to use the user-provided names
+    let altVarNames ←
+      match optIds with
+      | none => do
+        let id0 ← mkFreshAnonPropUserName
+        let id1 ← mkFreshAnonPropUserName
+        pure #[{ varNames := [id0, id1] }]
+      | some (id0, id1) => do
+        pure #[{ varNames := [id0, id1] }]
+    let newGoals ← goal.cases h.fvarId! altVarNames
+    -- There should be exactly one goal
+    match newGoals.toList with
+    | [ newGoal ] =>
+      -- Set the new goal
+      let goals ← getUnsolvedGoals
+      setGoals (newGoal.mvarId :: goals)
+      -- There should be exactly two fields
+      let fields := newGoal.fields
+      withMainContext do
+      k (fields.get! 0) (fields.get! 1)
+    | _ =>
+      throwError "Unreachable"
+  else
+    throwError "Not a conjunction"
+
+-- Tactic to fully split a conjunction
+partial def splitFullConjTacAux [Inhabited α] [Nonempty α] (keepCurrentName : Bool) (l : List Expr) (h : Expr) (k : List Expr → TacticM α)  : TacticM α := do
+  try
+    let ids ← do
+      if keepCurrentName then do
+        let cur := (← h.fvarId!.getDecl).userName
+        let nid ← mkFreshAnonPropUserName
+        pure (some (cur, nid))
+      else
+        pure none
+    splitConjTac h ids (λ h1 h2 =>
+      splitFullConjTacAux keepCurrentName l h1 (λ l1 =>
+        splitFullConjTacAux keepCurrentName l1 h2 (λ l2 =>
+          k l2)))
+  catch _ =>
+    k (h :: l)
+
+-- Tactic to fully split a conjunction
+-- `keepCurrentName`: if `true`, then the first conjunct has the name of the original assumption
+def splitFullConjTac [Inhabited α] [Nonempty α] (keepCurrentName : Bool) (h : Expr) (k : List Expr → TacticM α)  : TacticM α := do
+  splitFullConjTacAux keepCurrentName [] h (λ l => k l.reverse)
+
+syntax optAtArgs := ("at" ident)?
+def elabOptAtArgs (args : TSyntax `Utils.optAtArgs) : TacticM (Option Expr) := do
+  withMainContext do
+  let args := (args.raw.getArgs.get! 0).getArgs
+  if args.size > 0 then do
+    let n := (args.get! 1).getId
+    let decl ← Lean.Meta.getLocalDeclFromUserName n
+    let fvar := mkFVar decl.fvarId
+    pure (some fvar)
+  else
+    pure none
+
+elab "split_conj" args:optAtArgs : tactic => do
+  withMainContext do
+  match ← elabOptAtArgs args with
+  | some fvar => do
+    trace[Utils] "split at {fvar}"
+    splitConjTac fvar none (fun _ _ =>  pure ())
+  | none => do
+    trace[Utils] "split goal"
+    splitConjTarget
+
+elab "split_conjs" args:optAtArgs : tactic => do
+  withMainContext do
+  match ← elabOptAtArgs args with
+  | some fvar =>
+    trace[Utils] "split at {fvar}"
+    splitFullConjTac false fvar (fun _ =>  pure ())
+  | none =>
+    trace[Utils] "split goal"
+    repeatTac splitConjTarget
+
+elab "split_existsl" " at " n:ident : tactic => do
+  withMainContext do
+  let decl ← Lean.Meta.getLocalDeclFromUserName n.getId
+  let fvar := mkFVar decl.fvarId
+  splitAllExistsTac fvar [] (fun _ _ => pure ())
+
+example (h : a ∧ b) : a := by
+  split_existsl at h
+  split_conj at h
+  assumption
+
+example (h : ∃ x y z, x + y + z ≥ 0) : ∃ x, x ≥ 0 := by
+  split_existsl at h
+  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)
+  -- Add the equational theorem for the declarations to unfold
+  let simpThms ←
+    declsToUnfold.foldlM (fun thms decl => thms.addDeclToUnfold decl) simpThms
+  -- Add the hypotheses and the rewriting theorems
+  let simpThms ←
+    hypsToUse.foldlM (fun thms fvarId =>
+      -- post: TODO: don't know what that is
+      -- inv: invert the equality
+      thms.add (.fvar fvarId) #[] (mkFVar fvarId) (post := false) (inv := false)
+      -- thms.eraseCore (.fvar fvar)
+      ) simpThms
+  -- Add the rewriting theorems to use
+  let simpThms ←
+    thms.foldlM (fun thms thmName => do
+      let info ← getConstInfo thmName
+      if (← isProp info.type) then
+        -- post: TODO: don't know what that is
+        -- inv: invert the equality
+        thms.addConst thmName (post := false) (inv := false)
+      else
+        throwError "Not a proposition: {thmName}"
+      ) simpThms
+  let congrTheorems ← getSimpCongrTheorems
+  let ctx : Simp.Context := { simpTheorems := #[simpThms], congrTheorems }
+  -- Apply the simplifier
+  let _ ← Tactic.simpLocation ctx (discharge? := .none) loc
+
+end Utils