Exercises.lean

import Aeneas
open Aeneas.Std Result

set_option maxHeartbeats 1000000

namespace Primitives

end Primitives

namespace Tutorial.Solutions

/- [tutorial::mul2_add1]:
   Source: 'src/lib.rs', lines 11:0-11:31 -/
def mul2_add1 (x : U32) : Result U32 :=
  do
  let i ← x + x
  i + 1#u32

#check U32.add_spec

/-- Theorem about `mul2_add1`: without the `progress` tactic -/
-- @[progress]
theorem mul2_add1_spec (x : U32) (h : 2 * x.val + 1 ≤ U32.max)
  : ∃ y, mul2_add1 x = ok y ∧
  ↑y = 2 * ↑x + (1 : Int)
  := by
  unfold mul2_add1
  have ⟨ x1, hEq1, hPost1 ⟩ := @U32.add_spec x x (by scalar_tac)
  simp [hEq1]
  have ⟨ x2, hEq2, hPost2 ⟩ := @U32.add_spec x1 1#u32 (by scalar_tac)
  simp [hEq2]
  scalar_tac

/-- Theorem about `mul2_add1`: with the `progress` tactic -/
-- @[progress]
theorem mul2_add1_spec' (x : U32) (h : 2 * x.val + 1 ≤ U32.max)
  : ∃ y, mul2_add1 x = ok y ∧
  ↑y = 2 * ↑x + (1 : Int)
  := by
  unfold mul2_add1
  progress with U32.add_spec as ⟨ x1 ⟩
  progress as ⟨ x2 ⟩
  scalar_tac

/- [tutorial::mul2_add1_add]:
   Source: 'src/lib.rs', lines 15:0-15:43 -/
def mul2_add1_add (x : U32) (y : U32) : Result U32 :=
  do
  let i ← mul2_add1 x
  i + y

/-- Theorem about `mul2_add1_add` -/
theorem mul2_add1_add_spec (x : U32) (y : U32) (h : 2 * x.val + 1 + y.val ≤ U32.max) :
  ∃ z, mul2_add1_add x y = ok z ∧
  ↑z = 2 * ↑x + (1 : Int) + ↑y := by
  unfold mul2_add1_add
  progress with mul2_add1_spec as ⟨ x1 ⟩
  progress as ⟨ x2 ⟩
  scalar_tac

/- [tutorial::CList]
   Source: 'src/lib.rs', lines 32:0-32:17 -/
inductive CList (T : Type) where
| CCons : T → CList T → CList T
| CNil : CList T

open CList

/-- Convert a "custom" list to a standard Lean list.

    By putting this definition in the namespace `CList`, we give the possibility of using the `.`
    notation: if `x` has type `CList α` we can write `x.toList` instead of `toList x`.
 -/
@[simp] def CList.toList {α : Type} (x : CList α) : List α :=
  match x with
  | CNil => []
  | CCons hd tl => hd :: tl.toList

/- [tutorial::list_nth]:
   Source: 'src/lib.rs', lines 37:0-37:56 -/
def list_nth {T : Type} (l : CList T) (i : U32) : Result T :=
  match l with
  | CList.CCons x tl =>
    if i = 0#u32
    then Result.ok x
    else do
         let i1 ← i - 1#u32
         list_nth tl i1
  | CList.CNil => Result.fail .panic
partial_fixpoint

/-- Theorem about `list_nth` -/
theorem list_nth_spec {T : Type} [Inhabited T] (l : CList T) (i : U32)
  (h : i.val < l.toList.length) :
  ∃ x, list_nth l i = ok x ∧
  x = l.toList[i.val]!
  := by
  unfold list_nth
  split
  . split
    . simp_all
    . simp_all
      progress as ⟨ i1 ⟩
      progress as ⟨ x ⟩
      simp_all
  . simp_all

/- [tutorial::i32_id]:
   Source: 'src/lib.rs', lines 78:0-78:29 -/
def i32_id (i : I32) : Result I32 :=
  if i = 0#i32
  then Result.ok 0#i32
  else do
       let i1 ← i - 1#i32
       let i2 ← i32_id i1
       i2 + 1#i32
partial_fixpoint

/-- Theorem about `i32_id` -/
theorem i32_id_spec (n : I32) (h : 0 ≤ n.val) :
  i32_id n = ok n := by
  unfold i32_id
  split
  . simp [*]
  . progress as ⟨ n1 ⟩
    progress
    progress as ⟨ n2 ⟩
    scalar_tac
termination_by n.toNat
decreasing_by
  -- We always need to start termination proofs with this tactic (it massages the goal)
  simp_wf
  -- It is an arithmetic goal:
  scalar_tac
  -- Remark: as the pattern above is quite common (`simp_wf; scalar_tac`),
  -- we introduced the tactic `scalar_decr_tac` (it does the same)

/- [tutorial::even]:
   Source: 'src/lib.rs', lines 87:0-87:28 -/
mutual def even (i : U32) : Result Bool :=
  if i = 0#u32
  then Result.ok true
  else do
       let i1 ← i - 1#u32
       odd i1
partial_fixpoint

/- [tutorial::odd]:
   Source: 'src/lib.rs', lines 96:0-96:27 -/
def odd (i : U32) : Result Bool :=
  if i = 0#u32
  then Result.ok false
  else do
       let i1 ← i - 1#u32
       even i1
partial_fixpoint

end

mutual

/-- The proof about `even` -/
theorem even_spec (n : U32) :
  ∃ b, even n = ok b ∧ (b ↔ Even n.val) := by
  unfold even
  split
  . simp [*]
  . progress as ⟨ n' ⟩
    progress as ⟨ b ⟩
    simp [*]
    simp [Nat.even_add_one]
termination_by n.val
decreasing_by scalar_decr_tac

/-- The proof about `odd` -/
theorem odd_spec (n : U32) :
  ∃ b, odd n = ok b ∧ (b ↔ Odd n.val) := by
  unfold odd
  split
  . simp [*]
  . progress as ⟨ n' ⟩
    progress as ⟨ b ⟩
    simp [*]
    simp [Nat.odd_add_one]
termination_by n.val
decreasing_by scalar_decr_tac

end

/- [tutorial::list_nth_mut1]: loop 0:
   Source: 'src/lib.rs', lines 107:0-116:1 -/
def list_nth_mut1_loop
  {T : Type} (l : CList T) (i : U32) :
  Result (T × (T → CList T))
  :=
  match l with
  | CList.CCons x tl =>
    if i = 0#u32
    then
      let back := fun ret => CList.CCons ret tl
      Result.ok (x, back)
    else
      do
      let i1 ← i - 1#u32
      let (t, back) ← list_nth_mut1_loop tl i1
      let back1 :=
        fun ret => CList.CCons x (back ret)
      Result.ok (t, back1)
  | CList.CNil => Result.fail .panic
partial_fixpoint

/- [tutorial::list_nth_mut1]:
   Source: 'src/lib.rs', lines 107:0-107:77 -/
@[reducible]
def list_nth_mut1
  {T : Type} (l : CList T) (i : U32) :
  Result (T × (T → CList T))
  :=
  list_nth_mut1_loop l i

/-
  # #####################################################################
  # ###   THE EXERCISES START HERE     ##################################
  # #####################################################################
-/

/- # Notations

  You will need to type some special characters like `↑` or `∃`, `∧`, etc..

  In order to type the special characters, you need to type the character '\' then a
  specific string (note that typing an abbreviation often works, for instance "ex" instead
  of "exists"):
  - `↑` : \ + u    ("up")
  - `→` : \ + r    ("right")
  - `∧` : \ + and
  - `∨` : \ + or
  - `∀` : \ + forall
  - `∃` : \ + exists
-/

/- # Basic tactics -/

/- `simp` simplifies the goal by applying rewriting lemmas and simplifying definitions
   marked as `@[simp]`. For instance, the lemma `List.length_cons` which states that
   `(hd :: tl).length = tl.length + 1` has been marked as `simp`: -/
#check List.length_cons
example α (hd : α) (tl : List α) : (hd :: tl).length = tl.length + 1 := by
  -- This proves the goal automatically, by using `List.length_cons` above
  simp

/- You can see which lemmas are applied by `simp` by using `simp?`.
   This is often useful to build the intuition of what `simp` does, especially when
   it simplifies the goal a lot.

   Note that `simp [...]` allows you to specify additional rewriting lemmas to apply,
   and that `simp only` allows to forbid `simp` from applying any lemmas other than
   the ones you provide: this allows you to have quite fine-grained control on the
   simplifications

   There are ways of being even more precise, for instance by using conversions
   (https://leanprover.github.io/theorem_proving_in_lean4/conv.html) and the `rw`
   tactic, but you will not need to understand those for this tutorial.

   Just be aware that `rw` doesn't apply simplification lemmas recursively, while
   `simp` repeatedly simplifies the goal until it can't make progress anymore.
 -/
example α (hd : α) (tl : List α) : (hd :: tl).length = tl.length + 1 := by
  simp? -- This prints: `simp only [List.length_cons]`

/- Your turn!

   Note that we give **tips and tricks** in the solutions: we advise to always have a look at them
   after you completed a proof.
 -/
example α (n : Nat) (x y : α) (l0 l1 l2 : List α)
  (h0 : l1 = x :: l0)
  (h1 : l2 = y :: l1)
  (h2 : l0.length = n) :
  l2.length = n + 2 := by
  sorry

/- The `simp` tactic is quite flexible: -/
example α (a b c d e : α) (h0 : a = b) (h1 : b = c) (h2 : c = d) (h3 : d = e) : a = e := by
  -- You can give (named) assumptions to `simp` instead of theorems
  simp [h0]
  -- You can simplify an assumption rather than the goal
  simp [h2, h3] at h1
  /- Note that you can simplify the goal *and* all the assumptions with `simp at *`.

     Similarly, you can simplify the goal by usoing all the assumptions in the context with
    `simp [*]`:
   -/
  simp [*]

/- The `simp_all` tactic is quite powerful: it repeatedly simplifies the assumptions
   and the goal by using the simp lemmas and the assumptions themselves.

   Note that it supports a syntax similar to `simp`: `simp_all only [...]`
 -/
example α (a b c d e : α) (h0 : a = b) (h1 : b = c) (h2 : c = d) (h3 : d = e) : a = e := by
  simp_all

/- You may want to prove intermediate goals: you can do so with `have`. -/
example (a b c : Prop) (h0 : a → b → c) (h1 : a) (h2 : b) : c := by
  have h3 : c := by simp_all
  -- If you omit the name, Lean will use the name `this`
  have : c := by simp_all
  -- You can also use an instantiated lemma/assumption with `have`:
  have _h4 := h0 h1 h2
  -- If your lemma/assumption has a precondition (i.e., if it is an implication)
  -- you can use the keyword `by` to prove it
  have := h0 (by simp [*]) (by simp [*])
  -- The `apply` tactic allows to apply a theorem or an assumption to the goal, if
  -- the conclusion of the assumption/theorem matches the goal (here, it allows us
  -- to finish the goal)
  apply this

/- As a side note, `simp` simplifies implications, equivalences, etc. so you
   can actually use it to prove the goal below. -/
example (a b c : Prop) (h0 : a → b → c) (h1 : a) (h2 : b) : c := by
  simp_all

/- Your turn! -/
example (a b c d : Prop) (h0 : a → b → c) (h1 : c → d → e)
  (ha : a) (hb : b) (hd : d) : e := by
  sorry

/- You can apply a simplification lemma the other way around: -/
example α (a b c : α) (h0 : a = b) (h1 : b = c) : a = c := by
  simp [← h1]
  simp [← h0]

/- To prove a goal of the shape `∀ x0 x1, ...` use `intro y0 y1 ...` to introduce
   the quantified variables in the context. -/
example α (p q : α → Prop) (h0 : ∀ x, p x) (h1 : ∀ x, p x → q x) : ∀ x, q x := by
  intro x
  have h2 := h0 x
  have h3 := h1 x h2
  apply h3

/- If the goal is an linear arithmetic goal, you can use `scalar_tac` to discharge it -/
example (x y z : Int) (h0 : x + y ≤ 3) (h1 : 2 * z ≤ 4) : x + y + z ≤ 5
  := by scalar_tac

/- Note that `scalar_tac` is aware of the bounds of the machine integers -/
example (x : U32) : 0 ≤ x.val ∧ x.val ≤ 2 ^ 32 - 1 := by scalar_tac

/- Renaming: note that the variables which appear in grey are variables which you can't
   reference directly, either because Lean automatically introduced fresh names for them,
   or because they are shadowed. You can use the `rename` and `rename_i` tactics to rename$
   them. -/
example α (p q r : α → Prop) (h : ∀ x, p x) (h : ∀ x, p x → q x) (h : ∀ x, q x → r x) :
  ∀ x, r x := by
  -- Rename all the unnamed variables/assumptions at once
  rename_i hp hpq
  -- The `rename` tactic allows to use a pattern to select the variable/assumption to rename
  rename ∀ (x : α), p x => h0
  rename ∀ (x : α), p x → q x => h1
  -- You can use holes in your pattern
  rename ∀ _, _ → _ => h2
  --
  simp_all

/- **Coercions**:
   The notation `↑` indicates a coercion. For instance:
   - if `x` is a machine scalar, `↑x : Int` desugars to `x.val`
   - if `v` is a vector (see exercises below), `↑v : List α` desugars to `v.val`

   Note that if you don't know what a **notation** which appears in the goal is exactly, just
   put your mouse over it.

   If you find coercions too confusing, you can set the option below to `false`.
   Note however that the goals will become significantly more verbose.
 -/
set_option pp.coercions true

/- # Reasoning about lists

   Small preparation for theorem `list_nth_mut1`.
 -/

/- The notation `l[i]!` stands for `getElem! l`, and is the `i`th element of list `l`.

   We deactivate the simp lemma below as it replaces terms of the shape `l[i]!` with more
   complicated terms: in the present case it is more annoying than anything. -/
attribute [-simp] List.getElem!_eq_getElem?_getD

/- Reasoning about `List.index`.

   You can use the following two lemmas.
 -/
#check List.getElem!_cons_zero
#check List.getElem!_cons_nzero

/- Example 1: indexing the first element of the list -/
example [Inhabited α] (i : U32) (hd : α) (tl : CList α)
  (hEq : i = 0#u32) :
  (hd :: tl.toList)[i.val]! = hd := by
  have hi : i.val = 0 := by scalar_tac
  simp only [hi]
  --
  have hIndex := @List.getElem!_cons_zero _ hd _ tl.toList
  simp only [hIndex]

/- Example 2: indexing in the tail -/
example [Inhabited α] (i : U32) (hd : α) (tl : CList α)
  (hEq : i ≠ 0#u32) :
  (hd :: tl.toList)[i.val]! = tl.toList[i.val - 1]! := by
  -- Note that `scalar_tac` is aware of `Arith.Nat.not_eq`
  have hIndex := List.getElem!_cons_nzero hd tl.toList i.val (by scalar_tac)
  simp only [hIndex]

/- Note that `List.index_zero_cons` and `List.index_cons_nzero` have been
   marked as `@[simp]` and are thus automatically applied. Also note
   that `simp` can automatically prove the premises of rewriting lemmas,
   if it has enough information.

   For this reason, `simp [*]` and `simp_all` can often do more work than
   you expect. -/
example [Inhabited α] (i : U32) (hd : α) (tl : CList α)
  (hEq : i = 0#u32) :
  (hd :: tl.toList)[i.val]! = hd := by
  simp [hEq]

/- Note that `simp_all` manages to automatically apply `List.index_cons_nzero` below,
   by using the fact that `i ≠ 0#u32`. -/
example [Inhabited α] (i : U32) (hd : α) (tl : CList α)
  (hEq : i ≠ 0#u32) :
  (hd :: tl.toList)[i.val]! = tl.toList[i.val - 1]! := by
  simp_all

/- Below, you will need to reason about `List.update`.
   You can use the following lemmas.

   Those lemmas have been marked as `@[simp]`, meaning that if `simp` is properly used,
   it will manage to apply them automatically.
 -/
#check List.set_cons_zero
#check List.set_cons_nzero

/- # Some proofs of programs -/

/-- Theorem about `list_nth_mut1`.

    **Hints**:
    - you can use `progress` to automatically apply a lemma, then refine it into
      `progress as ⟨ ... ⟩` to name the variables and the post-conditions(s) it introduces.
    - if there is a disjunction or branching (like an `if then else`) in the goal, use `split`
    - if the goal is a conjunction, you can use `split_conjs` to introduce one subgoal
      per disjunct
    - you should use `progress` for as long as you see a monadic `do` block, unless you
      see a branching, in which case you should use `split`.

    **Remark**: we wrote two versions of the solution in `Solutions.lean`:
    - a verbose one, where we attempt to be precise with the simplifer and
      precisely instantiate the lemmas we need. This is for pedagogical
      purpose.
    - a simpler one, which is closer to what one would write in a real
      development
 -/
theorem list_nth_mut1_spec {T: Type} [Inhabited T] (l : CList T) (i : U32)
  (h : i.val < l.toList.length) :
  ∃ x back, list_nth_mut1 l i = ok (x, back) ∧
  x = l.toList[i.val]! ∧
  -- Specification of the backward function
  ∀ x', (back x').toList = l.toList.set i.val x' := by
  unfold list_nth_mut1 list_nth_mut1_loop
  sorry

/- [tutorial::list_tail]: loop 0:
   Source: 'src/lib.rs', lines 118:0-123:1 -/
def list_tail_loop
  {T : Type} (l : CList T) :
  Result ((CList T) × (CList T → CList T))
  :=
  match l with
  | CList.CCons t tl =>
    do
    let (c, back) ← list_tail_loop tl
    let back1 :=
      fun ret => CList.CCons t (back ret)
    Result.ok (c, back1)
  | CList.CNil => Result.ok (CList.CNil, fun ret => ret)
partial_fixpoint

/- [tutorial::list_tail]:
   Source: 'src/lib.rs', lines 118:0-118:68 -/
@[reducible]
def list_tail
  {T : Type} (l : CList T) :
  Result ((CList T) × (CList T → CList T))
  :=
  list_tail_loop l

/- [tutorial::append_in_place]:
   Source: 'src/lib.rs', lines 125:0-125:67 -/
def append_in_place
  {T : Type} (l0 : CList T) (l1 : CList T) : Result (CList T) :=
  do
  let (_, list_tail_back) ← list_tail l0
  Result.ok (list_tail_back l1)


/-- Theorem about `list_tail`: exercise -/
@[progress]
theorem list_tail_spec {T : Type} (l : CList T) :
  ∃ back, list_tail l = ok (CList.CNil, back) ∧
  ∀ tl', (back tl').toList = l.toList ++ tl'.toList := by
  unfold list_tail list_tail_loop
  sorry

/-- Theorem about `append_in_place`: exercise -/
@[progress]
theorem append_in_place_spec {T : Type} (l0 l1 : CList T) :
  ∃ l2, append_in_place l0 l1 = ok l2 ∧
  l2.toList = l0.toList ++ l1.toList := by
  unfold append_in_place
  sorry

/- [tutorial::reverse]: loop 0:
   Source: 'src/lib.rs', lines 147:4-154:1 -/
def reverse_loop
  {T : Type} (l : CList T) (out : CList T) : Result (CList T) :=
  match l with
  | CList.CCons hd tl => reverse_loop tl (CList.CCons hd out)
  | CList.CNil => Result.ok out
partial_fixpoint

@[progress]
theorem reverse_loop_spec {T : Type} (l : CList T) (out : CList T) :
  ∃ l', reverse_loop l out = ok l' ∧
  True -- Leaving the post-condition as an exercise
  := by
  unfold reverse_loop
  sorry

/- [tutorial::reverse]:
   Source: 'src/lib.rs', lines 146:0-154:1 -/
def reverse {T : Type} (l : CList T) : Result (CList T) :=
  reverse_loop l CList.CNil

theorem reverse_spec {T : Type} (l : CList T) :
  ∃ l', reverse l = ok l' ∧
  True -- Leaving the post-condition as an exercise
  := by
  unfold reverse
  sorry

/-
  # BIG NUMBERS
 -/

/- Deactivating some simplification procedures which are not necessary and which sometimes
   makes the goal hard to understand by reducing 2 ^ 32 to 4294967296.

   Remark: as mentioned above, if the simplifier simplifies too much, you can display the
   simplification lemmas and the simplification procedures it uses with `simp?`, `simp_all?`,
   etc. and then use `simp only` to control the set of simplifications which gets applied.
 -/
attribute [-simp] Int.reducePow Nat.reducePow

/- [tutorial::zero]: loop 0:
   Source: 'src/lib.rs', lines 6:4-11:1 -/
def zero_loop
  (x : alloc.vec.Vec U32) (i : Usize) : Result (alloc.vec.Vec U32) :=
  let i1 := alloc.vec.Vec.len x
  if i < i1
  then
    do
    let (_, index_mut_back) ←
      alloc.vec.Vec.index_mut
        (core.slice.index.SliceIndexUsizeSliceInst U32) x i
    let i2 ← i + 1#usize
    let x1 := index_mut_back 0#u32
    zero_loop x1 i2
  else Result.ok x
partial_fixpoint

/-- Auxiliary definitions to interpret a vector of u32 as a mathematical integer -/
@[simp]
def toInt (l : List U32) : ℤ :=
  match l with
  | [] => 0
  | x :: l =>
    x + 2 ^ 32 * toInt l

/-- The theorem about `zero_loop`: exercise.

    Hints:
    - if you have lemma (or assumption): `h : ∀ x, P x → Q x`, you can instantiate it and introduce it in
      the context by doing:
      `have h' := h y (by tactic)`
    - for the proof of termination: look at `i32_id_spec` above
    - if you need to do a case disjunction, use `dcases`
      Ex.: `dcases x = y` will introduce two goals, one with the assumption `x = y` and the
      other with the assumption `x ≠ y`. You can name this assumption by writing: `dcases h : x = y`
 -/
@[progress]
theorem zero_loop_spec
  (x : alloc.vec.Vec U32) (i : Usize) (h : i.val ≤ x.length) :
  ∃ x',
    zero_loop x i = ok x' ∧
    x'.length = x.length ∧
    (∀ j, j < i.val → x'[j]! = x[j]!) ∧
    (∀ j, i.val ≤ j → j < x.length → x'[j]! = 0#u32) := by
  unfold zero_loop
  simp
  sorry

/- [tutorial::zero]:
   Source: 'src/lib.rs', lines 5:0-5:28 -/
def zero (x : alloc.vec.Vec U32) : Result (alloc.vec.Vec U32) :=
  zero_loop x 0#usize

/-- You will need this lemma for the proof of `zero_spec`

    Advice: do the proof of `zero_spec` first, then come back to prove this lemma.
-/
theorem all_nil_impl_toInt_eq_zero
  (l : List U32) (h : ∀ (j : ℕ), j < l.length → l[j]! = 0#u32) :
  toInt l = 0 := by
  /- There are two ways of proving this theorem.

     Either you use the induction tactic applied to `l` (*advised*):
     ```
     induction l
     . sorry
     . rename_i hd tl hInd -- To rename the variables introduced by the tactic
       sorry
     ```

     Or you write it as a recursive definition:
     ```
     match l with
     | [] => sorry
     | hd :: tl =>
       -- At some point you will need to introduce a recursive call:
       -- `have := all_nil_impl_toInt_eq_zero ...`
       sorry
     ```
   -/
  sorry

/-- The theorem about `zero` -/
theorem zero_spec (x : alloc.vec.Vec U32) :
  ∃ x',
    zero x = ok x' ∧
    x'.length = x.length ∧
    toInt x' = 0 := by
  unfold zero
  sorry

/- [tutorial::add_no_overflow]: loop 0:
   Source: 'src/lib.rs', lines 19:4-24:1 -/
def add_no_overflow_loop
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) (i : Usize) :
  Result (alloc.vec.Vec U32)
  :=
  let i1 := alloc.vec.Vec.len x
  if i < i1
  then
    do
    let i2 ←
      alloc.vec.Vec.index (core.slice.index.SliceIndexUsizeSliceInst U32) y i
    let (i3, index_mut_back) ←
      alloc.vec.Vec.index_mut
        (core.slice.index.SliceIndexUsizeSliceInst U32) x i
    let i4 ← i3 + i2
    let i5 ← i + 1#usize
    let x1 := index_mut_back i4
    add_no_overflow_loop x1 y i5
  else Result.ok x
partial_fixpoint

/-- You will need this lemma for the proof of `add_no_overflow_loop_spec`.

    Advice: do the proof of `add_no_overflow_loop_spec` first, then come back to prove this lemma.

    Hint: you will need to reason about non-linear arithmetic. You can use two tactics to do so:
    - `scalar_nf`, `scalar_nf at h`, `scalar_nf at *`:
      normalizes (simplifies) the expressions of the shape `a ^ k * x + ... + b ^ k * y `
    - `scalar_eq_nf`: similar, but tuned to prove goals of the shape: `... = ...`
    You can try both tactics and see their effect.
 -/
@[simp]
theorem toInt_drop (l : List U32) (i : Nat) (h0 : i < l.length) :
  toInt (l.drop i) = l[i]! + 2 ^ 32 * toInt (l.drop (i + 1)) := by
  sorry

/-- You will need this lemma for the proof of `add_no_overflow_loop_spec`.

    Advice: do the proof of `add_no_overflow_loop_spec` first, then come back to prove this lemma.

    Remark: this proof is slightly technical and requires to control rewritings precisely (we
    haven't explained how to do that above) to prove at some point that (beware of the conversions
    between ℕ and ℤ!):
    `2 ^ (i * 32) = ((2 ^ ((i - 1) * 32) * 2 ^ 32 : Int)`

    We recommend that you keep this proof until the very end then:
    - either ask for help
    - or go see the solution
 -/
@[simp]
theorem toInt_update (l : List U32) (i : Nat) (x : U32) (h0 : i < l.length) :
  toInt (l.set i x) = toInt l + 2 ^ (32 * i) * (x - l[i]!) := by
  sorry

/-- The proof about `add_no_overflow_loop`.

    Hint: you will need to reason about non-linear arithmetic with `scalar_nf` and
    `scalar_eq_nf` (see above).
 -/
@[progress]
theorem add_no_overflow_loop_spec
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) (i : Usize)
  (hLength : x.length = y.length)
  -- No overflow occurs when we add the individual thunks
  (hNoOverflow : ∀ (j : Nat), i.val ≤ j → j < x.length → x[j]!.val + y[j]!.val ≤ U32.max) :
  ∃ x', add_no_overflow_loop x y i = ok x' ∧
  x'.length = x.length ∧
  toInt x' = toInt x + 2 ^ (32 * i.val) * toInt (y.val.drop i.val) := by
  unfold add_no_overflow_loop
  simp
  sorry

/- [tutorial::add_no_overflow]:
   Source: 'src/lib.rs', lines 18:0-18:50 -/
def add_no_overflow
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) :
  Result (alloc.vec.Vec U32)
  :=
  add_no_overflow_loop x y 0#usize

/-- The proof about `add_no_overflow` -/
theorem add_no_overflow_spec (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32)
  (hLength : x.length = y.length)
  (hNoOverflow : ∀ (j : Nat), j < x.length → x[j]!.val + y[j]!.val ≤ U32.max) :
  ∃ x', add_no_overflow x y = ok x' ∧
  x'.length = y.length ∧
  toInt x' = toInt x + toInt y := by
  unfold add_no_overflow
  sorry

/- [tutorial::add_with_carry]: loop 0:
   Source: 'src/lib.rs', lines 39:4-50:1 -/
def add_with_carry_loop
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) (c0 : U8) (i : Usize) :
  Result (U8 × (alloc.vec.Vec U32))
  :=
  let i1 := alloc.vec.Vec.len x
  if i < i1
  then
    do
    let i2 ←
      alloc.vec.Vec.index (core.slice.index.SliceIndexUsizeSliceInst
        U32) x i
    let i3 := UScalar.cast .U32 c0
    let p := core.num.U32.overflowing_add i2 i3
    let (sum, c1) := p
    let i4 ←
      alloc.vec.Vec.index (core.slice.index.SliceIndexUsizeSliceInst U32) y i
    let p1 := core.num.U32.overflowing_add sum i4
    let (sum1, c2) := p1
    let i5 := UScalar.cast_fromBool .U8 c1
    let i6 := UScalar.cast_fromBool .U8 c2
    let c01 ← i5 + i6
    let (_, index_mut_back) ←
      alloc.vec.Vec.index_mut
        (core.slice.index.SliceIndexUsizeSliceInst U32) x i
    let i7 ← i + 1#usize
    let x1 := index_mut_back sum1
    add_with_carry_loop x1 y c01 i7
  else Result.ok (c0, x)
partial_fixpoint

/-- The proof about `add_with_carry_loop` -/
@[progress]
theorem add_with_carry_loop_spec
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) (c0 : U8) (i : Usize)
  (hLength : x.length = y.length)
  (hi : i.val ≤ x.length)
  (hCarryLe : c0.val ≤ 1) :
  ∃ x' c1, add_with_carry_loop x y c0 i = ok (c1, x') ∧
  x'.length = x.length ∧
  c1.val ≤ 1 ∧
  toInt x' + c1.val * 2 ^ (32 * x'.length) =
    toInt x + 2 ^ (32 * i.val) * toInt (y.val.drop i.val) + c0.val * 2 ^ (32 * i.val) := by
  unfold add_with_carry_loop
  simp
  sorry

/- [tutorial::add_with_carry]:
   Source: 'src/lib.rs', lines 37:0-37:55 -/
def add_with_carry
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) :
  Result (U8 × (alloc.vec.Vec U32))
  :=
  add_with_carry_loop x y 0#u8 0#usize

/-- The proof about `add_with_carry` -/
@[progress]
theorem add_with_carry_spec
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32)
  (hLength : x.length = y.length) :
  ∃ x' c, add_with_carry x y = ok (c, x') ∧
  x'.length = x.length ∧
  c.val ≤ 1 ∧
  toInt x' + c.val * 2 ^ (32 * x'.length) = toInt x + toInt y := by
  unfold add_with_carry
  sorry

/- Bonus exercises -/
/-- We leave it to you to state the theorems for the functions below -/

/- [tutorial::max]:
   Source: 'src/lib.rs', lines 26:0-26:37 -/
def max (x : Usize) (y : Usize) : Result Usize :=
  if x > y
  then Result.ok x
  else Result.ok y

/- [tutorial::get_or_zero]:
   Source: 'src/lib.rs', lines 30:0-30:45 -/
def get_or_zero (y : alloc.vec.Vec U32) (i : Usize) : Result U32 :=
  let i1 := alloc.vec.Vec.len y
  if i < i1
  then
    alloc.vec.Vec.index (core.slice.index.SliceIndexUsizeSliceInst
      U32) y i
  else Result.ok 0#u32

/- [tutorial::add]: loop 0:
   Source: 'src/lib.rs', lines 60:4-76:1 -/
def add_loop
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) (max1 : Usize) (c0 : U8)
  (i : Usize) :
  Result (alloc.vec.Vec U32)
  :=
  if i < max1
  then
    do
    let yi ← get_or_zero y i
    let i1 ←
      alloc.vec.Vec.index (core.slice.index.SliceIndexUsizeSliceInst U32) x i
    let i2 := UScalar.cast .U32 c0
    let p := core.num.U32.overflowing_add i1 i2
    let (sum, c1) := p
    let p1 := core.num.U32.overflowing_add sum yi
    let (sum1, c2) := p1
    let i3 := UScalar.cast_fromBool .U8 c1
    let i4 := UScalar.cast_fromBool .U8 c2
    let c01 ← i3 + i4
    let (_, index_mut_back) ←
      alloc.vec.Vec.index_mut
        (core.slice.index.SliceIndexUsizeSliceInst U32) x i
    let i5 ← i + 1#usize
    let x1 := index_mut_back sum1
    add_loop x1 y max1 c01 i5
  else
    if c0 != 0#u8
    then do
         let i1 := UScalar.cast .U32 c0
         alloc.vec.Vec.push x i1
    else Result.ok x
partial_fixpoint

/- [tutorial::add]:
   Source: 'src/lib.rs', lines 55:0-55:38 -/
def add
  (x : alloc.vec.Vec U32) (y : alloc.vec.Vec U32) :
  Result (alloc.vec.Vec U32)
  :=
  do
  let i := alloc.vec.Vec.len x
  let i1 := alloc.vec.Vec.len y
  let max1 ← max i i1
  let x1 ← alloc.vec.Vec.resize core.clone.CloneU32 x max1 0#u32
  add_loop x1 y max1 0#u8 0#usize

end Tutorial.Solutions