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-rw-r--r--tests/lean/lakefile.lean1
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+/- A tutorial about using Lean to verify properties of programs generated by Aeneas -/
+import Base
+
+open Primitives
+open Result
+
+namespace Tutorial
+
+/-#===========================================================================#
+ #
+ # Simple Arithmetic Example
+ #
+ #===========================================================================#-/
+
+/- As a first example, let's consider the function below.
+ -/
+
+def mul2_add1 (x : U32) : Result U32 := do
+ let x1 ← x + x
+ let x2 ← x1 + 1#u32
+ ret x2
+
+/- There are several things to note.
+
+ # Machine integers
+ ==================
+ Because Rust programs manipulate machine integers which occupy a fixed
+ size in memory, we model integers by using types like [U32], which is
+ the type of integers which take their values between 0 and 2^32 - 1 (inclusive).
+ [1#u32] is simply the constant 1 (seen as a [U32]).
+
+ You can see a definition or its type by using the [#print] and [#check] commands.
+ It is also possible to jump to definitions (right-click + "Go to Definition"
+ in VS Code).
+
+ For instance, you can see below that [U32] is defined in terms of a more generic
+ type [Scalar] (just move the cursor to the [#print] command below).
+
+ -/
+#print U32 -- This shows the definition of [U32]
+
+#check mul2_add1 -- This shows the type of [mul2_add1]
+#print mul2_add1 -- This show the full definition of [mul2_add1]
+
+/- # Syntax
+ ========
+ Because machine integers are bounded, arithmetic operations can fail, for instance
+ because of an overflow: this is the reason why the output of [mul2_add1] uses
+ the [Result] type. In particular, addition can fail.
+
+ We use a lightweight "do"-notation to write code which calls potentially failing
+ functions. In practice, all our function bodies start with a [do] keyword,
+ which enables using this lightweight syntax. After the [do], instead of writing
+ let-bindings as [let x1 := ...], we write them as: [let x1 ← ...]. We also
+ have lightweight notations for common operations like the addition.
+
+ For instance, in [let x1 ← x + x], the [x + x] expression desugars to
+ [Scalar.add x x] and the [let x1 ← ...] desugars to a call to [bind].
+
+ The definition of [bind x f] is worth investigating. It simply checks whether
+ [x : Result ...] successfully evaluates to some value, in which case it
+ calls [f] with this value, and propagates the error otherwise. See the output
+ of the [#print] command below.
+
+ *Remark:* in order to type the left-arrow symbol [←] you can type: [\l]. Generally
+ speaking, your editor can tell you how to type the symbols you see in Lean
+ code. For instance in VS Code, you can simply hover your mouse over the
+ symbol and a pop-up window will open displaying all the information you need.
+ -/
+#print Primitives.bind
+
+/- We show a desugared version of [mul2_add1] below: we remove the syntactic
+ sugar, and inline the definition of [bind] to make the matches over the
+ results explicit.
+ -/
+def mul2_add1_desugared (x : U32) : Result U32 :=
+ match Scalar.add x x with
+ | ret x1 => -- Success case
+ match Scalar.add x1 (U32.ofInt 1) with
+ | ret x2 => ret x2
+ | error => error
+ | error => error -- Propagating the errors
+
+/- Now that we have seen how [mul2_add1] is defined precisely, we can prove
+ simple properties about it. For instance, what about proving that it evaluates
+ to [2 * x + 1]?
+
+ We advise writing specifications in a Hoare-logic style, that is with
+ preconditions (requirements which must be satisfied by the inputs upon
+ calling the function) and postconditions (properties that we know about
+ the output after the function call).
+
+ In the case of [mul2_add1] we could state the theorem as follows.
+ -/
+
+theorem mul2_add1_spec
+ -- The input
+ (x : U32)
+ /- The precondition (we give it the name "h" to be able to refer to it in the proof).
+ We simply state that [2 * x + 1] must not overflow.
+
+ The ↑ notation ("\u") is used to coerce values. Here, we coerce [x], which is
+ a bounded machine integer, to an unbounded mathematical integer, which is
+ easier to work with. Note that writing [↑x] is the same as writing [x.val].
+ -/
+ (h : 2 * ↑x + 1 ≤ U32.max)
+ /- The postcondition -/
+ : ∃ y, mul2_add1 x = ret y ∧ -- The call succeeds
+ ↑ y = 2 * ↑x + (1 : Int) -- The output has the expected value
+ := by
+ /- The proof -/
+ -- Start by a call to the rewriting tactic to reveal the body of [mul2_add1]
+ rw [mul2_add1]
+ /- Here we use the fact that if [x + x] doesn't overflow, then the addition
+ succeeds and returns the value we expect, as given by the theorem [U32.add_spec].
+ Doing this properly requires a few manipulations: we need to instantiate
+ the theorem, introduce it in the context, destruct it to introduce [x1], etc.
+ We automate this with the [progress] tactic: [progress with th as ⟨ x1 .. ⟩]
+ uses theorem [th], instantiates it properly by looking at the goal, renames
+ the output to [x1] and further decomposes the postcondition of [th].
+
+ Note that it is possible to provide more inputs to name the assumptions
+ introduced by the postcondition (for instance: [as ⟨ x1, h ⟩]).
+
+ If you look at the goal after the call to [progress], you wil see that:
+ - there is a new variable [x1] and an assumption stating that [↑x1 = ↑x + ↑x]
+ - the call [x + x] disappeared from the goal: we "progressed" by one step
+
+ Remark: the theorem [U32.add_spec] actually has a precondition, namely that
+ the addition doesn't overflow.
+ In the present case, [progress] manages to prove this automatically by using
+ the fact that [2 * x + 1 < U32.max]. In case [progress] fails to prove a
+ precondition, it leaves it as a subgoal.
+ -/
+ progress with U32.add_spec as ⟨ x1 ⟩
+ /- We can call [progress] a second time for the second addition -/
+ progress with U32.add_spec as ⟨ x2 ⟩
+ /- We are now left with the remaining goal. We do this by calling the simplifier
+ then [scalar_tac], a tactic to solve arithmetic problems:
+ -/
+ simp at *
+ scalar_tac
+
+/- The proof above works, but it can actually be simplified a bit. In particular,
+ it is a bit tedious to specify that [progress] should use [U32.add_spec], while
+ in most situations the theorem to use is obvious by looking at the function.
+
+ For this reason, we provide the possibility of registering theorems in a database
+ so that [progress] can automatically look them up. This is done by marking
+ theorems with custom attributes, like [pspec] below.
+
+ Theorems in the standard library like [U32.add_spec] have already been marked with such
+ attributes, meaning we don't need to tell [progress] to use them.
+ -/
+@[pspec] -- the [pspec] attribute saves the theorem in a database, for [progress] to use it
+theorem mul2_add1_spec2 (x : U32) (h : 2 * ↑x + 1 ≤ U32.max)
+ : ∃ y, mul2_add1 x = ret y ∧
+ ↑ y = 2 * ↑x + (1 : Int)
+ := by
+ rw [mul2_add1]
+ progress as ⟨ x1 .. ⟩ -- [progress] automatically lookups [U32.add_spec]
+ progress as ⟨ x2 .. ⟩ -- same
+ simp at *; scalar_tac
+
+/- Because we marked [mul2_add1_spec2] theorem with [pspec], [progress] can
+ now automatically look it up. For instance, below:
+ -/
+-- A dummy function which uses [mul2_add1]
+def use_mul2_add1 (x : U32) (y : U32) : Result U32 := do
+ let x1 ← mul2_add1 x
+ x1 + y
+
+@[pspec]
+theorem use_mul2_add1_spec (x : U32) (y : U32) (h : 2 * ↑x + 1 + ↑y ≤ U32.max) :
+ ∃ z, use_mul2_add1 x y = ret z ∧
+ ↑z = 2 * ↑x + (1 : Int) + ↑y := by
+ rw [use_mul2_add1]
+ -- Here we use [progress] on [mul2_add1]
+ progress as ⟨ x1 .. ⟩
+ progress as ⟨ z .. ⟩
+ simp at *; scalar_tac
+
+
+/-#===========================================================================#
+ #
+ # Recursion
+ #
+ #===========================================================================#-/
+
+/- We can have a look at more complex examples, for example recursive functions. -/
+
+/- A custom list type.
+
+ Original Rust code:
+ ```
+ pub enum CList<T> {
+ CCons(T, Box<CList<T>>),
+ CNil,
+ }
+ ```
+-/
+inductive CList (T : Type) :=
+| CCons : T → CList T → CList T
+| CNil : CList T
+
+-- Open the [CList] namespace, so that we can write [CCons] instead of [CList.CCons]
+open CList
+
+/- A function accessing the ith element of a list.
+
+ Original Rust code:
+ ```
+ pub fn list_nth<'a, T>(l: &'a CList<T>, i: u32) -> &'a T {
+ match l {
+ List::CCons(x, tl) => {
+ if i == 0 {
+ return x;
+ } else {
+ return list_nth(tl, i - 1);
+ }
+ }
+ List::CNil => {
+ panic!()
+ }
+ }
+ }
+ ```
+ -/
+divergent def list_nth (T : Type) (l : CList T) (i : U32) : Result T :=
+ match l with
+ | CCons x tl =>
+ if i = 0#u32
+ then ret x
+ else do
+ let i1 ← i - 1#u32
+ list_nth T tl i1
+ | CNil => fail Error.panic
+
+/- Conversion to Lean's standard list type.
+
+ Note that because we use the suffix "CList.", we can use the notation [l.to_list]
+ if [l] has type [CList ...].
+ -/
+def CList.to_list {α : Type} (x : CList α) : List α :=
+ match x with
+ | CNil => []
+ | CCons hd tl => hd :: tl.to_list
+
+/- Let's prove that [list_nth] indeed accesses the ith element of the list.
+
+ Remark: the parameter [Inhabited T] tells us that we must have an instance of the
+ typeclass [Inhabited] for the type [T]. As of today we can only use [index] with
+ inhabited types, that is to say types which are not empty (i.e., for which it is
+ possible to construct a value - for instance, [Int] is inhabited because we can exhibit
+ the value [0: Int]). This is a technical detail.
+
+ Remark: we didn't mention it before, but we advise always writing inequalities
+ in the same direction (that is: use [<] and not [>]), because it helps the simplifier.
+ More specifically, if you have the assumption that [x > y] in the context, the simplifier
+ may not be able to rewrite [y < x] to [⊤].
+ -/
+theorem list_nth_spec {T : Type} [Inhabited T] (l : CList T) (i : U32)
+ -- Precondition: the index is in bounds
+ (h : ↑i < l.to_list.len)
+ -- Postcondition
+ : ∃ x, list_nth T l i = ret x ∧
+ -- [x] is the ith element of [l] after conversion to [List]
+ x = l.to_list.index ↑i
+ := by
+ -- Here we have to be careful when unfolding the body of [list_nth]: we could
+ -- use the [simp] tactic, but it will sometimes loop on recursive definitions.
+ rw [list_nth]
+ -- Let's simply follow the structure of the function, by first matching on [l]
+ match l with
+ | CNil =>
+ -- We can't get there: we can derive a contradiction from the precondition:
+ -- we have that [i < 0] (because [i < CNil.to_list.len]) and at the same
+ -- time [0 ≤ i] (because [i] is a [U32] unsigned integer).
+ -- First, let's simplify [to_list CNil] to [0]
+ simp [CList.to_list] at h
+ -- Proving we have a contradiction
+ scalar_tac
+ | CCons hd tl =>
+ -- Simplify the match
+ simp only []
+ -- Perform a case disjunction on [i].
+ -- The notation [hi : ...] allows us to introduce an assumption in the
+ -- context, to remember the fact that in the branches we have [i = 0#u32]
+ -- and [¬ i = 0#u32].
+ if hi: i = 0#u32 then
+ -- We can finish the proof simply by using the simplifier.
+ -- We decompose the proof into several calls on purpose, so that it is
+ -- easier to understand what is going on.
+ -- Simplify the condition and the [if then else]
+ simp [hi]
+ -- Prove the final equality
+ simp [CList.to_list]
+ else
+ -- The interesting branch
+ -- Simplify the condition and the [if then else]
+ simp [hi]
+ -- i0 := i - 1
+ progress as ⟨ i1, hi1 ⟩
+ -- [progress] can handle recursion
+ simp [CList.to_list] at h -- we need to simplify this inequality to prove the precondition
+ progress as ⟨ l1 ⟩
+ -- Proving the postcondition
+ -- We need this to trigger the simplification of [index to.to_list i.val]
+ --
+ -- Among other things, the call to [simp] below will apply the theorem
+ -- [List.index_nzero_cons], which has the precondition [i.val ≠ 0]. [simp]
+ -- can automatically use the assumptions/theorems we give it to prove
+ -- preconditions when applying rewriting lemmas. In the present case,
+ -- by giving it [*] as argument, we tell [simp] to use all the assumptions
+ -- to perform rewritings. In particular, it will use [i.val ≠ 0] to
+ -- apply [List.index_nzero_cons].
+ have : i.val ≠ 0 := by scalar_tac -- Remark: [simp at hi] also works
+ simp [CList.to_list, *]
+
+/-#===========================================================================#
+ #
+ # Partial Functions
+ #
+ #===========================================================================#-/
+
+/- Recursive functions may not terminate on all inputs.
+
+ For instance, the function below only terminates on positive inputs (note
+ that we switched to signed integers), in which cases it behaves like the
+ identity. When we need to define such a potentially partial function,
+ we use the [divergent] keyword, which means that the function may diverge
+ (i.e., infinitely loop).
+
+ We will skip the details of how [divergent] precisely handles non-termination.
+ All you need to know is that the [Result] type has actually 3 cases (we saw
+ the first 2 cases in the examples above):
+ - [ret]: successful computation
+ - [fail]: failure (panic because of overflow, etc.)
+ - [div]: the computation doesn't terminate
+
+ If in a theorem we state and prove that:
+ ```
+ ∃ y, i32_id x = ret x
+ ```
+ we not only prove that the function doesn't fail, but also that it terminates.
+
+ *Remark*: in practice, whenever Aeneas generates a recursive function, it
+ annotates it with the [divergent] keyword.
+ -/
+divergent def i32_id (x : I32) : Result I32 :=
+ if x = 0#i32 then ret 0#i32
+ else do
+ let x1 ← x - 1#i32
+ let x2 ← i32_id x1
+ x2 + 1#i32
+
+/- We can easily prove that [i32_id] behaves like the identity on positive inputs -/
+theorem i32_id_spec (x : I32) (h : 0 ≤ x.val) :
+ ∃ y, i32_id x = ret y ∧ x.val = y.val := by
+ rw [i32_id]
+ if hx : x = 0#i32 then
+ simp_all
+ else
+ simp [hx]
+ -- x - 1
+ progress as ⟨ x1 ⟩
+ -- Recursive call
+ progress as ⟨ x2 ⟩
+ -- x2 + 1
+ progress
+ -- Postcondition
+ simp; scalar_tac
+-- Below: we have to prove that the recursive call performed in the proof terminates.
+-- Otherwise, we could prove any result we want by simply writing a theorem which
+-- uses itself in the proof.
+--
+-- We first specify a decreasing value. Here, we state that [x], seen as a natural number,
+-- decreases at every recursive call.
+termination_by i32_id_spec x _ => x.val.toNat
+-- And we now have to prove that it indeed decreases - you can skip this for now.
+decreasing_by
+ -- We first need to "massage" the goal (in practice, all the proofs of [decreasing_by]
+ -- should start with a call to [simp_wf]).
+ simp_wf
+ -- Finish the proof
+ have : 1 ≤ x.val := by scalar_tac
+ simp [Int.toNat_sub_of_le, *]
+
+end Tutorial
diff --git a/tests/lean/lakefile.lean b/tests/lean/lakefile.lean
index cc63c48f..8acf6973 100644
--- a/tests/lean/lakefile.lean
+++ b/tests/lean/lakefile.lean
@@ -8,6 +8,7 @@ require Base from "../../backends/lean"
package «tests» {}
+@[default_target] lean_lib tutorial
@[default_target] lean_lib betreeMain
@[default_target] lean_lib constants
@[default_target] lean_lib external