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wkrozowski merged 2 commits into from
Sep 22, 2025
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chore: add comments to Fixpoints.lean #19

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chore: add comments
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wkrozowski Sep 22, 2025
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106 changes: 106 additions & 0 deletions LeroyCompilerVerificationCourse/Fixpoints.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -10,6 +10,16 @@ import Batteries.Data.List.Perm

universe u

/-
9. More about fixpoints

9.1 From Knaster-Tarski to effective computation of fixpoints
-/

/-
Consider a type `α` equipped with a decidable equality `eq` and a
transitive ordering `le`.
-/
@[grind] class OrderStruct (α : Sort u) where
eq : α → α → Prop
le : α → α → Prop
Expand All @@ -20,14 +30,24 @@ universe u
gt_wf : WellFounded (fun x y : α => le y x ∧ ¬eq y x)
open OrderStruct

/-
This is the strict order induced by `le`. We assume it is well-founded:
all strictly ascending chains are finite.
-/
@[grind] def gt {α : Sort u} [OrderStruct α] (x y : α) : Prop := le y x ∧ ¬eq y x

def gt_well_founded {α : Sort u} [OrderStruct α] : WellFounded (gt : α → α → Prop) := by apply @gt_wf

/-
Let `bot` be a smallest element of `α`.
-/
class OrderWithBot (α : Sort u) extends OrderStruct α where
bot : α
bot_smallest : ∀ x, le bot x

/-
Let `F` be a monotonic function from `α` to `α`.
-/
class Monotone (α : Sort u) (F : α → α) [OrderStruct α] where
F_mon : ∀ {x y : α}, le x y → le (F x) (F y)

Expand All @@ -38,6 +58,10 @@ section FixpointExistence
variable (α : Sort u) (F : α → α) [OrderWithBot α]

open OrderStruct OrderWithBot
/-
Let's show the existence of a fixpoint, as in the Knaster-Tarski theorem.
The proof is by well-founded induction.
-/
theorem fixpoint_exists_1 [Monotone α F] : ∃ x : α, eq x (F x) := by
have REC : ∀ x : α, le x (F x) -> ∃ y : α , eq y (F y) := by
intro x
Expand All @@ -59,6 +83,32 @@ theorem fixpoint_exists_1 [Monotone α F] : ∃ x : α, eq x (F x) := by
apply REC
apply bot_smallest

/-
Unfortunately, we cannot extract the value of the fixpoint.
-/
/--
error: Tactic `cases` failed with a nested error:
Tactic `induction` failed: recursor `Exists.casesOn` can only eliminate into `Prop`

α : Sort u
F : α → α
inst✝ : OrderWithBot α
motive : (∃ x, eq x (F x)) → Sort ?u.1192
h_1 : (x : α) → (P : eq x (F x)) → motive ⋯
fixpoint✝ : ∃ x, eq x (F x)
⊢ motive fixpoint✝ after processing
_
the dependent pattern matcher can solve the following kinds of equations
- <var> = <term> and <term> = <var>
- <term> = <term> where the terms are definitionally equal
- <constructor> = <constructor>, examples: List.cons x xs = List.cons y ys, and List.cons x xs = List.nil
-/
#guard_msgs in
def cannotExtractFixpoint [Monotone α F] : α :=
let fixpoint := fixpoint_exists_1 α F
match fixpoint with
| ⟨x, P⟩ => x

end FixpointExistence

section Iterate
Expand All @@ -71,12 +121,29 @@ instance : WellFoundedRelation α where
rel := gt
wf := gt_wf

/-
We can alternatively write the iteration algorithm explicitly,
proving later that it is correct.

The algorithm is simple: we iterate `F` starting from a pre-fixpoint `x`

The `iterate` function takes as argument not just `x`, but also two proofs:
- that `x` is a pre-fixpoint, i.e. `le x (F x)`
- that `x` is below any post-fixpoint `z`.

Likewise, it returns as result not just the fixpoint `y`, but also two proofs:
- that `y` is a fixpoint, i.e. `eq y (F y)`
- that `y` is below any post-fixpoint `z`.
-/
@[grind] def iterate (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) : α :=
if beq x (F x) then x else iterate (F x) (by apply F_mon; exact PRE) (by intro z hyp; specialize SMALL z hyp; apply le_trans; apply F_mon; exact SMALL; exact hyp)
termination_by x
decreasing_by
grind [beq_false']

/-
We then prove that the algorithm implemented above indeed calculates the least fixpoint of `F`.
-/
@[grind] theorem iterate_correct (x : α) (PRE : le x (F x)) (SMALL : ∀ z, le (F z) z -> le x z) (heq : y = iterate _ F x PRE SMALL ) : eq y (F y) ∧ ∀ z, le (F z) z → le y z := by
fun_induction iterate
case case1 x' PRE SMALL isTrue =>
Expand All @@ -96,8 +163,14 @@ section Fixpoint
open OrderWithBot
variable {α : Sort u} [i : OrderWithBot α] (F : α → α) [Monotone α F]

/-
The fixpoint is obtained by iterating from `bot`.
-/
@[grind] def fixpoint' : α := iterate α F bot (by apply bot_smallest) (by grind [bot_smallest])

/-
It is therefore both a fixpoint and the smallest post-fixpoint.
-/
theorem fixpoint_correct :
eq (fixpoint' F) (F (fixpoint' F)) ∧ ∀ z : α, le (F z) z → le (fixpoint' F) z := by
unfold fixpoint'
Expand All @@ -109,6 +182,13 @@ theorem fixpoint_correct :
· rfl
end Fixpoint

/-
9.2 Application to constant propagation analysis

First, we equip the type of abstract stores with the required equality
and ordering predicates. `le` and `beq` are defined in `Constprop.lean`,
under the names `Le` and `Equal`.
-/
section Constprop

open Std.HashMap
Expand Down Expand Up @@ -136,6 +216,9 @@ def Eq'_sym : ∀ S1 S2, Eq' S1 S2 → Eq' S2 S1 := by

@[grind] def Gt (S1 S2 : Store) := Le S2 S1 ∧ ¬ Eq' S2 S1

/-
We will use monotonically increasing functions a lot.
-/
@[grind] def Increasing (F : Store → Store) := ∀ x y, Le x y → Le (F x) (F y)

theorem hash_set_incl_size_leq (S1 S2 : Store) : Le S2 S1 → List.Subperm (S1.toList) (S2.toList) := by
Expand All @@ -150,6 +233,12 @@ theorem hash_set_incl_size_leq (S1 S2 : Store) : Le S2 S1 → List.Subperm (S1.t
specialize LE k v (by grind)
grind

/-
The really hard proof is to show that the strict order `Gt` is
well-founded. For this we reason on the cardinal of the finite maps
representing abstract stores, that is, the number of "`x` is mapped to `n`" facts
contained in abstract stores.
-/
@[grind] theorem Le_cardinal :
∀ S T : Store,
Le T S ->
Expand Down Expand Up @@ -197,6 +286,11 @@ noncomputable instance : OrderStruct Store where
gt_wf := Gt_wf
end Constprop

/-
Another difficulty is that our type of abstract stores does not have
a smallest element `bot`. But for the kind of functions we take fixpoints of,
we know a pre-fixpoint we can start iterating with.
-/
section FixpointJoin
variable (Init : Store)
variable (F : Store → Store) [Monotone Store F]
Expand Down Expand Up @@ -263,6 +357,14 @@ theorem fixpoint_join_smallest :
· unfold fixpoint_join
dsimp

/-
Now we can try to use the `fixpoint_join` function above in the `Cexec`
static analyzer. Howver, we need to simulateneously define the `Cexec` function, while
showing that it is increasing.

This can be done, but we'll need a lot of lemmas about increasing
functions first.
-/
@[grind] theorem Join_increasing :
∀ S1 S2 S3 S4,
Le S1 S2 -> Le S3 S4 -> Le (Join S1 S3) (Join S2 S4) := by grind
Expand Down Expand Up @@ -312,6 +414,10 @@ theorem fixpoint_join_increasing (_ : Store) (F : Store → Store) (F_mon : ∀
· exact hyp
· grind

/-
At long last, we can define `Cexec` while at the same time showing that
it is increasing.
-/
noncomputable def Cexec' : (c : com) → {F : Store → Store // ∀ x y, le x y → le (F x) (F y)}
| .SKIP => ⟨(fun S => S), by grind⟩
| .ASSIGN x a => ⟨(fun S => Update x (Aeval S a) S), by
Expand Down