Lean migration: Phase 6 (forward analysis framework)
- Spa.Analysis.Forward.Lattices: VariableValues/StateVariables (FiniteMap instantiations), fixed heights, variablesAt, joinForKey/joinAll, interpV and its sup/foldr lemmas - Spa.Analysis.Forward.Evaluation: StmtEvaluator/ExprEvaluator + validity (the Agda Valid* instance records become plain Props) - Spa.Analysis.Forward.Adapters: expr-to-stmt evaluator adapter + validity - Spa.Analysis.Forward: updateAll, analyze, result (least fixpoint via the gas-based Fixedpoint), walkTrace, analyze_correct — the framework's main soundness theorem Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>
This commit is contained in:
153
lean/Spa/Analysis/Forward/Lattices.lean
Normal file
153
lean/Spa/Analysis/Forward/Lattices.lean
Normal file
@@ -0,0 +1,153 @@
|
||||
/-
|
||||
Port of `Analysis/Forward/Lattices.agda`.
|
||||
|
||||
The Agda module instantiates `Lattice.FiniteMap` twice (variables ↦ abstract
|
||||
values, states ↦ variable maps) and re-exports everything with ᵛ/ᵐ suffixes.
|
||||
In Lean the two instantiations are `abbrev`s and the FiniteMap API is used
|
||||
directly; the module parameters (the finite-height lattice `L`, the program)
|
||||
become section variables.
|
||||
|
||||
Correspondence:
|
||||
VariableValues, StateVariables ↦ VariableValues, StateVariables
|
||||
isLatticeᵛ/isLatticeᵐ, ⊔ᵛ, ≼ᵛ … ↦ (the FiniteMap Lattice instances)
|
||||
fixedHeightᵛ ↦ varsFixedHeight
|
||||
⊥ᵛ, ⊥ᵛ-contains-bottoms ↦ botV, FiniteMap.bot_contains_bots
|
||||
states-in-Map ↦ states_memKey
|
||||
variablesAt ↦ variablesAt
|
||||
variablesAt-∈ ↦ variablesAt_mem
|
||||
variablesAt-≈ ↦ (congruence, trivial with `=`)
|
||||
joinForKey, joinForKey-Mono ↦ joinForKey, joinForKey_mono
|
||||
joinAll, joinAll-Mono,
|
||||
joinAll-k∈ks-≡ ↦ joinAll, joinAll_mono, joinAll_mem_eq
|
||||
variablesAt-joinAll ↦ variablesAt_joinAll
|
||||
⟦_⟧ᵛ ↦ interpV
|
||||
⟦⊥ᵛ⟧ᵛ∅ ↦ interpV_botV_nil
|
||||
⟦⟧ᵛ-respects-≈ᵛ ↦ (trivial with `=`)
|
||||
⟦⟧ᵛ-⊔ᵛ-∨ ↦ interpV_sup
|
||||
⟦⟧ᵛ-foldr ↦ interpV_foldr
|
||||
-/
|
||||
import Spa.Language
|
||||
import Spa.Lattice.FiniteMap
|
||||
|
||||
namespace Spa
|
||||
|
||||
variable (L : Type) [Lattice L] (prog : Program)
|
||||
|
||||
/-- Agda: `VariableValues`. -/
|
||||
abbrev VariableValues : Type := FiniteMap String L prog.vars
|
||||
|
||||
/-- Agda: `StateVariables`. -/
|
||||
abbrev StateVariables : Type := FiniteMap prog.State (VariableValues L prog) prog.states
|
||||
|
||||
variable {h : ℕ}
|
||||
|
||||
/-- Agda: `fixedHeightᵛ`. -/
|
||||
def varsFixedHeight (fhL : FixedHeight L h) :
|
||||
FixedHeight (VariableValues L prog) (prog.vars.length * h) :=
|
||||
FiniteMap.fixedHeight fhL prog.vars
|
||||
|
||||
/-- Agda: `⊥ᵛ`. -/
|
||||
def botV (fhL : FixedHeight L h) : VariableValues L prog :=
|
||||
(varsFixedHeight L prog fhL).bot
|
||||
|
||||
/-- Agda: `fixedHeight` on `StateVariables` (assembled in `Forward.agda`'s
|
||||
fixpoint call; named here for reuse). -/
|
||||
def statesFixedHeight (fhL : FixedHeight L h) :
|
||||
FixedHeight (StateVariables L prog) (prog.states.length * (prog.vars.length * h)) :=
|
||||
FiniteMap.fixedHeight (varsFixedHeight L prog fhL) prog.states
|
||||
|
||||
variable {L prog}
|
||||
|
||||
omit [Lattice L] in
|
||||
/-- Agda: `states-in-Map`. -/
|
||||
theorem states_memKey (s : prog.State) (sv : StateVariables L prog) :
|
||||
FiniteMap.MemKey s sv :=
|
||||
FiniteMap.memKey_iff.mpr (prog.states_complete s)
|
||||
|
||||
/-- Agda: `variablesAt`. -/
|
||||
def variablesAt (s : prog.State) (sv : StateVariables L prog) :
|
||||
VariableValues L prog :=
|
||||
(FiniteMap.locate (states_memKey s sv)).1
|
||||
|
||||
omit [Lattice L] in
|
||||
/-- Agda: `variablesAt-∈`. -/
|
||||
theorem variablesAt_mem (s : prog.State) (sv : StateVariables L prog) :
|
||||
(s, variablesAt s sv) ∈ sv :=
|
||||
(FiniteMap.locate (states_memKey s sv)).2
|
||||
|
||||
/-- Agda: `m₁≼m₂⇒m₁[k]ᵐ≼m₂[k]ᵐ`, specialized the way `Forward.agda` uses it. -/
|
||||
theorem variablesAt_le {sv₁ sv₂ : StateVariables L prog} (hle : sv₁ ≤ sv₂)
|
||||
(s : prog.State) : variablesAt s sv₁ ≤ variablesAt s sv₂ :=
|
||||
FiniteMap.le_of_mem_mem prog.states_nodup hle
|
||||
(variablesAt_mem s sv₁) (variablesAt_mem s sv₂)
|
||||
|
||||
variable (fhL : FixedHeight L h)
|
||||
|
||||
/-- Agda: `joinForKey`. -/
|
||||
def joinForKey (k : prog.State) (sv : StateVariables L prog) :
|
||||
VariableValues L prog :=
|
||||
(sv.valuesAt (prog.incoming k)).foldr (· ⊔ ·) (botV L prog fhL)
|
||||
|
||||
/-- Agda: `joinForKey-Mono`. -/
|
||||
theorem joinForKey_mono (k : prog.State) :
|
||||
Monotone (joinForKey fhL k) := by
|
||||
intro sv₁ sv₂ hle
|
||||
exact foldr_mono _ (FiniteMap.valuesAt_le hle (prog.incoming k)) (le_refl _)
|
||||
(fun b _ _ hab => sup_le_sup_right hab b)
|
||||
(fun a _ _ hab => sup_le_sup_left hab a)
|
||||
|
||||
/-- Agda: `joinAll` (the "Exercise 4.26" generalized update with `f = id`). -/
|
||||
def joinAll (sv : StateVariables L prog) : StateVariables L prog :=
|
||||
FiniteMap.generalizedUpdate id (joinForKey fhL) prog.states sv
|
||||
|
||||
/-- Agda: `joinAll-Mono`. -/
|
||||
theorem joinAll_mono : Monotone (joinAll (prog := prog) fhL) :=
|
||||
FiniteMap.generalizedUpdate_monotone monotone_id (joinForKey_mono fhL)
|
||||
|
||||
/-- Agda: `joinAll-k∈ks-≡`. -/
|
||||
theorem joinAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
|
||||
{sv : StateVariables L prog} (h : (s, vs) ∈ joinAll fhL sv) :
|
||||
vs = joinForKey fhL s sv :=
|
||||
FiniteMap.generalizedUpdate_mem_eq (prog.states_complete s) h
|
||||
|
||||
/-- Agda: `variablesAt-joinAll`. -/
|
||||
theorem variablesAt_joinAll (s : prog.State) (sv : StateVariables L prog) :
|
||||
variablesAt s (joinAll fhL sv) = joinForKey fhL s sv :=
|
||||
joinAll_mem_eq fhL (variablesAt_mem s (joinAll fhL sv))
|
||||
|
||||
/-! ### Lifting an interpretation to variable maps -/
|
||||
|
||||
variable (I : LatticeInterpretation L)
|
||||
|
||||
/-- Agda: `⟦_⟧ᵛ`. -/
|
||||
def interpV (vs : VariableValues L prog) (ρ : Env) : Prop :=
|
||||
∀ (k : String) (l : L), (k, l) ∈ vs →
|
||||
∀ (v : Value), Env.Mem (k, v) ρ → I.interp l v
|
||||
|
||||
/-- Agda: `⟦⊥ᵛ⟧ᵛ∅`. -/
|
||||
theorem interpV_botV_nil : interpV I (botV L prog fhL) [] := by
|
||||
intro k l _ v hmem
|
||||
cases hmem
|
||||
|
||||
/-- Agda: `⟦⟧ᵛ-⊔ᵛ-∨`. -/
|
||||
theorem interpV_sup {vs₁ vs₂ : VariableValues L prog} {ρ : Env}
|
||||
(h : interpV I vs₁ ρ ∨ interpV I vs₂ ρ) : interpV I (vs₁ ⊔ vs₂) ρ := by
|
||||
intro k l hmem v hv
|
||||
obtain ⟨l₁, l₂, rfl, h₁, h₂⟩ := FiniteMap.mem_sup hmem
|
||||
rcases h with h | h
|
||||
· exact I.interp_sup v (Or.inl (h _ _ h₁ _ hv))
|
||||
· exact I.interp_sup v (Or.inr (h _ _ h₂ _ hv))
|
||||
|
||||
/-- Agda: `⟦⟧ᵛ-foldr`. -/
|
||||
theorem interpV_foldr {vs : VariableValues L prog}
|
||||
{vss : List (VariableValues L prog)} {ρ : Env}
|
||||
(hvs : interpV I vs ρ) (hmem : vs ∈ vss) :
|
||||
interpV I (vss.foldr (· ⊔ ·) (botV L prog fhL)) ρ := by
|
||||
induction vss with
|
||||
| nil => cases hmem
|
||||
| cons vs' vss' ih =>
|
||||
rcases List.mem_cons.mp hmem with rfl | hmem'
|
||||
· exact interpV_sup I (Or.inl hvs)
|
||||
· exact interpV_sup I (Or.inr (ih hmem'))
|
||||
|
||||
end Spa
|
||||
Reference in New Issue
Block a user