agda-spa/lean/Spa/Analysis/Forward.lean

import Spa.Analysis.Forward.Lattices
import Spa.Analysis.Forward.Evaluation
import Spa.Analysis.Forward.Adapters
import Spa.Fixedpoint

namespace Spa

namespace Forward

variable {L : Type} [FiniteHeightLattice L] {prog : Program} [E : StmtEvaluator L prog]

def updateVariablesForState (s : prog.State) (sv : StateVariables L prog) :
    VariableValues L prog := E.eval s (variablesAt s sv)

lemma updateVariablesForState_mono (s : prog.State) :
    Monotone (updateVariablesForState (L := L) s) := fun _ _ hle =>
  E.eval_mono s (variablesAt_le hle s)

def updateAll (sv : StateVariables L prog) : StateVariables L prog :=
  FiniteMap.generalizedUpdate id updateVariablesForState
    prog.states sv

lemma updateAll_mono : Monotone (updateAll (L := L) (prog := prog)) :=
  FiniteMap.generalizedUpdate_monotone monotone_id updateVariablesForState_mono

lemma updateAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
    {sv : StateVariables L prog} (hmem : (s, vs) ∈ updateAll sv) :
    vs = updateVariablesForState s sv :=
  FiniteMap.generalizedUpdate_mem_eq (prog.states_complete s) hmem

lemma variablesAt_updateAll (s : prog.State) (sv : StateVariables L prog) :
    variablesAt s (updateAll sv) = updateVariablesForState s sv :=
  updateAll_mem_eq (variablesAt_mem s (updateAll sv))

def analyze (sv : StateVariables L prog) : StateVariables L prog :=
  updateAll (joinAll sv)

lemma analyze_mono : Monotone (analyze (L := L) (prog := prog)) := fun _ _ hle =>
  updateAll_mono (joinAll_mono hle)

variable [DecidableEq L]

variable (L prog) in
def result : StateVariables L prog :=
  Fixedpoint.aFix analyze analyze_mono

variable (L prog) in
lemma result_eq : result L prog = analyze (result L prog) :=
  Fixedpoint.aFix_eq analyze analyze_mono

lemma joinForKey_initialState :
    joinForKey prog.initialState (result L prog) = botV L prog := by
  rw [joinForKey, prog.incoming_initialState_eq_nil]
  rfl

class ValidStateEvaluator (L : Type) [FiniteHeightLattice L] (prog : Program)
    [E : StmtEvaluator L prog] [S : StateInterpretation L prog] where
  valid : ∀ (s₁ s₂ : prog.State) {ρ₁ ρ₂ ρ₃: Env}
    {vs : VariableValues L prog},
    (tr : Traceₗ prog.cfg s₁ s₂ ρ₁ ρ₂) →
    (hbs : EvalBasicStmtOpt ρ₂ (prog.cfg.nodes s₂) ρ₃) → ⟦ vs ⟧ (S.Pre tr) →
    ⟦ E.eval s₂ vs ⟧ (S.Post (tr ++ hbs))
  botV_init : ⟦ botV L prog ⟧ (S.Pre (Traceₗ.single prog.cfg prog.initialState []))

instance [LatticeInterpretation L] [ValidStmtEvaluator L prog] :
    ValidStateEvaluator L prog where
  valid := by intro _ _ _ _ _ _ tr hbs hvs; exact ValidStmtEvaluator.valid hbs hvs
  botV_init := by intro k l _ v hmem; cases hmem

section
variable [S : StateInterpretation L prog] [V : ValidStateEvaluator L prog]

omit [DecidableEq L] in
lemma updateAll_matches {s₁ s₂ : prog.State} {sv : StateVariables L prog}
    {ρ₁ ρ₂ ρ₃ : Env}
    (tr : Traceₗ prog.cfg s₁ s₂ ρ₁ ρ₂)
    (hnode : EvalBasicStmtOpt ρ₂ (prog.code s₂) ρ₃)
    (hvs : ⟦ variablesAt s₂ sv ⟧ (S.Pre tr)) :
    ⟦ variablesAt s₂ (updateAll sv) ⟧ (S.Post (tr ++ hnode)) := by
  rw [variablesAt_updateAll]
  exact V.valid s₁ s₂ tr hnode hvs

lemma stepTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
    (tr : Traceₗ prog.cfg s₁ s₂ ρ₁ ρ₂)
    (hjoin : ⟦ joinForKey s₂ (result L prog) ⟧ (S.Pre tr))
    (hnode : EvalBasicStmtOpt ρ₂ (prog.code s₂) ρ₃) :
    ⟦ variablesAt s₂ (result L prog) ⟧ (S.Post (tr ++ hnode)) := by
  rw [result_eq L prog]
  refine updateAll_matches tr hnode ?_
  rw [variablesAt_joinAll]
  exact hjoin

/-- Soundness propagates along an execution prefix: if the analysis is sound at
    `s₂` for the run so far (`trₗ`), then it is sound wherever the further prefix
    `mid` ends up. -/
lemma walkPrefix : ∀ {s₂ s : prog.State} {ρ₂ ρin : Env}
    (mid : Traceₗ prog.cfg s₂ s ρ₂ ρin) {s₁ : prog.State} {ρ₁ : Env}
    (trₗ : Traceₗ prog.cfg s₁ s₂ ρ₁ ρ₂),
    ⟦ joinForKey s₂ (result L prog) ⟧ (S.Pre trₗ) →
    ⟦ joinForKey s (result L prog) ⟧ (S.Pre (trₗ ++ mid)) := by
  intro s₂ s ρ₂ ρin mid
  induction mid with
  | nil => intro s₁ ρ₁ trₗ hjoin; simpa [HAppend.hAppend, Traceₗ.append] using hjoin
  | cons hnode hedge rest ih =>
      intro s₁ ρ₁ trₗ hjoin
      have hstep := stepTrace trₗ hjoin hnode
      have hmem := FiniteMap.mem_valuesAt prog.states_nodup
        (prog.mem_incoming_of_edge hedge) (variablesAt_mem _ (result L prog))
      simpa [HAppend.hAppend, Traceₗ.append] using
        ih ((trₗ ++ hnode).addEdge hedge)
          (interp_foldr (S.post_pre (trₗ ++ hnode) hedge hstep) hmem)

omit [DecidableEq L] in
/-- The final node of a trace is always reached, with the environment/state the trace
    ends in. Used to recover the final-state soundness theorem from `walkPrefix`. -/
def reaches_final {s : prog.State} {ρ : Env}
    (tr : Trace prog.cfg prog.initialState s [] ρ) : Σ ρin, Reaches s ρin ρ :=
  ⟨_, ⟨tr.split.2.1, tr.split.2.2⟩⟩

omit [DecidableEq L] in
@[simp] lemma reaches_final_post {s : prog.State} {ρ : Env}
    (tr : Trace prog.cfg prog.initialState s [] ρ) :
    (reaches_final tr).2.post = tr := Trace.split_append tr

variable (L prog) in
/-- Soundness at every program point an execution actually visits: the analysis
    over-approximates both the environment entering that point and the one leaving
    it. -/
theorem analyze_correct_at {s : prog.State} {ρin ρout : Env} (hr : Reaches s ρin ρout) :
    ⟦ joinForKey s (result L prog) ⟧ (S.Pre hr.pre)
      ∧ ⟦ variablesAt s (result L prog) ⟧ (S.Post hr.post) :=
  have hpre := walkPrefix hr.pre Traceₗ.nil
    (by rw [joinForKey_initialState]; exact ValidStateEvaluator.botV_init)
  ⟨hpre, stepTrace hr.pre hpre hr.step⟩

variable (L prog) in
theorem analyze_correct'
    {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
    ⟦ variablesAt prog.finalState (result L prog) ⟧ (S.Post (prog.trace hrun)) := by
  have h := (analyze_correct_at L prog (reaches_final (prog.trace hrun)).2).2
  rwa [reaches_final_post] at h

end

variable (L prog) in
theorem analyze_correct [LatticeInterpretation L] [ValidStmtEvaluator L prog]
    {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
    ⟦ variablesAt prog.finalState (result L prog) ⟧ ρ :=
  analyze_correct' L prog hrun

end Forward

end Spa