199 lines
9.3 KiB
Lean4
199 lines
9.3 KiB
Lean4
import Spa.Analysis.Forward.Lattices
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import Spa.Analysis.Forward.Evaluation
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import Spa.Analysis.Forward.Adapters
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import Spa.Fixedpoint
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namespace Spa
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namespace Forward
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variable {L : Type} [FiniteHeightLattice L] {prog : Program} [E : StmtEvaluator L prog]
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def updateVariablesForState (s : prog.State) (sv : StateVariables L prog) :
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VariableValues L prog := E.eval s (variablesAt s sv)
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lemma updateVariablesForState_mono (s : prog.State) :
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Monotone (updateVariablesForState (L := L) s) := fun _ _ hle =>
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E.eval_mono s (variablesAt_le hle s)
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def updateAll (sv : StateVariables L prog) : StateVariables L prog :=
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FiniteMap.generalizedUpdate id updateVariablesForState
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prog.states sv
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lemma updateAll_mono : Monotone (updateAll (L := L) (prog := prog)) :=
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FiniteMap.generalizedUpdate_monotone monotone_id updateVariablesForState_mono
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lemma updateAll_mem_eq {s : prog.State} {vs : VariableValues L prog}
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{sv : StateVariables L prog} (hmem : (s, vs) ∈ updateAll sv) :
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vs = updateVariablesForState s sv :=
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FiniteMap.generalizedUpdate_mem_eq (prog.states_complete s) hmem
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lemma variablesAt_updateAll (s : prog.State) (sv : StateVariables L prog) :
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variablesAt s (updateAll sv) = updateVariablesForState s sv :=
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updateAll_mem_eq (variablesAt_mem s (updateAll sv))
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def analyze (sv : StateVariables L prog) : StateVariables L prog :=
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updateAll (joinAll sv)
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lemma analyze_mono : Monotone (analyze (L := L) (prog := prog)) := fun _ _ hle =>
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updateAll_mono (joinAll_mono hle)
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variable [DecidableEq L]
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variable (L prog) in
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def result : StateVariables L prog :=
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Fixedpoint.aFix analyze analyze_mono
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variable (L prog) in
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lemma result_eq : result L prog = analyze (result L prog) :=
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Fixedpoint.aFix_eq analyze analyze_mono
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lemma joinForKey_initialState :
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joinForKey prog.initialState (result L prog) = botV L prog := by
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rw [joinForKey, prog.incoming_initialState_eq_nil]
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rfl
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class ValidStateEvaluator (L : Type) [FiniteHeightLattice L] (prog : Program)
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[E : StmtEvaluator L prog] [S : StateInterpretation L prog] where
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step : (s : prog.State) → {ρ₁ ρ₂ : Env} → EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂ → S.St ρ₁ → S.St ρ₂
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valid : ∀ (s : prog.State) {ρ₁ ρ₂ : Env}
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{vs : VariableValues L prog} {st : S.St ρ₁},
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(hbs : EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂) → ⟦ vs ⟧ ρ₁ st →
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⟦ E.eval s vs ⟧ ρ₂ (step s hbs st)
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botV_init : ⟦ botV L prog ⟧ [] S.init
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instance [LatticeInterpretation L] [ValidStmtEvaluator L prog] :
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ValidStateEvaluator L prog where
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step := by intro _ _ _ _ _; exact PUnit.unit
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valid := by intro _ _ _ _ _ hbs hvs; exact ValidStmtEvaluator.valid hbs hvs
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botV_init := by intro k l _ v hmem; cases hmem
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section
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variable [S : StateInterpretation L prog] [V : ValidStateEvaluator L prog]
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noncomputable def stepNode (s : prog.State) {ρ₁ ρ₂ : Env}
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(h : EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂) (st : S.St ρ₁) : S.St ρ₂ :=
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V.step s h st
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noncomputable def stepTraceState :
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{s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
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Trace prog.cfg s₁ s₂ ρ₁ ρ₂ → S.St ρ₁ → S.St ρ₂
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| s₁, _, _, _, .single hnode, st => stepNode s₁ hnode st
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| s₁, _, _, _, .edge hnode _ subtr, st =>
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stepTraceState subtr (stepNode s₁ hnode st)
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/-- `Reaches tr st₁ s ρin ρout stin stout` witnesses that, when the trace `tr`
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(starting at state `st₁`) is executed, node `s` is visited at some point: `ρin`
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and `ρout` are the concrete environments just before and after `s`'s basic block
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runs, and `stin`/`stout` are the corresponding abstract execution states. A node
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inside a loop is reached once per iteration, each with its own environments. -/
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inductive Reaches : {s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
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Trace prog.cfg s₁ s₂ ρ₁ ρ₂ → S.St ρ₁ →
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(s : prog.State) → (ρin ρout : Env) → S.St ρin → S.St ρout → Prop
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| single_here {s₁ : prog.State} {ρ₁ ρ₂ : Env} {st₁ : S.St ρ₁}
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(hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂) :
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Reaches (.single hnode) st₁ s₁ ρ₁ ρ₂ st₁ (stepNode s₁ hnode st₁)
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| edge_here {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env} {st₁ : S.St ρ₁}
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(hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂)
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(hedge : (s₁, s₂) ∈ prog.cfg.edges) (rest : Trace prog.cfg s₂ s₃ ρ₂ ρ₃) :
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Reaches (.edge hnode hedge rest) st₁ s₁ ρ₁ ρ₂ st₁ (stepNode s₁ hnode st₁)
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| edge_there {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env} {st₁ : S.St ρ₁}
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(hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂)
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(hedge : (s₁, s₂) ∈ prog.cfg.edges) (rest : Trace prog.cfg s₂ s₃ ρ₂ ρ₃)
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{s : prog.State} {ρin ρout : Env} {stin : S.St ρin} {stout : S.St ρout} :
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Reaches rest (stepNode s₁ hnode st₁) s ρin ρout stin stout →
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Reaches (.edge hnode hedge rest) st₁ s ρin ρout stin stout
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omit [DecidableEq L] in
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lemma updateAll_matches {s : prog.State} {sv : StateVariables L prog}
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{ρ₁ ρ₂ : Env} {st : S.St ρ₁}
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(hnode : EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂)
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(hvs : ⟦ variablesAt s sv ⟧ ρ₁ st) :
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⟦ variablesAt s (updateAll sv) ⟧ ρ₂ (stepNode s hnode st) := by
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rw [variablesAt_updateAll]
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exact V.valid s hnode hvs
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lemma stepTrace {s₁ : prog.State} {ρ₁ ρ₂ : Env} {st : S.St ρ₁}
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(hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁ st)
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(hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂) :
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⟦ variablesAt s₁ (result L prog) ⟧ ρ₂ (stepNode s₁ hnode st) := by
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rw [result_eq L prog]
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refine updateAll_matches hnode ?_
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rw [variablesAt_joinAll]
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exact hjoin
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/-- Soundness at *every* visited node: if the analysis result over-approximates the
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incoming environment at the start of the trace, then at each node reached along the
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way it over-approximates both the environment entering that node (via `joinForKey`)
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and the environment leaving it (via `variablesAt`). The intermediate `variablesAt`
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evidence used to be computed and discarded inside `walkTrace`; here it is returned. -/
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lemma walkTrace_reaches {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env} {st₁ : S.St ρ₁}
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{s : prog.State} {ρin ρout : Env} {stin : S.St ρin} {stout : S.St ρout}
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{tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂}
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(hr : Reaches tr st₁ s ρin ρout stin stout)
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(hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁ st₁) :
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⟦ joinForKey s (result L prog) ⟧ ρin stin
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∧ ⟦ variablesAt s (result L prog) ⟧ ρout stout := by
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induction hr with
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| single_here hnode => exact ⟨hjoin, stepTrace hjoin hnode⟩
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| edge_here hnode hedge rest => exact ⟨hjoin, stepTrace hjoin hnode⟩
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| edge_there hnode hedge rest hr' ih =>
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have hstep := stepTrace hjoin hnode
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have hmem := FiniteMap.mem_valuesAt prog.states_nodup
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(prog.mem_incoming_of_edge hedge) (variablesAt_mem _ (result L prog))
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exact ih (interp_foldr hstep hmem)
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omit [DecidableEq L] in
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/-- The final node of a trace is always reached, with the environment/state the trace
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ends in. Used to recover the final-state soundness theorem from `walkTrace_reaches`. -/
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lemma reaches_final {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env} (st₁ : S.St ρ₁)
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(tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
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∃ ρin, ∃ stin : S.St ρin,
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Reaches tr st₁ s₂ ρin ρ₂ stin (stepTraceState tr st₁) := by
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induction tr with
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| single hnode => exact ⟨_, _, .single_here hnode⟩
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| edge hnode hedge rest ih =>
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obtain ⟨ρin, stin, hr⟩ := ih (stepNode _ hnode st₁)
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exact ⟨ρin, stin, .edge_there hnode hedge rest hr⟩
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lemma walkTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env} {st₁ : S.St ρ₁}
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(hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁ st₁)
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(tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
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⟦ variablesAt s₂ (result L prog) ⟧ ρ₂ (stepTraceState tr st₁) := by
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obtain ⟨_, _, hr⟩ := reaches_final st₁ tr
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exact (walkTrace_reaches hr hjoin).2
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variable (L prog) in
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theorem analyze_correct_state {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
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⟦ variablesAt prog.finalState (result L prog) ⟧ ρ
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(stepTraceState (prog.trace hrun) S.init) := by
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refine walkTrace ?_ (prog.trace hrun)
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rw [joinForKey_initialState]
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exact ValidStateEvaluator.botV_init
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variable (L prog) in
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/-- Soundness at every program point reached during execution: for any node `s` visited
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by the run `hrun` (witnessed by `hr`), the analysis result over-approximates both the
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environment entering `s` and the one leaving it. The final-state theorem
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`analyze_correct_state` is the special case where `s` is `prog.finalState`. -/
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theorem analyze_correct_at {ρf : Env} (hrun : EvalStmt [] prog.rootStmt ρf)
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{s : prog.State} {ρin ρout : Env} {stin : S.St ρin} {stout : S.St ρout}
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(hr : Reaches (prog.trace hrun) S.init s ρin ρout stin stout) :
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⟦ joinForKey s (result L prog) ⟧ ρin stin
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∧ ⟦ variablesAt s (result L prog) ⟧ ρout stout := by
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refine walkTrace_reaches hr ?_
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rw [joinForKey_initialState]
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exact ValidStateEvaluator.botV_init
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end
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variable (L prog) in
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theorem analyze_correct [LatticeInterpretation L] [ValidStmtEvaluator L prog]
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{ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
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⟦ variablesAt prog.finalState (result L prog) ⟧ ρ () :=
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analyze_correct_state L prog hrun
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end Forward
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end Spa
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