Remove trace suffix from Reaching type

lean/Spa/Analysis/Constant.lean

@@ -137,12 +137,11 @@ theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
     ⟦ variablesAt prog.finalState (result ConstLattice prog) ⟧ ρ :=
   Forward.analyze_correct ConstLattice prog hrun
 
-theorem analyze_correct_at {ρf : Env} (hrun : EvalStmt [] prog.rootStmt ρf)
-    {s : prog.State} {ρin ρout : Env}
-    (hr : Reaches (prog.trace hrun) s ρin ρout) :
+theorem analyze_correct_at {s : prog.State} {ρin ρout : Env}
+    (hr : Reaches s ρin ρout) :
     ⟦ joinForKey s (result ConstLattice prog) ⟧ ρin
       ∧ ⟦ variablesAt s (result ConstLattice prog) ⟧ ρout :=
-  Forward.analyze_correct_at ConstLattice prog hrun hr
+  Forward.analyze_correct_at ConstLattice prog hr
 
 end ConstAnalysis
 

lean/Spa/Analysis/Forward.lean

@@ -90,75 +90,54 @@ lemma stepTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
   rw [variablesAt_joinAll]
   exact hjoin
 
-/-- Soundness at *every* visited node: if the analysis result over-approximates the
-    incoming environment at the start of the trace, then at each node reached along the
-    way it over-approximates both the environment entering that node (via `joinForKey`)
-    and the environment leaving it (via `variablesAt`). The intermediate `variablesAt`
-    evidence used to be computed and discarded inside `walkTrace`; here it is returned. -/
-lemma walkTrace_reaches {s₁ s₂ s₃: prog.State} {ρ₁ ρ₂ ρ₃: Env}
-    {s : prog.State} {ρin ρout : Env}
-    {tr : Trace prog.cfg s₂ s₃ ρ₂ ρ₃}
-    (hr : Reaches tr s ρin ρout)
-    (trₗ : Traceₗ prog.cfg s₁ s₂ ρ₁ ρ₂)
-    (hjoin : ⟦ joinForKey s₂ (result L prog) ⟧ (S.Pre trₗ)) :
-    ⟦ joinForKey s (result L prog) ⟧ (S.Pre (trₗ ++ hr.pre))
-      ∧ ⟦ variablesAt s (result L prog) ⟧ (S.Post (trₗ ++ hr.post)) := by
-  induction hr with
-  | single_here hnode =>
-    simp [Reaches.pre, Reaches.post]
-    refine ⟨?_, ?_⟩ <;> try simpa [HAppend.hAppend]
-    exact stepTrace trₗ hjoin hnode
-  | edge_here hnode hedge rest =>
-    simp [Reaches.pre, Reaches.post]
-    refine ⟨?_, ?_⟩ <;> try simpa [HAppend.hAppend]
-    exact stepTrace trₗ hjoin hnode
-  | edge_there hnode hedge rest hr' ih =>
+/-- 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 [Reaches.pre, Reaches.post, HAppend.hAppend] using
+      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 `walkTrace_reaches`. -/
-def reaches_final {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
-    (tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
-    Σ ρin, Reaches tr s₂ ρin ρ₂ :=
-  match tr with
-  | .single hnode => ⟨_, .single_here hnode⟩
-  | .edge hnode hedge rest =>
-    let ⟨ρin, r'⟩ := reaches_final rest; ⟨ρin, .edge_there hnode hedge _ r'⟩
+    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
-/-- Reaching the final node covers the whole trace. -/
-@[simp] lemma reaches_final_post {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
-    (tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
-    (reaches_final tr).2.post = tr := by
-  induction tr with
-  | single hnode => rfl
-  | edge hnode hedge rest ih => simp [reaches_final, Reaches.post, ih]
+@[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 reached during execution: for any node `s` visited
-    by the run `hrun` (witnessed by `hr`), the analysis result over-approximates both the
-    environment entering `s` and the one leaving it. The final-state theorem
-    `analyze_correct_state` is the special case where `s` is `prog.finalState`. -/
-theorem analyze_correct_at {ρf : Env} (hrun : EvalStmt [] prog.rootStmt ρf)
-    {s : prog.State} {ρin ρout : Env}
-    (hr : Reaches (prog.trace hrun) s ρin ρout) :
+/-- 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) := by
-  refine walkTrace_reaches hr (Traceₗ.single _ _ []) ?_
-  rw [joinForKey_initialState]
-  exact ValidStateEvaluator.botV_init
+      ∧ ⟦ 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 hrun (reaches_final (prog.trace hrun)).2).2
+  have h := (analyze_correct_at L prog (reaches_final (prog.trace hrun)).2).2
   rwa [reaches_final_post] at h
 
 end

lean/Spa/Analysis/Reaching.lean

@@ -118,12 +118,11 @@ theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
       (runOfTrace prog (prog.trace hrun)) :=
   Forward.analyze_correct' (DefSet prog) prog hrun
 
-theorem analyze_correct_at {ρf : Env} (hrun : EvalStmt [] prog.rootStmt ρf)
-    {s : prog.State} {ρin ρout : Env}
-    (hr : Reaches (prog.trace hrun) s ρin ρout) :
+theorem analyze_correct_at {s : prog.State} {ρin ρout : Env}
+    (hr : Reaches s ρin ρout) :
     ⟦ joinForKey s (result (DefSet prog) prog) ⟧ (runOfTraceₗ prog hr.pre)
       ∧ ⟦ variablesAt s (result (DefSet prog) prog) ⟧ (runOfTrace prog hr.post) :=
-  Forward.analyze_correct_at (DefSet prog) prog hrun hr
+  Forward.analyze_correct_at (DefSet prog) prog hr
 
 end ReachingAnalysis
 

lean/Spa/Analysis/Sign.lean

@@ -228,12 +228,11 @@ theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
     ⟦ variablesAt prog.finalState (result SignLattice prog) ⟧ ρ :=
   Forward.analyze_correct SignLattice prog hrun
 
-theorem analyze_correct_at {ρf : Env} (hrun : EvalStmt [] prog.rootStmt ρf)
-    {s : prog.State} {ρin ρout : Env}
-    (hr : Reaches (prog.trace hrun) s ρin ρout) :
+theorem analyze_correct_at {s : prog.State} {ρin ρout : Env}
+    (hr : Reaches s ρin ρout) :
     ⟦ joinForKey s (result SignLattice prog) ⟧ ρin
       ∧ ⟦ variablesAt s (result SignLattice prog) ⟧ ρout :=
-  Forward.analyze_correct_at SignLattice prog hrun hr
+  Forward.analyze_correct_at SignLattice prog hr
 
 end SignAnalysis
 

lean/Spa/Language/Traces.lean

@@ -217,50 +217,30 @@ inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Type
       (idx₂ : g.Index) (idx₂_mem : idx₂ ∈ g.outputs)
       (trace : Trace g idx₁ idx₂ ρ₁ ρ₂) : EndToEndTrace g ρ₁ ρ₂
 
-inductive Reaches {prog : Program} : {s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
-    Trace prog.cfg s₁ s₂ ρ₁ ρ₂ →
-    (s : prog.State) → (ρin ρout : Env) → Type
-  | single_here {s₁ : prog.State} {ρ₁ ρ₂ : Env}
-      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂) :
-      Reaches (.single hnode) s₁ ρ₁ ρ₂
-  | edge_here {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env}
-      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂)
-      (hedge : (s₁, s₂) ∈ prog.cfg.edges) (rest : Trace prog.cfg s₂ s₃ ρ₂ ρ₃) :
-      Reaches (.edge hnode hedge rest) s₁ ρ₁ ρ₂
-  | edge_there {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env}
-      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂)
-      (hedge : (s₁, s₂) ∈ prog.cfg.edges) (rest : Trace prog.cfg s₂ s₃ ρ₂ ρ₃)
-      {s : prog.State} {ρin ρout : Env} :
-      Reaches rest s ρin ρout →
-      Reaches (.edge hnode hedge rest) s ρin ρout
+/-- Every trace splits into the prefix that arrives at its last node and that node's own step. -/
+def Trace.split {g : Graph} {i₁ i₂ : g.Index} {ρ₁ ρ₂ : Env} :
+    Trace g i₁ i₂ ρ₁ ρ₂ → Σ ρ, Traceₗ g i₁ i₂ ρ₁ ρ × EvalBasicStmtOpt ρ (g.nodes i₂) ρ₂
+  | .single hnode => ⟨_, .nil, hnode⟩
+  | .edge hnode hedge rest =>
+      let ⟨ρ, pre, step⟩ := rest.split
+      ⟨ρ, .cons hnode hedge pre, step⟩
 
-def Reaches.pre {prog : Program} {s₁ s₂ s: prog.State}
-    {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : Reaches tr s ρin ρout) → Traceₗ prog.cfg s₁ s ρ₁ ρin
-  | .single_here _ => .nil
-  | .edge_here _ _ _ => .nil
-  | .edge_there hnode hedge _ r => .cons hnode hedge r.pre
+@[simp] lemma Trace.split_append {g : Graph} {i₁ i₂ : g.Index} {ρ₁ ρ₂ : Env}
+    (tr : Trace g i₁ i₂ ρ₁ ρ₂) : tr.split.2.1 ++ tr.split.2.2 = tr := by
+  induction tr with
+  | single hnode => rfl
+  | edge hnode hedge rest ih =>
+      show Traceₗ.appendStep _ _ = _
+      simpa [Trace.split, Traceₗ.appendStep, Traceₗ.appendTrace] using ih
 
-def Reaches.post {prog : Program} {s₁ s₂ s: prog.State}
-    {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : Reaches tr s ρin ρout) → Trace prog.cfg s₁ s ρ₁ ρout
-  | .single_here hnode => .single hnode
-  | .edge_here hnode _ _ => .single hnode
-  | .edge_there hnode hedge _ r => .edge hnode hedge r.post
+structure Reaches {prog : Program} (s : prog.State) (ρin ρout : Env) : Type where
+  pre : Traceₗ prog.cfg prog.initialState s [] ρin
+  step : EvalBasicStmtOpt ρin (prog.code s) ρout
 
-def Reaches.first {prog : Program} {s₁ s₂ s: prog.State}
-    {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : Reaches tr s ρin ρout) → Σ ρ₁', Reaches tr s₁ ρ₁ ρ₁'
-  | .single_here hnode => ⟨_, .single_here hnode⟩
-  | .edge_here hnode hedge hrest => ⟨_, .edge_here hnode hedge hrest⟩
-  | .edge_there hnode hedge hrest tmp' => ⟨_, .edge_here hnode hedge hrest⟩
-
-def Reaches.step {prog : Program} {s₁ s₂ s: prog.State}
-    {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : Reaches tr s ρin ρout) → EvalBasicStmtOpt ρin (prog.code s) ρout
-  | .single_here hnode => hnode
-  | .edge_here hnode hedge hrest => hnode
-  | .edge_there hnode hedge hrest tmp' => tmp'.step
+/-- Forget the environment before the last evaluated state. -/
+def Reaches.post {prog : Program} {s : prog.State} {ρin ρout : Env}
+    (r : Reaches s ρin ρout) : Trace prog.cfg prog.initialState s [] ρout :=
+  r.pre ++ r.step
 
 
 end Spa