Migrate most of the codebase (sans Reaching.lean / LICM left) to projections

lean/Spa/Analysis/Constant.lean

@@ -134,14 +134,14 @@ instance eval_valid : ValidExprEvaluator ConstLattice prog := by
     exact minus_valid h₁ h₂
 
 theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
-    ⟦ variablesAt prog.finalState (result ConstLattice prog) ⟧ 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} {stin stout : PUnit}
-    (hr : Reaches (L := ConstLattice) (prog.trace hrun) PUnit.unit s ρin ρout stin stout) :
-    ⟦ joinForKey s (result ConstLattice prog) ⟧ ρin stin
-      ∧ ⟦ variablesAt s (result ConstLattice prog) ⟧ ρout stout :=
+    {s : prog.State} {ρin ρout : Env}
+    (hr : Reaches (prog.trace hrun) s ρin ρout) :
+    ⟦ joinForKey s (result ConstLattice prog) ⟧ ρin
+      ∧ ⟦ variablesAt s (result ConstLattice prog) ⟧ ρout :=
   Forward.analyze_correct_at ConstLattice prog hrun hr
 
 end ConstAnalysis

lean/Spa/Analysis/Forward.lean

@@ -55,70 +55,38 @@ lemma joinForKey_initialState :
 
 class ValidStateEvaluator (L : Type) [FiniteHeightLattice L] (prog : Program)
     [E : StmtEvaluator L prog] [S : StateInterpretation L prog] where
-  step : (s : prog.State) → {ρ₁ ρ₂ : Env} → EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂ → S.St ρ₁ → S.St ρ₂
-  valid : ∀ (s : prog.State) {ρ₁ ρ₂ : Env}
-    {vs : VariableValues L prog} {st : S.St ρ₁},
-    (hbs : EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂) → ⟦ vs ⟧ ρ₁ st →
-    ⟦ E.eval s vs ⟧ ρ₂ (step s hbs st)
-  botV_init : ⟦ botV L prog ⟧ [] S.init
+  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
-  step := by intro _ _ _ _ _; exact PUnit.unit
-  valid := by intro _ _ _ _ _ hbs hvs; exact ValidStmtEvaluator.valid hbs hvs
+  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]
 
-noncomputable def stepNode (s : prog.State) {ρ₁ ρ₂ : Env}
-    (h : EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂) (st : S.St ρ₁) : S.St ρ₂ :=
-  V.step s h st
-
-noncomputable def stepTraceState :
-    {s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
-    Trace prog.cfg s₁ s₂ ρ₁ ρ₂ → S.St ρ₁ → S.St ρ₂
-  | s₁, _, _, _, .single hnode, st => stepNode s₁ hnode st
-  | s₁, _, _, _, .edge hnode _ subtr, st =>
-      stepTraceState subtr (stepNode s₁ hnode st)
-
-/-- `Reaches tr st₁ s ρin ρout stin stout` witnesses that, when the trace `tr`
-    (starting at state `st₁`) is executed, node `s` is visited at some point: `ρin`
-    and `ρout` are the concrete environments just before and after `s`'s basic block
-    runs, and `stin`/`stout` are the corresponding abstract execution states. A node
-    inside a loop is reached once per iteration, each with its own environments. -/
-inductive Reaches : {s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
-    Trace prog.cfg s₁ s₂ ρ₁ ρ₂ → S.St ρ₁ →
-    (s : prog.State) → (ρin ρout : Env) → S.St ρin → S.St ρout → Prop
-  | single_here {s₁ : prog.State} {ρ₁ ρ₂ : Env} {st₁ : S.St ρ₁}
-      (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂) :
-      Reaches (.single hnode) st₁ s₁ ρ₁ ρ₂ st₁ (stepNode s₁ hnode st₁)
-  | edge_here {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env} {st₁ : S.St ρ₁}
-      (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) st₁ s₁ ρ₁ ρ₂ st₁ (stepNode s₁ hnode st₁)
-  | edge_there {s₁ s₂ s₃ : prog.State} {ρ₁ ρ₂ ρ₃ : Env} {st₁ : S.St ρ₁}
-      (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} {stin : S.St ρin} {stout : S.St ρout} :
-      Reaches rest (stepNode s₁ hnode st₁) s ρin ρout stin stout →
-      Reaches (.edge hnode hedge rest) st₁ s ρin ρout stin stout
-
 omit [DecidableEq L] in
-lemma updateAll_matches {s : prog.State} {sv : StateVariables L prog}
-    {ρ₁ ρ₂ : Env} {st : S.St ρ₁}
-    (hnode : EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂)
-    (hvs : ⟦ variablesAt s sv ⟧ ρ₁ st) :
-    ⟦ variablesAt s (updateAll sv) ⟧ ρ₂ (stepNode s hnode st) := by
+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 s hnode hvs
+  exact V.valid s₁ s₂ tr hnode hvs
 
-lemma stepTrace {s₁ : prog.State} {ρ₁ ρ₂ : Env} {st : S.St ρ₁}
-    (hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁ st)
-    (hnode : EvalBasicStmtOpt ρ₁ (prog.code s₁) ρ₂) :
-    ⟦ variablesAt s₁ (result L prog) ⟧ ρ₂ (stepNode s₁ hnode st) := by
+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 hnode ?_
+  refine updateAll_matches tr hnode ?_
   rw [variablesAt_joinAll]
   exact hjoin
 
@@ -127,49 +95,41 @@ lemma stepTrace {s₁ : prog.State} {ρ₁ ρ₂ : Env} {st : S.St ρ₁}
     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 : prog.State} {ρ₁ ρ₂ : Env} {st₁ : S.St ρ₁}
-    {s : prog.State} {ρin ρout : Env} {stin : S.St ρin} {stout : S.St ρout}
-    {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂}
-    (hr : Reaches tr st₁ s ρin ρout stin stout)
-    (hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁ st₁) :
-    ⟦ joinForKey s (result L prog) ⟧ ρin stin
-      ∧ ⟦ variablesAt s (result L prog) ⟧ ρout stout := by
+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 => exact ⟨hjoin, stepTrace hjoin hnode⟩
-  | edge_here hnode hedge rest => exact ⟨hjoin, stepTrace hjoin hnode⟩
+  | 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 =>
-    have hstep := stepTrace hjoin hnode
+    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))
-    exact ih (interp_foldr hstep hmem)
+    simpa [Reaches.pre, Reaches.post, HAppend.hAppend] 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`. -/
-lemma reaches_final {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env} (st₁ : S.St ρ₁)
+def reaches_final {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}
     (tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
-    ∃ ρin, ∃ stin : S.St ρin,
-      Reaches tr st₁ s₂ ρin ρ₂ stin (stepTraceState tr st₁) := by
-  induction tr with
-  | single hnode => exact ⟨_, _, .single_here hnode⟩
-  | edge hnode hedge rest ih =>
-    obtain ⟨ρin, stin, hr⟩ := ih (stepNode _ hnode st₁)
-    exact ⟨ρin, stin, .edge_there hnode hedge rest hr⟩
-
-lemma walkTrace {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env} {st₁ : S.St ρ₁}
-    (hjoin : ⟦ joinForKey s₁ (result L prog) ⟧ ρ₁ st₁)
-    (tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) :
-    ⟦ variablesAt s₂ (result L prog) ⟧ ρ₂ (stepTraceState tr st₁) := by
-  obtain ⟨_, _, hr⟩ := reaches_final st₁ tr
-  exact (walkTrace_reaches hr hjoin).2
-
-variable (L prog) in
-theorem analyze_correct_state {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
-    ⟦ variablesAt prog.finalState (result L prog) ⟧ ρ
-      (stepTraceState (prog.trace hrun) S.init) := by
-  refine walkTrace ?_ (prog.trace hrun)
-  rw [joinForKey_initialState]
-  exact ValidStateEvaluator.botV_init
+    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'⟩
 
 variable (L prog) in
 /-- Soundness at every program point reached during execution: for any node `s` visited
@@ -177,21 +137,30 @@ variable (L prog) in
     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} {stin : S.St ρin} {stout : S.St ρout}
-    (hr : Reaches (prog.trace hrun) S.init s ρin ρout stin stout) :
-    ⟦ joinForKey s (result L prog) ⟧ ρin stin
-      ∧ ⟦ variablesAt s (result L prog) ⟧ ρout stout := by
-  refine walkTrace_reaches hr ?_
+    {s : prog.State} {ρin ρout : Env}
+    (hr : Reaches (prog.trace hrun) 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
 
+variable (L prog) in
+theorem analyze_correct'
+    {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
+    ⟦ variablesAt prog.finalState (result L prog) ⟧ (S.Post (reaches_final (prog.trace hrun)).2.post) := by
+  let idk₀ := prog.trace hrun
+  have ⟨_, idk₁⟩ := reaches_final idk₀
+  have ⟨_, idk₂⟩ := analyze_correct_at L prog hrun idk₁
+  assumption
+
 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_state L prog hrun
+    ⟦ variablesAt prog.finalState (result L prog) ⟧ ρ :=
+  analyze_correct' L prog hrun
 
 end Forward
 

lean/Spa/Analysis/Forward/Evaluation.lean

@@ -17,12 +17,12 @@ class ExprEvaluator where
 class ValidExprEvaluator [ExprEvaluator L prog] [I : LatticeInterpretation L] :
     Prop where
   valid : ∀ {vs : VariableValues L prog} {ρ : Env} {e : Expr} {v : Value},
-    EvalExpr ρ e v → ⟦ vs ⟧ ρ () → I.interp (ExprEvaluator.eval e vs) v
+    EvalExpr ρ e v → ⟦ vs ⟧ ρ → I.interp (ExprEvaluator.eval e vs) v
 
 class ValidStmtEvaluator [E : StmtEvaluator L prog] [LatticeInterpretation L] :
     Prop where
   valid : ∀ {s : prog.State} {vs : VariableValues L prog} {ρ₁ ρ₂ : Env},
-    EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂ → ⟦ vs ⟧ ρ₁ () → ⟦ E.eval s vs ⟧ ρ₂ ()
+    EvalBasicStmtOpt ρ₁ (prog.code s) ρ₂ → ⟦ vs ⟧ ρ₁ → ⟦ E.eval s vs ⟧ ρ₂
 
 end Forward
 

lean/Spa/Analysis/Forward/Lattices.lean

@@ -65,22 +65,28 @@ lemma variablesAt_joinAll (s : prog.State) (sv : StateVariables L prog) :
   joinAll_mem_eq (variablesAt_mem s (joinAll sv))
 
 class StateInterpretation (L : Type) [Lattice L] (prog : Program) where
-  St : Env → Type
-  init : St []
-  interp : VariableValues L prog → (ρ : Env) → St ρ → Prop
-  interp_sup : ∀ {vs₁ vs₂ : VariableValues L prog} {ρ : Env} {st : St ρ},
-    interp vs₁ ρ st ∨ interp vs₂ ρ st → interp (vs₁ ⊔ vs₂) ρ st
-  interp_inf : ∀ {vs₁ vs₂ : VariableValues L prog} {ρ : Env} {st : St ρ},
-    interp vs₁ ρ st ∧ interp vs₂ ρ st → interp (vs₁ ⊓ vs₂) ρ st
+  Proj : Type
+  Pre : ∀ {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}, Traceₗ prog.cfg s₁ s₂ ρ₁ ρ₂ → Proj
+  Post : ∀ {s₁ s₂ : prog.State} {ρ₁ ρ₂ : Env}, Trace prog.cfg s₁ s₂ ρ₁ ρ₂ → Proj
+
+  interp : VariableValues L prog → (p : Proj) → Prop
+  interp_sup : ∀ {vs₁ vs₂ : VariableValues L prog} {p : Proj},
+    interp vs₁ p ∨ interp vs₂ p → interp (vs₁ ⊔ vs₂) p
+  interp_inf : ∀ {vs₁ vs₂ : VariableValues L prog} {p : Proj},
+    interp vs₁ p ∧ interp vs₂ p → interp (vs₁ ⊓ vs₂) p
+
+  post_pre : ∀ {vs} {s₁ s₂ s₃: prog.State} {ρ₁ ρ₂ : Env}
+    (tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂) (hedge : (s₂, s₃) ∈ prog.cfg.edges),
+    interp vs (Post tr) → interp vs (Pre (tr.addEdge hedge))
 
 instance [S : StateInterpretation L prog] :
-    Interp (VariableValues L prog) ((ρ : Env) → S.St ρ → Prop) :=
+    Interp (VariableValues L prog) (S.Proj → Prop) :=
   ⟨S.interp⟩
 
 lemma interp_foldr [S : StateInterpretation L prog]
     {vs : VariableValues L prog} {vss : List (VariableValues L prog)}
-    {ρ : Env} {st : S.St ρ} (hvs : ⟦ vs ⟧ ρ st) (hmem : vs ∈ vss) :
-    ⟦ vss.foldr (· ⊔ ·) (botV L prog) ⟧ ρ st := by
+    {p : S.Proj} (hvs : ⟦ vs ⟧ p) (hmem : vs ∈ vss) :
+    ⟦ vss.foldr (· ⊔ ·) (botV L prog) ⟧ p := by
   induction vss with
   | nil => cases hmem
   | cons vs' vss' ih =>
@@ -91,20 +97,24 @@ lemma interp_foldr [S : StateInterpretation L prog]
 variable [I : LatticeInterpretation L]
 
 instance : StateInterpretation L prog where
-  St := fun _ => PUnit
-  init := PUnit.unit
-  interp vs ρ _ := ∀ (k : String) (l : L), (k, l) ∈ vs →
+  Proj := Env
+  Pre := fun {_ _ _ ρ₂} _ => ρ₂
+  Post := fun {_ _ _ ρ₂} _ => ρ₂
+
+  interp vs ρ := ∀ (k : String) (l : L), (k, l) ∈ vs →
     ∀ (v : Value), Env.Mem (k, v) ρ → I.interp l v
   interp_sup := by
-    intro vs₁ vs₂ ρ st h k l hmem v hv
+    intro vs₁ vs₂ ρ h 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))
   interp_inf := by
-    intro vs₁ vs₂ ρ st h k l hmem v hv
+    intro vs₁ vs₂ ρ h k l hmem v hv
     obtain ⟨l₁, l₂, rfl, h₁, h₂⟩ := FiniteMap.mem_inf hmem
     exact I.interp_inf v ⟨h.1 _ _ h₁ _ hv, h.2 _ _ h₂ _ hv⟩
+  post_pre := by simp
+
 
 end Forward
 

lean/Spa/Analysis/Sign.lean

@@ -210,14 +210,14 @@ instance eval_valid : ValidExprEvaluator SignLattice prog := by
     exact minus_valid h₁ h₂
 
 theorem analyze_correct {ρ : Env} (hrun : EvalStmt [] prog.rootStmt ρ) :
-    ⟦ variablesAt prog.finalState (result SignLattice prog) ⟧ 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} {stin stout : PUnit}
-    (hr : Reaches (L := SignLattice) (prog.trace hrun) PUnit.unit s ρin ρout stin stout) :
-    ⟦ joinForKey s (result SignLattice prog) ⟧ ρin stin
-      ∧ ⟦ variablesAt s (result SignLattice prog) ⟧ ρout stout :=
+    {s : prog.State} {ρin ρout : Env}
+    (hr : Reaches (prog.trace hrun) s ρin ρout) :
+    ⟦ joinForKey s (result SignLattice prog) ⟧ ρin
+      ∧ ⟦ variablesAt s (result SignLattice prog) ⟧ ρout :=
   Forward.analyze_correct_at SignLattice prog hrun hr
 
 end SignAnalysis

lean/Spa/Language/Program.lean

@@ -50,6 +50,7 @@ lemma states_nodup : p.states.Nodup := p.cfg.nodup_indices
     At this time, for convenience of proofs, the CFGs have at most
     one basic statement, and multi-statement basic blocks are encoded
     as chains of blocks. Thus, this returns at most one `Spa.BasicStmt`. -/
+@[reducible]
 def code (st : p.State) : Option BasicStmt := p.cfg.nodes st
 
 /-- Get the predecessors of a particular CFG node / program state. -/

lean/Spa/Language/Traces.lean

@@ -93,12 +93,17 @@ def Trace.addEdge {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ : En
   | .single hnode, hedge => .cons hnode hedge .nil
   | .edge hnode hedge' rest, hedge => .cons hnode hedge' (rest.addEdge hedge)
 
+@[aesop simp]
 def Traceₗ.append {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
     Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → Traceₗ g idx₂ idx₃ ρ₂ ρ₃ →
     Traceₗ g idx₁ idx₃ ρ₁ ρ₃
   | .nil, rhs => rhs
   | .cons hnode hedge rest, rhs => .cons hnode hedge (rest.append rhs)
 
+@[simp] def traceₗ_append_nil {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ : Env}
+    {trₗ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂} : trₗ.append Traceₗ.nil = trₗ := by
+  induction trₗ <;> aesop
+
 def Traceₗ.appendTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
     Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → Trace g idx₂ idx₃ ρ₂ ρ₃ →
     Trace g idx₁ idx₃ ρ₁ ρ₃
@@ -107,9 +112,7 @@ def Traceₗ.appendTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ
 
 def Traceₗ.appendStep {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
     Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃ →
-    Trace g idx₁ idx₂ ρ₁ ρ₃
-  | .nil, rhs => .single rhs
-  | .cons hnode hedge rest, rhs => .edge hnode hedge (rest.appendStep rhs)
+    Trace g idx₁ idx₂ ρ₁ ρ₃ := fun trₗ hbs => trₗ.appendTrace (Trace.single hbs)
 
 def Trace.appendRight {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
     Trace g idx₁ idx₂ ρ₁ ρ₂ → Traceᵣ g idx₂ idx₃ ρ₂ ρ₃ →
@@ -133,53 +136,73 @@ instance instHAppendTraceTraceR {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ
     HAppend (Trace g idx₁ idx₂ ρ₁ ρ₂) (Traceᵣ g idx₂ idx₃ ρ₂ ρ₃) (Trace g idx₁ idx₃ ρ₁ ρ₃) where
   hAppend := Trace.appendRight
 
+@[simp] lemma Traceₗ.append_addEdge {g : Graph}
+    {idx₁ idx₂ idx₃ idx₄ : g.Index} {ρ₁ ρ₂ ρ₃ ρ₄ : Env}
+    (trₗ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
+    (hnode : EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃)
+    (hedge : (idx₂, idx₃) ∈ g.edges)
+    (rest : Traceₗ g idx₃ idx₄ ρ₃ ρ₄) :
+    trₗ.append (Traceₗ.cons hnode hedge rest) =
+      (Trace.addEdge (trₗ.appendStep hnode) hedge).append rest := by
+  induction trₗ <;> simp [Traceₗ.append, Traceₗ.appendStep, Traceₗ.appendTrace, Trace.addEdge, *]
+
+@[simp] lemma Traceₗ.appendTrace_addEdge {g : Graph}
+    {idx₁ idx₂ idx₃ idx₄ : g.Index} {ρ₁ ρ₂ ρ₃ ρ₄ : Env}
+    (trₗ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
+    (hnode : EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃)
+    (hedge : (idx₂, idx₃) ∈ g.edges)
+    (rest : Trace g idx₃ idx₄ ρ₃ ρ₄) :
+    trₗ.appendTrace (Trace.edge hnode hedge rest) =
+      (Trace.addEdge (trₗ.appendStep hnode) hedge).appendTrace rest := by
+  induction trₗ <;> simp [Traceₗ.appendTrace, Traceₗ.appendStep, Trace.addEdge, *]
+
 /-- A beginning-to-end trace corresponding to the CFG `g`. -/
 inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Type
   | intro (idx₁ : g.Index) (idx₁_mem : idx₁ ∈ g.inputs)
       (idx₂ : g.Index) (idx₂_mem : idx₂ ∈ g.outputs)
       (trace : Trace g idx₁ idx₂ ρ₁ ρ₂) : EndToEndTrace g ρ₁ ρ₂
 
-inductive MyReaches {prog : Program} : {s₁ s₂ : prog.State} → {ρ₁ ρ₂ : Env} →
+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) :
-      MyReaches (.single hnode) s₁ ρ₁ 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) :
-      MyReaches (.edge hnode hedge rest) s₁ ρ₁ ρ₂
+      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} :
-      MyReaches rest s ρin ρout →
-      MyReaches (.edge hnode hedge rest) s ρin ρout
+      Reaches rest s ρin ρout →
+      Reaches (.edge hnode hedge rest) s ρin ρout
 
-def MyReaches.pre {prog : Program} {s₁ s₂ s: prog.State}
+def Reaches.pre {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : MyReaches tr s ρin ρout) → Traceₗ prog.cfg s₁ s ρ₁ ρin
+    (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
 
-def MyReaches.post {prog : Program} {s₁ s₂ s: prog.State}
+def Reaches.post {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : MyReaches tr s ρin ρout) → Trace prog.cfg s₁ s ρ₁ ρout
+    (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
 
-def MyReaches.first {prog : Program} {s₁ s₂ s: prog.State}
+def Reaches.first {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : MyReaches tr s ρin ρout) → Σ ρ₁', MyReaches tr s₁ ρ₁ ρ₁'
+    (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 MyReaches.step {prog : Program} {s₁ s₂ s: prog.State}
+def Reaches.step {prog : Program} {s₁ s₂ s: prog.State}
     {ρ₁ ρ₂ ρin ρout : Env} {tr : Trace prog.cfg s₁ s₂ ρ₁ ρ₂} :
-    (r : MyReaches tr s ρin ρout) → EvalBasicStmtOpt ρin (prog.code s) ρout
+    (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