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Remove trace suffix from Reaching type

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This commit is contained in:
Danila Fedorin
2026-08-09 21:23:21 -05:00
parent df4d072f22
commit fd371ba175
5 changed files with 65 additions and 109 deletions
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62
lean/Spa/Language/Traces.lean
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@@ -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