247 lines
12 KiB
Lean4
247 lines
12 KiB
Lean4
import Spa.Language.Semantics
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import Spa.Language.Graphs
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import Spa.Language.Program
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/-!
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# Program Traces
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This module defines program traces tied to Control Flow Graphs, or CFGs
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(see `Spa.GGraph` and `Spa.Graph`). These traces boil town to sequences of
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basic-block executions (really, `Spa.BasicStmt` executions), each of which must
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have an actual basic block in the graph _and_ be connected to the previous
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basic block by an edge. In this way, traces encode executions admitted
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by the CFG.
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While the regular `Trace` is just _any_ path through the graph, an
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`EndToEndTrace` is a path from the entry node to the exit node, denoting
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full program execution.
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Properties about graphs and language semantics (especially,
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the fact that the graph contains the proper basic block and edges
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to represent any program execution according to the
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language's big-step semantics `EvalStmt`) is found
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in `Spa/Language/Properties.lean`.
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-/
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namespace Spa
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/-- A partial trace through a graph `g`, starting right before
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the execution of the basic block at the first index, and
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ending right after the execution of the basic block at the last index. -/
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inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Type
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| single {ρ₁ ρ₂ : Env} {idx : g.Index} :
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EvalBasicStmtOpt ρ₁ (g.nodes idx) ρ₂ → Trace g idx idx ρ₁ ρ₂
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| edge {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : g.Index} :
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EvalBasicStmtOpt ρ₁ (g.nodes idx₁) ρ₂ → (idx₁, idx₂) ∈ g.edges →
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Trace g idx₂ idx₃ ρ₂ ρ₃ → Trace g idx₁ idx₃ ρ₁ ρ₃
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/-!
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## Open Traces
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A normal `Trace` starts right before one state, and ends right after another.
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This is convenient for inductively proving correctness / sufficience, but
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awkward because 1) no empty traces exist and 2) concatenation requires an extra
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edge.
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However, when attempting an "empty" trace, two types are equally possible:
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traces that end _right before_ executing a state (`Traceₗ`) and
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traces that begin _right after_ executing a state (`Traceᵣ`). They
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are symmetric and can be concatenated with full traces on the left
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and right, respectively. -/
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/-- Left-open trace, representing execution that ends right before `idx₂`. -/
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inductive Traceₗ (g : Graph) : g.Index → g.Index → Env → Env → Type where
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| nil {idx : g.Index} {ρ : Env} : Traceₗ g idx idx ρ ρ
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| cons {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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EvalBasicStmtOpt ρ₁ (g.nodes idx₁) ρ₂ →
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(idx₁, idx₂) ∈ g.edges →
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Traceₗ g idx₂ idx₃ ρ₂ ρ₃ → Traceₗ g idx₁ idx₃ ρ₁ ρ₃
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def Traceₗ.single (g : Graph) (idx : g.Index) (ρ : Env) : Traceₗ g idx idx ρ ρ := .nil
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/-- Right-open trace, representing execution that starts right after `idx₁`. -/
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inductive Traceᵣ (g : Graph) : g.Index → g.Index → Env → Env → Type where
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| nil {idx : g.Index} {ρ : Env} : Traceᵣ g idx idx ρ ρ
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| cons {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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Traceᵣ g idx₁ idx₂ ρ₁ ρ₂ →
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(idx₂, idx₃) ∈ g.edges →
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EvalBasicStmtOpt ρ₂ (g.nodes idx₃) ρ₃ → Traceᵣ g idx₁ idx₃ ρ₁ ρ₃
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def Traceᵣ.single (g : Graph) (idx : g.Index) (ρ : Env) : Traceᵣ g idx idx ρ ρ := .nil
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/-- Sequence two traces together. Since the endpoint of the first trace
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is _after_ its last basic block's execution, and the beginning of
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the next trace is _before_ its first basic block's execution,
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there must be an edge to connect the two. -/
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def Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
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{ρ₁ ρ₂ ρ₃ : Env} (tr₁ : Trace g idx₁ idx₂ ρ₁ ρ₂)
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(he : (idx₂, idx₃) ∈ g.edges) (tr₂ : Trace g idx₃ idx₄ ρ₂ ρ₃) :
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Trace g idx₁ idx₄ ρ₁ ρ₃ :=
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match tr₁ with
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| single hbs => edge hbs he tr₂
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| edge hbs he' tr₁' => edge hbs he' (tr₁'.concat he tr₂)
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scoped notation:65 tr₁:66 " ++< " he " >++ " tr₂:65 => Trace.concat tr₁ he tr₂
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def Trace.addEdge {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ : Env} :
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Trace g idx₁ idx₂ ρ₁ ρ₂ →
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(idx₂, idx₃) ∈ g.edges →
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Traceₗ g idx₁ idx₃ ρ₁ ρ₂
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| .single hnode, hedge => .cons hnode hedge .nil
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| .edge hnode hedge' rest, hedge => .cons hnode hedge' (rest.addEdge hedge)
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@[aesop simp]
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def Traceₗ.append {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → Traceₗ g idx₂ idx₃ ρ₂ ρ₃ →
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Traceₗ g idx₁ idx₃ ρ₁ ρ₃
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| .nil, rhs => rhs
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| .cons hnode hedge rest, rhs => .cons hnode hedge (rest.append rhs)
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@[simp] def traceₗ_append_nil {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ : Env}
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{trₗ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂} : trₗ.append Traceₗ.nil = trₗ := by
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induction trₗ <;> aesop
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def Traceₗ.appendTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → Trace g idx₂ idx₃ ρ₂ ρ₃ →
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Trace g idx₁ idx₃ ρ₁ ρ₃
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| .nil, rhs => rhs
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| .cons hnode hedge rest, rhs => .edge hnode hedge (rest.appendTrace rhs)
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def Traceₗ.appendStep {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃ →
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Trace g idx₁ idx₂ ρ₁ ρ₃ := fun trₗ hbs => trₗ.appendTrace (Trace.single hbs)
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def Trace.appendRight {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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Trace g idx₁ idx₂ ρ₁ ρ₂ → Traceᵣ g idx₂ idx₃ ρ₂ ρ₃ →
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Trace g idx₁ idx₃ ρ₁ ρ₃
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| lhs, .nil => lhs
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| lhs, .cons rest hedge hnode => Trace.concat (lhs.appendRight rest) hedge (.single hnode)
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instance instHAppendTraceLTraceL {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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HAppend (Traceₗ g idx₁ idx₂ ρ₁ ρ₂) (Traceₗ g idx₂ idx₃ ρ₂ ρ₃) (Traceₗ g idx₁ idx₃ ρ₁ ρ₃) where
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hAppend := Traceₗ.append
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instance instHAppendTraceLTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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HAppend (Traceₗ g idx₁ idx₂ ρ₁ ρ₂) (Trace g idx₂ idx₃ ρ₂ ρ₃) (Trace g idx₁ idx₃ ρ₁ ρ₃) where
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hAppend := Traceₗ.appendTrace
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instance instHAppendTraceLStep {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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HAppend (Traceₗ g idx₁ idx₂ ρ₁ ρ₂) (EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃) (Trace g idx₁ idx₂ ρ₁ ρ₃) where
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hAppend := Traceₗ.appendStep
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instance instHAppendTraceTraceR {g : Graph} {idx₁ idx₂ idx₃ : g.Index} {ρ₁ ρ₂ ρ₃ : Env} :
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HAppend (Trace g idx₁ idx₂ ρ₁ ρ₂) (Traceᵣ g idx₂ idx₃ ρ₂ ρ₃) (Trace g idx₁ idx₃ ρ₁ ρ₃) where
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hAppend := Trace.appendRight
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/-!
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## Trace Steps
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Analyses that care about *which statements executed* (e.g. reaching
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definitions) need to project a trace down to its list of executed statements.
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Defining that projection here, once, as a chronological mathlib `List` means
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all the re-association facts about concatenating traces come for free from
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`List.append_assoc` and friends, instead of being re-proven per analysis. -/
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/-- The (index, statement) pairs executed by a single optional-statement step:
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none if the node is empty, and the node's statement otherwise. -/
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def EvalBasicStmtOpt.steps {α : Type*} (idx : α) {ρ₁ ρ₂ : Env} {obs : Option BasicStmt} :
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EvalBasicStmtOpt ρ₁ obs ρ₂ → List (α × BasicStmt)
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| .none => []
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| .some (bs := bs) _ => [(idx, bs)]
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/-- The statements executed by a left-open trace, in chronological order. -/
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def Traceₗ.steps {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ : Env} :
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Traceₗ g idx₁ idx₂ ρ₁ ρ₂ → List (g.Index × BasicStmt)
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| .nil => []
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| .cons (idx₁ := idx) hnode _ rest => hnode.steps idx ++ rest.steps
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/-- The statements executed by a trace, in chronological order. -/
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def Trace.steps {g : Graph} {idx₁ idx₂ : g.Index} {ρ₁ ρ₂ : Env} :
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Trace g idx₁ idx₂ ρ₁ ρ₂ → List (g.Index × BasicStmt)
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| .single (idx := idx) hnode => hnode.steps idx
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| .edge (idx₁ := idx) hnode _ rest => hnode.steps idx ++ rest.steps
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@[simp] lemma Traceₗ.steps_append {g : Graph} {idx₁ idx₂ idx₃ : g.Index}
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{ρ₁ ρ₂ ρ₃ : Env} (tr₁ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
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(tr₂ : Traceₗ g idx₂ idx₃ ρ₂ ρ₃) :
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(tr₁ ++ tr₂).steps = tr₁.steps ++ tr₂.steps := by
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show (tr₁.append tr₂).steps = _
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induction tr₁ <;> simp [Traceₗ.append, Traceₗ.steps, *]
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@[simp] lemma Traceₗ.steps_appendTrace {g : Graph} {idx₁ idx₂ idx₃ : g.Index}
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{ρ₁ ρ₂ ρ₃ : Env} (tr₁ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
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(tr₂ : Trace g idx₂ idx₃ ρ₂ ρ₃) :
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(tr₁ ++ tr₂).steps = tr₁.steps ++ tr₂.steps := by
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show (tr₁.appendTrace tr₂).steps = _
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induction tr₁ <;> simp [Traceₗ.appendTrace, Traceₗ.steps, Trace.steps, *]
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@[simp] lemma Traceₗ.steps_appendStep {g : Graph} {idx₁ idx₂ : g.Index}
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{ρ₁ ρ₂ ρ₃ : Env} (tr : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
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(hbs : EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃) :
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(tr ++ hbs).steps = tr.steps ++ hbs.steps idx₂ :=
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Traceₗ.steps_appendTrace tr (Trace.single hbs)
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@[simp] lemma Trace.steps_addEdge {g : Graph} {idx₁ idx₂ idx₃ : g.Index}
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{ρ₁ ρ₂ : Env} (tr : Trace g idx₁ idx₂ ρ₁ ρ₂)
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(hedge : (idx₂, idx₃) ∈ g.edges) :
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(tr.addEdge hedge).steps = tr.steps := by
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induction tr <;> simp [Trace.addEdge, Trace.steps, Traceₗ.steps, *]
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@[simp] lemma Traceₗ.append_addEdge {g : Graph}
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{idx₁ idx₂ idx₃ idx₄ : g.Index} {ρ₁ ρ₂ ρ₃ ρ₄ : Env}
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(trₗ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
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(hnode : EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃)
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(hedge : (idx₂, idx₃) ∈ g.edges)
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(rest : Traceₗ g idx₃ idx₄ ρ₃ ρ₄) :
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trₗ.append (Traceₗ.cons hnode hedge rest) =
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(Trace.addEdge (trₗ.appendStep hnode) hedge).append rest := by
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induction trₗ <;> simp [Traceₗ.append, Traceₗ.appendStep, Traceₗ.appendTrace, Trace.addEdge, *]
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@[simp] lemma Traceₗ.appendTrace_addEdge {g : Graph}
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{idx₁ idx₂ idx₃ idx₄ : g.Index} {ρ₁ ρ₂ ρ₃ ρ₄ : Env}
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(trₗ : Traceₗ g idx₁ idx₂ ρ₁ ρ₂)
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(hnode : EvalBasicStmtOpt ρ₂ (g.nodes idx₂) ρ₃)
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(hedge : (idx₂, idx₃) ∈ g.edges)
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(rest : Trace g idx₃ idx₄ ρ₃ ρ₄) :
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trₗ.appendTrace (Trace.edge hnode hedge rest) =
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(Trace.addEdge (trₗ.appendStep hnode) hedge).appendTrace rest := by
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induction trₗ <;> simp [Traceₗ.appendTrace, Traceₗ.appendStep, Trace.addEdge, *]
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/-- A beginning-to-end trace corresponding to the CFG `g`. -/
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inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Type
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| intro (idx₁ : g.Index) (idx₁_mem : idx₁ ∈ g.inputs)
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(idx₂ : g.Index) (idx₂_mem : idx₂ ∈ g.outputs)
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(trace : Trace g idx₁ idx₂ ρ₁ ρ₂) : EndToEndTrace g ρ₁ ρ₂
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/-- Every trace splits into the prefix that arrives at its last node and that node's own step. -/
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def Trace.split {g : Graph} {i₁ i₂ : g.Index} {ρ₁ ρ₂ : Env} :
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Trace g i₁ i₂ ρ₁ ρ₂ → Σ ρ, Traceₗ g i₁ i₂ ρ₁ ρ × EvalBasicStmtOpt ρ (g.nodes i₂) ρ₂
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| .single hnode => ⟨_, .nil, hnode⟩
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| .edge hnode hedge rest =>
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let ⟨ρ, pre, step⟩ := rest.split
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⟨ρ, .cons hnode hedge pre, step⟩
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@[simp] lemma Trace.split_append {g : Graph} {i₁ i₂ : g.Index} {ρ₁ ρ₂ : Env}
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(tr : Trace g i₁ i₂ ρ₁ ρ₂) : tr.split.2.1 ++ tr.split.2.2 = tr := by
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induction tr with
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| single hnode => rfl
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| edge hnode hedge rest ih =>
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show Traceₗ.appendStep _ _ = _
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simpa [Trace.split, Traceₗ.appendStep, Traceₗ.appendTrace] using ih
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structure Reaches {prog : Program} (s : prog.State) (ρin ρout : Env) : Type where
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pre : Traceₗ prog.cfg prog.initialState s [] ρin
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step : EvalBasicStmtOpt ρin (prog.code s) ρout
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/-- Forget the environment before the last evaluated state. -/
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def Reaches.post {prog : Program} {s : prog.State} {ρin ρout : Env}
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(r : Reaches s ρin ρout) : Trace prog.cfg prog.initialState s [] ρout :=
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r.pre ++ r.step
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end Spa
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