agda-spa/lean/Spa/Language/Traces.lean

import Spa.Language.Semantics
import Spa.Language.Graphs

/-!

# Program Traces

This module defines program traces tied to Control Flow Graphs, or CFGs
(see `Spa.GGraph` and `Spa.Graph`). These traces boil town to sequences of
basic-block executions (really, `Spa.BasicStmt` executions), each of which must
have an actual basic block in the graph _and_ be connected to the previous
basic block by an edge. In this way, traces encode executions admitted
by the CFG.

While the regular `Trace` is just _any_ path through the graph, an
`EndToEndTrace` is a path from the entry node to the exit node, denoting
full program execution.

Properties about graphs and language semantics (especially,
the fact that the graph contains the proper basic block and edges
to represent any program execution according to the
language's big-step semantics `EvalStmt`) is found
in `Spa/Language/Properties.lean`.

-/

namespace Spa

/-- A partial trace through a graph `g`, starting right before
    the execution of the basic block at the first index, and
    ending right after the execution of the basic block at the last index. -/
inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Type
  | single {ρ₁ ρ₂ : Env} {idx : g.Index} :
      EvalBasicStmtOpt ρ₁ (g.nodes idx) ρ₂ → Trace g idx idx ρ₁ ρ₂
  | edge {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : g.Index} :
      EvalBasicStmtOpt ρ₁ (g.nodes idx₁) ρ₂ → (idx₁, idx₂) ∈ g.edges →
      Trace g idx₂ idx₃ ρ₂ ρ₃ → Trace g idx₁ idx₃ ρ₁ ρ₃

/-- Sequence two traces together. Since the endpoint of the first trace
    is _after_ its last basic block's execution, and the beginning of
    the next trace is _before_ its first basic block's execution,
    there must be an edge to connect the two. -/
noncomputable def Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
    {ρ₁ ρ₂ ρ₃ : Env} (tr₁ : Trace g idx₁ idx₂ ρ₁ ρ₂)
    (he : (idx₂, idx₃) ∈ g.edges) (tr₂ : Trace g idx₃ idx₄ ρ₂ ρ₃) :
    Trace g idx₁ idx₄ ρ₁ ρ₃ := by
  induction tr₁ with
  | single hbs => exact Trace.edge hbs he tr₂
  | edge hbs he' _ ih => exact Trace.edge hbs he' (ih he tr₂)

scoped notation:65 tr₁:66 " ++< " he " >++ " tr₂:65 => Trace.concat tr₁ he tr₂

/-- 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

end Spa