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Add proof of reaching definition analysis

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This requires a few pieces:

* Make node tags use `Fin n` intead of natural numbers. This makes
  it possible to build a finite lattice over AST nodes, and also
  ensure automatic, total indexing from CFG nodes into the AST that
  created them. For this, use the elaborator to derive the ordering
  statements etc. where possible.
* Adjust the forward framework to enable proofs that don't just state
  correctness on the environment, but also on an arbitrary additional
  state accumulated from traversing the trace.
* State the reaching definition analysis's correctness in terms
  of this new framework.

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
This commit is contained in:
Danila Fedorin
2026-06-27 16:29:16 -05:00
parent 5737805125
commit b6b30958aa
20 changed files with 678 additions and 197 deletions
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10
lean/Spa/Language/Traces.lean
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@@ -3,14 +3,14 @@ import Spa.Language.Graphs
namespace Spa
inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Prop
inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Type
| single {ρ₁ ρ₂ : Env} {idx : g.Index} :
EvalBasicStmts ρ₁ (g.nodes idx) ρ₂ → Trace g idx idx ρ₁ ρ₂
EvalBasicStmtOpt ρ₁ (g.nodes idx) ρ₂ → Trace g idx idx ρ₁ ρ₂
| edge {ρ₁ ρ₂ ρ₃ : Env} {idx₁ idx₂ idx₃ : g.Index} :
EvalBasicStmts ρ₁ (g.nodes idx₁) ρ₂ → (idx₁, idx₂) ∈ g.edges →
EvalBasicStmtOpt ρ₁ (g.nodes idx₁) ρ₂ → (idx₁, idx₂) ∈ g.edges →
Trace g idx₂ idx₃ ρ₂ ρ₃ → Trace g idx₁ idx₃ ρ₁ ρ₃
lemma Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
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
@@ -18,7 +18,7 @@ lemma Trace.concat {g : Graph} {idx₁ idx₂ idx₃ idx₄ : g.Index}
| single hbs => exact Trace.edge hbs he tr₂
| edge hbs he' _ ih => exact Trace.edge hbs he' (ih he tr₂)
inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Prop
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