Danila Fedorin
b6b30958aa
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>
27 lines
1.2 KiB
Lean4
27 lines
1.2 KiB
Lean4
import Spa.Language.Semantics
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import Spa.Language.Graphs
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namespace Spa
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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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noncomputable 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 := by
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induction tr₁ with
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| single hbs => exact Trace.edge hbs he tr₂
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| edge hbs he' _ ih => exact Trace.edge hbs he' (ih he tr₂)
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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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end Spa
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