39 lines
1.6 KiB
Lean4
39 lines
1.6 KiB
Lean4
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/-
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Port of `Language/Traces.agda`.
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Correspondence:
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Trace ↦ Trace (a `Prop`-valued inductive; only used in proofs)
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_++⟨_⟩_ ↦ Trace.concat
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EndToEndTrace ↦ EndToEndTrace (a `Prop`-valued structure, like `∃`; its
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fields are accessed by destructuring inside proofs)
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-/
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import Spa.Language.Semantics
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import Spa.Language.Graphs
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namespace Spa
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/-- Agda: `Trace`. -/
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inductive Trace (g : Graph) : g.Index → g.Index → Env → Env → Prop
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| single {ρ₁ ρ₂ : Env} {idx : g.Index} :
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EvalBasicStmts 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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EvalBasicStmts 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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/-- Agda: `_++⟨_⟩_`. -/
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theorem 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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/-- Agda: `EndToEndTrace` (an existential package, destructured in proofs). -/
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inductive EndToEndTrace (g : Graph) (ρ₁ ρ₂ : Env) : Prop
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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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