Danila Fedorin
94278a6389
Introduce a finite-height lattice instance for Bool, then build the
reaching-definitions analysis on top of the forward framework:
* Spa/Lattice/Bool.lean: FiniteHeightLattice Bool (the two-element
lattice false ≤ true), making FiniteMap A Bool ks a finite-height
"power set" lattice for free.
* Spa/Analysis/Reaching.lean: DefSet prog = FiniteMap prog.State Bool
prog.states as the per-variable lattice of definition sites, with a
StmtEvaluator whose transfer function performs a strong update
(assignment to k at node s sets k's def-set to {s}).
The analysis computes a least fixed point and produces correct
reaching-definitions sets. Soundness (relating def-sets to actual
execution provenance) is deferred; not yet exposed in Spa.lean.
Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>
42 lines
1.0 KiB
Lean4
42 lines
1.0 KiB
Lean4
import Spa.Analysis.Forward
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import Spa.Lattice.Bool
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import Spa.Showable
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namespace Spa
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open Forward
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instance : Showable Bool := ⟨fun b => if b then "true" else "false"⟩
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abbrev DefSet (prog : Program) : Type := FiniteMap prog.State Bool prog.states
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namespace ReachingAnalysis
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variable (prog : Program)
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def genSet (s : prog.State) : DefSet prog :=
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FiniteMap.updating (⊥ : DefSet prog) [s] (fun _ => true)
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def eval (s : prog.State) :
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BasicStmt → VariableValues (DefSet prog) prog → VariableValues (DefSet prog) prog
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| .assign k _, vs =>
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FiniteMap.generalizedUpdate id (fun _ _ => genSet prog s) [k] vs
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| .noop, vs => vs
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theorem eval_mono (s : prog.State) (bs : BasicStmt) :
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Monotone (eval prog s bs) := by
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cases bs with
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| assign k e =>
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exact FiniteMap.generalizedUpdate_monotone monotone_id (fun _ => monotone_const)
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| noop => exact monotone_id
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instance stmtEvaluator : StmtEvaluator (DefSet prog) prog :=
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⟨eval prog, eval_mono prog⟩
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def output : String :=
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show' (result (DefSet prog) prog)
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end ReachingAnalysis
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end Spa
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