78 lines
3.3 KiB
Lean4
78 lines
3.3 KiB
Lean4
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/-
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Port of `Analysis/Forward/Adapters.agda` (`ExprToStmtAdapter`).
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Correspondence:
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updateVariablesFromExpression ↦ updateVariablesFromExpression
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updateVariablesFromExpression-Mono ↦ updateVariablesFromExpression_mono
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(the -k∈ks-≡ / -k∉ks-backward renames ↦ used directly from FiniteMap)
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evalᵇ, evalᵇ-Monoʳ ↦ evalB, evalB_mono
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stmtEvaluator (instance) ↦ ExprEvaluator.toStmtEvaluator
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evalᵇ-valid, validStmtEvaluator ↦ ExprEvaluator.toStmtEvaluator_valid
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(the Agda `k ≟ˢ k'` case split is
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subsumed by `cases` on `Env.Mem`,
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whose `here` case forces `k' = k`)
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-/
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import Spa.Analysis.Forward.Evaluation
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namespace Spa
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variable {L : Type} [Lattice L] {prog : Program}
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/-- Agda: `updateVariablesFromExpression` — set the single key `k` to the
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value of `e` (the `GeneralizedUpdate` with `ks = [k]`). -/
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def updateVariablesFromExpression (E : ExprEvaluator L prog) (k : String)
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(e : Expr) (vs : VariableValues L prog) : VariableValues L prog :=
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FiniteMap.generalizedUpdate id (fun _ vs => E.eval e vs) [k] vs
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/-- Agda: `updateVariablesFromExpression-Mono`. -/
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theorem updateVariablesFromExpression_mono (E : ExprEvaluator L prog)
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(k : String) (e : Expr) :
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Monotone (updateVariablesFromExpression E k e) :=
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FiniteMap.generalizedUpdate_monotone monotone_id (fun _ => E.eval_mono e)
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/-- Agda: `evalᵇ`. -/
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def evalB (E : ExprEvaluator L prog) (_ : prog.State) (bs : BasicStmt)
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(vs : VariableValues L prog) : VariableValues L prog :=
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match bs with
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| .assign k e => updateVariablesFromExpression E k e vs
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| .noop => vs
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/-- Agda: `evalᵇ-Monoʳ`. -/
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theorem evalB_mono (E : ExprEvaluator L prog) (s : prog.State) (bs : BasicStmt) :
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Monotone (evalB E s bs) := by
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cases bs with
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| assign k e => exact updateVariablesFromExpression_mono E k e
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| noop => exact monotone_id
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/-- Agda: the `stmtEvaluator` instance of `ExprToStmtAdapter`. -/
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def ExprEvaluator.toStmtEvaluator (E : ExprEvaluator L prog) :
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StmtEvaluator L prog :=
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⟨evalB E, evalB_mono E⟩
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/-- Agda: `evalᵇ-valid` / the `validStmtEvaluator` instance. -/
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theorem ExprEvaluator.toStmtEvaluator_valid (E : ExprEvaluator L prog)
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{I : LatticeInterpretation L} (hE : IsValidExprEvaluator E I) :
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IsValidStmtEvaluator E.toStmtEvaluator I := by
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intro s vs ρ₁ ρ₂ bs hbs hvs
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cases hbs with
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| noop => exact hvs
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| assign k e v hev =>
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intro k' l hk'l v' hv'
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cases hv' with
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| here =>
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have hk'l₀ : (k, l) ∈ FiniteMap.generalizedUpdate (ks := prog.vars) id
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(fun _ vs => E.eval e vs) [k] vs := hk'l
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have hl := FiniteMap.generalizedUpdate_mem_eq (f := id)
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(g := fun _ vs => E.eval e vs) (List.mem_singleton_self k) hk'l₀
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rw [hl]
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exact hE hev hvs
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| there _ _ _ _ _ hne hmem' =>
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have hk'l₀ : (k', l) ∈ FiniteMap.generalizedUpdate (ks := prog.vars) id
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(fun _ vs => E.eval e vs) [k] vs := hk'l
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have hk'l' : (k', l) ∈ (id vs : VariableValues L prog) :=
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FiniteMap.generalizedUpdate_not_mem_backward
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(fun hmem => hne (List.mem_singleton.mp hmem)) hk'l₀
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exact hvs _ _ hk'l' _ hmem'
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end Spa
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