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feat: decidable LcAt and LC #572

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chenson2018 merged 1 commit into from
May 20, 2026
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feat: decidable LcAt and LC #572

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10 changes: 7 additions & 3 deletions Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/LcAt.lean
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Expand Up @@ -27,10 +27,10 @@ variable {Var : Type u}

/-- `LcAt k M` is satisfied when all bound indices of M are smaller than `k`. -/
@[simp, scoped grind =]
def LcAt (k : ℕ) : Term Var → Prop
def LcAt (k : ℕ) : Term Var → Bool
| bvar i => i < k
| fvar _ => True
| app t₁ t₂ => LcAt k t₁ ∧ LcAt k t₂
| fvar _ => true
| app t₁ t₂ => LcAt k t₁ && LcAt k t₂
| abs t => LcAt (k + 1) t

/-- `depth` counts the maximum number of the lambdas that are enclosing variables. -/
Expand Down Expand Up @@ -83,6 +83,10 @@ theorem lcAt_iff_LC (M : Term Var) [HasFresh Var] : LcAt 0 M ↔ M.LC := by
grind [fresh_exists L]
| _ => grind [cases LC]

instance [HasFresh Var] (t : Term Var) : Decidable t.LC := by
rw [← lcAt_iff_LC]
infer_instance

/- Opening for some term at i-th bound variable increments `LcAt` by one -/
lemma lcAt_openRec_lcAt (M N : Term Var) (i : ℕ) :
LcAt i (M⟦i ↝ N⟧) → LcAt (i + 1) M := by
Expand Down
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