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feat: generalize lcAt_openRec_iff_lcAt #833

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feat: generalize lcAt_openRec_iff_lcAt #833

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19 changes: 11 additions & 8 deletions Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/LcAt.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -64,15 +64,21 @@ lemma depth_openRec_fvar_eq_depth (M : Term Var) (x : Var) (i : ℕ) :
theorem depth_open_fvar_eq_depth (M : Term Var) (x : Var) : depth (M ^ fvar x) = depth M :=
depth_openRec_fvar_eq_depth M x 0

/-- Opening for some free variable at i-th bound variable, increments `LcAt`. -/
lemma lcAt_le (M : Term Var) (i j : ℕ) (h : i ≤ j) (lc : LcAt i M) : LcAt j M := by
induction M generalizing i j <;> grind

@[simp, scoped grind =]
theorem lcAt_openRec_fvar_iff_lcAt (M : Term Var) (x : Var) (i : ℕ) :
LcAt i (M⟦i ↝ fvar x⟧) = LcAt (i + 1) M := by
induction M generalizing i <;> grind
theorem lcAt_openRec_iff_lcAt (M N : Term Var) (i : ℕ) (h : LcAt i N) :
LcAt i (M⟦i ↝ N⟧) = LcAt (i + 1) M := by
induction M generalizing i with
| bvar _ => grind
| fvar _ => grind
| app _ _ _ _ => grind
| abs _ ih => exact ih _ (lcAt_le _ _ _ (by omega) h)

/-- Opening for some free variable is locally closed if and only if `M` is `LcAt 1`. -/
theorem lcAt_open_fvar_iff_lcAt (M : Term Var) (x : Var) : LcAt 0 (M ^ fvar x) = LcAt 1 M :=
lcAt_openRec_fvar_iff_lcAt M x 0
lcAt_openRec_iff_lcAt M (fvar x) 0 (lcAt_le _ 0 0 (by omega) (by grind))

/-- Locally closed terms. -/
inductive LC : Term Var → Prop
Expand Down Expand Up @@ -135,7 +141,4 @@ lemma lcAt_openRec_above_lcAt (M N : Term Var) (i j : ℕ) (h : i ≤ j) (lc : L
M⟦j ↝ N⟧ = M := by
induction M generalizing i j <;> grind

lemma lcAt_le (M : Term Var) (i j : ℕ) (h : i ≤ j) (lc : LcAt i M) : LcAt j M := by
induction M generalizing i j <;> grind

end Cslib.LambdaCalculus.LocallyNameless.Untyped.Term
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