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feat: precise close_var_not_fvar #585

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feat: precise close_var_not_fvar #585

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2 changes: 1 addition & 1 deletion Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/FullBeta.lean
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Expand Up @@ -104,7 +104,7 @@ lemma step_not_fv (step : M ⭢βᶠ N) (hw : w ∉ M.fv) : w ∉ N.fv := by
| abs =>
have ⟨x, _⟩ := fresh_exists <| free_union [fv] Var
have := open_close x
grind [close_preserve_not_fvar, open_preserve_not_fvar]
grind [open_preserve_not_fvar]
| _ => grind

/-- Abstracting then closing preserves a single reduction. -/
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2 changes: 1 addition & 1 deletion Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/FullEta.lean
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Expand Up @@ -69,7 +69,7 @@ lemma step_not_fv (step : M ⭢ηᶠ M') (hw : w ∉ M.fv) : w ∉ M'.fv := by
| abs =>
have ⟨x, _⟩ := fresh_exists <| free_union [fv] Var
have := open_close x
grind [close_preserve_not_fvar, open_preserve_not_fvar]
grind [open_preserve_not_fvar]
| _ => grind

/-- Substitution of a fresh variable preserves an η-reduction step. -/
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17 changes: 7 additions & 10 deletions Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/Properties.lean
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Expand Up @@ -52,9 +52,14 @@ lemma swap_open_fvar_close (k n : ℕ) (x y : Var) (m : Term Var) (neq₁ : k
induction m generalizing k n <;> grind

/-- Closing preserves free variables. -/
lemma close_preserve_not_fvar {k y} (m : Term Var) : (m⟦k ↜ y⟧).fv = m.fv.erase y := by
@[scoped grind =]
lemma close_rec_fv {k y} (m : Term Var) : (m⟦k ↜ y⟧).fv = m.fv.erase y := by
induction m generalizing k <;> grind

/-- Specializes `close_var_not_fvar_rec` to first closing. -/
@[scoped grind =]
lemma close_var_not_fvar (x) (t : Term Var) : (t ^* x).fv = t.fv.erase x := close_rec_fv t

/-- Opening preserves free variables. -/
lemma open_preserve_not_fvar {k x} (m n : Term Var) (nmem_m : x ∉ m.fv) (nmem_n : x ∉ n.fv) :
x ∉ (m⟦k ↝ n⟧).fv := by
Expand All @@ -65,14 +70,6 @@ lemma subst_preserve_not_fvar {x y : Var} (m n : Term Var) (nmem : x ∉ m.fv
x ∉ (m [y := n]).fv := by
induction m <;> grind

/-- Closing removes a free variable. -/
@[scoped grind ←]
lemma close_var_not_fvar_rec (x) (k) (t : Term Var) : x ∉ (t⟦k ↜ x⟧).fv := by
induction t generalizing k <;> grind

/-- Specializes `close_var_not_fvar_rec` to first closing. -/
lemma close_var_not_fvar (x) (t : Term Var) : x ∉ (t ^* x).fv := close_var_not_fvar_rec x 0 t

variable [HasFresh Var]

omit [DecidableEq Var] in
Expand Down Expand Up @@ -139,7 +136,7 @@ lemma open_close_to_subst (m : Term Var) (x y : Var) (k : ℕ) (m_lc : LC m) :
grind [
swap_open, =_ swap_open_fvar_close,
open_close x' (t⟦k+1 ↜ x⟧⟦k+1 ↝ fvar y⟧) 0, open_close x' (t[x := fvar y]) 0,
open_preserve_not_fvar, close_preserve_not_fvar, subst_preserve_not_fvar]
open_preserve_not_fvar, close_rec_fv, subst_preserve_not_fvar]
| _ => grind

/-- Closing and opening are inverses. -/
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