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feat(FullBetaEta): redex_app_l_cong redex_app_r_cong #571

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1 change: 1 addition & 0 deletions Cslib.lean
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Original file line number Diff line number Diff line change
Expand Up @@ -116,6 +116,7 @@ public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.Basic
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.Congruence
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBeta
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaConfluence
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaEta
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaEtaConfluence
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullEta
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullEtaConfluence
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66 changes: 66 additions & 0 deletions Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/FullBetaEta.lean
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@@ -0,0 +1,66 @@
/-
Copyright (c) 2026 Maximiliano Onofre Martínez. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Maximiliano Onofre Martínez
-/

module

public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaConfluence
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullEtaConfluence

/-! # βη-Confluence for the λ-calculus

## Reference

* [T. Nipkow, *More Church-Rosser Proofs (in Isabelle/HOL)*][Nipkow2001]

-/

@[expose] public section

set_option linter.unusedDecidableInType false

namespace Cslib

universe u

variable {Var : Type u}

namespace LambdaCalculus.LocallyNameless.Untyped.Term

open Relation

/-- Full βη-reduction. -/
@[reduction_sys "βηᶠ"]
abbrev FullBetaEta : Term Var → Term Var → Prop := FullBeta ⊔ FullEta

namespace FullBetaEta

theorem redex_app_l_cong (redex : M ↠βηᶠ M') (lc_N : LC N) : app M N ↠βηᶠ app M' N := by
induction redex with
| refl => grind
| tail _ h ih => apply Relation.ReflTransGen.trans ih
apply Relation.ReflTransGen.single
cases h
· left
apply Xi.appR <;> grind
· right
apply Xi.appR <;> grind

theorem redex_app_r_cong (redex : M ↠βηᶠ M') (lc_N : LC N) : app N M ↠βηᶠ app N M' := by
induction redex with
| refl => grind
| tail _ h ih => apply Relation.ReflTransGen.trans ih
apply Relation.ReflTransGen.single
cases h
· left
apply Xi.appL <;> grind
· right
apply Xi.appL <;> grind

end FullBetaEta

end LambdaCalculus.LocallyNameless.Untyped.Term

end Cslib
5 changes: 1 addition & 4 deletions Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/FullBetaEtaConfluence.lean
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Expand Up @@ -8,6 +8,7 @@ module

public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaConfluence
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullEtaConfluence
public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaEta

/-! # βη-Confluence for the λ-calculus

Expand All @@ -32,10 +33,6 @@ namespace LambdaCalculus.LocallyNameless.Untyped.Term

open Relation

/-- Full βη-reduction. -/
@[reduction_sys "βηᶠ"]
abbrev FullBetaEta : Term Var → Term Var → Prop := FullBeta ⊔ FullEta

open FullEta FullBeta in
/-- η-reduction and β-reduction strongly commute. -/
lemma stronglyCommute_eta_beta : StronglyCommute (@FullEta Var) FullBeta := by
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