feat(untyped): the Leftmost Reduction Theorem

Cslib.lean

@@ -138,6 +138,7 @@ public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullBetaEta
 public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullEta
 public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.FullEtaConfluence
 public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.LcAt
+public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.LeftmostReduction
 public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.MultiApp
 public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.MultiSubst
 public import Cslib.Languages.LambdaCalculus.LocallyNameless.Untyped.Properties

Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/LcAt.lean

@@ -86,6 +86,11 @@ attribute [scoped grind .] LC.fvar LC.app
 inductive Value : Term Var → Prop
 | abs (e : Term Var) : e.abs.LC → e.abs.Value
 
+/-- `IsAbs m` holds when `m` is an abstraction. -/
+@[scoped grind]
+inductive IsAbs : Term Var → Prop
+| abs (m : Term Var) : IsAbs (abs m)
+
 set_option linter.tacticAnalysis.verifyGrindOnly false in
 /-- `M` is `LcAt 0` if and only if `M` is locally closed. -/
 theorem lcAt_iff_LC (M : Term Var) [HasFresh Var] : LcAt 0 M ↔ M.LC := by

Cslib/Languages/LambdaCalculus/LocallyNameless/Untyped/LeftmostReduction.lean

@@ -0,0 +1,141 @@
+/-
+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.StandardReduction
+
+/-! # The Leftmost Reduction Theorem
+
+## Reference
+
+* [M. Copes, *A machine-checked proof of the Standardization Theorem in λ-calculus*][Copes2018]
+
+-/
+
+@[expose] public section
+
+set_option linter.unusedDecidableInType false
+
+namespace Cslib
+
+universe u
+
+variable {Var : Type u}
+
+namespace LambdaCalculus.LocallyNameless.Untyped.Term
+
+/-- A term is in normal form when it contains no β-redexes. -/
+def BetaNormal (m : Term Var) : Prop := countRedexes m = 0
+
+/-- Leftmost reduction: a β-reduction contracting the redex at position 0. -/
+@[reduction_sys "ℓ"]
+abbrev Leftmost : Term Var → Term Var → Prop := BetaAt 0
+
+variable {L L' M M' N : Term Var} {i : Nat}
+
+/-- In a normal-form application, both sides are normal and the operator is not an
+    abstraction. -/
+lemma BetaNormal.app_inv (h : BetaNormal (app L M)) :
+    ¬IsAbs L ∧ BetaNormal L ∧ BetaNormal M := by
+  cases L <;> grind [BetaNormal, countRedexes]
+
+/-- The body of a normal-form abstraction opens to a normal form. -/
+lemma BetaNormal.abs_open {x : Var} (h : BetaNormal (abs M)) : BetaNormal (M ^ fvar x) := by
+  rw [BetaNormal, countRedexes_open_fvar]
+  exact h
+
+/-- Leftmost reduction preserves being an abstraction. -/
+lemma Leftmost.steps_isAbs_r (h : M ↠ℓ N) (ha : IsAbs M) : IsAbs N := by
+  induction h with
+  | refl => exact ha
+  | tail _ step ih => exact step.isAbs_r ih
+
+/-- Left congruence for leftmost reduction, provided the target is not an abstraction. -/
+lemma Leftmost.steps_app_l_cong (h : L ↠ℓ L') (hna : ¬IsAbs L') :
+    app L M ↠ℓ app L' M := by
+  induction h
+  case refl => rfl
+  case tail P _ _ step ih =>
+    have hnb : ¬IsAbs P := mt step.isAbs_r hna
+    exact (ih hnb).tail (step.appNoAbsL hnb)
+
+/-- Reducing the operand across a non-abstraction normal form keeps the position. -/
+lemma BetaAt.app_r_cong (h : BetaAt i M M') (hL : BetaNormal L) (hna : ¬IsAbs L) :
+    BetaAt i (app L M) (app L M') := by
+  have := h.appNoAbsR hna
+  rwa [hL] at this
+
+/-- Right congruence for leftmost reduction, provided the operator is a non-abstraction
+    normal form. -/
+lemma Leftmost.steps_app_r_cong (h : M ↠ℓ M') (hL : BetaNormal L) (hna : ¬IsAbs L) :
+    app L M ↠ℓ app L M' := by
+  induction h with
+  | refl => rfl
+  | tail _ step ih => exact ih.tail (step.app_r_cong hL hna)
+
+/-- Congruence for leftmost reduction on applications whose reduced operator is a
+    non-abstraction normal form. -/
+lemma Leftmost.steps_app_cong (hL : L ↠ℓ L') (hM : M ↠ℓ M')
+    (hnf : BetaNormal L') (hna : ¬IsAbs L') : app L M ↠ℓ app L' M' :=
+  (steps_app_l_cong hL hna).trans (steps_app_r_cong hM hnf hna)
+
+/-- Call-by-Name reduction is contained in leftmost reduction. -/
+lemma Leftmost.of_cbn (h : M ↠ₙ N) : M ↠ℓ N := by
+  induction h with
+  | refl => rfl
+  | tail _ step ih => exact ih.tail (BetaAt.of_cbn_step step)
+
+variable [DecidableEq Var] [HasFresh Var]
+
+/-- Leftmost reduction preserves local closure. -/
+lemma Leftmost.steps_lc_r (h : M ↠ℓ M') (lc : LC M) : LC M' := by
+  induction h with
+  | refl => exact lc
+  | tail _ step ih => exact step.lc_r ih
+
+/-- Leftmost reduction is preserved by closing a variable and abstracting. -/
+lemma Leftmost.steps_abs_close {x : Var} (h : M ↠ℓ M') (lc : LC M) :
+    (M⟦0 ↜ x⟧.abs) ↠ℓ (M'⟦0 ↜ x⟧.abs) := by
+  induction h with
+  | refl => rfl
+  | tail hs step ih => exact ih.tail (step.abs_close (steps_lc_r hs lc))
+
+/-- Cofinite congruence rule for leftmost reduction under an abstraction. -/
+lemma Leftmost.steps_abs_cong (xs : Finset Var)
+    (cofin : ∀ x ∉ xs, (M ^ fvar x) ↠ℓ (M' ^ fvar x)) (lc : LC (abs M)) :
+    abs M ↠ℓ abs M' := by
+  have ⟨w, _⟩ := fresh_exists <| free_union [fv] Var
+  rw [open_close w M 0 (by grind), open_close w M' 0 (by grind)]
+  have hstep := cofin w (by grind)
+  have hlc := beta_lc lc (.fvar w)
+  exact steps_abs_close hstep hlc
+
+/-- A standard reduction to a normal form is a leftmost reduction. -/
+theorem Leftmost.of_standard (h : M ⭢ₛ N) (hn : BetaNormal N) : M ↠ℓ N := by
+  induction h
+  case fvar x => rfl
+  case app _ _ ihL ihM =>
+    have ⟨hna, hL', hM'⟩ := hn.app_inv
+    exact steps_app_cong (ihL hL') (ihM hM') hL' hna
+  case abs xs h_body ih =>
+    have lc := (Standard.abs xs h_body).lc_l
+    apply steps_abs_cong xs _ lc
+    intro x hx
+    exact ih x hx hn.abs_open
+  case rdx M N M' _ lc_M lc_N cbn std_P ih =>
+    have s1 : M.app N ↠ℓ M'.abs.app N := of_cbn (CBN.steps_app_l_cong cbn lc_N)
+    have s2 : M'.abs.app N ⭢ℓ M' ^ N := .outer (CBN.steps_lc_r lc_M cbn) lc_N
+    exact (s1.tail s2).trans (ih hn)
+
+/-- The leftmost reduction theorem: if a term β-reduces to a normal form, then leftmost
+    reduction reaches it. -/
+theorem Leftmost.normalization (lc : LC M) (h : M ↠βᶠ N) (hn : BetaNormal N) : M ↠ℓ N :=
+  of_standard (.standardization lc h) hn
+
+end LambdaCalculus.LocallyNameless.Untyped.Term
+
+end Cslib