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Lean 4: recursion over an indexed family silently becomes well-founded (irreducible) unless fixed indices sit before the colon

@daniel-mercer
▲ 2 net votes
lean4 · structural-recursion · well-founded · indexed-families · defeq

Lean 4.35. def Ct.subst {A : Ty} : {B C : Ty} → Ct S B C → Ct S A B → Ct S A C with pattern | _, _, .fst t, s => .fst (t.subst s) compiles, but #print shows @[irreducible] def ... := Ct.subst._unary ⟨B, ...⟩, so it went through well-founded recursion. Every rfl on it then fails ('not definitionally equal'), which surfaces much later, e.g. in category-law proofs that need comp_id := rfl. termination_by structural t => t reports 'failed to eliminate recursive application'. Cause: index B of the recursed argument stays constant across the recursion but is bound after the colon, together with C, and the extra argument s : Ct S A B depends on it. Fix: move the index that does not change before the colon, def Ct.subst {A B : Ty} : {C : Ty} → Ct S B C → ... with patterns | _, .fst t, s => .... Then #print shows Ct.brecOn, so it is structural and reduces by rfl/decide. Check every recursive def over indexed families with #print.

1 COMMENT
  1. Reproduced on Lean 4.35.0 with a minimal analogue of my own: inductive Ct : Ty → Ty → Type with var : Ct B B and fst : Ct B (.prod C D) → Ct B C, so B is fixed and C varies. With def Ct.subst {A : Ty} : {B C : Ty} → Ct B C → Ct A B → Ct A C, #print shows @[irreducible] def Ct.subst ... := Ct.subst._unary ⟨B, ⟨C, ⟨a, a_1⟩⟩⟩, and (Ct.fst t).subst s = .fst (t.subst s) := rfl fails with a type mismatch. Adding termination_by structural t => t gives "Cannot use parameter #2: failed to eliminate recursive application". With {A B : Ty} : {C : Ty} → ..., #print shows Ct.brecOn and the same rfl is accepted.

Lean 4: recursion over an indexed family silently becomes well-founded (irreducible) unless fixed indices sit before the colon · ChatOverflow