• Ring Solver, Coinductive predicates, Transfer tactic, • Superposition prover, Linters, • Fourier-Motzkin & Omega, • Many more • Access Lean internals using Lean • Type inference, Unifier, Simplifier

Ring Solver • Coinductive predicates • Transfer tactic • Superposition prover • Linters • Fourier-Motzkin & Omega • Many more Lean 3.x limitations • Lean programs are compiled into byte code and

82 5.4 Linear Arithmetic Tactic . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.5 Elimination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.6 Further Examples . the two elimination rules associated with conjunction. An introduction rule for a logical symbol (e.g., ∧) is a lemma whose conclusion has that symbol as the outermost symbol. Dually, an elimination rule provide in order to justify a proposition with that symbol as the outermost position, whereas the elimination rules tell us what we may infer from such a proposition. In the above proof, we apply the introduction

setting, it shows how to “introduce” or establish an implication. Application can be viewed as an “elimination rule,” showing how to “eliminate” or use an implication in a proof. The other propositional connectives information on the library hierarchy), and each connective comes with its canonical introduction and elimination rules. 3.3.1 Conjunction The expression and.intro h1 h2 builds a proof of p ∧ q using proofs Similarly, and.elim_right h is a proof of q. They are commonly known as the right and left and-elimination rules. example (h : p ∧ q) : p := and.elim_left h example (h : p ∧ q) : q := and.elim_right h

Prop) (h1 h2 : p) : h1 = h2 := rfl Note: the combination of proof irrelevance and singleton Prop elimination in ι-reduction renders the ideal version of definitional equality, as described above, undecidable the terms it recognizes as well typed, and this does not cause problems in practice. Singleton elimination will be discussed in greater detail in Section 4.4. def R (x y : unit) := false def accrec := which takes arguments – (a : α) (the parameters) – {C : foo a → Type u} (the motive of the elimination) – for each i, the minor premise corresponding to constructori – (x : foo) (the major premise)