Booleans true: 2 false: 2 Natural numbers zero: N suc: N N - PowerPoint PPT Presentation

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Booleans true: 2 false: 2 Natural numbers zero: N suc: N → N recursion!

Circle base: S1 loop: Id(base; base) Warning: another destroyer of harmony

S1 : U base: S1 loop: Id S1 (base; base) x: S1 ⊢ C : U M loop : ⁇ ? M base : C[base/x] N: S1 elim S1 [x.C](M base ; M loop ; N) : C[N/x]

C[b/x] C[a/x] path over p c a c a = p x.C c b c b entire C p b a A

Define x.C c b c a = p pa  ern match on p

S1 : U base: S1 loop: Id S1 (base; base) x: S1 ⊢ C : U M base : C[base/x] M loop : base = loop base N: S1 elim S1 [x.C](M base ; M loop ; N) : C[N/x]

(…) elim S1 [x.C](M base ; M loop ; base) ≡ M base : … (…) elim S1 [x.C](M base ; M loop ; loop) ≡ M loop : … This does not type check seems di ff icult

C[b/x] C[a/x] apd(f; p) f(a) f(b) p b a A

(…) elim S1 [x.C](M base ; M loop ; base) ≡ M base : … (…) apd( λ x.elim S1 [x.C](M base ; M loop ; x); loop) = M loop only a path

S1-Alg Σ X : U Σ base : X Id X (base; base) ⟨ S1 , ⟨ base , loop ⟩ ⟩ is the initial one