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

sli.do#cube

password: cube

Answers to 3-2-1 questions and private emails, Homework 1 regrading+solution, Homework 4…

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Natural numbers Booleans

true: 2 false: 2 zero: N suc: N → N recursion!

Circle

base: S1 loop: Id(base; base)

Warning: another destroyer of harmony

base: S1 S1 : U loop: IdS1(base; base) elimS1[x.C](Mbase; Mloop; N) : C[N/x] x: S1 ⊢ C : U Mbase : C[base/x] Mloop :⁇? N: S1

a b entire C C[a/x] C[b/x] A path over p p ca cb ca =p

x.C cb

ca =p

x.C cb

Define

paern match on p

base: S1 S1 : U loop: IdS1(base; base) elimS1[x.C](Mbase; Mloop; N) : C[N/x] x: S1 ⊢ C : U Mbase : C[base/x] N: S1 Mloop : base =loop base

elimS1[x.C](Mbase; Mloop; base) ≡ Mbase : … (…) elimS1[x.C](Mbase; Mloop; loop) ≡ Mloop : … (…) seems difficult This does not type check

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

elimS1[x.C](Mbase; Mloop; base) ≡ Mbase : … (…) apd(λx.elimS1[x.C](Mbase; Mloop; x); loop) = Mloop (…)

• nly a path

S1-Alg

ΣX : U Σbase : X IdX(base; base) ⟨ S1 , ⟨ base , loop ⟩ ⟩

is the initial one