Transfinite Constructions in Classical Type Theory Gert Smolka, - PowerPoint PPT Presentation

Transfinite Constructions in Classical Type Theory Gert Smolka, Steven Sch¨ afer, Christian Doczkal Saarland University ITP 2015, Nanjing, 26 August 2015

Existence theorems from set theory that carry over to type theory Zermelo 1904 Existence of well-orderings. Hausdorff 1914 Existence of maximal chains. Zorn 1935 Existence of maximal elements. Bourbaki-Witt 1950 Existence of fixed points. Gert Smolka et al. (Saarland University) 2 / 18

Existence theorems in type theory Need excluded middle and impredicative propositions. Zermelo Every extensional type with a CF can be well-ordered. Hausdorff Every e.p.o. type with a CF has a maximal chain. Zorn Every e.p.o. type with a CF has a maximal element provided every chain has an upper bound (corollary of Hausdorff). Bourbaki-Witt Every increasing function on an inhabited p.o. type has a FP provided every chain has a supremum. Analysis of these results in type theory is enlightening. Sheds light on type theory versus set theory as math. language. Formalization in Coq informative and elegant. Gert Smolka et al. (Saarland University) 3 / 18

Existence theorems mentioned can all be obtained with Bourbaki’s tower construction (1949) Formalizes transfinite iteration without ordinals. Matters since ordinals are not available in plain type theory. Not covered in set theory textbooks, do everything with ordinals. Originated with Zermelo’s usually neglected second proof of the well-ordering theorem (1908). Story of the paper: Chad E. Brown formalized Zermelo’s second proof in February 2014 in less than 200 lines of Coq. Starting from there, we developed the general tower construction, linearity theorem was challenging. Only after submission we discovered Bourbaki’s paper. Nicolas Bourbaki. Sur le th´ eor` eme de Zorn . Archiv der Mathematik, 2(6):434–437, 1949. Gert Smolka et al. (Saarland University) 4 / 18

Transfinite iteration, idea Given: P.o. type X , starting point a , increasing function f (i.e., x ≤ f x ). a < f a < f 2 a < · · · < � f i a < f ( � f i a ) < · · · Iterate f on a . Add supremum of infinite chain generated. Continue iterating f and taking suprema. Stop if fixed point of f is reached. Gert Smolka et al. (Saarland University) 5 / 18

Tower construction formalizes transfinite iteration Given: P.o. type X , starting point a , increasing function f . a < f a < f 2 a < · · · < � f i a < f ( � f i a ) < · · · Tower construction formalizes and generalizes transfinite iteration so that uncountable chains can be obtained (in set theory). Tower construction takes closure of a under f and suprema. In set theory, one may take set inclusion as order. Then suprema always exist (union). Starting from a := ∅ , closure under union and f x := x ∪ { x } yields the ordinals. f x := x ∪ P x yields the stages of the cumulative hierarchy. Gert Smolka et al. (Saarland University) 6 / 18

Sets in type theory Sets appear as unary predicates p : X → P . x ∈ p is notation for px . Sets are extensional if unary predicates are extensional. Sets belong to different types (only one type of sets in set theory). . . . ( X → P ) → P sets of sets X → P sets X Union can be defined on set types with ∨ and ∃ . Intersection can be defined on set types with ∧ and ∀ . Set inclusion is partial order on set types with suprema (union) and infima (intersection). Gert Smolka et al. (Saarland University) 7 / 18

Tower construction Given: p.o. type X , increasing function f , starting point a . Define inductive set T as follows: x ∈ T p ⊆ T p inhabited sup p x a ∈ T f x ∈ T x ∈ T T is least set containing a and closed under f and suprema. a is least element of T . x is greatest element of T iff sup T x . Bourbaki defines T as the intersection of all subsets of X containing A and closed under f and suprema. Gert Smolka et al. (Saarland University) 8 / 18

Tower theorem [Bourbaki 1949] Theorem If x , y ∈ T , then either x ≤ y ≤ f y or y ≤ f y ≤ x . Corollary 1 T is well-founded chain. Corollary 2 sup T x ↔ f x = x ∧ x ∈ T . Gert Smolka et al. (Saarland University) 9 / 18

Proof of tower theorem [Bourbaki 1949] Theorem If x , y ∈ T , then either x ≤ y or f y ≤ x . Follows with two lemmas and the notion of regular points . R := { y ∈ T | ∀ x ∈ T . x < y → f x ≤ y } Lemma 1 If x ∈ T and y ∈ R , then either x ≤ y or f y ≤ x . Lemma 2 T ⊆ R . Both lemmas are shown by induction on x ∈ T , proof of Lemma 2 uses Lemma 1 twice. Gert Smolka et al. (Saarland University) 10 / 18

Proof of well-foundedness p well-founded := every inhabited subset of p has a minimal element. Fact p is well-founded iff every element of p is p -inductive. The set Ip of p -inductive points is defined inductively by ∀ y ∈ p . y < x → y ∈ Ip x ∈ Ip Proving T ⊆ IT by induction on x ∈ T is straightforward. Gert Smolka et al. (Saarland University) 11 / 18

Bourbaki-Witt fixed point theorem Theorem Every increasing function on a p.o. type such that every well-founded chain has a supremum has a FP above a given point a . Consider tower T for a and f . Since T is well-founded chain (T. Theo.), T has supremum u . u ∈ T and u FP of f (Cor. 2). a ≤ u . Gert Smolka et al. (Saarland University) 12 / 18

Choice functions Definition Given a type B , a CF is a function γ : ( B → P ) → ( B → P ) from sets to sets such that γ p ⊆ p for every set p and γ p is a singleton whenever p is inhabited. Definition of CF is carefully chosen. Defines choice functions as choice predicates. Acknowledges that set-theoretic functions are relations. Ensures that well-ordered types have choice functions (least element). Avoids the assumption of a description operator. Gert Smolka et al. (Saarland University) 13 / 18

Specialized tower construction suffices for Zermelo and Hausdorff Assume extensional type B . Obtain tower over X := B → P with inclusion order and a := ∅ . Supremum of F ⊆ T is union � F (always exists). T can be obtained inductively with only two rules: p ∈ T F ⊆ T � F ∈ T f p ∈ T For Zermelo and Hausdorff, f adds at most one element. Proof of tower theorem is much easier for this case. Gert Smolka et al. (Saarland University) 14 / 18

Zermelo’s theorem Theorem Every extensional type B with a CF γ can be well-ordered. Specialized tower construction with f p := p ∪ γ { b | b / ∈ p } Construct injective embedding of B into T . Map b to b := � { p ∈ T | x / ∈ p } ∈ T (segment for b ). ∈ b and b ∈ f b . b / λ bc . b ⊆ c is well-ordering of B . Theorem can be strengthened: Every well-founded relation on B can be extended to a WO of B . Let CF select minimal elements of given relation. Proof can be massaged so that extensionality of B can be weakened to extensionality of CF. Gert Smolka et al. (Saarland University) 15 / 18

Hausdorff’s theorem Theorem Every e.p.o. type B with a CF has a maximal chain. Specialized tower construction with X := B → P and f p := p ∪ γ { b | b / ∈ p ∧ chain( p ∪ { b } ) } Every p ∈ T is a chain since chains are closed under tower rules. every subset of T is a chain (since T is a chain). union of chain of chains is a chain. � T ∈ T is maximal chain. Theorem can be strengthened so that a given chain is extended to a maximal chain. Gert Smolka et al. (Saarland University) 16 / 18

Related work in type theory Isabelle’s standard library contains proofs of Zorn’s lemma and Zermelo’s well-ordering theorem (classical extensional type theory with choice). Danko Ilik , Types 2006, proves Zermelo’s well-ordering theorem in Agda Light with an extensional choice principle entailing excluded middle. Based on Zermelo’s first proof from 1904. Andrej Bauer and Peter Lefanu Lumsdaine , 2013, Math. Proc. Cambridge Phil. Soc.: Study Bourbaki-Witt FP principle in toposes. Gert Smolka et al. (Saarland University) 17 / 18

Conclusions Bourbaki’s tower construction in type theory. Formalizes transfinite iteration without ordinals. Constructs well-founded chains as closure of starting point under successor function and suprema. Provides for proofs of Zermelo’s well-ordering theorem, Hausdorff’s maximal chain theorem, and the Bourbaki-Witt FP theorem. Type-theoretic formulation is natural. Formalization in Coq is elegant and instructive. Future Work: Construct uncountable tower in type theory. Gert Smolka et al. (Saarland University) 18 / 18