Bar Induction: The Good, the Bad and the Ugly Vincent Rahli, Mark Bickford, and Robert L. Constable June 22, 2017 Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 1/28
Bar induction? Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 2/28
What bar induction is not about? (source: https://get.taphunter.com/blog/4-ways-to-ensure-your-bar-rocks/ ) Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 3/28
Intuitionism § First act : Intuitionistic logic is based on our inner consciousness of time , which gives rise to the two-ity . § As opposed to Platonism, it’s about constructions in the mind and not objects that exist independently of us. There are no mathematical truths outside human thought: “all mathematical truths are experienced truths” (Brouwer) § A statement is true when we have an appropriate construction, and false when no construction is possible. Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 4/28
Intuitionism § Second act : New mathematical entities can be created through more or less freely proceeding sequences of mathematical entities. § Also by defining new mathematical species (types, sets) that respect equality of mathematical entities. § Gives rise to (never finished) choice sequences. Could be lawlike or lawless. Laws can be 1st order, 2nd order. . . § The continuum is captured by choice sequences of nested rational intervals. Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 5/28
Intuitionism— Continuity What can we do with these never finished sequences? Brouwer’s answer: one never needs the whole sequence. His continuity axiom for numbers says that functions from sequences to numbers only need initial segments @ F : N B . @ α : B . D n : N . @ β : B . α “ B n β Ñ F p α q “ N F p β q From which his uniform continuity theorem follows: Let f be of type r α, β s Ñ R , then @ ǫ ą 0 . D δ ą 0 . @ x , y : r α, β s . | x ´ y | ď δ Ñ | f p x q ´ f p y q| ď ǫ ( B “ N N & B n “ N N n ) Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 6/28
Intuitionism— Continuity False (Kreisel 62, Troelstra 77, Escardó & Xu 2015): � Π F : N B . Π α : B . Σ n : N . Π β : B .α “ B n β Ñ F p α q “ N F p β q (no continuous way of finding a modulus of continuity of a given function F at a point α ) ( D F “ G : N B . F and G have different moduli of continuity) True in Nuprl (see our CPP 2016 paper): Π F : B Ñ N . Π α : B . å Σ n : N . Π β : B .α “ B n β Ñ F p α q “ N F p β q Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 7/28
Intuitionism— Bar induction To prove his uniform continuity theorem , Brouwer also used the Fan theorem . Which follows from bar induction . The fan theorem says that if for each branch α of a binary tree T , a property A is true about some initial segment of α , then there is a uniform bound on the depth at which A is met. Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 8/28
Bar Induction— The intuition Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 9/28
What is this talk about? Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 10/28
What this talk is not about Not about the philosophical foundations of intuitionism Not about which foundation is best About useful constructions (source: https://sententiaeantiquae.com/2014/10/23 ) Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 11/28
What is this talk about? Non-truncated Some bar induction Minor restriction: monotone bar principles are valid sequences have to induction is false in in Nuprl be name-free Nuprl (source: http://cinetropolis.net/scene-is-believing-the-good-the-bad-and-the-ugly/ ) Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 12/28
Nuprl? Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 13/28
Nuprl in a Nutshell Became operational in 1984 (Constable & Bates) Similar to Coq and Agda Extensional Constructive Type Theory with partial functions Types are interpreted as Partial Equivalence Relations on terms (PERs) Consistency proof in Coq (see our ITP 2014): https://github.com/vrahli/NuprlInCoq Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 14/28
Extensional CTT with partial functions? Extensional p@ a : A . f p a q “ g p a q P B q Ñ f “ g P A Ñ B Constructive p A Ñ A q true because inhabited by p λ x . x q Partial functions fix p λ x . x q inhabits N Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 15/28
Nuprl Types— Martin-Löf’s extensional type theory Equality a “ b P T Dependent product a : A Ñ B r a s or Π a : A . B r a s Dependent sum a : A ˆ B r a s or Σ a : A . B r a s Universe U i Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 16/28
Nuprl Types— Less “conventional types” Partial : A Domain : Base Disjoint union : A ` B Simulation : t 1 ď t 2 Intersection : X a : A . B r a s ( Void “ 0 ď 1 and Unit “ 0 ď 0) Union : Y a : A . B r a s Bisimulation : t 1 „ t 2 Set : t a : A | B r a su Image : Img p A , f q PER : per p R q Quotient : T {{ E Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 17/28
Nuprl Types— Squashing/Truncation t Unit | T u Ó T Proof erasure: Img p T , λ _ . ‹q Proof irrelevance: å T T {{ True Π P : P . p P _ � P q ✗ For example: Π P : P . åp P _ � P q ✗ Π P : P . Óp P _ � P q ✓ Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 18/28
Nuprl PER Semantics Implemented in Coq Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 19/28
Bar Induction in Nuprl Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 20/28
Bar Induction— Non-intuitionistic in Coq H $ Ó P p 0 , ‚ q BY [BID] p wfd q H , n : N , s : B n $ B p n , s q P Type p bar q H , s : B $ ÓD n : N . B p n , s q p imp q H , n : N , s : B n , m : B p n , s q $ P p n , s q p ind q H , n : N , s : B n , x : p@ m : N . P pp n ` 1 q , s ‘ n m qq $ P p n , s q Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 21/28
Bar Induction— On decidable bars in Nuprl H $ P p 0 , ‚ q BY [BID] p dec q H , n : N , s : B n $ B p n , s q _ � B p n , s q p bar q H , s : B $ ÓD n : N . B p n , s q p imp q H , n : N , s : B n , m : B p n , s q $ P p n , s q p ind q H , n : N , s : B n , x : p@ m : N . P pp n ` 1 q , s ‘ n m qq $ P p n , s q Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 22/28
Bar Induction— On monotone bars in Nuprl H $ å P p 0 , ‚ q BY [BIM] p mon q H , n : N , s : B n $ @ m : N . B p n , s q ñ B p n ` 1 , s ‘ n m q p bar q H , s : B $ åD n : N . B p n , s q p imp q H , n : N , s : B n , m : B p n , s q $ P p n , s q p ind q H , n : N , s : B n , x : p@ m : N . P pp n ` 1 q , s ‘ n m qq $ P p n , s q Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 23/28
Bar Induction— Why the squashing operator? Continuity is false in Martin-Löf-like type theories when not å -squashed p A q Π F : N B . Π f : B . å Σ n : N . Π g : B . f “ B n g Ñ F p f q “ N F p g q p B q � Π F : N B . Π f : B . Σ n : N . Π g : B . f “ B n g Ñ F p f q “ N F p g q From which we derived: BIM is false when not å -squashed otherwise we could derive Π F : N B . Π f : B . Σ n : N . Π g : B . f “ B n g Ñ F p f q “ N F p g q from BIM & (A) Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 24/28
Bar Induction— Sequences of numbers We derived BID/BIM for sequences of numbers (easy) We added “choice sequences” of numbers to Nuprl’s model: all Coq functions from N to N What about sequences of terms? Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 25/28
Bar Induction— Sequences of terms We derived BID for sequences of closed name-free terms Harder because we turned our terms into a big W type: Coq functions from N to terms are now terms! Why without names? ν picks fresh names and we can’t compute the collection of all names anymore Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 26/28
Bar Induction— Questions Can we prove continuity for sequences of terms instead of B ? What does that give us? “ proof-theoretic strength? Can we hope to prove BID/BIM in Coq without LEM/AC? We’re working on this: Can we derive BID/BIM for sequences of terms with names? Vincent Rahli Bar Induction: The Good, the Bad and the Ugly June 22, 2017 27/28
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