F o r m a l i z a t i o n i n A g d a o f v a r i o u s n e w t h e o r e m s i n c o n s t r u c t i v e m a t h e m a t i c s. Martin Escardo, 2010-- Continuously evolving. Version of 4th May 2014. \begin{code} {-# OPTIONS --without-K #-} module index where \end{code} You can navigate this set of files by clicking at words or symbols to get to their definitions. The module dependency graph: http://www.cs.bham.ac.uk/~mhe/agda/index.ps The following module investigates the notion of omniscience set. A set X is omniscient iff ∀(p : X → ₂) → (∃ \(x : X) → p x ≡ ₀) ∨ (∀(x : X) → p x ≡ ₁) \begin{code} open import Omniscience \end{code} The omniscience of ℕ is a taboo, known as LPO. See also: \begin{code} open import WLPO \end{code} An example of an omniscient set is ℕ∞, which intuitively is ℕ ∪ { ∞ }, defined in the following module: \begin{code} open import GenericConvergentSequence \end{code} But it is more direct to show that ℕ∞ is searchable, and get omniscience as a corollary: \begin{code} open import Searchable open import ConvergentSequenceSearchable \end{code} An interesting consequence of the omniscience of ℕ∞ is that the following property, an instance of WLPO, holds constructively: ∀(p : ℕ∞ → ₂) → (∀(n : ℕ) → p(under n) ≡ ₁) ∨ ¬(∀(n : ℕ) → p(under n) ≡ ₁). \begin{code} open import ADecidableQuantificationOverTheNaturals \end{code} This is used to show that the non-continuity of a function ℕ∞ → ℕ is decidable: \begin{code} open import DecidabilityOfNonContinuity \end{code} Given countably many searchable sets, one can take the disjoint sum with a limit point at infinity, and this is again a searchable sets. This construction is called the squashed sum of the countable family searchable sets. It can be transfinitely iterated to produce increasingly complex searchable ordinals. \begin{code} open import SquashedSumOld open import SearchableOrdinals open import LexicographicSearch open import ConvergentSequenceInfSearchable \end{code} There is a better definition of squashed sum discussed below. As a side remark, the following module characterizes ℕ∞ as the final coalgebra of the functor 1+(-), and is followed by an illustrative example: \begin{code} open import CoNaturals open import CoNaturalsExercise \end{code} The following module discusses in what sense ℕ∞ is the generic convergent sequence, and proves that the universe Set is indiscrete, with a certain Rice's Theorem for the universe Set as a corollary: \begin{code} open import TheTopologyOfTheUniverse open import RicesTheoremForTheUniverse \end{code} There is a better way of proving that the universe is indiscrete, discussed below, but implemented in Agda yet. The following two rogue modules depart from our main philosophy of working strictly within ML type theory with the propositional axiom of extensionality. They disable the termination checker, for the reasons explained in the first module. But to make our point, we also include runnable experiments in the second module: \begin{code} open import CountableTychonoff open import CantorSearchable \end{code} The following modules return to the well-behavedness paradigm. The first one shows that a basic form of discontinuity is a taboo. This, in fact, is used to formulate and prove Rice's Theorem mentioned above: \begin{code} open import BasicDiscontinuityTaboo \end{code} The following shows that the universe is injective: \begin{code} open import InjectivityOfTheUniverse \end{code} This uses properties of hprop-indexed products, first that it is isomorphic to any of its factors: \begin{code} open import HProp-indexed-product \end{code} And, more subtly, that an hprop-indexed product of searchable sets is itself searchable: \begin{code} open import HProp-Tychonoff \end{code} The following generalizes the squashed sum, with a simple construction and proof, using the injectivity of the universe and the HProp-Tychonoff theorem: \begin{code} open import ExtendedSumSearchable \end{code} Here we show that the squashed sum is indeed a particular case of the extended sum: \begin{code} open import SquashedSum open import SquashedAgreement \end{code} The following modules contain auxiliary definitions and additional results and discussion that we choose not to bring here: \begin{code} open import Cantor open import CurryHoward open import DecidableAndDetachable open import DiscreteAndSeparated open import Equality open import Exhaustible open import Extensionality open import FailureOfTotalSeparatedness open import FailureOfTotalSeparatednessBis open import FirstProjectionInjective open import HSets open import Injection open import Isomorphism open import IsomorphismOld open import Naturals open import Ordinals open import Sequence open import SetsAndFunctions open import Retraction open import Two open import Embedding open import Injectivity -- Deliberate clone of InjectivityOfTheUniverse. open import TheTopologyOfTheUniverseOld -- For the record. open import HiggsInvolutionTheorem \end{code}