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. September 2017. This version removes CurryHoward.lagda, so that we use the symbols Σ and + rather than ∃ and ∨. This is to be compatible with univalent logic. We also make our development more compatible with the philosophy of univalent mathematics and tried to streamline it a bit. The original version remains at http://www.cs.bham.ac.uk/~mhe/agda/ for the record and to avoid broken incoming links. \begin{code} {-# OPTIONS --without-K --exact-split --safe #-} 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-new/manual.pdf The following module investigates the notion of omniscience set. A set X is omniscient iff (p : X → 𝟚) → (Σ \(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 (and under classical logic) 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). where under : ℕ → ℕ∞ is the embedding. (The name for the embedding comes from the fact that in published papers we used an underlined symbol n to denote the copy of n : ℕ in ℕ∞.) \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 SquashedSum open import SearchableOrdinals open import LexicographicSearch open import ConvergentSequenceInfSearchable \end{code} 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 U of types is indiscrete, with a certain Rice's Theorem for the universe U as a corollary: \begin{code} open import TheTopologyOfTheUniverse open import RicesTheoremForTheUniverse \end{code} 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 products indexed by univalent propositions, first that it is isomorphic to any of its factors: \begin{code} open import Prop-indexed-product \end{code} And, more subtly, that a product of searchable sets indexed by a univalent proposition is itself searchable: \begin{code} open import Prop-Tychonoff \end{code} The following generalizes the squashed sum, with a simple construction and proof, using the injectivity of the universe and the Prop-Tychonoff theorem: \begin{code} open import ExtendedSumSearchable \end{code} The following modules contain auxiliary definitions and additional results and discussion that we choose not to bring here: \begin{code} open import SpartanMLTT open import DecidableAndDetachable open import DiscreteAndSeparated open import Exhaustible open import FailureOfTotalSeparatedness open import FirstProjectionInjective open import Sets open import Injection open import Equivalence open import Naturals open import OrdinalCodes open import Sequence open import Retraction open import Two open import Embedding open import InjectivityOfTheUniverse open import HiggsInvolutionTheorem \end{code}