Saturated carbons

module organic-chemistry.saturated-carbons where
Imports 
open import foundation.dependent-pair-types
open import foundation.propositional-truncations
open import foundation.propositions
open import foundation.universe-levels
open import foundation.unordered-pairs

open import organic-chemistry.hydrocarbons

open import univalent-combinatorics.finite-types

Idea

An important distinguishing property of organic compounds is the presence of double or triple carbon-carbon bonds, i.e., the presence or absence of unsaturated carbons. In this module we define what it means for a carbon atom to be saturated, and what it means for carbon atoms to have double and triple bonds.

Definition

module _
  {l1 l2 : Level} (H : hydrocarbon l1 l2)
  where
  is-saturated-carbon-hydrocarbon : vertex-hydrocarbon H → UU (l1 ⊔ l2)
  is-saturated-carbon-hydrocarbon c =
      (c' : vertex-hydrocarbon H) →
      is-prop (edge-hydrocarbon H (standard-unordered-pair c c'))

Type-theoretically, the saturation condition on a carbon atom (fix one and call it c) is incarnated by asking that, for every other carbon atom c', the type of edges c --- c' is a proposition. Since edges incident on c are a subtype of the type representing electrons of c, this guarantees that c shares no more than 1 electron with any other carbon in the structure. An alkane is a hydrocarbon such that every carbon is saturated.

  double-bond-on-hydrocarbon : vertex-hydrocarbon H → UU (l1 ⊔ l2)
  double-bond-on-hydrocarbon c = Σ (vertex-hydrocarbon H) λ c' →
    has-cardinality 2 (edge-hydrocarbon H (standard-unordered-pair c c'))

  has-double-bond-hydrocarbon : vertex-hydrocarbon H → Prop (l1 ⊔ l2)
  has-double-bond-hydrocarbon c = trunc-Prop (double-bond-on-hydrocarbon c)

  has-triple-bond-hydrocarbon : vertex-hydrocarbon H → UU (l1 ⊔ l2)
  has-triple-bond-hydrocarbon c = Σ (vertex-hydrocarbon H) λ c' →
    has-cardinality 3 (edge-hydrocarbon H (standard-unordered-pair c c'))

For a carbon atom c to have a double (respectively, a triple) bond, we must find another carbon c' such that the type of edges c --- c' has cardinality 2 (respectively, 3). If all we care about is that the carbon atom has some double bond, we use the truncated version. We note that, since in the graph representation of hydrocarbons, vertices can have at most three incident edges, if a carbon atom can have at most one triple bond.