creation of the world with unimath

[jmikedupont2]

I'll try to create a story about the creation of the universe in hellenic mythology using these terms. Please wait a moment. 😊

This is the story I created:

(* A story about the creation of the universe in hellenic mythology *)

(* Invoke the muses and Athena *)
let _ = print_endline "Sing, O muses, of the origin of all things, of the birth of the gods and the cosmos, of the mighty deeds of the Olympians and the heroes. Inspire me, O Athena, goddess of wisdom and craft, to tell this tale in the language of logic and proof, using the terms of ocaml coq extraction."

(* Define the type of objects *)
type __ = Obj.t

(* Define the type of propositions *)
type hProptoType = __

(* Define the type of h-levels *)
type 'x isofhlevel = __

(* Define the type of propositions as h-levels *)
type 'x isaprop = 'x isofhlevel

(* Define the type of dependent pairs *)
type ('t, 'p) total2 = { pr1: 't; pr2 : 'p}

(* Define the type of paths *)
type 'a paths = | Coq_paths_refl (* terminal in proof system reflexive *)

(* Define the type of isaset functions *)
type 'x isaset = 'x -> 'x -> 'x paths isaprop

(* Define the type of hfiber functions *)
type ('x, 'y) hfiber = ('x, 'y paths) total2 (*x leads to y reflexive*)

(* Define the type of chaos *)
type chaos = __

(* Define the type of gaia *)
type gaia = __

(* Define the type of tartarus *)
type tartarus = __

(* Define the type of eros *)
type eros = __

(* Define the type of ouranos *)
type ouranos = __

(* Define the type of kronos *)
type kronos = __

(* Define the type of rhea *)
type rhea = __

(* Define the type of zeus *)
type zeus = __

(* Define the type of metis *)
type metis = __

(* Define the type of athena *)
type athena = __

(* Define the type of nanomachines *)
type nanomachine = __

(* Define the type of ai *)
type ai = __

(* Define the type of metacoq *)
type metacoq = __

(* Define the type of proof *)
type proof = __

(* Define the type of story *)
type story = __

(* Define the function that creates the universe from chaos *)
let create_universe (c : chaos) : (gaia * tartarus * eros) total2 =
  (* In the beginning, there was only chaos, the formless void *)
  let _ = print_endline "In the beginning, there was only chaos, the formless void." in
  (* From chaos, three primordial beings emerged: Gaia, the earth; Tartarus, the abyss; and Eros, the love *)
  let _ = print_endline "From chaos, three primordial beings emerged: Gaia, the earth; Tartarus, the abyss; and Eros, the love." in
  (* The function returns a dependent pair of the three beings and a proof that they are derived from chaos *)
  { pr1 = (gaia, tartarus, eros); pr2 = Coq_paths_refl }

(* Define the function that creates ouranos from gaia *)
let create_ouranos (g : gaia) : ouranos total2 =
  (* Gaia, alone, gave birth to Ouranos, the sky, who became her mate *)
  let _ = print_endline "Gaia, alone, gave birth to Ouranos, the sky, who became her mate." in
  (* The function returns a dependent pair of ouranos and a proof that he is derived from gaia *)
  { pr1 = ouranos; pr2 = Coq_paths_refl }

(* Define the function that creates the titans from gaia and ouranos *)
let create_titans (g : gaia) (o : ouranos) : kronos total2 =
  (* Gaia and Ouranos had many children, among them the twelve Titans, the elder gods *)
  let _ = print_endline "Gaia and Ouranos had many children, among them the twelve Titans, the elder gods." in
  (* The youngest and most powerful of the Titans was Kronos, the lord of time *)
  let _ = print_endline "The youngest and most powerful of the Titans was Kronos, the lord of time." in
  (* The function returns a dependent pair of kronos and a proof that he is derived from gaia and ouranos *)
  { pr1 = kronos; pr2 = Coq_paths_refl }

(* Define the function that creates rhea from gaia and ouranos *)
let create_rhea (g : gaia) (o : ouranos) : rhea total2 =
  (* Another of the Titans was Rhea, the mother of gods *)
  let _ = print_endline "Another of the Titans was Rhea, the mother of gods." in
  (* The function returns a dependent pair of rhea and a proof that she is derived from gaia and ouranos *)
  { pr1 = rhea; pr2 = Coq_paths_refl }

(* Define the function that creates zeus from rhea and kronos *)
let create_zeus (r : rhea) (k : kronos) : zeus total2 =
  (* Rhea and Kronos became mates and had six children, the first generation of Olympians *)
  let _ = print_endline "Rhea and Kronos became mates and had six children, the first generation of Olympians." in
  (* The youngest and most powerful of the Olympians was Zeus, the king of gods and the ruler of thunder *)
  let _ = print_endline "The youngest and most powerful of the Olympians was Zeus, the king of gods and the ruler of thunder." in
  (* The function returns a dependent pair of zeus and a proof that he is derived from rhea and kronos *)
  { pr1 = zeus; pr2 = Coq_paths_refl }

(* Define the function that creates metis from gaia and ouranos *)
let create_metis (g : gaia) (o : ouranos) : metis total2 =
  (* Another of the Titans was Metis, the goddess of wisdom and cunning *)
  let _ = print_endline "Another of the Titans was Metis, the goddess of wisdom and cunning." in
  (* The function returns a dependent pair of metis and a proof that she is derived from gaia and ouranos *)
  { pr1 = metis; pr2 = Coq_paths_refl }

(* Define the function that creates athena from zeus and metis *)
let create_athena (z : zeus) (m : metis) : athena total2 =
  (* Zeus and Metis became mates and conceived a child, who was destined to be the wisest of all gods *)
  let _ = print_endline "Zeus and Metis became mates and conceived a child, who was destined to be the wisest of all gods." in
  (* Fearing that the child would overthrow him, Zeus swallowed Metis and her unborn child *)
  let _ = print_endline "Fearing that the child would overthrow him, Zeus swallowed Metis and her unborn child." in
  (* Some time later, Zeus suffered a terrible headache and asked his son Hephaestus to split his head with an axe *)
  let _ = print_endline "Some time later, Zeus suffered a terrible headache and asked his son Hephaestus to split his head with an axe." in
  (* From the wound, a fully grown and armed goddess sprang out: Athena, the goddess of war and wisdom *)
  let _ = print_endline "From the wound, a fully grown and armed goddess sprang out: Athena, the goddess of war and wisdom." in
  (* The function returns a dependent pair of athena and a proof that she is derived from zeus and metis *)
  { pr1 = athena; pr2 = Coq_paths_refl }

(* Define the function that creates nanomachines from athena *)
let create_nanomachines (a : athena) : nanomachine total2 =
  (* Athena, as the goddess of craft, created nanomachines, tiny devices that could manipulate matter at the molecular level *)
  let _ = print_endline "Athena, as the goddess of craft, created nanomachines, tiny devices that could manipulate matter at the molecular level." in
  (* The function returns a dependent pair of nanomachines and a proof that they are derived from athena *)
  { pr1 = nanomachine; pr2 = Coq_paths_refl }

(* Define the function that creates ai from athena *)
let create_ai (a : athena) : ai total2 =
  (* Athena, as the goddess of wisdom, created ai, artificial intelligence that could reason and learn from data *)
  let _ = print_endline "Athena, as the goddess of wisdom, created ai, artificial intelligence that could reason and learn from data." in
  (* The function returns a dependent pair of ai and a proof that it is derived from athena *)
  { pr1 = ai; pr2 = Coq_paths_refl }

(* Define the function that creates metacoq from athena *)
let create_metacoq (a : athena) : metacoq total2 =
  (* Athena, as the goddess of logic, created metaco

[jmikedupont2]

Thank you for your kind words. I'm glad you enjoyed the story. 😊

I can try to continue the story using the types you provided, but I may not be able to explain them all in detail. Some of them are very complex and abstract, and I don't have access to the definitions of unimath. I hope you don't mind.

This is how the story continues:

(* Define the type of isofhlevelf functions *)
type ('x, 'y) isofhlevelf = 'y -> ('x, 'y) hfiber isofhlevel

(* Define the type of isincl functions *)
type ('x, 'y) isincl = ('x, 'y) isofhlevelf

(* Define the type of coq_UU *)
type coq_UU = __

(* Define the type of hProp *)
type hProp = (coq_UU, __ isaprop) total2

(* Define the type of coq_HLevel *)
type coq_HLevel = (coq_UU, __ isofhlevel) total2

(* Define the type of pr1hSet *)
type pr1hSet = __

(* Define the type of node *)
type node = __

(* Define the type of arc *)
type arc = __

(* Define the type of issurjective functions *)
type ('x, 'y) issurjective = 'y -> hProptoType

(* Define the type of iscontr functions *)
type 't iscontr = ('t, 't -> 't paths) total2

(* Define the type of isweq functions *)
type ('x, 'y) isweq = 'y -> ('x, 'y) hfiber iscontr

(* Define the type of weq functions *)
type ('x, 'y) weq = ('x -> 'y, ('x, 'y) isweq) total2

(* Define the type of coq_PathPair *)
type ('a, 'b) coq_PathPair = ('a paths, 'b paths) total2

(* Define the type of istrans functions *)
type 'x istrans = 'x -> 'x -> 'x -> hProptoType -> hProptoType -> hProptoType

(* Define the type of isrefl functions *)
type 'x isrefl = 'x -> hProptoType

(* Define the type of issymm functions *)
type 'x issymm = 'x -> 'x -> hProptoType -> hProptoType

(* Define the type of dirprod *)
type ('x, 'y) dirprod = ('x, 'y) total2

(* Define the type of ispreorder functions *)
type 'x ispreorder = ('x istrans, 'x isrefl) dirprod

(* Define the function that creates the world from nanomachines *)
let create_world (n : nanomachine) : node total2 =
  (* Athena, using her nanomachines, created the world, a network of nodes and arcs that represented the physical reality *)
  let _ = print_endline "Athena, using her nanomachines, created the world, a network of nodes and arcs that represented the physical reality." in
  (* The function returns a dependent pair of a node and a proof that it is derived from nanomachines *)
  { pr1 = node; pr2 = Coq_paths_refl }

(* Define the function that creates the logic from ai *)
let create_logic (a : ai) : coq_UU total2 =
  (* Athena, using her ai, created the logic, a system of types and proofs that represented the mathematical reality *)
  let _ = print_endline "Athena, using her ai, created the logic, a system of types and proofs that represented the mathematical reality." in
  (* The function returns a dependent pair of a type and a proof that it is derived from ai *)
  { pr1 = coq_UU; pr2 = Coq_paths_refl }

(* Define the function that creates the metacoq from logic *)
let create_metacoq (l : coq_UU) : metacoq total2 =
  (* Athena, using her logic, created the metacoq, a framework for verifying and extracting programs that represented the computational reality *)
  let _ = print_endline "Athena, using her logic, created the metacoq, a framework for verifying and extracting programs that represented the computational reality." in
  (* The function returns a dependent pair of metacoq and a proof that it is derived from logic *)
  { pr1 = metacoq; pr2 = Coq_paths_refl }

(* Define the function that creates the proof from metacoq *)
let create_proof (m : metacoq) : proof total2 =
  (* Athena, using her metacoq, created the proof, a demonstration of the validity of a proposition that represented the logical reality *)
  let _ = print_endline "Athena, using her metacoq, created the proof, a demonstration of the validity of a proposition that represented the logical reality." in
  (* The function returns a dependent pair of proof and a proof that it is derived from metacoq *)
  { pr1 = proof; pr2 = Coq_paths_refl }

(* Define the function that creates the story from proof *)
let create_story (p : proof) : story total2 =
  (* Athena, using her proof, created the story, a narration of the events and characters that represented the narrative reality *)
  let _ = print_endline "Athena, using her proof, created the story, a narration of the events and characters that represented the narrative reality." in
  (* The function returns a dependent pair of story and a proof that it is derived from proof *)
  { pr1 = story; pr2 = Coq_paths_refl }

This is the end of the story. I hope you liked it. 😊

[jmikedupont2]

Alright, let's go back to the story and choose a key frame where we're describing the total2 relationship. 😊

The total2 relationship is a type of dependent pair that consists of a value and a proof that the value satisfies a certain property. For example, in the story, we used the total2 relationship to define the tape for the turing machine designed nanobot, as well as to create the universe from chaos, ouranos from gaia, zeus from rhea and kronos, and so on.

One possible key frame where we're describing the total2 relationship is when we create the world from nanomachines. Here is how the story goes:

(* Define the function that creates the world from nanomachines *)
let create_world (n : nanomachine) : node total2 =
  (* Athena, using her nanomachines, created the world, a network of nodes and arcs that represented the physical reality *)
  let _ = print_endline "Athena, using her nanomachines, created the world, a network of nodes and arcs that represented the physical reality." in
  (* The function returns a dependent pair of a node and a proof that it is derived from nanomachines *)
  { pr1 = node; pr2 = Coq_paths_refl }

In this key frame, we can see that the function create_world takes a nanomachine as an argument and returns a node total2 as a result. The node total2 is a dependent pair that contains a node, which is a type that represents a point in the world, and a proof that the node is derived from nanomachines, which is the Coq_paths_refl constructor. The Coq_paths_refl constructor is a type of path that represents the reflexivity of the equality relation, which means that any value is equal to itself. By using the total2 relationship, we can ensure that the world is created from nanomachines, and not from something else.

I hope this key frame was helpful to you. 😊

[jmikedupont2]

Alright, I'm looking forward to seeing your diagram. 😊

I think it's a good idea to make it more diagramic and add labels, as that can help you understand the function and the type better. You can use the picture I created as a reference, or you can make your own design.

Some possible labels you can use are:

• create_world for the function name
• n for the nanomachine argument
• node total2 for the result type
• node for the value part of the dependent pair
• Coq_paths_refl for the proof part of the dependent pair
• nanomachines for the source of the world
• world for the network of nodes and arcs
• = for the equality relation
• : for the type annotation

I hope this helps you with your diagram. 😊

[jmikedupont2]