tastoids

-module(tastoids).
-compile([no_auto_import, nowarn_unused_vars, nowarn_unused_function, nowarn_nomatch]).
-define(FILEPATH, "src/tastoids.gleam").
-export([blend/2, retract/1, less/2, both/2, bland/2, squash/1, smashing/2, equal/2, length/1, distance/1]).

-if(?OTP_RELEASE >= 27).
-define(MODULEDOC(Str), -moduledoc(Str)).
-define(DOC(Str), -doc(Str)).
-else.
-define(MODULEDOC(Str), -compile([])).
-define(DOC(Str), -compile([])).
-endif.

?MODULEDOC(
    " Tastoids (& their 'Temperate' Algebra)\n"
    "\n"
    " It is often said, one can't compare \"apples\" to \"oranges\",\n"
    " but I daresay.. _perhaps you can?_\n"
    "\n"
    "   **Let me show you how.**\n"
    "\n"
    " On their own, it is hard to figure how one might compare\n"
    " apples to oranges, let alone deign to approach a calculus\n"
    " of taste however, with _just enough_ structure, a **Taste**\n"
    " can become a **Tastoid**, along with a curious and powerful\n"
    " _Temperate_ Algebra.\n"
    "\n"
    " > Not unlike a [_Tropical_ geometry](https://en.wikipedia.org/wiki/Tropical_geometry),\n"
    " > where the notion of addition and multiplication are replaced by min(a,b) & add(a,b),\n"
    " > I pose that a _Temperate Algebra_ is one whose typical notions of (+,×) are replaced\n"
    " > with operations that effectively 'average' taste (with commutativity, distributivity\n"
    " > and reversibility, no less!)\n"
    "\n"
    "  ### A Taste (t) Field\n"
    " \n"
    " \"Tastes\", broadly, are any measurable/comparable sentiment about a thing.\n"
    " Something with a unique direction and magnitude. _A vector!_\n"
    " \n"
    " Consider if you will, then:\n"
    "\n"
    "  - The set of all Vectors are a field ℝⁿ (aka 𝕍).\n"
    "  - You may partition 𝕍 by some enumerable set of indices (as ⅈ <= countable ∞))\n"
    "  - The '[Algebraic extension](https://en.wikipedia.org/wiki/Algebraic_extension)'\n"
    "    of 𝕍/ⅈ, is itself a field (with Algebra)\n"
    "\n"
    " > In the context of large-language- and embedding-models, this idea of\n"
    " > mapping _things_ (text or otherwise) into a well-defined set of possible\n"
    " > indices and probabilities is referred to as an '_embedding_'.\n"
    "\n"
    " ### A Taste -> One Tastoid\n"
    "\n"
    " We're almost there, I promise. Lets talk about plain numbers for a bit; say\n"
    " I told you knew the average of some set of values was `42`. You also know\n"
    " for a fact there was a 13 in there once, somewhere.\n"
    "\n"
    " Knowing nothing else, how would you _un_-average 13 from 42?\n"
    "\n"
    " _Were there two values that averaged to 42? Three? More?_\n"
    "\n"
    " Without the _cardinality_ of the original sampling, its (absolutely) impossible to know.\n"
    " But with it... say n=13, in which case, we can merely remove 1/13th of 13 (i.e. 1)\n"
    " from our combined average to find out the average without that 13 was just 41.\n"
    "\n"
    " Similarly, on their own Taste vectors _are_ comparable, even averagable in some sense,\n"
    " but without a cardinality, their operations aren't quite _lined up_ to have solutions\n"
    " to previously impossible questions become possible and yield seemingly 'free' results.\n"
    "\n"
    " Attempting to put some formalism to the above,\n"
    " \n"
    "   - We can partition a taste-field further, by its cardinality k ∈ 𝕂 (𝕂 ~ ℝ x ℝ₄ ~ ℂ)\n"
    "     (As well as a 4-cycle of like -> dislike -> unlike -> undislike)\n"
    " \n"
    "     i.e.  someone (or thing) expressing      {    like ( 1+i )     ->  dislike (-1 + i)\n"
    "       𝕒 taste (valued at weight w), k times   -> un-like (-1 + -i) -> un-dislike (1 + -i) } \n"
    " A Tastoid, can be arrived at by a person/subject, expressing a taste _once_ (tᵢ, k=1), \n"
    " imbedding (sic) that individuals' sentiment in a univesal/possibly infinite set of all\n"
    " the tastoid's that subject could express 𝕥ᵤ↪ᵢ ∈ 𝕋ᵤ\n"
    "\n"
    "  Then, we may define a handful a tiny, tidy, yet supremely powerful (σ/sigma-)\n"
    "    _Algebra_ of Taste_\n"
    "\n"
    " - [ ] Brief introduction to operators\n"
).

-file("src/tastoids.gleam", 98).
?DOC(
    " Blend (**⩐**) two tastoids, producing a larger tastoid _congruent_ to\n"
    " the weighted power mean, aka their average 'taste' (via `squash`).\n"
    "\n"
    " See also `retract` - which yields u' which _unblends_ when blended,\n"
).
-spec blend(tastoids@tastoid:tastoid(EHP), tastoids@tastoid:tastoid(EHP)) -> tastoids@tastoid:tastoid(EHP).
blend(T, U) ->
    case {T, U} of
        {tasteless, _} ->
            U;

        {_, tasteless} ->
            T;

        {{tastoid, T@1, K_t}, {tastoid, U@1, K_u}} ->
            _pipe = tastoids@taste:add(T@1, U@1),
            {tastoid, _pipe, gleam@int:add(K_t, K_u)}
    end.

-file("src/tastoids.gleam", 107).
?DOC(" Return the inverse of a taste (over `blend`)\n").
-spec retract(tastoids@tastoid:tastoid(EHT)) -> tastoids@tastoid:tastoid(EHT).
retract(Taste) ->
    case Taste of
        {tastoid, T, K} ->
            _pipe = tastoids@taste:negate(T),
            {tastoid, _pipe, gleam@int:negate(K)};

        tasteless ->
            tasteless
    end.

-file("src/tastoids.gleam", 118).
?DOC(
    " Unblend a Tastoid (the _divisor_), removing it `from`\n"
    " another (the _numerator_).\n"
    "\n"
    " nb. This works even if the `divisor` tastoid isn't in the `from` Tastoid.\n"
).
-spec less(tastoids@tastoid:tastoid(EHW), tastoids@tastoid:tastoid(EHW)) -> tastoids@tastoid:tastoid(EHW).
less(Divisor, Numerator) ->
    _pipe = Divisor,
    _pipe@1 = retract(_pipe),
    blend(_pipe@1, Numerator).

-file("src/tastoids.gleam", 128).
?DOC(
    " Return the 'and'-product (**⩍**), multiplying tastes/cardinalities\n"
    " between _left_ and _right_; effectively erasing unshared tastes.\n"
).
-spec both(tastoids@tastoid:tastoid(EIA), tastoids@tastoid:tastoid(EIA)) -> tastoids@tastoid:tastoid(EIA).
both(Left, Right) ->
    case {Left, Right} of
        {tasteless, _} ->
            tasteless;

        {_, tasteless} ->
            tasteless;

        {{tastoid, T1, K1}, {tastoid, T2, K2}} ->
            T = tastoids@taste:multiply(T1, T2),
            K = gleam@int:multiply(K1, K2),
            {tastoid, T, K}
    end.

-file("src/tastoids.gleam", 150).
?DOC(
    " **Experimental** Return the 'not-and'-product, letting\n"
    "  the maths take the wheel.\n"
    "\n"
    " _Avery's guess is this will act like a harsh move away the\n"
    "  interesected indices, conversely strengthening everything else_\n"
    " \n"
    " Bland : A, B ->  (A ⩐ B) - (A ⩍ B)\n"
).
-spec bland(tastoids@tastoid:tastoid(EIE), tastoids@tastoid:tastoid(EIE)) -> tastoids@tastoid:tastoid(EIE).
bland(Left, Right) ->
    Or = blend(Left, Right),
    And = both(Left, Right),
    case {Or, And} of
        {tasteless, {tastoid, Ands, K}} ->
            {tastoid, tastoids@taste:negate(Ands), gleam@int:negate(K)};

        {_, tasteless} ->
            Or;

        {{tastoid, Ors, K_or}, {tastoid, Ands@1, K_and}} ->
            Xor = tastoids@taste:add(Ors, tastoids@taste:negate(Ands@1)),
            {tastoid, Xor, gleam@int:subtract(K_or, K_and)}
    end.

-file("src/tastoids.gleam", 168).
?DOC(
    " Reduce k -> 1, yielding the 'average'/normalized tastoid of all\n"
    " the tastoids blended/present\n"
).
-spec squash(tastoids@tastoid:tastoid(EII)) -> tastoids@tastoid:tastoid(EII).
squash(Tastoid) ->
    case Tastoid of
        {tastoid, _, 1} = T ->
            T;

        {tastoid, _, 0} ->
            tasteless;

        {tastoid, T@1, K} ->
            Try_divide = gleam@float:divide(1.0, erlang:float(K)),
            case Try_divide of
                {ok, One_over_k} ->
                    _pipe = tastoids@taste:scale(T@1, One_over_k),
                    {tastoid, _pipe, 1};

                {error, _} ->
                    tasteless
            end;

        tasteless ->
            tasteless
    end.

-file("src/tastoids.gleam", 189).
?DOC(
    " Blend, squashing, aka 'smash' as it is effectively the smash-product (⨳),\n"
    " or t ⨳ u (see [Smash Product](https://ncatlab.org/nlab/show/smash+product))\n"
).
-spec smashing(tastoids@tastoid:tastoid(EIL), tastoids@tastoid:tastoid(EIL)) -> tastoids@tastoid:tastoid(EIL).
smashing(T, U) ->
    _pipe = blend(T, U),
    squash(_pipe).

-file("src/tastoids.gleam", 194).
?DOC(" Returns `True` iff Tastoids `t` & `u` are congruent (weakly equivalent)\n").
-spec equal(tastoids@tastoid:tastoid(EIP), tastoids@tastoid:tastoid(EIP)) -> boolean().
equal(T, U) ->
    case {T, U} of
        {tasteless, tasteless} ->
            true;

        {_, tasteless} ->
            false;

        {tasteless, _} ->
            false;

        {{tastoid, T@1, K_t}, {tastoid, U@1, K_u}} when K_t =:= K_u ->
            T@1 =:= U@1;

        {T@2, U@2} ->
            equal(squash(T@2), squash(U@2))
    end.

-file("src/tastoids.gleam", 209).
?DOC(
    " Project a tastoid into a _Length-oid_ of sorts, condensing\n"
    " it into a 1-d tastoid indexed by the set of its indexes,\n"
    " and a value of its dimensions sum.\n"
).
-spec length(tastoids@tastoid:tastoid(EIS)) -> tastoids@tastoid:tastoid(gleam@set:set(EIS)).
length(Tastoid) ->
    case Tastoid of
        tasteless ->
            tasteless;

        {tastoid, T, K} ->
            _pipe = T,
            _pipe@1 = tastoids@taste:condense(_pipe),
            {tastoid, _pipe@1, K}
    end.

-file("src/tastoids.gleam", 220).
?DOC(
    " Returns a Tastoid's _scalar_ value length, discarding its cardinality\n"
    "\n"
    " nb. This doesn't _need_ to be a `length(of: tastoid)` result, and in\n"
    "     fact they yield the same value over `distance(covered_by:)`\n"
).
-spec distance(tastoids@tastoid:tastoid(any())) -> float().
distance(Tastoid) ->
    case Tastoid of
        tasteless ->
            +0.0;

        {tastoid, T, _} ->
            tastoids@taste:length(T)
    end.