extendable trees

trees.ts

type Grammer = {
    "expression": unknown
}

type Expr<α extends Grammer> = α["expression"]

type Plus<α extends Grammer> = { 
    kind: "plus"
    left: Expr<α>
    right: Expr<α>
}

type Times<α extends Grammer> = { 
    kind: "times"
    left: Expr<α>
    right: Expr<α>
}

type Variable<α> = { 
    kind: "variable"
    name: string
}

type AbelianGroupBaseGrammer<α extends Grammer> = { 
    "expression": Variable<α> | Plus<α>
}

type AbelianGroupGrammer = { 
    "expression": Expr<AbelianGroupBaseGrammer<AbelianGroupGrammer>>
}

type RingBaseGrammer<α extends Grammer> = { 
    "expression": Expr<AbelianGroupBaseGrammer<α>> | Times<α>
}

type RingGrammer = {
    "expression": Expr<RingBaseGrammer<RingGrammer>>
}

const x : Expr<AbelianGroupGrammer> = {kind: "variable", name: "x"}
const y : Expr<AbelianGroupGrammer> = {kind: "variable", name: "y"}
const x_plus_y : Expr<AbelianGroupGrammer> = {kind:"plus", left: x, right: y}
const x_times_x_plus_y : Expr<RingGrammer> = {kind:"times", left: x, right: x_plus_y}

function group2ring(a : Expr<AbelianGroupGrammer>) : Expr<RingGrammer> { 
    return a
}