joins-a-side-whose-void-the-fabricated-formation-holds.yaml

# SPDX-FileCopyrightText: Copyright (c) 2025 Objectionary.com
# SPDX-License-Identifier: MIT
---
# yamllint disable rule:line-length
# The two sides of 'L_fork' differ in shape, so the firing fabricates a
# formation (#2024). The left side leaves 'tu' void, so the fabricated
# formation leaves 'tu' void too and holds that side applied to 'tu' under
# 'left🌵', reading it off the fabricated formation (#2029). The program asks
# the formation for 'φ' and the copy of 'L_join' it finds there for 'k', which
# asks both copies for 'φ' through its ρ. Nobody handed 'tu' over, so the left
# copy binds it to ⊥, which its 'φ' never reads, and the join meets the two
# symbols the sides hold under 'k'.
symbolic: |
  - λ: L_fork
    dataize:
      𝛿1: $.c
    join:
      on: 𝛿1
      left: $.a
      right: $.b
      otherwise: L_join
  - λ: L_join
    join:
      on: $.on🌵
      left: $.left🌵
      right: $.right🌵
      otherwise: L_join
location: 'Q.y'
input: '⟦ y ↦ ⟦ c ↦ ⟦ λ ⤍ 𝜎1 ⟧, a ↦ ⟦ φ ↦ ⟦ k ↦ ⟦ λ ⤍ 𝜎2 ⟧ ⟧, tu ↦ ∅ ⟧, b ↦ ⟦ φ ↦ ⟦ k ↦ ⟦ λ ⤍ 𝜎3 ⟧ ⟧ ⟧, λ ⤍ L_fork ⟧.φ.k ⟧'
result: '𝜎4:λ'
protocol: |
  𝕄(Φ.y):
    𝔼(L_fork):
      𝛿1·1 := 𝔻(𝜎1:λ)  # 𝔻(ξ.c)
      fabricated 𝑛·1·1 := ⟦ φ ↦ 𝜎2:λ:k, tu ↦ ∅ ⟧ and 𝜎3:λ:k:φ  # ⟦ tu ↦ ∅, left🌵 ↦ ⟦ φ ↦ 𝜎2:λ:k, tu ↦ ∅ ⟧( tu ↦ tu ), right🌵 ↦ 𝜎3:λ:k:φ, φ ↦ ⟦ λ ⤍ L_join, on🌵 ↦ 𝜎1:λ, left🌵 ↦ ρ.left🌵.φ, right🌵 ↦ ρ.right🌵.φ ⟧ ⟧
    𝔼(L_join):
      𝑛·2·1 := ⟦ φ ↦ 𝜎2:λ:k, tu ↦ ∅ ⟧( tu ↦ ⊥ )  # 𝕄(Φ.a🌵3)
      joined 𝜎4 := ξ.on🌵 ? 𝜎2 : 𝜎3
      𝑛·2·2 := 𝜎4:λ:k  # 𝕄(ξ.left🌵), 𝕄(ξ.right🌵)