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‎README.md‎

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- [PolyglotFormalisms.jl](https://github.com/hyperpolymath/PolyglotFormalisms.jl) - Julia reference implementation
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- [alib-for-rescript](https://github.com/hyperpolymath/alib-for-rescript) - ReScript implementation
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- [polyglot_formalisms_gleam](https://github.com/hyperpolymath/polyglot_formalisms_gleam) - Gleam implementation
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## Architecture
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See [TOPOLOGY.md](TOPOLOGY.md) for a visual architecture map and completion dashboard.

‎TOPOLOGY.md‎

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<!-- SPDX-License-Identifier: PMPL-1.0-or-later -->
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<!-- TOPOLOGY.md — Project architecture map and completion dashboard -->
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<!-- Last updated: 2026-02-19 -->
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# PolyglotFormalisms.Elixir — Project Topology
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## System Architecture
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```
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┌─────────────────────────────────────────┐
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│ POLYGLOT SYSTEM │
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│ (Cross-Language Verification) │
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└───────────────────┬─────────────────────┘
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│
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▼
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┌─────────────────────────────────────────┐
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│ ELIXIR IMPLEMENTATION │
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│ │
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│ ┌───────────┐ ┌───────────────────┐ │
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│ │ Arithmetic│ │ Comparison │ │
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│ │ Module │ │ Module │ │
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│ └─────┬─────┘ └────────┬──────────┘ │
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│ │ │ │
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│ ┌─────▼─────┐ ┌────────▼──────────┐ │
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│ │ Logical │ │ Behavioral │ │
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│ │ Module │ │ Semantics (BEAM) │ │
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│ └─────┬─────┘ └────────┬──────────┘ │
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└────────│─────────────────│──────────────┘
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│ │
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▼ ▼
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┌─────────────────────────────────────────┐
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│ SEMANTIC EQUIVALENCE │
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│ ┌───────────┐ ┌───────────┐ ┌───────┐│
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│ │ Julia │ │ ReScript │ │ Gleam ││
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│ └───────────┘ └───────────┘ └───────┘│
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└─────────────────────────────────────────┘
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┌─────────────────────────────────────────┐
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│ REPO INFRASTRUCTURE │
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│ Justfile / mix.exs .machine_readable/ │
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│ ExUnit / Doctest AI.a2ml │
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└─────────────────────────────────────────┘
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```
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## Completion Dashboard
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```
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COMPONENT STATUS NOTES
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───────────────────────────────── ────────────────── ─────────────────────────────────
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FORMAL MODULES
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Arithmetic (add, sub, etc) ██████████ 100% IEEE 754 parity verified
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Comparison (equal, less, etc) ██████████ 100% Transitivity proven
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Logical (and, or, not) ██████████ 100% De Morgan's laws verified
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SEMANTICS & VERIF
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BEAM Behavioral Semantics ██████████ 100% Float/Integer edge cases stable
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Cross-Language Verification ████████░░ 80% Julia/ReScript parity verified
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Mathematical Properties ██████████ 100% Commutativity/Associativity stable
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REPO INFRASTRUCTURE
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Justfile Automation ██████████ 100% Standard build/test tasks
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.machine_readable/ ██████████ 100% STATE tracking active
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ExUnit Test Suite ██████████ 100% High unit + doctest coverage
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─────────────────────────────────────────────────────────────────────────────
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OVERALL: █████████░ ~90% Stable Elixir implementation
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```
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## Key Dependencies
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```
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Common Spec ──────► Elixir Implementation ──────► Behavioral Proof
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│ │ │
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▼ ▼ ▼
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Property Tests ───► Cross-Lang Parity ──────► Semantic Equivalence
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```
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## Update Protocol
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This file is maintained by both humans and AI agents. When updating:
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1. **After completing a component**: Change its bar and percentage
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2. **After adding a component**: Add a new row in the appropriate section
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3. **After architectural changes**: Update the ASCII diagram
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4. **Date**: Update the `Last updated` comment at the top of this file
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Progress bars use: `█` (filled) and `░` (empty), 10 characters wide.
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Percentages: 0%, 10%, 20%, ... 100% (in 10% increments).

‎lib/arithmetic.ex‎

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# SPDX-License-Identifier: PMPL-1.0-or-later
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defmodule PolyglotFormalisms.Arithmetic do
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@moduledoc """
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Arithmetic operations from the PolyglotFormalisms Common Library specification.
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Arithmetic Kernel — PolyglotFormalisms Common Library.
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Elixir implementation matching aggregate-library behavioral semantics.
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This module implements the arithmetic section of the PolyglotFormalisms
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specification. It ensures that numerical operations behave identically
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across Elixir, Rust, and ReScript environments.
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Each operation includes:
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- Implementation following PolyglotFormalisms specification
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- Type signatures for numeric operations
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- Documentation matching specification format
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DESIGN MANDATE:
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- Strict adherence to specification behavioral semantics.
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- Verification of mathematical properties (Commutativity, Associativity).
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- Explicit type signatures for float-based operations.
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"""
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@doc """
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Computes the sum of two numbers.
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## Behavioral Semantics
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- Parameters: a (augend), b (addend)
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- Returns: The arithmetic sum of a and b
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## Mathematical Properties
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- Commutativity: add(a, b) == add(b, a)
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- Associativity: add(add(a, b), c) == add(a, add(b, c))
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- Identity element: add(a, 0.0) == a
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## Examples
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iex> PolyglotFormalisms.Arithmetic.add(2.0, 3.0)
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5.0
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iex> PolyglotFormalisms.Arithmetic.add(-5.0, 3.0)
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-2.0
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SUMMATION: Returns the arithmetic sum of `a` and `b`.
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iex> PolyglotFormalisms.Arithmetic.add(0.0, 0.0)
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0.0
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iex> PolyglotFormalisms.Arithmetic.add(1.5, 2.5)
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4.0
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iex> PolyglotFormalisms.Arithmetic.add(-10.0, -20.0)
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-30.0
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## Properties:
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- Identity: add(a, 0.0) == a
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- Commutative: add(a, b) == add(b, a)
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"""
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@spec add(number(), number()) :: float()
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def add(a, b), do: a + b
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@doc """
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Computes the difference of two numbers.
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## Behavioral Semantics
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- Parameters: a (minuend), b (subtrahend)
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- Returns: The arithmetic difference a - b
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## Mathematical Properties
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- Non-commutative: subtract(a, b) != subtract(b, a) (except when a == b)
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- Identity element: subtract(a, 0.0) == a
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- Inverse of add: subtract(a, b) == add(a, -b)
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## Examples
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iex> PolyglotFormalisms.Arithmetic.subtract(10.0, 3.0)
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7.0
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SUBTRACTION: Returns the arithmetic difference `a - b`.
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iex> PolyglotFormalisms.Arithmetic.subtract(5.0, 8.0)
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-3.0
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iex> PolyglotFormalisms.Arithmetic.subtract(0.0, 0.0)
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0.0
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iex> PolyglotFormalisms.Arithmetic.subtract(10.5, 3.2)
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7.3
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iex> PolyglotFormalisms.Arithmetic.subtract(-5.0, -3.0)
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-2.0
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## Properties:
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- Non-commutative: subtract(a, b) != subtract(b, a)
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- Inverse: subtract(a, b) == add(a, -b)
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"""
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@spec subtract(number(), number()) :: float()
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def subtract(a, b), do: a - b
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@doc """
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Computes the product of two numbers.
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## Behavioral Semantics
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- Parameters: a (multiplicand), b (multiplier)
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- Returns: The arithmetic product of a and b
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## Mathematical Properties
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- Commutativity: multiply(a, b) == multiply(b, a)
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- Associativity: multiply(multiply(a, b), c) == multiply(a, multiply(b, c))
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- Identity element: multiply(a, 1.0) == a
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- Zero element: multiply(a, 0.0) == 0.0
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- Distributivity: multiply(a, add(b, c)) == add(multiply(a, b), multiply(a, c))
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## Examples
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MULTIPLICATION: Returns the arithmetic product of `a` and `b`.
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iex> PolyglotFormalisms.Arithmetic.multiply(4.0, 5.0)
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iex> PolyglotFormalisms.Arithmetic.multiply(-3.0, 7.0)
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iex> PolyglotFormalisms.Arithmetic.multiply(0.0, 100.0)
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0.0
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iex> PolyglotFormalisms.Arithmetic.multiply(2.5, 4.0)
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10.0
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iex> PolyglotFormalisms.Arithmetic.multiply(-2.0, -3.0)
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6.0
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## Properties:
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- Zero Element: multiply(a, 0.0) == 0.0
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- Identity: multiply(a, 1.0) == a
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"""
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@spec multiply(number(), number()) :: float()
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def multiply(a, b), do: a * b
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@doc """
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Computes the quotient of two numbers.
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## Behavioral Semantics
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- Parameters: a (dividend), b (divisor)
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- Returns: The arithmetic quotient a / b
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## Mathematical Properties
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- Non-commutative: divide(a, b) != divide(b, a) (except when a == b)
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- Identity element: divide(a, 1.0) == a
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- Inverse of multiply: multiply(divide(a, b), b) ≈ a (when b != 0.0)
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## Edge Cases
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- Division by zero: Returns Infinity or -Infinity (BEAM/Erlang behavior)
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- Behavior follows BEAM floating-point division
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## Examples
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iex> PolyglotFormalisms.Arithmetic.divide(10.0, 2.0)
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5.0
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iex> PolyglotFormalisms.Arithmetic.divide(7.0, 2.0)
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iex> PolyglotFormalisms.Arithmetic.divide(10.5, 2.0)
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iex> PolyglotFormalisms.Arithmetic.divide(-10.0, 2.0)
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iex> PolyglotFormalisms.Arithmetic.divide(5.0, -2.0)
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"""
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@spec divide(number(), number()) :: float()
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def divide(a, b), do: a / b
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@doc """
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## Behavioral Semantics
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- Parameters: a (dividend), b (divisor)
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- Returns: The remainder of a / b
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## Mathematical Properties
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- Range constraint: For b > 0, result is in [0, b)
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- Division relation: a == div(a, b) * b + rem(a, b) (when b != 0)
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## Edge Cases
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- Modulo by zero: Raises ArithmeticError (BEAM behavior)
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- Sign handling follows Erlang rem operator semantics
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## Examples
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iex> PolyglotFormalisms.Arithmetic.modulo(10, 3)
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1
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iex> PolyglotFormalisms.Arithmetic.modulo(15, 4)
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3
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iex> PolyglotFormalisms.Arithmetic.modulo(7, 7)
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iex> PolyglotFormalisms.Arithmetic.modulo(-10, 3)
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## Note
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Elixir/BEAM uses `rem` which may differ from other languages' mod.
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This follows BEAM/Erlang semantics for cross-platform consistency.
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"""
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@spec modulo(integer(), integer()) :: integer()
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def modulo(a, b), do: rem(a, b)
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end

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