Mathematical proof

Design to Code Theorem: A Mathematical Formulation

Let D be the set of all designs, where each design d is represented as a rectangle. Each rectangle in D may contain sub-rectangles and content.

Let C be the set of all code, where each code c is a rendering process that transforms a design rectangle into a corresponding visual representation.

Statements:

1. Design Representation:

   • Designs, denoted as d ∈ D, are rectangles, where d = (width, height), and may contain sub-designs or content.

2. Content in Designs:

   • Content within a rectangle in the set D can be image, video, or text.

3. Code Rendering Process:

   • Code, represented as c ∈ C, is a function that maps a design rectangle d to its visual representation, denoted as c(d).

Mathematical Formulation:

1. Existence of Designs:

   • For any design d ∈ D, there exists a rectangle representing a visual concept:
     • (d = (width, height))

2. Design Hierarchy:

   • Designs can be hierarchical, where a design d may contain sub-designs or content:
     • (d = (width, height, {sub_designs}, {content}))

3. Code Rendering Process:

   • The rendering process c is a function that maps a design rectangle d to its visual representation:
     • (c: D \rightarrow Visual\ Representation)

Examples:

1. Simple Design:

   • Let (d_1) be a simple design: (d_1 = (100px, 50px))

2. Hierarchical Design:

   • Let (d_2) be a design with sub-designs and content:
     • (d_2 = (200px, 100px, {d_{2.1}, d_{2.2}}, {image_1, text_1}))

Proof:

1. Existence Proof:

   • Given any design d, its existence is confirmed by the definition of designs as rectangles.

2. Rendering Process:

   • The code rendering process c transforms a design d into its visual representation, satisfying (c: D \rightarrow Visual\ Representation).

3. Hierarchy Handling:

   • The hierarchical nature of designs is accommodated by the definition of designs in D.

Conclusion:

The Design to Code Theorem provides a mathematical foundation for the systematic transformation of designs into visual representations through a code rendering process. This formulation, supported by examples and proofs, establishes the conceptual framework for translating designs, represented as rectangles, into executable code.