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| <title>Mathematics Processes - Programming Framework Analysis</title> | |
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| <h1>Mathematics Processes - Programming Framework Analysis</h1> | |
| <p>This document presents mathematics processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.</p> | |
| <h2>1. Mathematical Induction Proof Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| graph TD | |
| A1[Peano Axioms] --> B1[Axiom Processing] | |
| C1[Given n in Natural Numbers] --> D1[Input Validation] | |
| E1[Goal: Prove P(n)] --> F1[Target Identification] | |
| B1 --> G1[Mathematical Universe Setup] | |
| D1 --> H1[Variable Declaration] | |
| F1 --> I1[Proof Strategy Selection] | |
| G1 --> J1[Induction Hypothesis P(k)] | |
| H1 --> K1[Base Case Analysis] | |
| I1 --> L1[Inductive Step Planning] | |
| K1 --> M1[P(0) Verification] | |
| M1 --> N1[Base Case Success] | |
| N1 --> O1[Induction Foundation] | |
| L1 --> P1[Assume P(k) for k in Natural Numbers] | |
| P1 --> Q1[Show P(k+1) follows] | |
| Q1 --> R1[Inductive Step Execution] | |
| R1 --> S1[Algebraic Manipulation] | |
| S1 --> T1[Logical Deduction] | |
| T1 --> U1[Theorem Application] | |
| U1 --> V1[Sub-proof Construction] | |
| V1 --> W1[Lemma Application] | |
| W1 --> X1[Contradiction Analysis] | |
| X1 --> Y1[Logical Consistency Check] | |
| Y1 --> Z1[Mathematical Rigor Verification] | |
| Z1 --> AA1[Proof Completeness Assessment] | |
| AA1 --> BB1{Proof Complete?} | |
| BB1 -->|No| CC1[Identify Gap] | |
| BB1 -->|Yes| DD1[Proof Validated] | |
| CC1 --> EE1[Additional Lemma Needed] | |
| EE1 --> FF1[Sub-proof Construction] | |
| FF1 --> GG1[Gap Resolution] | |
| GG1 --> Y1 | |
| DD1 --> HH1[Theorem P(n) Proven] | |
| HH1 --> II1[Mathematical Truth Established] | |
| II1 --> JJ1[Proof Tree Complete] | |
| style A1 fill:#ff6b6b,color:#fff | |
| style C1 fill:#ff6b6b,color:#fff | |
| style E1 fill:#ff6b6b,color:#fff | |
| style J1 fill:#ffd43b,color:#000 | |
| style P1 fill:#ffd43b,color:#000 | |
| style Q1 fill:#ffd43b,color:#000 | |
| style S1 fill:#51cf66,color:#fff | |
| style T1 fill:#51cf66,color:#fff | |
| style U1 fill:#51cf66,color:#fff | |
| style V1 fill:#51cf66,color:#fff | |
| style W1 fill:#51cf66,color:#fff | |
| style X1 fill:#51cf66,color:#fff | |
| style B1 fill:#74c0fc,color:#fff | |
| style D1 fill:#74c0fc,color:#fff | |
| style F1 fill:#74c0fc,color:#fff | |
| style G1 fill:#74c0fc,color:#fff | |
| style H1 fill:#74c0fc,color:#fff | |
| style I1 fill:#74c0fc,color:#fff | |
| style K1 fill:#74c0fc,color:#fff | |
| style L1 fill:#74c0fc,color:#fff | |
| style M1 fill:#74c0fc,color:#fff | |
| style N1 fill:#74c0fc,color:#fff | |
| style O1 fill:#74c0fc,color:#fff | |
| style R1 fill:#74c0fc,color:#fff | |
| style Y1 fill:#74c0fc,color:#fff | |
| style Z1 fill:#74c0fc,color:#fff | |
| style AA1 fill:#74c0fc,color:#fff | |
| style BB1 fill:#74c0fc,color:#fff | |
| style CC1 fill:#74c0fc,color:#fff | |
| style DD1 fill:#74c0fc,color:#fff | |
| style EE1 fill:#74c0fc,color:#fff | |
| style FF1 fill:#74c0fc,color:#fff | |
| style GG1 fill:#74c0fc,color:#fff | |
| style HH1 fill:#b197fc,color:#fff | |
| style II1 fill:#b197fc,color:#fff | |
| style JJ1 fill:#b197fc,color:#fff | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Logical Structures & Hypotheses | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Deductions & Theorem Applications | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 1.</strong> Mathematical Induction Proof Process. This mathematics process visualization demonstrates formal mathematical reasoning. The flowchart shows axioms and given conditions, logical structures and hypotheses, deduction steps and theorem applications, intermediate calculations and sub-proofs, and final proven theorems. | |
| </div> | |
| </div> | |
| <h2>2. Euclidean Algorithm Process</h2> | |
| <div class="figure"> | |
| <div class="mermaid"> | |
| graph TD | |
| A2[Integer a] --> B2[Input Validation] | |
| C2[Integer b] --> D2[Input Validation] | |
| E2[Goal: Find GCD(a,b)] --> F2[Problem Statement] | |
| B2 --> G2[Set r₀ = a] | |
| D2 --> H2[Set r₁ = b] | |
| F2 --> I2[Algorithm Selection] | |
| G2 --> J2[Division Algorithm] | |
| H2 --> K2[Division Algorithm] | |
| I2 --> L2[Iterative Process] | |
| J2 --> M2[r₀ = q₁r₁ + r₂] | |
| K2 --> N2[Calculate q₁ and r₂] | |
| L2 --> O2[Initialize iteration counter] | |
| M2 --> P2{Is r₂ = 0?} | |
| N2 --> Q2[Store r₂] | |
| O2 --> R2[Increment counter] | |
| P2 -->|No| S2[Set r₀ = r₁, r₁ = r₂] | |
| P2 -->|Yes| T2[GCD Found: r₁] | |
| Q2 --> U2[Update remainders] | |
| R2 --> V2[Track iterations] | |
| S2 --> W2[Next Division Step] | |
| U2 --> X2[Prepare for next iteration] | |
| V2 --> Y2[Check termination condition] | |
| T2 --> Z2[GCD(a,b) = r₁] | |
| W2 --> AA2[Repeat division process] | |
| X2 --> BB2[Update variables] | |
| Y2 --> CC2{Continue?} | |
| Z2 --> DD2[Result Validation] | |
| AA2 --> P2 | |
| BB2 --> P2 | |
| CC2 -->|Yes| AA2 | |
| CC2 -->|No| T2 | |
| DD2 --> EE2[GCD Calculation Complete] | |
| EE2 --> FF2[Mathematical Proof of Correctness] | |
| FF2 --> GG2[Algorithm Efficiency Analysis] | |
| style A2 fill:#ff6b6b,color:#fff | |
| style C2 fill:#ff6b6b,color:#fff | |
| style E2 fill:#ff6b6b,color:#fff | |
| style G2 fill:#ffd43b,color:#000 | |
| style H2 fill:#ffd43b,color:#000 | |
| style I2 fill:#ffd43b,color:#000 | |
| style J2 fill:#ffd43b,color:#000 | |
| style K2 fill:#ffd43b,color:#000 | |
| style L2 fill:#51cf66,color:#fff | |
| style M2 fill:#51cf66,color:#fff | |
| style N2 fill:#51cf66,color:#fff | |
| style O2 fill:#51cf66,color:#fff | |
| style S2 fill:#51cf66,color:#fff | |
| style W2 fill:#51cf66,color:#fff | |
| style AA2 fill:#51cf66,color:#fff | |
| style B2 fill:#74c0fc,color:#fff | |
| style D2 fill:#74c0fc,color:#fff | |
| style F2 fill:#74c0fc,color:#fff | |
| style P2 fill:#74c0fc,color:#fff | |
| style Q2 fill:#74c0fc,color:#fff | |
| style R2 fill:#74c0fc,color:#fff | |
| style T2 fill:#74c0fc,color:#fff | |
| style U2 fill:#74c0fc,color:#fff | |
| style V2 fill:#74c0fc,color:#fff | |
| style X2 fill:#74c0fc,color:#fff | |
| style Y2 fill:#74c0fc,color:#fff | |
| style CC2 fill:#74c0fc,color:#fff | |
| style DD2 fill:#74c0fc,color:#fff | |
| style Z2 fill:#b197fc,color:#fff | |
| style EE2 fill:#b197fc,color:#fff | |
| style FF2 fill:#b197fc,color:#fff | |
| style GG2 fill:#b197fc,color:#fff | |
| </div> | |
| <div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;"> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Mathematical Methods & Algorithms | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Computational Operations | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates | |
| </div> | |
| <div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;"> | |
| <span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products | |
| </div> | |
| </div> | |
| <div class="figure-caption"> | |
| <strong>Figure 2.</strong> Euclidean Algorithm Process. This mathematics process visualization demonstrates algorithmic computation. The flowchart shows integer inputs, mathematical methods and algorithms, computational operations, intermediate calculations, and final GCD results. | |
| </div> | |
| </div> | |
| <p><strong>Generated using the Programming Framework methodology</strong></p> | |
| <p>This collection demonstrates the computational nature of mathematical processes and systems</p> | |
| <p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p> | |
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