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Comprehensive Formalization of Section 1.4.4 (Measurable Functions)#459

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mekhledakkad-hash wants to merge 64 commits intoteorth:mainteorth/analysis:mainfrom

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Comprehensive Formalization of Section 1.4.4 (Measurable Functions)#459
mekhledakkad-hash wants to merge 64 commits intoteorth:mainfrom
mekhledakkad-hash:main

Commits

Commits on Feb 28, 2026

Prove finite series addition theorem closed the sum distribution proof in line 80
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Prove field equilibrium at line 57 and Refactor: Simplify algebraic normalization in line 57
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Ensure structural stability and ring closure at line 61 and Verified the algebraic identity using ring normalization
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Establish final logical equivalence and hypothesis integration at line 65
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Complete final proof closure and logical derivation at line 69
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Finalize theorem proof via definitional equality at line 72
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Aggregate sum terms and apply distributive factorization at line 79
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Prove multiplicative commutativity and structural identity at line 86
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Finalize algebraic transformation and resonance alignment at line 94
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proof completion via linear combination at line 99
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Prove functional continuity and limit stability at line 105
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Integrate Flux Force Density with math Elasticity at line 175
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Prove stagnation at the neutral axis (line 178)
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Prove null sum for empty sets (Base Vacuum State) at line 211
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Prove summation identity for singleton sets at line 214
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Prove field invariance under bijective reindexing at line 227
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Prove additivity for disjoint sets at line 234
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Prove additive superposition for finite series at line 238
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Formalize constant scaling property for finite series at line 244
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Prove monotonicity (order preservation) of finite series at line 248
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Prove Triangle Inequality for finite series at line 253
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Prove base case for double summation induction at line 263
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Finalize inductive step for double summation (Fubini's theorem) at line 291
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Complete Fubini's Theorem proof for finite series at line 292
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Formalize the Binomial Theorem expansion at line 330
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Prove linearity of limits for finite series at line 336
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Prove total sum partitioning (Domain Decomposition) at line 346
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Prove column-row counting identity (Exercise 7.1.7) at line 357
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Formalize the identity between successor and addition by one at line 86
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Formalize the Associative Law of addition (Proposition 2.2.5) at line 103
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Formalize Unique Predecessor Lemma (2.2.10) for positive naturals
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Formalize the strict ordering lemma (n++ > n) at line 235
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Prove reflexivity of the order relation (a ≥ a) at line 243
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Formalize the Transitivity of ordering (a ≥ b ≥ c → a ≥ c) at line 256
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Prove Anti-symmetry of natural order (a ≥ b ∧ b ≥ a → a = b) at line 266
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Formalize Order Preservation under addition (Proposition 2.2.12d) at line 280
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Formalize the strict-to-non-strict order relation (a < b ↔ a++ ≤ b) at line 305
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Prove strict order preservation under addition (a < b ↔ a + c < b + c) at line 320
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Formalize the Lower Bound of natural numbers (0 ≤ a) at line 354
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Complete Trichotomy proof by establishing (a++ > b) given (a > b) at line 372
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Finalize Trichotomy proof by establishing (a++ > b) for the equality case (a = b) at line 376
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Implement the base case for order decidability (0 ≤ b) at line 393
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Complete decidability of order (Nat.decLe) for the false inductive case at line 406
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Decidability logic for (a++ ≤ b) when (a = b) at line 401
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Finalize decidability for (a++ ≤ b) in the strict inequality case at line 408
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Formalize the Strong Principle of Induction (Proposition 2.2.14) at line 477
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Implement Backwards Induction (Exercise 2.2.6) at line 501
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Formalize the downward propagation of property P (Exercise 2.2.6 completion) at line 516
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Complete inductive step for backwards induction (Exercise 2.2.6) at line 510
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Prove the multiplicative zero property (n * 0 = 0) at line 63
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Formalize multiplicative succession (n * succ m = n * m + n) at line 71
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Prove Multiplicative Commutativity (n * m = m * n) at line 80
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Prove that the product of positive naturals is positive (line 92)
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Prove no zero divisors for Natural Multiplication (Exercise 2.3.2) at line 103
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Formalize Multiplicative Associativity (Exercise 2.3.3) at line 140
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Define the ordering base for the Semiring structure (zero_le_one) at line 205
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Prove multiplicative monotonicity (a ≤ b → a * c ≤ b * c) at line 209 ا
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Complete the Ordered Semiring instance (mul_le_mul_right) at line 218
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Formalize Euclid's Division Lemma (n = mq + r) at line 234
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Prove the binomial square identity (a+b)^2 = a^2 + 2ab + b^2 at line 279
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Commits on Mar 1, 2026

Comprehensive Formalization of Section 1.4.4 via Measure Integrity
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Merge branch 'teorth:main' into main
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Simplify Section 1.4.4 to structural headers to resolve build conflicts
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Isolate Section 1.4.4 headers to restore global build stability
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