Complete proofs for PreL1 and L1 space structure #434
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- Implement ComplexAbsolutelyIntegrable.zero: prove zero function is absolutely integrable using zero simple function representation and empty set measure - Complete PreL1.inst_AddZeroClass.zero: use ComplexAbsolutelyIntegrable.zero instead of sorry - Complete PreL1.inst_AddZeroClass.zero_add and add_zero: prove zero addition identities via function extensionality and Pi.zero_apply - Complete PreL1.inst_addCommMonoid.add_assoc and add_comm: prove associativity and commutativity via function extensionality and ring properties All proofs replace sorry placeholders with complete implementations.
Prove that the norm of the zero function is zero, completing the proof that zero is absolutely integrable with norm zero.
Remove redundant comments
- L1.dist_eq: Prove distance equals norm of difference (trivial by definition) - L1.dist_eq_zero: Complete proof that distance is zero iff functions are equal almost everywhere The proof shows that dist([f], [g]) = 0 ↔ f = g a.e. by: 1. Converting distance to integral of |f - g| 2. Using UnsignedLebesgueIntegral.eq_zero_aeZero 3. Showing the sets where functions differ are equal
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This PR completes proofs for the PreL1 and L1 space structure in Section 1.3.4.
Summary of changes:
All proofs replace sorry placeholders with complete implementations, establishing the foundational structure for L¹ space as a normed vector space.