Complete proofs for simple functions and AlmostAlways lemmas #423
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…ories - Add **/.env pattern to ignore environment files - Add **/.aider.conf.yml to ignore aider configuration files - Add **/docker/ pattern to ignore docker directories
- Prove UnsignedSimpleFunction.support_measurable: support of simple function is measurable via union of positive-coefficient sets - Prove AlmostAlways.ofAlways: always-true properties are almost always true (empty complement has measure zero) - Prove AlmostAlways.mp: modus ponens for almost-always properties via subset monotonicity - Prove AlmostAlways.countable: countable conjunction preserves almost-always via countable union of null sets
- Prove ComplexSimpleFunction.re: real part of complex simple function is real simple function via Re(c_i) coefficients - Prove ComplexSimpleFunction.im: imaginary part of complex simple function is real simple function via Im(c_i) coefficients - Both proofs extract representation, apply Re/Im to coefficients, and use complex arithmetic properties (re_sum, im_sum, re_mul_ofReal, im_mul_ofReal)
…roofs - Add RealSimpleFunction.DisjointRepr namespace with atom-based disjoint representation machinery - Prove RealSimpleFunction.disjoint_representation: any simple function has equivalent representation with pairwise disjoint measurable sets - Complete RealSimpleFunction.pos proof: uses disjoint representation and max(v_i, 0) coefficients - Complete RealSimpleFunction.neg proof: reduces to pos via neg_fun f = pos_fun (-f)
- Move singleAtom, singleAtom_pairwiseDisjoint, mem_singleAtom_iff, exists_unique_singleAtom, and atomValue from RealSimpleFunction.DisjointRepr to UnsignedSimpleFunction.IntegralWellDef - Update references in RealSimpleFunction.disjoint_representation to use new locations - Simplify comments and remove redundant explanations - Better organization: general atom machinery in general namespace, measure-specific lemmas remain in DisjointRepr
- Prove RealSimpleFunction.toComplex: real simple function coerces to complex simple function - Extract representation, apply Complex.ofReal to coefficients, use Complex.ofReal_sum and Complex.ofReal_mul to show equality
- Add AlmostEverywhereEqual.refl, symm, trans lemmas (proofs pending) - Add AlmostEverywhereEqual.equivalence theorem combining them into Equivalence instance - Sets up structure to prove AlmostEverywhereEqual is an equivalence relation
- Prove AlmostEverywhereEqual.refl: uses AlmostAlways.ofAlways (empty set is null) - Prove AlmostEverywhereEqual.symm: uses ne_comm to show symmetric sets have same measure - Prove AlmostEverywhereEqual.trans: uses subset inclusion and countable subadditivity to show union of null sets is null
…nces) - Add UnsignedMeasurable.TFAE_helpers namespace with statement abbreviations (i)-(xi) - Implement helper lemmas: complementation proofs (v↔vi↔vii↔viii, x↔xi), definitional (i↔ii), pointwise convergence (ii→iii) - Add TFAE proof structure using tfae_have and tfae_finish to assemble implication graph - Several technical lemmas marked with sorry (limsup, monotone convergence, interval preimages, approximating sequence construction)
- Complete iv_imp_ii: use tendsto_atTop_iSup for monotone sequences - Complete v_imp_vi: case analysis for ⊥, ⊤, and finite values using countable intersections - Complete vi_imp_v: use countable union over rationals with EReal.exists_rat_btwn_of_lt - Complete v_to_viii_imp_ix: all interval preimage cases using EReal.image_coe_Ixx lemmas - Fix x_imp_viii → x_imp_vii: use open set Iio instead of closed Iic - Update documentation to reflect corrected implication paths
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Summary
This PR completes several proofs in Section 1.3.1 and adds structure for Lemma 1.3.9:
All proofs replace
sorrystatements with complete formal proofs.