{ "case_id": "MG-C003", "version": "0.1.1", "items": [ {"id":"CH03-D01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Const"},{"file":"Main.lean","name":"Chapter3.Ident"},{"file":"Main.lean","name":"Chapter3.Active"},{"file":"Main.lean","name":"Chapter3.component_partition"}],"representation":"任意分量按常值优先定义三类;证明互斥及覆盖。","dependencies":[]}, {"id":"CH03-D02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Fiber"},{"file":"Main.lean","name":"Chapter3.SingleFiber"},{"file":"Main.lean","name":"Chapter3.subfamily_zero"},{"file":"Main.lean","name":"Chapter3.fiber_representation"}],"representation":"子族逐分量零距定义纤维;证明等价于复合伪度量零距。","dependencies":[]}, {"id":"CH03-D03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.PointRel"},{"file":"Main.lean","name":"Chapter3.ZeroRel"},{"file":"Main.lean","name":"Chapter3.imageSetoid"},{"file":"Main.lean","name":"Chapter3.ImageQuotient"},{"file":"Main.lean","name":"Chapter3.imageClass"},{"file":"Main.lean","name":"Chapter3.FiberLE"},{"file":"Main.lean","name":"Chapter3.zeroRel_fiber"}],"representation":"点对关系限于U×U;实际像集取零距商;预序为关系包含,并证明纤维对应。","dependencies":["CH03-D02"]}, {"id":"CH03-D04","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Legal"},{"file":"Main.lean","name":"Chapter3.radius"},{"file":"Main.lean","name":"Chapter3.legal_ball"},{"file":"Main.lean","name":"Chapter3.le_radius_iff"}],"representation":"半径为合法有限正半径的上确界;球包含与逐点公式等价。","dependencies":[]}, {"id":"CH03-D05","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.uniform"},{"file":"Main.lean","name":"Chapter3.N"}],"representation":"uniform在指定基点集合取下确界;N的首指标为输出行。","dependencies":["CH03-D04"]}, {"id":"CH03-D06","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Scale"},{"file":"Main.lean","name":"Chapter3.diameter"},{"file":"Main.lean","name":"Chapter3.diameterScale"},{"file":"Main.lean","name":"Chapter3.thresholdType"},{"file":"Main.lean","name":"Chapter3.channelType"},{"file":"Main.lean","name":"Chapter3.threshold_bits"}],"representation":"尺度定义在非空集合且单调;默认直径;Bool对精确表示两个阈值判断。","dependencies":["CH03-D05"]}, {"id":"CH03-D07","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.TotallyBoundedFor"},{"file":"Main.lean","name":"Chapter3.covering_representation"},{"file":"formal_oet/vol_01/ch_01_foundations/metric_entropy.lean","name":"formal_oet.vol_01.ch_01_foundations.covering_number_attained"},{"file":"formal_oet/vol_01/ch_01_foundations/metric_entropy.lean","name":"formal_oet.vol_01.ch_01_foundations.metric_entropy"}],"representation":"冻结内部覆盖数以内部有限中心及开球定义;covering_representation证明球并集等价及无有限覆盖边界,covering_number_attained证明有限时取得最小中心数;冻结metric_entropy为其log2,TotallyBoundedFor遍历全部有限正精度。","dependencies":[]}, {"id":"CH03-D08","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.DeltaDefined"},{"file":"Main.lean","name":"Chapter3.delta"},{"file":"Main.lean","name":"Chapter3.delta_guard"},{"file":"Main.lean","name":"Chapter3.finite_delta"},{"file":"Main.lean","name":"Chapter3.delta_infinite_cases"}],"representation":"两端至少一覆盖数有限才定义EReal熵差;证明排除双无穷及有限实数差对应;输入熵无穷而输出有限时差为正无穷,反向为负无穷。","dependencies":["CH03-D07"]}, {"id":"CH03-P01-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.const_zero_row"}],"representation":"常值条件给全部点对零行求和上界。","dependencies":["CH03-D01"]}, {"id":"CH03-P01-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.const_colip_zero"}],"representation":"任意正输入距离见证迫使指定co-Lip条目为零。","dependencies":["CH03-D01"]}, {"id":"CH03-P01-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.ident_distance"}],"representation":"由逐点零距双向传递证明所有输入点对保距。","dependencies":["CH03-D01"]}, {"id":"CH03-P02-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.compose_active"}],"representation":"在实际中间像域证明两步有效分量包含。","dependencies":["CH03-D01","CH03-P01-03","CH03-P02-03","CH03-P04-01"]}, {"id":"CH03-P02-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.chain"},{"file":"Main.lean","name":"Chapter3.activeSteps"},{"file":"Main.lean","name":"Chapter3.chain_active"},{"file":"Main.lean","name":"Chapter3.ValidChain"},{"file":"Main.lean","name":"Chapter3.chain_domains"}],"representation":"列表按时间顺序复合;递推实际像域,证明全部域合法及任意长度包含,含零步。","dependencies":["CH03-D01","CH03-P02-01"]}, {"id":"CH03-P02-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.compose_ident"}],"representation":"复合逐点零距且排除常值优先分支。","dependencies":["CH03-D01","CH03-P01-03"]}, {"id":"CH03-P03-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Swap.d"},{"file":"Main.lean","name":"Chapter3.Swap.f"},{"file":"Main.lean","name":"Chapter3.Swap.U"},{"file":"Main.lean","name":"Chapter3.Swap.valid"},{"file":"Main.lean","name":"Chapter3.Swap.domain"},{"file":"Main.lean","name":"Chapter3.Swap.image"},{"file":"Main.lean","name":"Chapter3.Swap.active"},{"file":"Main.lean","name":"Chapter3.Swap.composite"},{"file":"Main.lean","name":"Chapter3.Swap.reverse_failures"}],"representation":"Option(Fin2)表示三点空间;完整验证换位及两条逆包含失败。","dependencies":["CH03-D01"]}, {"id":"CH03-P04-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.outer_const"}],"representation":"外侧在实际像域常值,代入像点得复合常值。","dependencies":["CH03-D01"]}, {"id":"CH03-P04-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Seven.d"},{"file":"Main.lean","name":"Chapter3.Seven.f"},{"file":"Main.lean","name":"Chapter3.Seven.g"},{"file":"Main.lean","name":"Chapter3.Seven.U"},{"file":"Main.lean","name":"Chapter3.Seven.valid"},{"file":"Main.lean","name":"Chapter3.Seven.domain"},{"file":"Main.lean","name":"Chapter3.Seven.inner"},{"file":"Main.lean","name":"Chapter3.Seven.outer_fails"}],"representation":"Option(Fin6)表示七点;0,1为a,b,2,3为p,q,4,5为r,s;双分量合法并给出反例。","dependencies":["CH03-D01"]}, {"id":"CH03-P05-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.inner_const_lip"}],"representation":"指定输出行仅同分量输入时内侧常值传给复合;不要求系数有限。","dependencies":["CH03-D01"]}, {"id":"CH03-P06-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_target"}],"representation":"任意零距目标对应的子族纤维相等。","dependencies":["CH03-D02"]}, {"id":"CH03-P06-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_cover"}],"representation":"证明每个像目标纤维非空且纤维并集恰为U。","dependencies":["CH03-D02"]}, {"id":"CH03-P06-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_equal_or_disjoint"},{"file":"Main.lean","name":"Chapter3.fiber_cover"}],"representation":"全部纤维相等或不交;连同非空和覆盖给出去重分划。","dependencies":["CH03-D02","CH03-P06-01"]}, {"id":"CH03-P06-04","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_intersection"}],"representation":"非空子族纤维等于全部单分量纤维的交。","dependencies":["CH03-D02"]}, {"id":"CH03-P06-05","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_antitone"}],"representation":"更多输出分量要求给出更小纤维。","dependencies":["CH03-D02"]}, {"id":"CH03-P06-06","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_singleton"},{"file":"Main.lean","name":"Chapter3.fiber_all_subset"}],"representation":"单元素子族等式及全子族向每分量的包含。","dependencies":["CH03-D02"]}, {"id":"CH03-P06-07","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.ident_fiber"}],"representation":"恒等分量对任意目标的纤维等于输入零距类。","dependencies":["CH03-D01","CH03-D02"]}, {"id":"CH03-P06-08","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.const_fiber"},{"file":"Main.lean","name":"Chapter3.const_fiber_empty"}],"representation":"像内目标纤维为U;与之正距离的任意目标纤维为空。","dependencies":["CH03-D01","CH03-D02"]}, {"id":"CH03-P07-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Factors"},{"file":"Main.lean","name":"Chapter3.factors_iff"}],"representation":"实际像域上的因子映射存在推出点值关系包含。","dependencies":["CH03-D03"]}, {"id":"CH03-P07-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Factors"},{"file":"Main.lean","name":"Chapter3.factors_iff"}],"representation":"从实际像点选择输入代表元构造因子映射并验证公式;空像域也适用。","dependencies":["CH03-D03"]}, {"id":"CH03-P07-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.factors_both"}],"representation":"双向点值变换等价于点值关系相等。","dependencies":["CH03-D03","CH03-P07-01","CH03-P07-02"]}, {"id":"CH03-P07-04","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.factor_lip"}],"representation":"在实际像域证明任意规定T的L/K界;K正有限且L有限。","dependencies":["CH03-D03"]}, {"id":"CH03-P08-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.QuotientFactors"},{"file":"Main.lean","name":"Chapter3.quotient_factors_iff"}],"representation":"实际像零距商;构造代表元无关的Quotient.lift并双向验证规定公式。","dependencies":["CH03-D03"]}, {"id":"CH03-P08-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.quotient_factors_both"}],"representation":"双向规定商映射等价于零距关系相等。","dependencies":["CH03-D03","CH03-P08-01"]}, {"id":"CH03-P08-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.SeparatesOn"},{"file":"Main.lean","name":"Chapter3.relations_equal_on_separated_image"},{"file":"Main.lean","name":"Chapter3.separated_factor_criteria"}],"representation":"分别假设两个实际像集分离,证明两种关系及单向、双向判据一致。","dependencies":["CH03-D03","CH03-P07-01","CH03-P07-02","CH03-P08-01"]}, {"id":"CH03-P09-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiberLE_refl"},{"file":"Main.lean","name":"Chapter3.fiberLE_trans"}],"representation":"固定U,关系包含给出预序的自反及传递。","dependencies":["CH03-D03"]}, {"id":"CH03-P09-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.relation_chain"}],"representation":"对任意自然数端点a≤b归纳证明逐步包含的复合;适用于任意有限截断。","dependencies":[]}, {"id":"CH03-P09-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.relation_chain_strict"}],"representation":"任一步真包含给端点真包含,并实际给出端点差集元素。","dependencies":["CH03-P09-02"]}, {"id":"CH03-P10-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.realGauge"},{"file":"Main.lean","name":"Chapter3.realOp"},{"file":"Main.lean","name":"Chapter3.realSet"},{"file":"Main.lean","name":"Chapter3.Step.a"},{"file":"Main.lean","name":"Chapter3.Step.f"},{"file":"Main.lean","name":"Chapter3.Step.U"},{"file":"Main.lean","name":"Chapter3.Step.valid"},{"file":"Main.lean","name":"Chapter3.Step.domain"}],"representation":"Option实线以none为吸收元;通常距离及阶跃算子;验证固定U非空且在定义域内。","dependencies":[]}, {"id":"CH03-P10-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Step.point_radius"},{"file":"Main.lean","name":"Chapter3.Step.legal_radii"}],"representation":"固定U中每个实基点,合法有限正半径恰满足r≤|y|,上确界等于|y|且正。","dependencies":["CH03-D04","CH03-P10-01"]}, {"id":"CH03-P10-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Step.uniform_zero"}],"representation":"在同一固定U构造任意接近零的异号基点,证明统一半径为零。","dependencies":["CH03-D04","CH03-D05","CH03-P10-01","CH03-P18-02"]}, {"id":"CH03-P11-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.const_subfamily_fiber"}],"representation":"子族各分量常值时任一输入基点的像目标纤维为U。","dependencies":["CH03-D01","CH03-D02"]}, {"id":"CH03-P11-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.const_subfamily_radius"}],"representation":"子族输出两两零距,证明所有基点及统一半径均为无穷。","dependencies":["CH03-D01","CH03-D04","CH03-D05","CH03-D02"]}, {"id":"CH03-P11-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Infinite.d"},{"file":"Main.lean","name":"Chapter3.Infinite.U"},{"file":"Main.lean","name":"Chapter3.Infinite.valid"},{"file":"Main.lean","name":"Chapter3.Infinite.domain"},{"file":"Main.lean","name":"Chapter3.Infinite.uniform_infinite"},{"file":"Main.lean","name":"Chapter3.Infinite.not_const"}],"representation":"合法两点无穷输入距离模型;输入0、输出j遍历Fin2,涵盖同分量与异分量两分支。","dependencies":["CH03-D01","CH03-D04","CH03-D05"]}, {"id":"CH03-P12-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.ident_image_fiber"}],"representation":"利用恒等分量保距,像目标纤维恰为输入基点零距类。","dependencies":["CH03-D01","CH03-D02","CH03-P01-03"]}, {"id":"CH03-P12-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.ident_radius_lower"}],"representation":"有限正r的球中只有输入零距点时,证明r是合法半径并给出下界。","dependencies":["CH03-D01","CH03-D04","CH03-P01-03"]}, {"id":"CH03-P13-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.radius_subfamily"},{"file":"Main.lean","name":"Chapter3.finite_inf_attained"}],"representation":"点态子族半径为各单分量半径下确界;有限非空时证明最小值取得。","dependencies":["CH03-D02","CH03-D04"]}, {"id":"CH03-P13-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.uniform_subfamily"},{"file":"Main.lean","name":"Chapter3.finite_inf_attained"}],"representation":"交换基点和分量下确界,结合最小值取得给出统一半径最小值公式。","dependencies":["CH03-D04","CH03-D05","CH03-P13-01"]}, {"id":"CH03-P13-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.uniform_subfamily_antitone"}],"representation":"子族包含经分量下确界公式推出统一半径反单调。","dependencies":["CH03-D05","CH03-P13-02"]}, {"id":"CH03-P14-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.fiber_truncate"}],"representation":"纤维成员的输出零距由对称性转为原文方向。","dependencies":["CH03-D02"]}, {"id":"CH03-P14-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.below_radius"}],"representation":"严格小于上确界时从合法集提取更大的有限正半径,再应用球内条件。","dependencies":["CH03-D04"]}, {"id":"CH03-P14-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.composite_truncate"}],"representation":"显式保留上游定义域、两像点在下游域及纤维条件,证明复合输出零距。","dependencies":["CH03-D02","CH03-P14-01"]}, {"id":"CH03-P15-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.radius_compose"}],"representation":"下游在实际像域保持零距时,给出复合统一半径下界。","dependencies":["CH03-D04","CH03-D05","CH03-P18-01"]}, {"id":"CH03-P15-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.radius_compose_lip"}],"representation":"有限正L拉回下游实际像域的统一半径;扩展数除法涵盖右端0及无穷。","dependencies":["CH03-D04","CH03-D05","CH03-P18-01"]}, {"id":"CH03-P16-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.threshold_cases"},{"file":"Main.lean","name":"Chapter3.threshold_bits"}],"representation":"四种Bool组合逐一等价于互补不等式,故互斥且穷尽。","dependencies":["CH03-D06"]}, {"id":"CH03-P16-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.threshold_scale_mono"}],"representation":"尺度严格小于阈值的判断随尺度阈值增大保持。","dependencies":[]}, {"id":"CH03-P16-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.threshold_radius_antitone"}],"representation":"半径大于等于阈值的判断随零化阈值增大反单调。","dependencies":[]}, {"id":"CH03-P17-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.scale_restrict"}],"representation":"任意单调尺度在非空域缩小时不增,并保留严格阈值判断。","dependencies":["CH03-D06"]}, {"id":"CH03-P17-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.radius_restrict"}],"representation":"固定任意基点,限制评估域使合法半径上确界不减。","dependencies":["CH03-D04"]}, {"id":"CH03-P17-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.uniform_restrict"}],"representation":"同时缩小评估域及下确界基点集合,统一半径不减。","dependencies":["CH03-D04","CH03-D05","CH03-P17-02"]}, {"id":"CH03-P17-04","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.scale_reverse_counterexample"}],"representation":"带吸收元通常实线,A={0,1}、B={0}及阈值1/2,证明域条件、两直径及反向失败。","dependencies":["CH03-D06"]}, {"id":"CH03-P18-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.below_uniform"}],"representation":"输入距离严格小于统一下确界则小于该基点半径,输出归零。","dependencies":["CH03-D04","CH03-D05","CH03-P14-02"]}, {"id":"CH03-P18-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.separated_input"}],"representation":"由正输出距离及单步判据逆否得到输入距离下界。","dependencies":["CH03-D05","CH03-P18-01"]}, {"id":"CH03-P18-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.separation_witness"}],"representation":"每个规定有限正α,从下确界及非法球提取两点,再取有限正ε;输出无穷亦适用。","dependencies":["CH03-D04","CH03-D05"]}, {"id":"CH03-P19-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.threshold_fiber"}],"representation":"统一半径达到阈值时,任意基点的阈值球交评估域包含在纤维中。","dependencies":["CH03-D02","CH03-D05","CH03-P18-01"]}, {"id":"CH03-P19-02","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.push_net"},{"file":"Main.lean","name":"Chapter3.push_cover"},{"file":"Main.lean","name":"Chapter3.threshold_cover"}],"representation":"任意B子集的内部输入中心映到内部输出中心且基数不增;下确界证明涵盖空集及无有限覆盖。","dependencies":["CH03-D07","CH03-D05","CH03-P18-01"]}, {"id":"CH03-P19-03","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.log2_mono"},{"file":"Main.lean","name":"Chapter3.entropy_mono_cover"},{"file":"Main.lean","name":"Chapter3.threshold_entropy"}],"representation":"非空B下证明扩展对数单调,从覆盖数界得到熵界,允许无穷。","dependencies":["CH03-D07","CH03-P19-02"]}, {"id":"CH03-P20-01","status":"complete","declarations":[{"file":"Main.lean","name":"Chapter3.Step.cover_counterexample"}],"representation":"在同一固定阶跃U中,对每个有限正τ取0