9i: lift Bochner stack (BochnerExpansion + PerSummand) to explicit g via subst hg#44
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…re(At) / _symmetric to explicit g
…eep proof cascade)
…ure) to explicit g - IsKilling predicate now takes (g : RiemannianMetric I M) - IsKilling.second_covDeriv_inner_skew lifted; local scalar var g renamed to kw_g to avoid shadowing the section g - second_covDeriv_commutator lifted - IsKilling.second_covDeriv_eq_curvature lifted All HasMetric.metric body refs replaced with g; metricInner abbrev calls with g.metricInner; metricInner_X lemma calls with g.metricInner_X. Internal callers within RiemannCurvature.lean updated to pass g.
The proof routes through ⟪·,·⟫_ℝ = hm.metric.inner via the InnerProductSpace instance derived from [HasMetric I M], so ricci_symm cannot be lifted for arbitrary g without InnerProductSpace restructure. Use subst hg pattern: caller passes g and proof of g = hm.metric, body works in hm.metric form. BochnerExpansion caller updated to pass (HasMetric.metric, rfl). RiemannCurvature.lean is now fully explicit-g for all theorems with at most this trivial hg sidecar.
- Section variable adds (g : RiemannianMetric I M) (hg : g = hm.metric) + include g hg - Each theorem signature uses g.X form - Each proof body starts with subst hg, then existing typeclass-form proof unchanged - Internal callers (ricciTensor_eq_sum_inner_orthonormal, smoothOrthoFrame_cov_skew) pass HasMetric.metric rfl - PerSummand callers updated to pass HasMetric.metric rfl Bochner stack body retains typeclass-bound infrastructure (smoothOrthoFrame + LinearMap.trace_eq_sum_inner via InnerProductSpace) — the subst hg bridges between the g-parametric public API and the typeclass-bound proof internals. Caller passes g = hm.metric and rfl to consume.
Same approach as BochnerExpansion: (g, hg : g = hm.metric) section variable + include g hg + subst hg at proof body start. Statement uses g.X; body unchanged (typeclass form). Internal callers within PerSummand pass HasMetric.metric rfl (since after subst hg, g and hg are gone from context). Bochner.lean's bochner_weitzenboeck now passes HasMetric.metric rfl to bochner_connectionLaplacian_grad_decomposition (lifted in PerSummand). Fixed h_id_LCQW/_LCBW statements: HasMetric.metric.metricInner (dot form) to match goal after subst hg.
This was referenced May 19, 2026
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Summary
9i — lift the remaining Bochner stack (
BochnerExpansion.lean+PerSummand.lean) to explicit(g : RiemannianMetric I M)parameters using the subst hg pattern introduced forricci_symmin 9h.Pattern
Each Bochner file now has section variable
Every public theorem signature uses
g.X(g-parametric API). Each proof body starts withsubst hg, then the existing typeclass-form proof closes the (now hm.metric-form) goal.Files
BochnerExpansion.lean: 11 theorems lifted (ricciTensor_eq_sum_inner_orthonormal,metricInner_secondCovDerivAt_grad_swap_of_hess_eventual_sym,hessianBilin_eventually_symm_of_strict_interior,hessianBilin_section_eventually_symm_of_strict_interior,metricInner_secondCovDerivAt_grad_eq_swap_add_curvature,heart_of_bochner_curvature_term,heart_curvature_orthonormal_sum_eq_ricci,sum_hessianBilin_smoothOrthoFrame_eventuallyEq_laplacian,smoothOrthoFrame_cov_skew,sum_hessianBilin_smoothOrthoFrame_cov_eq_zero,sum_inner_secondCovDerivAt_grad_smoothOrthoFrame_of_inner_form)PerSummand.lean: 4 theorems lifted (bochner_per_summand_swap,bochner_per_summand_riemann_form,bochner_per_summand_assembled,bochner_connectionLaplacian_grad_decomposition)Bochner.lean:bochner_weitzenboeckcaller updated to passHasMetric.metric rflto liftedbochner_connectionLaplacian_grad_decomposition.Why subst hg (not full g-parametric body)
Bochner proofs route through
LinearMap.trace_eq_sum_innerover⟪·,·⟫_ℝ(=hm.metric.innerviaInnerProductSpaceinstance from[HasMetric I M]). Without restructuring InnerProductSpace, full body lift requiresg = hm.metricto bridgeg.X↔hm.metric.X.subst hgdoes this bridging at proof start — body works in typeclass form, statement exposes g.End users invoke with
HasMetric.metricandrflforgandhg(typical case where the metric is the ambient typeclass metric).Test plan
lake buildclean