chore: generalize more materials about linear independence over semirings #20497
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PR summary ebfaf15931
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| File | Base Count | Head Count | Change |
|---|---|---|---|
| Mathlib.RingTheory.AlgebraTower | 886 | 885 | -1 (-0.11%) |
Import changes for all files
| Files | Import difference |
|---|---|
4 filesMathlib.RingTheory.AlgebraTower Mathlib.LinearAlgebra.Matrix.ToLin Mathlib.Data.Complex.Module Mathlib.RingTheory.TensorProduct.Free |
-1 |
Declarations diff
+ Finsupp.linearCombination_one_tmul
+ linearCombination_smul
+ linearIndependent_one_tmul
You can run this locally as follows
## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>
## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>The doc-module for script/declarations_diff.sh contains some details about this script.
No changes to technical debt.
You can run this locally as
./scripts/technical-debt-metrics.sh pr_summary
- The
relativevalue is the weighted sum of the differences with weight given by the inverse of the current value of the statistic. - The
absolutevalue is therelativevalue divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).
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This PR/issue depends on:
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alreadydone
commented
Jan 6, 2025
Comment on lines
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| theorem LinearIndependent.of_isLocalizedModule {ι : Type*} {v : ι → M} | ||
| (hv : LinearIndependent R v) : LinearIndependent Rₛ (f ∘ v) := by | ||
| rw [linearIndependent_iff'] at hv ⊢ | ||
| intro t g hg i hi | ||
| choose! a g' hg' using IsLocalization.exist_integer_multiples S t g | ||
| have h0 : f (∑ i ∈ t, g' i • v i) = 0 := by | ||
| apply_fun ((a : R) • ·) at hg | ||
| rw [smul_zero, Finset.smul_sum] at hg | ||
| rw [map_sum, ← hg] | ||
| refine Finset.sum_congr rfl fun i hi => ?_ | ||
| rw [← smul_assoc, ← hg' i hi, map_smul, Function.comp_apply, algebraMap_smul] | ||
| obtain ⟨s, hs⟩ := (IsLocalizedModule.eq_zero_iff S f).mp h0 | ||
| simp_rw [Finset.smul_sum, Submonoid.smul_def, smul_smul] at hs | ||
| specialize hv t _ hs i hi | ||
| rw [← (IsLocalization.map_units Rₛ a).mul_right_eq_zero, ← Algebra.smul_def, ← hg' i hi] | ||
| exact (IsLocalization.map_eq_zero_iff S _ _).2 ⟨s, hv⟩ | ||
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| variable [Module Rₛ M] [IsScalarTower R Rₛ M] | ||
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| theorem LinearIndependent.localization {ι : Type*} {b : ι → M} (hli : LinearIndependent R b) : | ||
| rw [linearIndependent_iff'ₛ] at hv ⊢ | ||
| intro t g₁ g₂ eq i hi | ||
| choose! a fg hfg using IsLocalization.exist_integer_multiples S (t.disjSum t) (Sum.elim g₁ g₂) | ||
| simp_rw [Sum.forall, Finset.inl_mem_disjSum, Sum.elim_inl, Finset.inr_mem_disjSum, Sum.elim_inr, | ||
| Subtype.forall'] at hfg | ||
| apply_fun ((a : R) • ·) at eq | ||
| simp_rw [← t.sum_coe_sort, Finset.smul_sum, ← smul_assoc, ← hfg, | ||
| algebraMap_smul, Function.comp_def, ← map_smul, ← map_sum, | ||
| t.sum_coe_sort (f := fun x ↦ fg (Sum.inl x) • v x), | ||
| t.sum_coe_sort (f := fun x ↦ fg (Sum.inr x) • v x)] at eq | ||
| have ⟨s, eq⟩ := IsLocalizedModule.exists_of_eq (S := S) eq | ||
| simp_rw [Finset.smul_sum, Submonoid.smul_def, smul_smul] at eq | ||
| have := congr(algebraMap R Rₛ $(hv t _ _ eq i hi)) | ||
| simpa only [map_mul, (IsLocalization.map_units Rₛ s).mul_right_inj, hfg.1 ⟨i, hi⟩, hfg.2 ⟨i, hi⟩, | ||
| Algebra.smul_def, (IsLocalization.map_units Rₛ a).mul_right_inj] using this |
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This proof can be golfed using that localizations are flat, but that would mean hundreds more transitive imports ...
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Also add
Finsupp.linearCombination_one_tmulandlinearIndependent_one_tmulthat connects linear independence to flatness.