Retractions with bounded rank on the primal inputs

Lax342547.PrimalRetractions · concepts/Lax342547/PrimalRetractions.lean · lax-342547

proven

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Natural Language Statement

Theorem

When the table together with the primal space spans the nominal space, one can retract onto table plus keys while preserving the primal space. The rank on primal inputs costs only table-primal directions and keys, and does not count the channel dimension.

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Proven claimDefinitionThis conceptRelated conceptA → B: B builds on ADescendants are omitted for concepts with more than 10 descendants.

Evidence

Each proof establishes this claim relative to its assumptions.

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thm✓Lax342547.PrimalRetractions

⊢Lax342547Proofs.PrimalRetractions.bounded_retraction

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1	import Lax342547.FrozenBaselines
2	import Mathlib.LinearAlgebra.FiniteDimensional.Lemmas
3
4	/-!
5	---
6	title: Retractions with bounded rank on the primal inputs
7	type: theorem
8	---
9	When the table together with the primal space spans the nominal space,
10	one can retract onto table plus keys while preserving the primal space.
11	The rank on primal inputs costs only table-primal directions and keys,
12	and does not count the channel dimension.
13	-/
14
15	namespace Lax342547.PrimalRetractions
16
17	open Lax342547.MomentSpace
18
19	axiom bounded_retraction {V : Type} [AddCommGroup V] [Module Binary V]
20	[FiniteDimensional Binary V] (J Keys U : Submodule Binary V)
21	(hspan : J ⊔ U = ⊤) (hKeys : Keys ≤ U) :
22	∃ p : V →ₗ[Binary] V,
23	(∀ v, p v ∈ J ⊔ Keys) ∧ (∀ v ∈ J ⊔ Keys, p v = v) ∧
24	(∀ v ∈ U, p v ∈ U) ∧
25	Module.finrank Binary (LinearMap.range (p.comp U.subtype)) ≤
26	Module.finrank Binary ↥(J ⊓ U) + Module.finrank Binary Keys
27
28	end Lax342547.PrimalRetractions
29