ExampleConstruction: AI-generatedVerification: AI-adaptedPipeline-generatedprecheck passjudge pass (gpt-6-luna)

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Non-strided, subgroup, and coset pooling on p4 features

Example

Let $G = P_{C_{4}}$ be the p4 group and let $H = {(0, ρ^{j}) : j = 0, 1, 2, 3}$ be its four-element rotation subgroup. For a scalar group feature map $F : G \to R$ , max pooling with $U = {(0, I), (0, ρ)}$ and no subsampling remains $G$ -equivariant. Restricting $F$ to $H$ retains only $H$ -equivariance. Left-coset pooling $Q_{H} F (g H) = \max_{h \in H} F (g h)$ has one value for each translation position: $(t, ρ^{r}) H = {(t, ρ^{r + j}) : j = 0, 1, 2, 3}$ , so the four orientation indices are pooled while the result remains equivariant on $G / H$ . In the separate finite case $H = G$ , $G / H$ has one element and the pooled value is invariant.

Example references

pooling and subgroup operations

the p4 group and its rotation subgroup

subgroup identity and closure conditions

pooling and subgroup equivariance conclusions

Cohen and Welling, “Group Equivariant Convolutional Networks,” §6.3 and Appendix A

Goodfellow, Bengio, and Courville, Deep Learning, Chapter 9 §9.3

Facts & Assumptions

Given: The p4 group, its rotation subgroup, and the pooling operators in the Example. Verify their equivariance and the quotient indexing.

[F1]

Pointwise nonlinearities, neighborhood pooling, subgroup sampling, and coset pooling: Non-strided pooling is $(P_{U} F) (g) = \max_{u \in U} F (g u)$ , subgroup sampling is restriction, and left-coset pooling is $(Q_{H} F) (g H) = \max_{h \in H} F (g h)$ .

[F2]

The p4 and p4m square-lattice symmetry groups: p4 elements are pairs $(t, ρ^{r})$ with product $(t, ρ^{r}) (s, ρ^{q}) = (t + ρ^{r} s, ρ^{r + q})$ .

[F3]

Group cross-correlation and compatible layers are equivariant: Non-strided pooling commutes with the left action, subgroup restriction preserves the subgroup action, and finite coset pooling is equivariant on the coset set.

[F4]

Subgroup: A subgroup contains the identity and is closed under the group operation and inverses.

Verification

1.1givenF1F3

The set $U$ is finite and nonempty, so the non-strided max pool is defined; [F3] gives $P_{U} (L_{a} F) = L_{a} (P_{U} F)$ for every $a \in G$ .

1.2givenF2F3F4algebra

The p4 multiplication in [F2] gives $(0, I) \in H$ , $(0, ρ^{j}) (0, ρ^{k}) = (0, ρ^{j + k}) \in H$ , and $(0, ρ^{j})^{- 1} = (0, ρ^{- j}) \in H$ , so [F4] confirms $H \leq G$ . For $a, h \in H$ , $a^{- 1} h \in H$ , and restricting $L_{a} F$ to $H$ equals applying the left $H$ -action to $F ∣_{H}$ . The theorem therefore retains exactly the subgroup equivariance claimed after restriction.

1.3F2algebra

By the p4 pair product in [F2], $(t, ρ^{r}) H = {(t, ρ^{r + j}) : j = 0, 1, 2, 3}$ , so each left coset fixes $t$ and includes all four rotations.

2.1givenF1F3step 1.3algebra∎

The coset-pooling theorem [F3] makes $Q_{H} F$ equivariant on $G / H$ . If $H = G$ is finite, there is one coset, and the induced left action on this singleton is the identity, so the pooled value is invariant.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Taco S. Cohen and Max Welling, Group Equivariant Convolutional Networks (standard reference, not scraped)
Ian Goodfellow, Yoshua Bengio, and Aaron Courville, Deep Learning (standard reference, not scraped)