feat: nested induced-subgraph graph equality — #2965#2990
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Add nestedSubtypeEquiv and inducedGraph_induce_preimage_map_eq: for S ⊆ T,
((inducedGraph G T).induce {x : ↥T | x.val ∈ S}).map nestedSubtypeEquiv =
inducedGraph G S — inducing box_k inside inducedGraph G box_{k+1} equals, up to
the nested-subtype relabeling {x : ↥T // x.val ∈ S} ≃ ↥S, the direct induced
subgraph on S. An edge survives iff its deep endpoints are G-adjacent and both
in S.
Graph-level foundation for instantiating the per-stage increment
(correlation_pair_sub_inducedGraph_le_derivBound, #2989) on cubic exhaustion
stages box_k ⊆ box_{k+1}. The correlation-level transport across this iso is a
follow-up (it currently hits a whnf elaboration wall in correlation_map_equiv on
the nested-subtype graph, needing a lighter sum-level encoding).
Part of #2965
Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com>
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#2991) * feat: nested-induce correlation transport (Part of #2965) * feat: nested-induce correlation transport (whnf wall bypassed) Add correlation_inducedGraph_induce_preimage: correlation companion of inducedGraph_induce_preimage_map_eq (#2990): correlation ((inducedGraph G T).induce {x | x.val ∈ S}) p (A.map nestedSubtypeEquiv.symm) = correlation (inducedGraph G S) p A. The earlier whnf wall (correlation_map_equiv on the nested-subtype graph) is bypassed by applying correlation_map_equiv to the DIRECT graph inducedGraph G S with the inverse relabeling nestedSubtypeEquiv.symm — so the heavy nested-subtype graph appears only as the map RESULT, never as the graph correlation_map_equiv operates on — then bridging via correlation_congr_all and the graph equality #2990. This is the c_k-side identity for the cubic-exhaustion per-stage increment: correlation on inducedGraph (box_k inside box_{k+1}) = correlation on the direct inducedGraph box_k. Part of #2965 Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> * docs: fix stale follow-up comment per codex (correlation transport now done) Co-Authored-By: Claude Opus 4.7 <noreply@anthropic.com> --------- Co-authored-by: Claude Opus 4.7 <noreply@anthropic.com>
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Part of #2965 (cubic-exhaustion connection, graph-level foundation).
Adds
nestedSubtypeEquivandinducedGraph_induce_preimage_map_eq(InducedUnion.lean): forS ⊆ T,((inducedGraph G T).induce {x : ↥T | x.val ∈ S}).map nestedSubtypeEquiv = inducedGraph G S.Inducing
box_kinsideinducedGraph G box_{k+1}equals, up to the nested-subtype relabeling{x : ↥T // x.val ∈ S} ≃ ↥S, the direct induced subgraph onS. Proved by adjacency: an edge survives iff its deep endpoints areG-adjacent and both inS.Graph-level foundation for instantiating the per-stage increment (#2989) on cubic exhaustion stages. The correlation-level transport across this iso is a documented follow-up (it hits a
whnfelaboration wall incorrelation_map_equivon the nested-subtype graph; needs a lighter sum-level encoding — see #2965).lake build IsingModel.AmbientLatticeSum.InducedUniongreen, sorry 0, file linter-clean, tex compiles.🤖 Generated with Claude Code