English
Reindexing a pre-submersive presentation along equivalences e and f yields a jacobiMatrix equal to the reindexed original matrix with a rename: jacobiMatrix(P.reindex e f) = (jacobiMatrix(P)).reindexed by f^{-1} and renamed by e^{-1}.
Русский
Перебиндинг пред-submersive presentation по эквивалентностям e и f даёт jacobiMatrix равен вновь индексированной матрице исходного jacobiMatrix с переименованием: jacobiMatrix(P^{e,f}) = (jacobiMatrix(P))^{f^{-1}}^{rename}_{e^{-1}}.
LaTeX
$$$\\operatorname{jacobiMatrix}(P^{e,f}) = \\big(\\operatorname{jacobiMatrix}(P)\\big)^{\\mathrm{reindex}(f^{-1}, f^{-1})} \\mapsto \\mathrm{rename}(e^{-1}).$$$
Lean4
/-- Given a pre-submersive presentation `P` and equivalences `ι' ≃ ι` and
`σ' ≃ σ`, this is the induced pre-submersive presentation with variables indexed
by `ι` and relations indexed by `κ -/
@[simps toPresentation, simps -isSimp map]
noncomputable def reindex (P : PreSubmersivePresentation R S ι σ) {ι' σ' : Type*} (e : ι' ≃ ι) (f : σ' ≃ σ) :
PreSubmersivePresentation R S ι' σ'
where
__ := P.toPresentation.reindex e f
map := e.symm ∘ P.map ∘ f
map_inj := by
rw [Function.Injective.of_comp_iff e.symm.injective, Function.Injective.of_comp_iff P.map_inj]
exact f.injective
