[Merged by Bors] - feat(ring_theory/polynomial/basic): Isomorphism between polynomials over a quotient and quotient over polynomials #3847
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| left_inv := by { | ||
| intro f, | ||
| apply polynomial.induction_on' f, | ||
| { simp_intros p q hp hq, | ||
| rw [hp, hq] }, | ||
| { rintros n ⟨x⟩, | ||
| simp [monomial_eq_smul_X, C_mul'] } | ||
| }, |
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The style guide recommends using begin ... end for any multiline proof, and also to not put braces on their own line.
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Could you fix the formatting suggestion, and then merge? bors d+ |
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…ver a quotient and quotient over polynomials (#3847) The main statement is `polynomial_quotient_equiv_quotient_polynomial`, which gives that If `I` is an ideal of `R`, then the ring polynomials over the quotient ring `I.quotient` is isomorphic to the quotient of `polynomial R` by the ideal `map C I`. Also, `mem_map_C_iff` shows that `map C I` contains exactly the polynomials whose coefficients all lie in `I`
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The main statement is
polynomial_quotient_equiv_quotient_polynomial, which gives that IfIis an ideal ofR, then the ring polynomials over the quotient ringI.quotientis isomorphic to the quotient ofpolynomial Rby the idealmap C I.Also,
mem_map_C_iffshows thatmap C Icontains exactly the polynomials whose coefficients all lie inII couldn't find this equivalence anywhere in mathlib already. I created
quotient_map_C_eq_zeroandeval₂_C_mk_eq_zerojust so they could be passed toquotient.liftas proofs that functions being defined are well defined. If they don't seem generally useful, I can inline them into the definition of the isomorphism, but the definition got very messy when I originally did that.