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[Merged by Bors] - refactor(ring_theory/ideal/quotient): extract a ring_con structure
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…, to be used for `ring_quot` too
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eric-wieser
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| section data | ||
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| section add_mul | ||
| variables [has_add R] [has_mul R] (c : ring_con R) | ||
| instance : has_add c.quotient := c.to_add_con.has_add | ||
| @[simp, norm_cast] lemma coe_add (x y : R) : (↑(x + y) : c.quotient) = ↑x + ↑y := rfl | ||
| instance : has_mul c.quotient := c.to_con.has_mul | ||
| @[simp, norm_cast] lemma coe_mul (x y : R) : (↑(x * y) : c.quotient) = ↑x * ↑y := rfl | ||
| end add_mul | ||
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| section zero | ||
| variables [add_zero_class R] [has_mul R] (c : ring_con R) | ||
| instance : has_zero c.quotient := c.to_add_con^.quotient.has_zero | ||
| @[simp, norm_cast] lemma coe_zero : (↑(0 : R) : c.quotient) = 0 := rfl | ||
| end zero | ||
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| section one | ||
| variables [has_add R] [mul_one_class R] (c : ring_con R) | ||
| instance : has_one c.quotient := c.to_con^.quotient.has_one | ||
| @[simp, norm_cast] lemma coe_one : (↑(1 : R) : c.quotient) = 1 := rfl | ||
| end one | ||
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| section neg_sub_zsmul | ||
| variables [add_group R] [has_mul R] (c : ring_con R) | ||
| instance : has_neg c.quotient := c.to_add_con^.has_neg | ||
| @[simp, norm_cast] lemma coe_neg (x : R) : (↑(-x) : c.quotient) = -x := rfl | ||
| instance : has_sub c.quotient := c.to_add_con^.has_sub | ||
| @[simp, norm_cast] lemma coe_sub (x y : R) : (↑(x - y) : c.quotient) = x - y := rfl | ||
| instance has_zsmul : has_smul ℤ c.quotient := c.to_add_con^.quotient.has_zsmul | ||
| @[simp, norm_cast] lemma coe_zsmul (z : ℤ) (x : R) : (↑(z • x) : c.quotient) = z • x := rfl | ||
| end neg_sub_zsmul | ||
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| section nsmul | ||
| variables [add_monoid R] [has_mul R] (c : ring_con R) | ||
| instance has_nsmul : has_smul ℕ c.quotient := c.to_add_con^.quotient.has_nsmul | ||
| @[simp, norm_cast] lemma coe_nsmul (n : ℕ) (x : R) : (↑(n • x) : c.quotient) = n • x := rfl | ||
| end nsmul | ||
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| section pow | ||
| variables [has_add R] [monoid R] (c : ring_con R) | ||
| instance : has_pow c.quotient ℕ := c.to_con^.nat.has_pow | ||
| @[simp, norm_cast] lemma coe_pow (x : R) (n : ℕ) : (↑(x ^ n) : c.quotient) = x ^ n := rfl | ||
| end pow | ||
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| section nat_cast | ||
| variables [add_monoid_with_one R] [has_mul R] (c : ring_con R) | ||
| instance : has_nat_cast c.quotient := ⟨λ n, ↑(n : R)⟩ | ||
| @[simp, norm_cast] lemma coe_nat_cast (n : ℕ) : (↑(n : R) : c.quotient) = n := rfl | ||
| end nat_cast | ||
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| section int_cast | ||
| variables [add_group_with_one R] [has_mul R] (c : ring_con R) | ||
| instance : has_int_cast c.quotient := ⟨λ z, ↑(z : R)⟩ | ||
| @[simp, norm_cast] lemma coe_int_cast (n : ℕ) : (↑(n : R) : c.quotient) = n := rfl | ||
| end int_cast | ||
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| instance [inhabited R] [has_add R] [has_mul R] (c : ring_con R) : inhabited c.quotient := | ||
| ⟨↑(default : R)⟩ | ||
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| end data |
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I know that we'd normally put blank lines here, but there's so little actualy going on (copy an existing instance, write the lemma unfolding it) that it feels like a waste of space.
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ring_con structurering_con structure
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I don't immediately see how this would make it easier to work with ring_quot: it looks like this is basically just a reimplementation of ring_quot that doesn't yet have as many lemmas. Is the point to (nearly) replace all of ring_quot with quotient (ring_con.r _)?
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Co-authored-by: Anne Baanen <[email protected]>
Co-authored-by: Anne Baanen <[email protected]>
The main advantages are that:
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Thanks! bors d+ |
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…17833) The intent here is to remove some duplication with `ring_quot`. I've only copied across the basic lemmas from `group_theory/congruence`; I imagine most of the rest won't be that useful, and that if we do want them, we should look for a means to deduplicate them. Note that `ring_con.quotient` can be used to take quotients of semirings.
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ring_con structurering_con structure
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The intent here is to remove some duplication with
ring_quot.I've only copied across the basic lemmas from
group_theory/congruence; I imagine most of the rest won't be that useful, and that if we do want them, we should look for a means to deduplicate them.Note that
ring_con.quotientcan be used to take quotients of semirings.