CoveringSystem.lean`

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4 declarations, 0 contain sorry, 0 declared as axioms

Imports (3)

Imported by 1 file

FormalConjecturesForMathlib.lean

structure CoveringSystem A covering system of a semiring R is a finite set of cosets of non-zero ideals whose union covers the whole ring line 31 source
Source documentation
A covering system of a semiring R is a finite set of cosets of non-zero ideals whose union covers the whole ring. In the case where R = ℤ, this corresponds to a choice of finitely many congruence relations such that every integer satisfies at least one of the relations.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring
```
structure CoveringSystem (R : Type*) [CommSemiring R] where
…
```
structure StrictCoveringSystem We say a covering system is strict if all the congruence relations that define it are take modulo a different ideal (or number) line 47 source
Source documentation
We say a covering system is strict if all the congruence relations that define it are take modulo a different ideal (or number).
Note: this corresponds to the notion of a covering system that Erdos was using in [ErGr80] Erdős, P. and Graham, R., Old and new problems and results in combinatorial number theory. Monographies de L'Enseignement Mathematique (1980).

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring
```
structure StrictCoveringSystem (R : Type*) [CommSemiring R] extends CoveringSystem R where
…
```
Mentions: CoveringSystem
def CoveringSystem.coset line 54 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, CoveringSystem, CoveringSystem.ι, Set
Proof uses: HAdd.hAdd, Singleton.singleton, CoveringSystem.residue, Ideal, CoveringSystem.moduli
10 instances, coercions and auxiliary constants left out.
theorem CoveringSystem.iUnion_cosets line 58 source
standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: CommSemiring, CoveringSystem, Eq, Set, Set.iUnion, CoveringSystem.ι, CoveringSystem.coset, Set.univ
Proof uses: CoveringSystem.unionCovers