fact stringlengths 9 34.3k | type stringclasses 3
values | library stringclasses 2
values | imports listlengths 0 227 | filename stringlengths 22 99 | symbolic_name stringlengths 1 57 | docstring stringclasses 1
value |
|---|---|---|---|---|---|---|
is-category-representing-arrow-Category : is-category-Precategory representing-arrow-Precategory | function | src | [
"open import category-theory.categories",
"open import category-theory.isomorphisms-in-precategories",
"open import category-theory.precategories",
"open import foundation.booleans",
"open import foundation.decidable-propositions",
"open import foundation.dependent-pair-types",
"open import foundation.e... | src/category-theory/representing-arrow-category.lagda.md | is-category-representing-arrow-Category | |
representing-arrow-Category : Category lzero lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.isomorphisms-in-precategories",
"open import category-theory.precategories",
"open import foundation.booleans",
"open import foundation.decidable-propositions",
"open import foundation.dependent-pair-types",
"open import foundation.e... | src/category-theory/representing-arrow-category.lagda.md | representing-arrow-Category | |
extensions : the identity map gives a right extension (with the identity natural | function | src | [
"open import category-theory.functors-precategories",
"open import category-theory.natural-transformations-functors-precategories",
"open import category-theory.precategories",
"open import foundation.action-on-equivalences-functions",
"open import foundation.action-on-identifications-functions",
"open im... | src/category-theory/right-extensions-precategories.lagda.md | extensions | |
rigid-obj-Category : {l1 l2 : Level} (C : Category l1 l2) → UU (l1 ⊔ l2) | function | src | [
"open import category-theory.categories",
"open import category-theory.rigid-objects-precategories",
"open import foundation.propositions",
"open import foundation.universe-levels"
] | src/category-theory/rigid-objects-categories.lagda.md | rigid-obj-Category | |
rigid-obj-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2) | function | src | [
"open import category-theory.isomorphisms-in-precategories",
"open import category-theory.precategories",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.universe-levels"
] | src/category-theory/rigid-objects-precategories.lagda.md | rigid-obj-Precategory | |
Set-Magmoid : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.truncated-types",
"open import f... | src/category-theory/set-magmoids.lagda.md | Set-Magmoid | |
total-hom-Set-Magmoid : {l1 l2 : Level} (M : Set-Magmoid l1 l2) → UU (l1 ⊔ l2) | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.truncated-types",
"open import f... | src/category-theory/set-magmoids.lagda.md | total-hom-Set-Magmoid | |
obj-simplex-Category : UU lzero | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.precategories",
"open import elementary-number-theory.inequality-standard-finite-types",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"ope... | src/category-theory/simplex-category.lagda.md | obj-simplex-Category | |
hom-set-simplex-Category : obj-simplex-Category → obj-simplex-Category → Set lzero | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.precategories",
"open import elementary-number-theory.inequality-standard-finite-types",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"ope... | src/category-theory/simplex-category.lagda.md | hom-set-simplex-Category | |
hom-simplex-Category : obj-simplex-Category → obj-simplex-Category → UU lzero | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.precategories",
"open import elementary-number-theory.inequality-standard-finite-types",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"ope... | src/category-theory/simplex-category.lagda.md | hom-simplex-Category | |
associative-composition-operation-simplex-Category : associative-composition-operation-binary-family-Set hom-set-simplex-Category | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.precategories",
"open import elementary-number-theory.inequality-standard-finite-types",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"ope... | src/category-theory/simplex-category.lagda.md | associative-composition-operation-simplex-Category | |
id-hom-simplex-Category : (n : obj-simplex-Category) → hom-simplex-Category n n | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.precategories",
"open import elementary-number-theory.inequality-standard-finite-types",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"ope... | src/category-theory/simplex-category.lagda.md | id-hom-simplex-Category | |
simplex-Precategory : Precategory lzero lzero | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.precategories",
"open import elementary-number-theory.inequality-standard-finite-types",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"ope... | src/category-theory/simplex-category.lagda.md | simplex-Precategory | |
Strict-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | function | src | [
"open import category-theory.categories",
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.isomorphisms-in-precategories",
"open import category-theory.nonunital-precategories",
"open import category-theory.precategories",
"open import category-... | src/category-theory/strict-categories.lagda.md | Strict-Category | |
Strongly-Preunivalent-Category : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.isomorphisms-in-precategories",
"open import category-theory.precategories",
"open import category-theory.preunivalent-categories",
"open import foundation.1-types",
"open import foundation.cartes... | src/category-theory/strongly-preunivalent-categories.lagda.md | Strongly-Preunivalent-Category | |
total-hom-Strongly-Preunivalent-Category : {l1 l2 : Level} (𝒞 : Strongly-Preunivalent-Category l1 l2) → UU (l1 ⊔ l2) | function | src | [
"open import category-theory.composition-operations-on-binary-families-of-sets",
"open import category-theory.isomorphisms-in-precategories",
"open import category-theory.precategories",
"open import category-theory.preunivalent-categories",
"open import foundation.1-types",
"open import foundation.cartes... | src/category-theory/strongly-preunivalent-categories.lagda.md | total-hom-Strongly-Preunivalent-Category | |
obj-terminal-Category : UU lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | obj-terminal-Category | |
hom-set-terminal-Category : obj-terminal-Category → obj-terminal-Category → Set lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | hom-set-terminal-Category | |
hom-terminal-Category : obj-terminal-Category → obj-terminal-Category → UU lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | hom-terminal-Category | |
id-hom-terminal-Category : {x : obj-terminal-Category} → hom-terminal-Category x x | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | id-hom-terminal-Category | |
terminal-Precategory : Precategory lzero lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | terminal-Precategory | |
is-category-terminal-Category : is-category-Precategory terminal-Precategory | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | is-category-terminal-Category | |
terminal-Category : Category lzero lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | terminal-Category | |
is-strongly-preunivalent-terminal-Category : is-strongly-preunivalent-Precategory terminal-Precategory | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | is-strongly-preunivalent-terminal-Category | |
terminal-Strongly-Preunivalent-Category : Strongly-Preunivalent-Category lzero lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | terminal-Strongly-Preunivalent-Category | |
is-strict-category-terminal-Category : is-strict-category-Precategory terminal-Precategory | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | is-strict-category-terminal-Category | |
terminal-Strict-Category : Strict-Category lzero lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | terminal-Strict-Category | |
is-gaunt-terminal-Category : is-gaunt-Category terminal-Category | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | is-gaunt-terminal-Category | |
terminal-Gaunt-Category : Gaunt-Category lzero lzero | function | src | [
"open import category-theory.categories",
"open import category-theory.constant-functors",
"open import category-theory.functors-categories",
"open import category-theory.functors-precategories",
"open import category-theory.gaunt-categories",
"open import category-theory.isomorphisms-in-categories",
"o... | src/category-theory/terminal-category.lagda.md | terminal-Gaunt-Category | |
is-terminal-obj-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → obj-Precategory C → UU (l1 ⊔ l2) | function | src | [
"open import category-theory.precategories",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.identity-types",
"open import foundation.universe-levels"
] | src/category-theory/terminal-objects-precategories.lagda.md | is-terminal-obj-Precategory | |
terminal-obj-Precategory : {l1 l2 : Level} (C : Precategory l1 l2) → UU (l1 ⊔ l2) | function | src | [
"open import category-theory.precategories",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.identity-types",
"open import foundation.universe-levels"
] | src/category-theory/terminal-objects-precategories.lagda.md | terminal-obj-Precategory | |
is-boolean-Commutative-Ring : {l : Level} (A : Commutative-Ring l) → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.universe-levels",
"open import ring-theory.idempotent-elements-rings"
] | src/commutative-algebra/boolean-rings.lagda.md | is-boolean-Commutative-Ring | |
Boolean-Ring : (l : Level) → UU (lsuc l) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.universe-levels",
"open import ring-theory.idempotent-elements-rings"
] | src/commutative-algebra/boolean-rings.lagda.md | Boolean-Ring | |
is-large-category-Commutative-Ring-Large-Category : is-large-category-Large-Precategory Commutative-Ring-Large-Precategory | function | src | [
"open import category-theory.categories",
"open import category-theory.large-categories",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import commutative-algebra.precategory-of-commutative-rings",
"open import foundation.universe-levels"
] | src/commutative-algebra/category-of-commutative-rings.lagda.md | is-large-category-Commutative-Ring-Large-Category | |
Commutative-Ring-Large-Category : Large-Category lsuc (_⊔_) | function | src | [
"open import category-theory.categories",
"open import category-theory.large-categories",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import commutative-algebra.precategory-of-commutative-rings",
"open import foundation.universe-levels"
] | src/commutative-algebra/category-of-commutative-rings.lagda.md | Commutative-Ring-Large-Category | |
Commutative-Ring-Category : (l : Level) → Category (lsuc l) l | function | src | [
"open import category-theory.categories",
"open import category-theory.large-categories",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import commutative-algebra.precategory-of-commutative-rings",
"open import foundation.universe-levels"
] | src/commutative-algebra/category-of-commutative-rings.lagda.md | Commutative-Ring-Category | |
Commutative-Ring : ( l : Level) → UU (lsuc l) | function | src | [
"open import commutative-algebra.commutative-semirings",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",... | src/commutative-algebra/commutative-rings.lagda.md | Commutative-Ring | |
Commutative-Semiring : ( l : Level) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.interchange-law",
"open import foundation.iterated-dependent-product-type... | src/commutative-algebra/commutative-semirings.lagda.md | Commutative-Semiring | |
is-discrete-field-Commutative-Ring : {l : Level} → Commutative-Ring l → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.universe-levels",
"open import ring-theory.division-rings"
] | src/commutative-algebra/discrete-fields.lagda.md | is-discrete-field-Commutative-Ring | |
is-euclidean-domain-Integral-Domain : { l : Level} (R : Integral-Domain l) → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.integral-domains",
"open import commutative-algebra.trivial-commutative-rings",
"open import elementary-number-theory.addition-natural-numbers",
"open import eleme... | src/commutative-algebra/euclidean-domains.lagda.md | is-euclidean-domain-Integral-Domain | |
Euclidean-Domain : (l : Level) → UU (lsuc l) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.integral-domains",
"open import commutative-algebra.trivial-commutative-rings",
"open import elementary-number-theory.addition-natural-numbers",
"open import eleme... | src/commutative-algebra/euclidean-domains.lagda.md | Euclidean-Domain | |
formal-power-series-Commutative-Ring : {l : Level} (R : Commutative-Ring l) → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.convolution-sequences-commutative-rings",
"open import commutative-algebra.formal-power-series-commutative-semirings",
"open import commutative-algebra.function-commutative-rings",
"open import commutative-algebra.homomorp... | src/commutative-algebra/formal-power-series-commutative-rings.lagda.md | formal-power-series-Commutative-Ring | |
formal-power-series-Commutative-Semiring {l : Level} (R : Commutative-Semiring l) : UU l where constructor formal-power-series-coefficients-Commutative-Semiring field coefficient-formal-power-series-Commutative-Semiring : sequence (type-Commutative-Semiring R) | record | src | [
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.convolution-sequences-commutative-semirings",
"open import commutative-algebra.function-commutative-semirings",
"open import commutative-algebra.homomorphisms-commutative-semirings",
"open import commutative-algebra.po... | src/commutative-algebra/formal-power-series-commutative-semirings.lagda.md | formal-power-series-Commutative-Semiring | |
is-heyting-field-prop-Local-Commutative-Ring : {l : Level} → Local-Commutative-Ring l → Prop l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.local-commutative-rings",
"open import commutative-algebra.trivial-commutative-rings",
"open import foundation.apartness-relations",
"open import f... | src/commutative-algebra/heyting-fields.lagda.md | is-heyting-field-prop-Local-Commutative-Ring | |
is-heyting-field-Local-Commutative-Ring : {l : Level} → Local-Commutative-Ring l → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.local-commutative-rings",
"open import commutative-algebra.trivial-commutative-rings",
"open import foundation.apartness-relations",
"open import f... | src/commutative-algebra/heyting-fields.lagda.md | is-heyting-field-Local-Commutative-Ring | |
Heyting-Field : (l : Level) → UU (lsuc l) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.local-commutative-rings",
"open import commutative-algebra.trivial-commutative-rings",
"open import foundation.apartness-relations",
"open import f... | src/commutative-algebra/heyting-fields.lagda.md | Heyting-Field | |
ideal-Commutative-Ring : {l1 : Level} (l2 : Level) → Commutative-Ring l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.powers-of-elements-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foun... | src/commutative-algebra/ideals-commutative-rings.lagda.md | ideal-Commutative-Ring | |
left-ideal-Commutative-Ring : {l1 : Level} (l2 : Level) → Commutative-Ring l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.powers-of-elements-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foun... | src/commutative-algebra/ideals-commutative-rings.lagda.md | left-ideal-Commutative-Ring | |
right-ideal-Commutative-Ring : {l1 : Level} (l2 : Level) → Commutative-Ring l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.powers-of-elements-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foun... | src/commutative-algebra/ideals-commutative-rings.lagda.md | right-ideal-Commutative-Ring | |
ideal-Commutative-Semiring : {l1 : Level} (l2 : Level) → Commutative-Semiring l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.subsets-commutative-semirings",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.propositions",
"open import foundation.universe-levels",
"open import ... | src/commutative-algebra/ideals-commutative-semirings.lagda.md | ideal-Commutative-Semiring | |
cancellation-property-Commutative-Ring : {l : Level} (R : Commutative-Ring l) → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.trivial-commutative-rings",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import f... | src/commutative-algebra/integral-domains.lagda.md | cancellation-property-Commutative-Ring | |
Integral-Domain : (l : Level) → UU (lsuc l) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.trivial-commutative-rings",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import f... | src/commutative-algebra/integral-domains.lagda.md | Integral-Domain | |
iso-eq-Commutative-Ring : { l : Level} (A B : Commutative-Ring l) → A ＝ B → iso-Commutative-Ring A B | function | src | [
"open import category-theory.isomorphisms-in-large-precategories",
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.homomorphisms-commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.precategory-of-... | src/commutative-algebra/isomorphisms-commutative-rings.lagda.md | iso-eq-Commutative-Ring | |
Large-Commutative-Ring (α : Level → Level) (β : Level → Level → Level) : UUω where constructor make-Large-Commutative-Ring field large-ring-Large-Commutative-Ring : Large-Ring α β type-Large-Commutative-Ring : (l : Level) → UU (α l) type-Large-Commutative-Ring = type-Large-Ring large-ring-Large-Commutative-Ring set-Lar... | record | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.homomorphisms-commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.large-binary-relations",
"open import foundation.large-similarity-relation... | src/commutative-algebra/large-commutative-rings.lagda.md | Large-Commutative-Ring | |
is-local-prop-Commutative-Ring : {l : Level} (A : Commutative-Ring l) → Prop l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.universe-levels",
"open import ring-theory.local-rings",
"open import ring-theory.rings"
] | src/commutative-algebra/local-commutative-rings.lagda.md | is-local-prop-Commutative-Ring | |
is-local-Commutative-Ring : {l : Level} → Commutative-Ring l → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.universe-levels",
"open import ring-theory.local-rings",
"open import ring-theory.rings"
] | src/commutative-algebra/local-commutative-rings.lagda.md | is-local-Commutative-Ring | |
is-prop-is-local-Commutative-Ring : {l : Level} (A : Commutative-Ring l) → is-prop (is-local-Commutative-Ring A) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.universe-levels",
"open import ring-theory.local-rings",
"open import ring-theory.rings"
] | src/commutative-algebra/local-commutative-rings.lagda.md | is-prop-is-local-Commutative-Ring | |
Local-Commutative-Ring : (l : Level) → UU (lsuc l) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.dependent-pair-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.universe-levels",
"open import ring-theory.local-rings",
"open import ring-theory.rings"
] | src/commutative-algebra/local-commutative-rings.lagda.md | Local-Commutative-Ring | |
subset-nilradical-Commutative-Ring : {l : Level} (A : Commutative-Ring l) → subset-Commutative-Ring l A | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.ideals-commutative-rings",
"open import commutative-algebra.prime-ideals-commutative-rings",
"open import commutative-algebra.radical-ideals-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
... | src/commutative-algebra/nilradical-commutative-rings.lagda.md | subset-nilradical-Commutative-Ring | |
nilradical-Commutative-Ring : {l : Level} (A : Commutative-Ring l) → ideal-Commutative-Ring l A | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.ideals-commutative-rings",
"open import commutative-algebra.prime-ideals-commutative-rings",
"open import commutative-algebra.radical-ideals-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
... | src/commutative-algebra/nilradical-commutative-rings.lagda.md | nilradical-Commutative-Ring | |
is-in-nilradical-Commutative-Ring : {l : Level} (R : Commutative-Ring l) → type-Commutative-Ring R → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.ideals-commutative-rings",
"open import commutative-algebra.prime-ideals-commutative-rings",
"open import commutative-algebra.radical-ideals-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
... | src/commutative-algebra/nilradical-commutative-rings.lagda.md | is-in-nilradical-Commutative-Ring | |
subset-nilradical-Commutative-Semiring : {l : Level} (A : Commutative-Semiring l) → subset-Commutative-Semiring l A | function | src | [
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.subsets-commutative-semirings",
"open import foundation.existential-quantification",
"open import foundation.identity-types",
"open import foundation.universe-levels",
"open import ring-theory.nilpotent-elements-semi... | src/commutative-algebra/nilradicals-commutative-semirings.lagda.md | subset-nilradical-Commutative-Semiring | |
polynomial-Commutative-Semiring : {l : Level} → (R : Commutative-Semiring l) → UU l | function | src | [
"open import commutative-algebra.commutative-semirings",
"open import commutative-algebra.formal-power-series-commutative-semirings",
"open import commutative-algebra.homomorphisms-commutative-semirings",
"open import commutative-algebra.powers-of-elements-commutative-semirings",
"open import commutative-al... | src/commutative-algebra/polynomials-commutative-semirings.lagda.md | polynomial-Commutative-Semiring | |
Commutative-Ring-Full-Large-Subprecategory : Full-Large-Subprecategory (λ l → l) Ring-Large-Precategory | function | src | [
"open import category-theory.full-large-subprecategories",
"open import category-theory.large-precategories",
"open import category-theory.precategories",
"open import commutative-algebra.commutative-rings",
"open import foundation.universe-levels",
"open import ring-theory.precategory-of-rings"
] | src/commutative-algebra/precategory-of-commutative-rings.lagda.md | Commutative-Ring-Full-Large-Subprecategory | |
Commutative-Ring-Large-Precategory : Large-Precategory lsuc (_⊔_) | function | src | [
"open import category-theory.full-large-subprecategories",
"open import category-theory.large-precategories",
"open import category-theory.precategories",
"open import commutative-algebra.commutative-rings",
"open import foundation.universe-levels",
"open import ring-theory.precategory-of-rings"
] | src/commutative-algebra/precategory-of-commutative-rings.lagda.md | Commutative-Ring-Large-Precategory | |
Commutative-Ring-Precategory : (l : Level) → Precategory (lsuc l) l | function | src | [
"open import category-theory.full-large-subprecategories",
"open import category-theory.large-precategories",
"open import category-theory.precategories",
"open import commutative-algebra.commutative-rings",
"open import foundation.universe-levels",
"open import ring-theory.precategory-of-rings"
] | src/commutative-algebra/precategory-of-commutative-rings.lagda.md | Commutative-Ring-Precategory | |
Commutative-Semiring-Full-Large-Precategory : Full-Large-Subprecategory (λ l → l) Semiring-Large-Precategory | function | src | [
"open import category-theory.full-large-subprecategories",
"open import category-theory.large-precategories",
"open import category-theory.precategories",
"open import commutative-algebra.commutative-semirings",
"open import foundation.universe-levels",
"open import ring-theory.precategory-of-semirings"
] | src/commutative-algebra/precategory-of-commutative-semirings.lagda.md | Commutative-Semiring-Full-Large-Precategory | |
Commutative-Semiring-Large-Precategory : Large-Precategory lsuc (_⊔_) | function | src | [
"open import category-theory.full-large-subprecategories",
"open import category-theory.large-precategories",
"open import category-theory.precategories",
"open import commutative-algebra.commutative-semirings",
"open import foundation.universe-levels",
"open import ring-theory.precategory-of-semirings"
] | src/commutative-algebra/precategory-of-commutative-semirings.lagda.md | Commutative-Semiring-Large-Precategory | |
Commutative-Semiring-Precategory : (l : Level) → Precategory (lsuc l) l | function | src | [
"open import category-theory.full-large-subprecategories",
"open import category-theory.large-precategories",
"open import category-theory.precategories",
"open import commutative-algebra.commutative-semirings",
"open import foundation.universe-levels",
"open import ring-theory.precategory-of-semirings"
] | src/commutative-algebra/precategory-of-commutative-semirings.lagda.md | Commutative-Semiring-Precategory | |
prime-ideal-Commutative-Ring : {l1 : Level} (l2 : Level) → Commutative-Ring l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.full-ideals-commutative-rings",
"open import commutative-algebra.ideals-commutative-rings",
"open import commutative-algebra.powers-of-elements-commutative-rings",
"open import commutative-algebra.radical-ideals-commutativ... | src/commutative-algebra/prime-ideals-commutative-rings.lagda.md | prime-ideal-Commutative-Ring | |
radical-ideal-Commutative-Ring : {l1 : Level} (l2 : Level) → Commutative-Ring l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.ideals-commutative-rings",
"open import commutative-algebra.powers-of-elements-commutative-rings",
"open import commutative-algebra.subsets-commutative-rings",
"open import elementary-number-theory.natural-numbers",
"ope... | src/commutative-algebra/radical-ideals-commutative-rings.lagda.md | radical-ideal-Commutative-Ring | |
subset-Commutative-Ring : (l : Level) {l1 : Level} (A : Commutative-Ring l1) → UU (lsuc l ⊔ l1) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import foundation.identity-types",
"open import foundation.propositional-extensionality",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.subtypes",
"open import foundation.universe-levels",
"open ... | src/commutative-algebra/subsets-commutative-rings.lagda.md | subset-Commutative-Ring | |
subset-Commutative-Semiring : (l : Level) {l1 : Level} (A : Commutative-Semiring l1) → UU (lsuc l ⊔ l1) | function | src | [
"open import commutative-algebra.commutative-semirings",
"open import foundation.identity-types",
"open import foundation.propositions",
"open import foundation.sets",
"open import foundation.subtypes",
"open import foundation.universe-levels",
"open import ring-theory.subsets-semirings"
] | src/commutative-algebra/subsets-commutative-semirings.lagda.md | subset-Commutative-Semiring | |
is-trivial-commutative-ring-Prop : {l : Level} → Commutative-Ring l → Prop l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | is-trivial-commutative-ring-Prop | |
is-trivial-Commutative-Ring : {l : Level} → Commutative-Ring l → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | is-trivial-Commutative-Ring | |
is-nontrivial-commutative-ring-Prop : {l : Level} → Commutative-Ring l → Prop l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | is-nontrivial-commutative-ring-Prop | |
is-nontrivial-Commutative-Ring : {l : Level} → Commutative-Ring l → UU l | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | is-nontrivial-Commutative-Ring | |
trivial-Ring : Ring lzero | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | trivial-Ring | |
is-commutative-trivial-Ring : is-commutative-Ring trivial-Ring | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | is-commutative-trivial-Ring | |
trivial-Commutative-Ring : Commutative-Ring lzero | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | trivial-Commutative-Ring | |
is-trivial-trivial-Commutative-Ring : is-trivial-Commutative-Ring trivial-Commutative-Ring | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | is-trivial-trivial-Commutative-Ring | |
To-do : complete proof of uniqueness of the zero ring using SIP, ideally refactor | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import commutative-algebra.isomorphisms-commutative-rings",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundatio... | src/commutative-algebra/trivial-commutative-rings.lagda.md | To-do | |
add-ℂ : {l1 l2 : Level} → ℂ l1 → ℂ l2 → ℂ (l1 ⊔ l2) | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.dependent-pair-types",
"open import foundation... | src/complex-numbers/addition-complex-numbers.lagda.md | add-ℂ | |
large-apartness-relation-ℂ : Large-Apartness-Relation _⊔_ ℂ | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.apartness-relations",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.empty-types",
"open import foundation.functi... | src/complex-numbers/apartness-complex-numbers.lagda.md | large-apartness-relation-ℂ | |
apartness-relation-ℂ : (l : Level) → Apartness-Relation l (ℂ l) | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.apartness-relations",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.empty-types",
"open import foundation.functi... | src/complex-numbers/apartness-complex-numbers.lagda.md | apartness-relation-ℂ | |
tight-apartness-relation-ℂ : (l : Level) → Tight-Apartness-Relation l (ℂ l) | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.apartness-relations",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.empty-types",
"open import foundation.functi... | src/complex-numbers/apartness-complex-numbers.lagda.md | tight-apartness-relation-ℂ | |
ℂ : (l : Level) → UU (lsuc l) | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | ℂ | |
re-ℂ : {l : Level} → ℂ l → ℝ l | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | re-ℂ | |
im-ℂ : {l : Level} → ℂ l → ℝ l | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | im-ℂ | |
ℂ-Set : (l : Level) → Set (lsuc l) | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | ℂ-Set | |
eq-ℂ : {l : Level} → {a b c d : ℝ l} → a ＝ c → b ＝ d → (a , b) ＝ (c , d) | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | eq-ℂ | |
complex-ℝ : {l : Level} → ℝ l → ℂ l | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | complex-ℝ | |
im-complex-ℝ : {l : Level} → ℝ l → ℂ l | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | im-complex-ℝ | |
zero-ℂ : ℂ lzero | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | zero-ℂ | |
one-ℂ : ℂ lzero | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | one-ℂ | |
neg-one-ℂ : ℂ lzero | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | neg-one-ℂ | |
i-ℂ : ℂ lzero | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | i-ℂ | |
neg-ℂ : {l : Level} → ℂ l → ℂ l | function | src | [
"open import complex-numbers.gaussian-integers",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equality-cartesian-product-types",
"open import foundation.identity-types"... | src/complex-numbers/complex-numbers.lagda.md | neg-ℂ | |
conjugate-ℂ : {l : Level} → ℂ l → ℂ l | function | src | [
"open import complex-numbers.complex-numbers",
"open import foundation.identity-types",
"open import foundation.universe-levels",
"open import real-numbers.negation-real-numbers"
] | src/complex-numbers/conjugation-complex-numbers.lagda.md | conjugate-ℂ | |
eisenstein-int-ℤ : ℤ → ℤ[ω] | function | src | [
"open import commutative-algebra.commutative-rings",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"open import foundation.action-on-identifications-binary-functions",
"open import ... | src/complex-numbers/eisenstein-integers.lagda.md | eisenstein-int-ℤ |
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