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 |
|---|---|---|---|---|---|---|
of a type `D` [equipped](foundation.structure.md) with a [subcounting](univalent-combinatorics.subcounting.md), i.e., an [embedding](foundation-core.embeddings.md) `D ↪ Fin n`, and a [surjection](foundation.surjective-maps.md) `D ↠ X`. Note that the subcounting of `D` is _proof-relevant_, and hence having a subfinite i... | data | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.distance-natural-numbers",
"open import elementary-number-theory.maximum-natural-numbers",
"open import elementary-number-theory.minimum-natural-numbers",
"open import elementary-number-theory.natural-numb... | src/univalent-combinatorics/subfinite-indexing.lagda.md | of | |
subfinite-indexing : {l1 : Level} (l2 : Level) → UU l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.distance-natural-numbers",
"open import elementary-number-theory.maximum-natural-numbers",
"open import elementary-number-theory.minimum-natural-numbers",
"open import elementary-number-theory.natural-numb... | src/univalent-combinatorics/subfinite-indexing.lagda.md | subfinite-indexing | |
subfinite-indexing-subcount : {l : Level} {X : UU l} → subcount X → subfinite-indexing l X | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.distance-natural-numbers",
"open import elementary-number-theory.maximum-natural-numbers",
"open import elementary-number-theory.minimum-natural-numbers",
"open import elementary-number-theory.natural-numb... | src/univalent-combinatorics/subfinite-indexing.lagda.md | subfinite-indexing-subcount | |
answer : <https://mathoverflow.net/a/433318>. | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.distance-natural-numbers",
"open import elementary-number-theory.maximum-natural-numbers",
"open import elementary-number-theory.minimum-natural-numbers",
"open import elementary-number-theory.natural-numb... | src/univalent-combinatorics/subfinite-indexing.lagda.md | answer | |
is-subfinite-Prop : {l : Level} → UU l → Prop l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.discrete-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.injective-maps"... | src/univalent-combinatorics/subfinite-types.lagda.md | is-subfinite-Prop | |
is-subfinite : {l : Level} → UU l → UU l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.discrete-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.injective-maps"... | src/univalent-combinatorics/subfinite-types.lagda.md | is-subfinite | |
is-prop-is-subfinite : {l : Level} {X : UU l} → is-prop (is-subfinite X) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.discrete-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.injective-maps"... | src/univalent-combinatorics/subfinite-types.lagda.md | is-prop-is-subfinite | |
Subfinite-Type : (l : Level) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.discrete-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.injective-maps"... | src/univalent-combinatorics/subfinite-types.lagda.md | Subfinite-Type | |
Fin-Subfinite-Type : (n : ℕ) → Subfinite-Type lzero | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.discrete-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.injective-maps"... | src/univalent-combinatorics/subfinite-types.lagda.md | Fin-Subfinite-Type | |
answer : <https://mathoverflow.net/a/433318>. | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.discrete-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.injective-maps"... | src/univalent-combinatorics/subfinite-types.lagda.md | answer | |
is-subfinitely-enumerable-Prop : {l1 : Level} (l2 : Level) → UU l1 → Prop (l1 ⊔ lsuc l2) | function | src | [
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.existential-quantification",
"open import foundation.functoriality-propositional-truncation",
"open import found... | src/univalent-combinatorics/subfinitely-enumerable-types.lagda.md | is-subfinitely-enumerable-Prop | |
is-subfinitely-enumerable : {l1 : Level} (l2 : Level) → UU l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.existential-quantification",
"open import foundation.functoriality-propositional-truncation",
"open import found... | src/univalent-combinatorics/subfinitely-enumerable-types.lagda.md | is-subfinitely-enumerable | |
is-prop-is-subfinitely-enumerable : {l1 l2 : Level} {X : UU l1} → is-prop (is-subfinitely-enumerable l2 X) | function | src | [
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.existential-quantification",
"open import foundation.functoriality-propositional-truncation",
"open import found... | src/univalent-combinatorics/subfinitely-enumerable-types.lagda.md | is-prop-is-subfinitely-enumerable | |
Subfinitely-Enumerable-Type : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | function | src | [
"open import foundation.decidable-equality",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.equivalences",
"open import foundation.existential-quantification",
"open import foundation.functoriality-propositional-truncation",
"open import found... | src/univalent-combinatorics/subfinitely-enumerable-types.lagda.md | Subfinitely-Enumerable-Type | |
Surjection-Finite-Type : {l1 : Level} (l2 : Level) → Finite-Type l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import foundation.surjective-maps public",
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.decidable-embeddings",
"open import foundation.decidable-equality",
"open import foundation.decidable-types",
"open import fo... | src/univalent-combinatorics/surjective-maps.lagda.md | Surjection-Finite-Type | |
type-trivial-Σ-Decomposition-Finite-Type : {l1 l2 l3 : Level} (A : Finite-Type l1) → UU (l1 ⊔ lsuc l2 ⊔ lsuc l3) | function | src | [
"open import foundation.trivial-sigma-decompositions public",
"open import foundation.contractible-types",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.inhabited-types",
"open import foundation.propositions",
"open import foundation.subt... | src/univalent-combinatorics/trivial-sigma-decompositions.lagda.md | type-trivial-Σ-Decomposition-Finite-Type | |
inductively : - A π₀-finite type is a [finite type](univalent-combinatorics.finite-types.md). | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | inductively | |
is-truncated-π-finite-Prop : {l : Level} (k : ℕ) → UU l → Prop l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-Prop | |
is-truncated-π-finite : {l : Level} (k : ℕ) → UU l → UU l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite | |
is-prop-is-truncated-π-finite : {l : Level} (k : ℕ) {A : UU l} → is-prop (is-truncated-π-finite k A) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-prop-is-truncated-π-finite | |
Truncated-π-Finite-Type : (l : Level) (k : ℕ) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | Truncated-π-Finite-Type | |
is-truncated-π-finite-empty : (k : ℕ) → is-truncated-π-finite k empty | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-empty | |
empty-Truncated-π-Finite-Type : (k : ℕ) → Truncated-π-Finite-Type lzero k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | empty-Truncated-π-Finite-Type | |
is-truncated-π-finite-is-empty : {l : Level} (k : ℕ) {A : UU l} → is-empty A → is-truncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-is-empty | |
is-truncated-π-finite-is-contr : {l : Level} (k : ℕ) {A : UU l} → is-contr A → is-truncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-is-contr | |
is-truncated-π-finite-unit : (k : ℕ) → is-truncated-π-finite k unit | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-unit | |
unit-Truncated-π-Finite-Type : (k : ℕ) → Truncated-π-Finite-Type lzero k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | unit-Truncated-π-Finite-Type | |
is-truncated-π-finite-Fin : (k n : ℕ) → is-truncated-π-finite k (Fin n) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-Fin | |
Fin-Truncated-π-Finite-Type : (k : ℕ) (n : ℕ) → Truncated-π-Finite-Type lzero k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | Fin-Truncated-π-Finite-Type | |
is-truncated-π-finite-count : {l : Level} (k : ℕ) {A : UU l} → count A → is-truncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-count | |
is-truncated-π-finite-is-finite : {l : Level} (k : ℕ) {A : UU l} → is-finite A → is-truncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-is-finite | |
truncated-π-finite-type-Finite-Type : {l : Level} (k : ℕ) → Finite-Type l → Truncated-π-Finite-Type l k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | truncated-π-finite-type-Finite-Type | |
is-truncated-π-finite-Type-With-Cardinality-ℕ : {l : Level} (n : ℕ) → is-truncated-π-finite 1 (Type-With-Cardinality-ℕ l n) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | is-truncated-π-finite-Type-With-Cardinality-ℕ | |
Type-With-Cardinality-ℕ-Truncated-π-Finite-Type : (l : Level) (n : ℕ) → Truncated-π-Finite-Type (lsuc l) 1 | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-function-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.e... | src/univalent-combinatorics/truncated-pi-finite-types.lagda.md | Type-With-Cardinality-ℕ-Truncated-π-Finite-Type | |
Slice-Surjection-Finite-Type : (l : Level) {l1 : Level} (A : Finite-Type l1) → UU (lsuc l ⊔ l1) | function | src | [
"open import foundation.type-duality public",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.full-subtypes",
"open import foundation.functoriality-dependent-function-types",
"open import foun... | src/univalent-combinatorics/type-duality.lagda.md | Slice-Surjection-Finite-Type | |
is-unbounded-π-finite {l : Level} (X : UU l) : UU l where coinductive field has-finitely-many-connected-components-is-unbounded-π-finite : has-finitely-many-connected-components X is-unbounded-π-finite-Id-is-unbounded-π-finite : (x y : X) → is-unbounded-π-finite (x ＝ y) | record | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite | |
Unbounded-π-Finite-Type : (l : Level) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | Unbounded-π-Finite-Type | |
is-unbounded-π-finite-empty : is-unbounded-π-finite empty | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-empty | |
empty-Unbounded-π-Finite-Type : Unbounded-π-Finite-Type lzero | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | empty-Unbounded-π-Finite-Type | |
is-unbounded-π-finite-is-empty : {l : Level} {A : UU l} → is-empty A → is-unbounded-π-finite A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-is-empty | |
is-unbounded-π-finite-is-contr : {l : Level} {A : UU l} → is-contr A → is-unbounded-π-finite A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-is-contr | |
is-unbounded-π-finite-unit : is-unbounded-π-finite unit | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-unit | |
unit-Unbounded-π-Finite-Type : Unbounded-π-Finite-Type lzero | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | unit-Unbounded-π-Finite-Type | |
Maybe-Unbounded-π-Finite-Type : {l : Level} → Unbounded-π-Finite-Type l → Unbounded-π-Finite-Type l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | Maybe-Unbounded-π-Finite-Type | |
is-unbounded-π-finite-Fin : (n : ℕ) → is-unbounded-π-finite (Fin n) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-Fin | |
Fin-Unbounded-π-Finite-Type : (n : ℕ) → Unbounded-π-Finite-Type lzero | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | Fin-Unbounded-π-Finite-Type | |
is-unbounded-π-finite-count : {l : Level} {A : UU l} → count A → is-unbounded-π-finite A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-count | |
is-unbounded-π-finite-is-finite : {l : Level} {A : UU l} → is-finite A → is-unbounded-π-finite A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-is-finite | |
unbounded-π-finite-type-Finite-Type : {l : Level} → Finite-Type l → Unbounded-π-Finite-Type l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | unbounded-π-finite-type-Finite-Type | |
is-unbounded-π-finite-Type-With-Cardinality-ℕ : {l : Level} (n : ℕ) → is-unbounded-π-finite (Type-With-Cardinality-ℕ l n) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-unbounded-π-finite-Type-With-Cardinality-ℕ | |
is-finite-is-unbounded-π-finite : {l : Level} {A : UU l} → is-set A → is-unbounded-π-finite A → is-finite A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.empty-types",
"open import foundation.eq... | src/univalent-combinatorics/unbounded-pi-finite-types.lagda.md | is-finite-is-unbounded-π-finite | |
iterated-product : {l : Level} (n : ℕ) (A : Fin n → UU l) → UU l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.f... | src/univalent-combinatorics/universal-property-standard-finite-types.lagda.md | iterated-product | |
is-untruncated-π-finite-Prop : {l : Level} (k : ℕ) → UU l → Prop l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-Prop | |
is-untruncated-π-finite : {l : Level} (k : ℕ) → UU l → UU l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite | |
is-prop-is-untruncated-π-finite : {l : Level} (k : ℕ) (X : UU l) → is-prop (is-untruncated-π-finite k X) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-prop-is-untruncated-π-finite | |
Untruncated-π-Finite-Type : (l : Level) (k : ℕ) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | Untruncated-π-Finite-Type | |
type-Untruncated-π-Finite-Type : {l : Level} (k : ℕ) → Untruncated-π-Finite-Type l k → UU l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | type-Untruncated-π-Finite-Type | |
is-untruncated-π-finite-empty : (k : ℕ) → is-untruncated-π-finite k empty | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-empty | |
empty-Untruncated-π-Finite-Type : (k : ℕ) → Untruncated-π-Finite-Type lzero k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | empty-Untruncated-π-Finite-Type | |
is-untruncated-π-finite-is-empty : {l : Level} (k : ℕ) {A : UU l} → is-empty A → is-untruncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-is-empty | |
is-untruncated-π-finite-is-contr : {l : Level} (k : ℕ) {A : UU l} → is-contr A → is-untruncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-is-contr | |
is-untruncated-π-finite-unit : (k : ℕ) → is-untruncated-π-finite k unit | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-unit | |
unit-Untruncated-π-Finite-Type : (k : ℕ) → Untruncated-π-Finite-Type lzero k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | unit-Untruncated-π-Finite-Type | |
is-untruncated-π-finite-Fin : (k n : ℕ) → is-untruncated-π-finite k (Fin n) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-Fin | |
Fin-Untruncated-π-Finite-Type : (k : ℕ) (n : ℕ) → Untruncated-π-Finite-Type lzero k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | Fin-Untruncated-π-Finite-Type | |
is-untruncated-π-finite-count : {l : Level} (k : ℕ) {A : UU l} → count A → is-untruncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-count | |
is-untruncated-π-finite-is-finite : {l : Level} (k : ℕ) {A : UU l} → is-finite A → is-untruncated-π-finite k A | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | is-untruncated-π-finite-is-finite | |
untruncated-π-finite-type-Finite-Type : {l : Level} (k : ℕ) → Finite-Type l → Untruncated-π-Finite-Type l k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | untruncated-π-finite-type-Finite-Type | |
Type-With-Cardinality-ℕ-Untruncated-π-Finite-Type : (l : Level) (k n : ℕ) → Untruncated-π-Finite-Type (lsuc l) k | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.0-connected-types",
"open import foundation.action-on-identifications-functions",
"open import foundation.contractible-types",
"open import foundation.coproduct-types",
"open import foundation.decidable-propositions",
"open ... | src/univalent-combinatorics/untruncated-pi-finite-types.lagda.md | Type-With-Cardinality-ℕ-Untruncated-π-Finite-Type | |
Algebraic-Theory : {l1 : Level} (l2 : Level) → signature l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import foundation.dependent-pair-types",
"open import foundation.universe-levels",
"open import universal-algebra.abstract-equations-over-signatures",
"open import universal-algebra.signatures"
] | src/universal-algebra/algebraic-theories.lagda.md | Algebraic-Theory | |
group-ops : UU lzero where unit-group-op mul-group-op inv-group-op : group-ops group-signature : signature lzero pr1 group-signature = group-ops pr2 group-signature unit-group-op = 0 pr2 group-signature mul-group-op = 2 pr2 group-signature inv-group-op = 1 | data | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | group-ops | |
group-laws : UU lzero where associative-l-group-laws : group-laws invl-l-group-laws : group-laws invr-r-group-laws : group-laws idl-l-group-laws : group-laws idr-r-group-laws : group-laws algebraic-theory-Group : Algebraic-Theory lzero group-signature pr1 algebraic-theory-Group = group-laws pr2 algebraic-theory-Group =... | data | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | group-laws | |
group-signature : signature lzero | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | group-signature | |
algebraic-theory-Group : Algebraic-Theory lzero group-signature | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | algebraic-theory-Group | |
algebra-Group : (l : Level) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | algebra-Group | |
group-algebra-Group : {l : Level} → algebra-Group l → Group l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | group-algebra-Group | |
algebra-group-Group : {l : Level} → Group l → algebra-Group l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.equality-dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-extensionality",
"open import foundation.identity-types",
"open import fo... | src/universal-algebra/algebraic-theory-of-groups.lagda.md | algebra-group-Group | |
is-model-of-signature-type : {l1 l2 : Level} → signature l1 → UU l2 → UU (l1 ⊔ l2) | function | src | [
"open import foundation.action-on-identifications-functions",
"open import foundation.dependent-pair-types",
"open import foundation.function-extensionality",
"open import foundation.sets",
"open import foundation.structure-identity-principle",
"open import foundation.universe-levels",
"open import foun... | src/universal-algebra/models-of-signatures.lagda.md | is-model-of-signature-type | |
is-model-of-signature : {l1 l2 : Level} → signature l1 → Set l2 → UU (l1 ⊔ l2) | function | src | [
"open import foundation.action-on-identifications-functions",
"open import foundation.dependent-pair-types",
"open import foundation.function-extensionality",
"open import foundation.sets",
"open import foundation.structure-identity-principle",
"open import foundation.universe-levels",
"open import foun... | src/universal-algebra/models-of-signatures.lagda.md | is-model-of-signature | |
Model-Of-Signature : {l1 : Level} (l2 : Level) → signature l1 → UU (l1 ⊔ lsuc l2) | function | src | [
"open import foundation.action-on-identifications-functions",
"open import foundation.dependent-pair-types",
"open import foundation.function-extensionality",
"open import foundation.sets",
"open import foundation.structure-identity-principle",
"open import foundation.universe-levels",
"open import foun... | src/universal-algebra/models-of-signatures.lagda.md | Model-Of-Signature | |
signature : (l : Level) → UU (lsuc l) | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.identity-types",
"open import foundation.universe-levels"
] | src/universal-algebra/signatures.lagda.md | signature | |
operation-signature : {l : Level} → signature l → UU l | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.identity-types",
"open import foundation.universe-levels"
] | src/universal-algebra/signatures.lagda.md | operation-signature | |
arity-operation-signature : {l : Level} (σ : signature l) → operation-signature σ → ℕ | function | src | [
"open import elementary-number-theory.natural-numbers",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.embeddings",
"open import foundation.identity-types",
"open import foundation.universe-levels"
] | src/universal-algebra/signatures.lagda.md | arity-operation-signature | |
is-coinductive-iso-Noncoherent-Large-ω-Precategory {α : Level → Level} {β : Level → Level → Level} (𝒞 : Noncoherent-Large-ω-Precategory α β) {l1 : Level} {x : obj-Noncoherent-Large-ω-Precategory 𝒞 l1} {l2 : Level} {y : obj-Noncoherent-Large-ω-Precategory 𝒞 l2} (f : hom-Noncoherent-Large-ω-Precategory 𝒞 x y) : UU (β... | record | src | [
"open import foundation.dependent-pair-types",
"open import foundation.universe-levels",
"open import wild-category-theory.coinductive-isomorphisms-in-noncoherent-omega-precategories",
"open import wild-category-theory.noncoherent-large-omega-precategories"
] | src/wild-category-theory/coinductive-isomorphisms-in-noncoherent-large-omega-precategories.lagda.md | is-coinductive-iso-Noncoherent-Large-ω-Precategory | |
is-coinductive-iso-Noncoherent-ω-Precategory {l1 l2 : Level} (𝒞 : Noncoherent-ω-Precategory l1 l2) {x y : obj-Noncoherent-ω-Precategory 𝒞} (f : hom-Noncoherent-ω-Precategory 𝒞 x y) : UU l2 where coinductive field hom-section-is-coinductive-iso-Noncoherent-ω-Precategory : hom-Noncoherent-ω-Precategory 𝒞 y x unit-is-... | record | src | [
"open import foundation.dependent-pair-types",
"open import foundation.universe-levels",
"open import wild-category-theory.noncoherent-omega-precategories"
] | src/wild-category-theory/coinductive-isomorphisms-in-noncoherent-omega-precategories.lagda.md | is-coinductive-iso-Noncoherent-ω-Precategory | |
is-colax-functor-Noncoherent-Large-ω-Precategory {α1 α2 : Level → Level} {β1 β2 : Level → Level → Level} {γ : Level → Level} (𝒜 : Noncoherent-Large-ω-Precategory α1 β1) (ℬ : Noncoherent-Large-ω-Precategory α2 β2) (F : map-Noncoherent-Large-ω-Precategory γ 𝒜 ℬ) : UUω where constructor make-is-colax-functor-Noncoherent... | record | src | [
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.identity-types",
"open import foundation.universe-levels",
"open import globular-types.globular-maps",
"open import globular-types.globular-types",
"open import globular-types.large-colax-refl... | src/wild-category-theory/colax-functors-noncoherent-large-omega-precategories.lagda.md | is-colax-functor-Noncoherent-Large-ω-Precategory | |
colax-functor-Noncoherent-Large-ω-Precategory {α1 α2 : Level → Level} {β1 β2 : Level → Level → Level} (δ : Level → Level) (𝒜 : Noncoherent-Large-ω-Precategory α1 β1) (ℬ : Noncoherent-Large-ω-Precategory α2 β2) : UUω where constructor make-colax-functor-Noncoherent-Large-ω-Precategory ``` The underlying large globular ... | record | src | [
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.identity-types",
"open import foundation.universe-levels",
"open import globular-types.globular-maps",
"open import globular-types.globular-types",
"open import globular-types.large-colax-refl... | src/wild-category-theory/colax-functors-noncoherent-large-omega-precategories.lagda.md | colax-functor-Noncoherent-Large-ω-Precategory | |
is-colax-functor-Noncoherent-ω-Precategory {l1 l2 l3 l4 : Level} (𝒜 : Noncoherent-ω-Precategory l1 l2) (ℬ : Noncoherent-ω-Precategory l3 l4) (F : map-Noncoherent-ω-Precategory 𝒜 ℬ) : UU (l1 ⊔ l2 ⊔ l4) where constructor make-is-colax-functor-Noncoherent-ω-Precategory coinductive field is-reflexive-is-colax-functor-Non... | record | src | [
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.identity-types",
"open import foundation.universe-levels",
"open import globular-types.colax-reflexive-globular-maps",
"open import globular-types.colax-transitive-globular-maps",
"open import... | src/wild-category-theory/colax-functors-noncoherent-omega-precategories.lagda.md | is-colax-functor-Noncoherent-ω-Precategory | |
Noncoherent-Large-ω-Precategory (α : Level → Level) (β : Level → Level → Level) : UUω where ``` The underlying large globular type of a noncoherent large wild precategory: ```agda field large-globular-type-Noncoherent-Large-ω-Precategory : Large-Globular-Type α β ``` The type of objects of a noncoherent large ω-precate... | record | src | [
"open import category-theory.precategories",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.homotopies",
"open import foundation.identity-types",
"open import foundation.... | src/wild-category-theory/noncoherent-large-omega-precategories.lagda.md | Noncoherent-Large-ω-Precategory | |
Noncoherent-ω-Precategory : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | function | src | [
"open import category-theory.precategories",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.cartesian-product-types",
"open import foundation.dependent-pair-types",
"open import foundation.function-types",
"open import foundation.homotopies",
"open import fo... | src/wild-category-theory/noncoherent-omega-precategories.lagda.md | Noncoherent-ω-Precategory |
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