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 |
|---|---|---|---|---|---|---|
heyting-field-ℂ : (l : Level) → Heyting-Field (lsuc l) | function | src | [
"open import commutative-algebra.heyting-fields",
"open import commutative-algebra.homomorphisms-heyting-fields",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import co... | src/complex-numbers/field-of-complex-numbers.lagda.md | heyting-field-ℂ | |
hom-heyting-field-complex-ℝ : (l : Level) → hom-Heyting-Field (heyting-field-ℝ l) (heyting-field-ℂ l) | function | src | [
"open import commutative-algebra.heyting-fields",
"open import commutative-algebra.homomorphisms-heyting-fields",
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import co... | src/complex-numbers/field-of-complex-numbers.lagda.md | hom-heyting-field-complex-ℝ | |
gaussian-int-ℤ : ℤ → ℤ[i] | function | src | [
"open import commutative-algebra.commutative-rings",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.difference-integers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"open import foundatio... | src/complex-numbers/gaussian-integers.lagda.md | gaussian-int-ℤ | |
large-semigroup-add-ℂ : Large-Semigroup lsuc | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | large-semigroup-add-ℂ | |
large-monoid-add-ℂ : Large-Monoid lsuc (_⊔_) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | large-monoid-add-ℂ | |
large-commutative-monoid-add-ℂ : Large-Commutative-Monoid lsuc (_⊔_) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | large-commutative-monoid-add-ℂ | |
large-group-add-ℂ : Large-Group lsuc (_⊔_) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | large-group-add-ℂ | |
large-ab-add-ℂ : Large-Ab lsuc (_⊔_) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | large-ab-add-ℂ | |
ab-add-ℂ : (l : Level) → Ab (lsuc l) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | ab-add-ℂ | |
hom-add-ab-complex-ℝ : (l : Level) → hom-Ab (ab-add-ℝ l) (ab-add-ℂ l) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.raising-universe-levels-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.fun... | src/complex-numbers/large-additive-group-of-complex-numbers.lagda.md | hom-add-ab-complex-ℝ | |
large-ring-ℂ : Large-Ring lsuc (_⊔_) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.homomorphisms-commutative-rings",
"open import commutative-algebra.large-commutative-rings",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.large-additive-group-of-complex-numbers",
"open imp... | src/complex-numbers/large-ring-of-complex-numbers.lagda.md | large-ring-ℂ | |
large-commutative-ring-ℂ : Large-Commutative-Ring lsuc (_⊔_) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.homomorphisms-commutative-rings",
"open import commutative-algebra.large-commutative-rings",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.large-additive-group-of-complex-numbers",
"open imp... | src/complex-numbers/large-ring-of-complex-numbers.lagda.md | large-commutative-ring-ℂ | |
commutative-ring-ℂ : (l : Level) → Commutative-Ring (lsuc l) | function | src | [
"open import commutative-algebra.commutative-rings",
"open import commutative-algebra.homomorphisms-commutative-rings",
"open import commutative-algebra.large-commutative-rings",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.large-additive-group-of-complex-numbers",
"open imp... | src/complex-numbers/large-ring-of-complex-numbers.lagda.md | commutative-ring-ℂ | |
is-local-commutative-ring-ℂ : (l : Level) → is-local-Commutative-Ring (commutative-ring-ℂ l) | function | src | [
"open import commutative-algebra.local-commutative-rings",
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.addition-nonzero-complex-numbers",
"open import complex-numbers.large-ring-of-complex-numbers",
"open import complex-numbers.multiplicative-inverses-nonzero-complex... | src/complex-numbers/local-ring-of-complex-numbers.lagda.md | is-local-commutative-ring-ℂ | |
local-commutative-ring-ℂ : (l : Level) → Local-Commutative-Ring (lsuc l) | function | src | [
"open import commutative-algebra.local-commutative-rings",
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.addition-nonzero-complex-numbers",
"open import complex-numbers.large-ring-of-complex-numbers",
"open import complex-numbers.multiplicative-inverses-nonzero-complex... | src/complex-numbers/local-ring-of-complex-numbers.lagda.md | local-commutative-ring-ℂ | |
nonnegative-squared-magnitude-ℂ : {l : Level} → ℂ l → ℝ⁰⁺ l | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.conjugation-complex-numbers",
"open import complex-numbers.multiplication-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.action-on-identifications-functions",
"open import found... | src/complex-numbers/magnitude-complex-numbers.lagda.md | nonnegative-squared-magnitude-ℂ | |
squared-magnitude-ℂ : {l : Level} → ℂ l → ℝ l | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.conjugation-complex-numbers",
"open import complex-numbers.multiplication-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.action-on-identifications-functions",
"open import found... | src/complex-numbers/magnitude-complex-numbers.lagda.md | squared-magnitude-ℂ | |
nonnegative-magnitude-ℂ : {l : Level} → ℂ l → ℝ⁰⁺ l | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.conjugation-complex-numbers",
"open import complex-numbers.multiplication-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.action-on-identifications-functions",
"open import found... | src/complex-numbers/magnitude-complex-numbers.lagda.md | nonnegative-magnitude-ℂ | |
magnitude-ℂ : {l : Level} → ℂ l → ℝ l | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.conjugation-complex-numbers",
"open import complex-numbers.multiplication-complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.action-on-identifications-functions",
"open import found... | src/complex-numbers/magnitude-complex-numbers.lagda.md | magnitude-ℂ | |
mul-ℂ : {l1 l2 : Level} → ℂ l1 → ℂ l2 → ℂ (l1 ⊔ l2) | function | src | [
"open import complex-numbers.addition-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import founda... | src/complex-numbers/multiplication-complex-numbers.lagda.md | mul-ℂ | |
complex-inv-nonzero-ℂ : {l : Level} (z : nonzero-ℂ l) → ℂ l | function | src | [
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.conjugation-complex-numbers",
"open import complex-numbers.large-ring-of-complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open imp... | src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md | complex-inv-nonzero-ℂ | |
inv-nonzero-ℂ : {l : Level} → nonzero-ℂ l → nonzero-ℂ l | function | src | [
"open import commutative-algebra.invertible-elements-commutative-rings",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.conjugation-complex-numbers",
"open import complex-numbers.large-ring-of-complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open imp... | src/complex-numbers/multiplicative-inverses-nonzero-complex-numbers.lagda.md | inv-nonzero-ℂ | |
is-nonzero-prop-ℂ : {l : Level} → ℂ l → Prop l | function | src | [
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.propositions",
"open import founda... | src/complex-numbers/nonzero-complex-numbers.lagda.md | is-nonzero-prop-ℂ | |
is-nonzero-ℂ : {l : Level} → ℂ l → UU l | function | src | [
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.propositions",
"open import founda... | src/complex-numbers/nonzero-complex-numbers.lagda.md | is-nonzero-ℂ | |
nonzero-ℂ : (l : Level) → UU (lsuc l) | function | src | [
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.propositions",
"open import founda... | src/complex-numbers/nonzero-complex-numbers.lagda.md | nonzero-ℂ | |
complex-nonzero-ℂ : {l : Level} → nonzero-ℂ l → ℂ l | function | src | [
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.propositions",
"open import founda... | src/complex-numbers/nonzero-complex-numbers.lagda.md | complex-nonzero-ℂ | |
positive-squared-magnitude-nonzero-ℂ : {l : Level} (z : nonzero-ℂ l) → ℝ⁺ l | function | src | [
"open import complex-numbers.apartness-complex-numbers",
"open import complex-numbers.complex-numbers",
"open import complex-numbers.magnitude-complex-numbers",
"open import foundation.dependent-pair-types",
"open import foundation.disjunction",
"open import foundation.propositions",
"open import founda... | src/complex-numbers/nonzero-complex-numbers.lagda.md | positive-squared-magnitude-nonzero-ℂ | |
raise-ℂ : {l1 : Level} (l2 : Level) → ℂ l1 → ℂ (l1 ⊔ l2) | function | src | [
"open import complex-numbers.complex-numbers",
"open import complex-numbers.similarity-complex-numbers",
"open import foundation.action-on-identifications-functions",
"open import foundation.dependent-pair-types",
"open import foundation.negated-equality",
"open import foundation.universe-levels",
"open... | src/complex-numbers/raising-universe-levels-complex-numbers.lagda.md | raise-ℂ | |
sim-prop-ℂ : {l1 l2 : Level} → ℂ l1 → ℂ l2 → Prop (l1 ⊔ l2) | function | src | [
"open import complex-numbers.complex-numbers",
"open import foundation.conjunction",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.large-equivalence-relations",
"open import foundation.large-similarity-relations",
"open import foundation.... | src/complex-numbers/similarity-complex-numbers.lagda.md | sim-prop-ℂ | |
sim-ℂ : {l1 l2 : Level} → ℂ l1 → ℂ l2 → UU (l1 ⊔ l2) | function | src | [
"open import complex-numbers.complex-numbers",
"open import foundation.conjunction",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.large-equivalence-relations",
"open import foundation.large-similarity-relations",
"open import foundation.... | src/complex-numbers/similarity-complex-numbers.lagda.md | sim-ℂ | |
large-equivalence-relation-sim-ℂ : Large-Equivalence-Relation (_⊔_) ℂ | function | src | [
"open import complex-numbers.complex-numbers",
"open import foundation.conjunction",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.large-equivalence-relations",
"open import foundation.large-similarity-relations",
"open import foundation.... | src/complex-numbers/similarity-complex-numbers.lagda.md | large-equivalence-relation-sim-ℂ | |
large-similarity-relation-ℂ : Large-Similarity-Relation (_⊔_) ℂ | function | src | [
"open import complex-numbers.complex-numbers",
"open import foundation.conjunction",
"open import foundation.dependent-pair-types",
"open import foundation.identity-types",
"open import foundation.large-equivalence-relations",
"open import foundation.large-similarity-relations",
"open import foundation.... | src/complex-numbers/similarity-complex-numbers.lagda.md | large-similarity-relation-ℂ | |
Directed-Complete-Poset : (l1 l2 l3 : Level) → UU (lsuc l1 ⊔ lsuc l2 ⊔ lsuc l3) | function | src | [
"open import domain-theory.directed-families-posets",
"open import foundation.binary-relations",
"open import foundation.dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.function-types",
"open import foundation.logical-equivalences",
"open import foundation.propositi... | src/domain-theory/directed-complete-posets.lagda.md | Directed-Complete-Poset | |
directed-family-Poset : {l1 l2 : Level} (l3 : Level) → Poset l1 l2 → UU (l1 ⊔ l2 ⊔ lsuc l3) | function | src | [
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"open import foundation.conjunction",
"open import foundation.dependent-pair-types",
"open import foundation.equivalences",
"open import foundation.existential-quantification",
"open import fo... | src/domain-theory/directed-families-posets.lagda.md | directed-family-Poset | |
ω-Complete-Poset : (l1 l2 : Level) → UU (lsuc l1 ⊔ lsuc l2) | function | src | [
"open import elementary-number-theory.decidable-total-order-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.binary-relations",
"open import foundation.dependent-pair-types",
"open import fo... | src/domain-theory/omega-complete-posets.lagda.md | ω-Complete-Poset | |
max-abs-closed-interval-ℚ : closed-interval-ℚ → ℚ⁰⁺ | function | src | [
"open import elementary-number-theory.absolute-value-rational-numbers",
"open import elementary-number-theory.closed-intervals-rational-numbers",
"open import elementary-number-theory.inequality-nonnegative-rational-numbers",
"open import elementary-number-theory.inequality-rational-numbers",
"open import e... | src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md | max-abs-closed-interval-ℚ | |
rational-max-abs-closed-interval-ℚ : closed-interval-ℚ → ℚ | function | src | [
"open import elementary-number-theory.absolute-value-rational-numbers",
"open import elementary-number-theory.closed-intervals-rational-numbers",
"open import elementary-number-theory.inequality-nonnegative-rational-numbers",
"open import elementary-number-theory.inequality-rational-numbers",
"open import e... | src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md | rational-max-abs-closed-interval-ℚ | |
abs-ℤ : ℤ → ℕ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | abs-ℤ | |
int-abs-ℤ : ℤ → ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | int-abs-ℤ | |
abs-int-ℕ : (n : ℕ) → abs-ℤ (int-ℕ n) ＝ n | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | abs-int-ℕ | |
abs-neg-ℤ : (x : ℤ) → abs-ℤ (neg-ℤ x) ＝ abs-ℤ x | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | abs-neg-ℤ | |
int-abs-is-nonnegative-ℤ : (x : ℤ) → is-nonnegative-ℤ x → int-abs-ℤ x ＝ x | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | int-abs-is-nonnegative-ℤ | |
eq-abs-ℤ : (x : ℤ) → is-zero-ℕ (abs-ℤ x) → is-zero-ℤ x | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | eq-abs-ℤ | |
abs-eq-ℤ : (x : ℤ) → is-zero-ℤ x → is-zero-ℕ (abs-ℤ x) | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | abs-eq-ℤ | |
predecessor-law-abs-ℤ : (x : ℤ) → (abs-ℤ (pred-ℤ x)) ≤-ℕ (succ-ℕ (abs-ℤ x)) | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | predecessor-law-abs-ℤ | |
successor-law-abs-ℤ : (x : ℤ) → (abs-ℤ (succ-ℤ x)) ≤-ℕ (succ-ℕ (abs-ℤ x)) | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | successor-law-abs-ℤ | |
subadditive-abs-ℤ : (x y : ℤ) → (abs-ℤ (x +ℤ y)) ≤-ℕ ((abs-ℤ x) +ℕ (abs-ℤ y)) | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | subadditive-abs-ℤ | |
negative-law-abs-ℤ : (x : ℤ) → abs-ℤ (neg-ℤ x) ＝ abs-ℤ x | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | negative-law-abs-ℤ | |
is-positive-abs-ℤ : (x : ℤ) → is-positive-ℤ x → is-positive-ℤ (int-abs-ℤ x) | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | is-positive-abs-ℤ | |
is-nonzero-abs-ℤ : (x : ℤ) → is-positive-ℤ x → is-nonzero-ℕ (abs-ℤ x) | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"op... | src/elementary-number-theory/absolute-value-integers.lagda.md | is-nonzero-abs-ℤ | |
rational-abs-ℚ : ℚ → ℚ | function | src | [
"open import elementary-number-theory.addition-nonnegative-rational-numbers",
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers",
"ope... | src/elementary-number-theory/absolute-value-rational-numbers.lagda.md | rational-abs-ℚ | |
abs-ℚ : ℚ → ℚ⁰⁺ | function | src | [
"open import elementary-number-theory.addition-nonnegative-rational-numbers",
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers",
"ope... | src/elementary-number-theory/absolute-value-rational-numbers.lagda.md | abs-ℚ | |
ackermann-péter-ℕ : ℕ → ℕ → ℕ | function | src | [
"open import elementary-number-theory.natural-numbers"
] | src/elementary-number-theory/ackermann-function.lagda.md | ackermann-péter-ℕ | |
simplified-ackermann-ℕ : ℕ → ℕ | function | src | [
"open import elementary-number-theory.natural-numbers"
] | src/elementary-number-theory/ackermann-function.lagda.md | simplified-ackermann-ℕ | |
semigroup-add-closed-interval-ℚ : Semigroup lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.additive-group-of-rational-numbers",
"open import elementary-number-theory.closed-intervals-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-n... | src/elementary-number-theory/addition-closed-intervals-rational-numbers.lagda.md | semigroup-add-closed-interval-ℚ | |
monoid-add-closed-interval-ℚ : Monoid lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.additive-group-of-rational-numbers",
"open import elementary-number-theory.closed-intervals-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-n... | src/elementary-number-theory/addition-closed-intervals-rational-numbers.lagda.md | monoid-add-closed-interval-ℚ | |
commutative-monoid-add-closed-interval-ℚ : Commutative-Monoid lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.additive-group-of-rational-numbers",
"open import elementary-number-theory.closed-intervals-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-n... | src/elementary-number-theory/addition-closed-intervals-rational-numbers.lagda.md | commutative-monoid-add-closed-interval-ℚ | |
add-fraction-ℤ : fraction-ℤ → fraction-ℤ → fraction-ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"open import elementary-number-theory.multiplication-positive-and-negative-inte... | src/elementary-number-theory/addition-integer-fractions.lagda.md | add-fraction-ℤ | |
add-fraction-ℤ' : fraction-ℤ → fraction-ℤ → fraction-ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.multiplication-integers",
"open import elementary-number-theory.multiplication-positive-and-negative-inte... | src/elementary-number-theory/addition-integer-fractions.lagda.md | add-fraction-ℤ' | |
add-ℤ : ℤ → ℤ → ℤ | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-number-theory.positive-and-negative-integers",
"ope... | src/elementary-number-theory/addition-integers.lagda.md | add-ℤ | |
add-ℤ' : ℤ → ℤ → ℤ | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-number-theory.positive-and-negative-integers",
"ope... | src/elementary-number-theory/addition-integers.lagda.md | add-ℤ' | |
ap-add-ℤ : {x y x' y' : ℤ} → x ＝ x' → y ＝ y' → x +ℤ y ＝ x' +ℤ y' | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-number-theory.positive-and-negative-integers",
"ope... | src/elementary-number-theory/addition-integers.lagda.md | ap-add-ℤ | |
equiv-left-add-ℤ : ℤ → (ℤ ≃ ℤ) | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-number-theory.positive-and-negative-integers",
"ope... | src/elementary-number-theory/addition-integers.lagda.md | equiv-left-add-ℤ | |
equiv-right-add-ℤ : ℤ → (ℤ ≃ ℤ) | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-number-theory.positive-and-negative-integers",
"ope... | src/elementary-number-theory/addition-integers.lagda.md | equiv-right-add-ℤ | |
is-binary-equiv-left-add-ℤ : is-binary-equiv add-ℤ | function | src | [
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-number-theory.positive-and-negative-integers",
"ope... | src/elementary-number-theory/addition-integers.lagda.md | is-binary-equiv-left-add-ℤ | |
add-ℕ : ℕ → ℕ → ℕ | function | src | [
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"ope... | src/elementary-number-theory/addition-natural-numbers.lagda.md | add-ℕ | |
add-ℕ' : ℕ → ℕ → ℕ | function | src | [
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"ope... | src/elementary-number-theory/addition-natural-numbers.lagda.md | add-ℕ' | |
ap-add-ℕ : {m n m' n' : ℕ} → m ＝ m' → n ＝ n' → m +ℕ n ＝ m' +ℕ n' | function | src | [
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"ope... | src/elementary-number-theory/addition-natural-numbers.lagda.md | ap-add-ℕ | |
right-unit-law-add-ℕ : (x : ℕ) → x +ℕ zero-ℕ ＝ x | function | src | [
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"ope... | src/elementary-number-theory/addition-natural-numbers.lagda.md | right-unit-law-add-ℕ | |
left-unit-law-add-ℕ : (x : ℕ) → zero-ℕ +ℕ x ＝ x | function | src | [
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"ope... | src/elementary-number-theory/addition-natural-numbers.lagda.md | left-unit-law-add-ℕ | |
right-successor-law-add-ℕ : (x y : ℕ) → x +ℕ (succ-ℕ y) ＝ succ-ℕ (x +ℕ y) | function | src | [
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identifications-binary-functions",
"open import foundation.action-on-identifications-functions",
"open import foundation.cartesian-product-types",
"ope... | src/elementary-number-theory/addition-natural-numbers.lagda.md | right-successor-law-add-ℕ | |
add-positive-ℤ : positive-ℤ → positive-ℤ → positive-ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.negative-integers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-... | src/elementary-number-theory/addition-positive-and-negative-integers.lagda.md | add-positive-ℤ | |
add-nonnegative-ℤ : nonnegative-ℤ → nonnegative-ℤ → nonnegative-ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.negative-integers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-... | src/elementary-number-theory/addition-positive-and-negative-integers.lagda.md | add-nonnegative-ℤ | |
add-negative-ℤ : negative-ℤ → negative-ℤ → negative-ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.negative-integers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-... | src/elementary-number-theory/addition-positive-and-negative-integers.lagda.md | add-negative-ℤ | |
add-nonpositive-ℤ : nonpositive-ℤ → nonpositive-ℤ → nonpositive-ℤ | function | src | [
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.integers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.negative-integers",
"open import elementary-number-theory.nonnegative-integers",
"open import elementary-... | src/elementary-number-theory/addition-positive-and-negative-integers.lagda.md | add-nonpositive-ℤ | |
add-ℚ' : ℚ → ℚ → ℚ | function | src | [
"open import elementary-number-theory.addition-integer-fractions",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open imp... | src/elementary-number-theory/addition-rational-numbers.lagda.md | add-ℚ' | |
ap-add-ℚ : {x y x' y' : ℚ} → x ＝ x' → y ＝ y' → x +ℚ y ＝ x' +ℚ y' | function | src | [
"open import elementary-number-theory.addition-integer-fractions",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open imp... | src/elementary-number-theory/addition-rational-numbers.lagda.md | ap-add-ℚ | |
succ-ℚ : ℚ → ℚ | function | src | [
"open import elementary-number-theory.addition-integer-fractions",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open imp... | src/elementary-number-theory/addition-rational-numbers.lagda.md | succ-ℚ | |
pred-ℚ : ℚ → ℚ | function | src | [
"open import elementary-number-theory.addition-integer-fractions",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open imp... | src/elementary-number-theory/addition-rational-numbers.lagda.md | pred-ℚ | |
equiv-succ-ℚ : ℚ ≃ ℚ | function | src | [
"open import elementary-number-theory.addition-integer-fractions",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open imp... | src/elementary-number-theory/addition-rational-numbers.lagda.md | equiv-succ-ℚ | |
equiv-pred-ℚ : ℚ ≃ ℚ | function | src | [
"open import elementary-number-theory.addition-integer-fractions",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.integer-fractions",
"open import elementary-number-theory.integers",
"open imp... | src/elementary-number-theory/addition-rational-numbers.lagda.md | equiv-pred-ℚ | |
semigroup-add-ℚ : Semigroup lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | semigroup-add-ℚ | |
is-unital-add-ℚ : is-unital add-ℚ | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | is-unital-add-ℚ | |
monoid-add-ℚ : Monoid lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | monoid-add-ℚ | |
group-add-ℚ : Group lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | group-add-ℚ | |
commutative-monoid-add-ℚ : Commutative-Monoid lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | commutative-monoid-add-ℚ | |
abelian-group-add-ℚ : Ab lzero | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | abelian-group-add-ℚ | |
hom-add-rational-ℤ : hom-Ab ℤ-Ab abelian-group-add-ℚ | function | src | [
"open import elementary-number-theory.addition-rational-numbers",
"open import elementary-number-theory.difference-rational-numbers",
"open import elementary-number-theory.group-of-integers",
"open import elementary-number-theory.rational-numbers",
"open import foundation.dependent-pair-types",
"open impo... | src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md | hom-add-rational-ℤ | |
base-based-strong-ind-ℕ : {l : Level} (k : ℕ) (P : ℕ → UU l) → P k → based-□-≤-ℕ k P k | function | src | [
"open import elementary-number-theory.based-induction-natural-numbers",
"open import elementary-number-theory.equality-natural-numbers",
"open import elementary-number-theory.inequality-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import foundation.action-on-identification... | src/elementary-number-theory/based-strong-induction-natural-numbers.lagda.md | base-based-strong-ind-ℕ | |
bell-number-ℕ : ℕ → ℕ | function | src | [
"open import elementary-number-theory.binomial-coefficients",
"open import elementary-number-theory.multiplication-natural-numbers",
"open import elementary-number-theory.natural-numbers",
"open import elementary-number-theory.strict-inequality-natural-numbers",
"open import elementary-number-theory.strong-... | src/elementary-number-theory/bell-numbers.lagda.md | bell-number-ℕ | |
bezouts-lemma-ℤ : (x y : ℤ) → Σ ℤ (λ s → Σ ℤ (λ t → (s *ℤ x) +ℤ (t *ℤ y) ＝ gcd-ℤ x y)) | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.bezouts-lemma-natural-numbers",
"open import elementary-number-theory.difference-in... | src/elementary-number-theory/bezouts-lemma-integers.lagda.md | bezouts-lemma-ℤ | |
div-right-factor-coprime-ℤ : (x y z : ℤ) → (div-ℤ x (y *ℤ z)) → (gcd-ℤ x y ＝ one-ℤ) → div-ℤ x z | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.bezouts-lemma-natural-numbers",
"open import elementary-number-theory.difference-in... | src/elementary-number-theory/bezouts-lemma-integers.lagda.md | div-right-factor-coprime-ℤ | |
div-right-factor-coprime-ℕ : (x y z : ℕ) → (div-ℕ x (y *ℕ z)) → (gcd-ℕ x y ＝ 1) → div-ℕ x z | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.bezouts-lemma-natural-numbers",
"open import elementary-number-theory.difference-in... | src/elementary-number-theory/bezouts-lemma-integers.lagda.md | div-right-factor-coprime-ℕ | |
is-distance-between-multiples-ℕ : ℕ → ℕ → ℕ → UU lzero | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | is-distance-between-multiples-ℕ | |
is-distance-between-multiples-fst-coeff-ℕ : {x y z : ℕ} → is-distance-between-multiples-ℕ x y z → ℕ | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | is-distance-between-multiples-fst-coeff-ℕ | |
is-distance-between-multiples-snd-coeff-ℕ : {x y z : ℕ} → is-distance-between-multiples-ℕ x y z → ℕ | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | is-distance-between-multiples-snd-coeff-ℕ | |
is-decidable-is-distance-between-multiples-ℕ : (x y z : ℕ) → is-decidable (is-distance-between-multiples-ℕ x y z) | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | is-decidable-is-distance-between-multiples-ℕ | |
pos-distance-between-multiples : (x y z : ℕ) → UU lzero | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | pos-distance-between-multiples | |
is-inhabited-pos-distance-between-multiples : (x y : ℕ) → Σ ℕ (pos-distance-between-multiples x y) | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | is-inhabited-pos-distance-between-multiples | |
minimal-pos-distance-between-multiples : (x y : ℕ) → minimal-element-ℕ (pos-distance-between-multiples x y) | function | src | [
"open import elementary-number-theory.absolute-value-integers",
"open import elementary-number-theory.addition-integers",
"open import elementary-number-theory.addition-natural-numbers",
"open import elementary-number-theory.congruence-integers",
"open import elementary-number-theory.difference-integers",
... | src/elementary-number-theory/bezouts-lemma-natural-numbers.lagda.md | minimal-pos-distance-between-multiples |
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