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Original file line number Diff line number Diff line change
Expand Up @@ -29,7 +29,7 @@ Let `C` and `D` be two large precategories. Two functors `F : C → D` and
- for every pair of morhpisms `f : X₂ → X₁` and `g : Y₁ → Y₂` the following
square commutes:

```md
```text
ϕ X₁ Y₁
hom X₁ (G Y₁) --------> hom (F X₁) Y₁
| |
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Original file line number Diff line number Diff line change
Expand Up @@ -35,8 +35,8 @@ open import univalent-combinatorics.standard-finite-types
The **binomial theorem** in commutative rings asserts that for any two elements
`x` and `y` of a commutative ring `A` and any natural number `n`, we have

```md
(x + y)ⁿ = ∑\_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.
```text
(x + y)ⁿ = ∑_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.
```

## Definitions
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Original file line number Diff line number Diff line change
Expand Up @@ -35,8 +35,8 @@ The **binomial theorem** in commutative semirings asserts that for any two
elements `x` and `y` of a commutative semiring `A` and any natural number `n`,
we have

```md
(x + y)ⁿ = ∑\_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.
```text
(x + y)ⁿ = ∑_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.
```

## Definitions
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Original file line number Diff line number Diff line change
Expand Up @@ -25,7 +25,7 @@ family `P` of types over `ℕ` and any natural number `k : ℕ`, equipped with
1. An element `p0 : P k`
2. A function `pS : (x : ℕ) → k ≤-ℕ x → P x → P (x + 1)` there is a function

```md
```text
based-ind-ℕ k P p0 pS : (x : ℕ) → k ≤-ℕ x → P x
```

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Original file line number Diff line number Diff line change
Expand Up @@ -34,7 +34,7 @@ numbers equipped with
`pS : (x : ℕ) → k ≤-ℕ x → ((y : ℕ) → k ≤-ℕ y ≤-ℕ x → P y) → P (x + 1)` there
is a function

```md
```text
f := based-strong-ind-ℕ k P p0 pS : (x : ℕ) → k ≤-ℕ x → P k
```

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Original file line number Diff line number Diff line change
Expand Up @@ -49,7 +49,7 @@ Bezout's lemma shows that for any two natural numbers `x` and `y` there exist
`k` and `l` such that `dist-ℕ (kx,ly) = gcd(x,y)`. To prove this, note that the
predicate `P : ℕ → UU lzero` given by

```md
```text
P z := Σ (k : ℕ), Σ (l : ℕ), dist-ℕ (kx, ly) = z
```

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Original file line number Diff line number Diff line change
Expand Up @@ -32,7 +32,7 @@ open import univalent-combinatorics.standard-finite-types
The binomial theorem for the integers asserts that for any two integers `x` and
`y` and any natural number `n`, we have

```md
```text
(x + y)ⁿ = ∑_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.
```

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Original file line number Diff line number Diff line change
Expand Up @@ -31,7 +31,7 @@ open import univalent-combinatorics.standard-finite-types
The binomial theorem for natural numbers asserts that for any two natural
numbers `x` and `y` and any natural number `n`, we have

```md
```text
(x + y)ⁿ = ∑_{0 ≤ i < n+1} (n choose i) xⁱ yⁿ⁻ⁱ.
```

Expand Down
2 changes: 1 addition & 1 deletion src/elementary-number-theory/cofibonacci.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -31,7 +31,7 @@ open import foundation.universe-levels
The [**cofibonacci sequence**][1] is the unique function G : ℕ → ℕ satisfying
the property that

```md
```text
div-ℕ (G m) n ↔ div-ℕ m (Fibonacci-ℕ n).
```

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Original file line number Diff line number Diff line change
Expand Up @@ -37,7 +37,7 @@ An integer `m` is said to **divide** an integer `n` if there exists an integer
`k` equipped with an identification `km = n`. Using the Curry-Howard
interpretation of logic into type theory, we express divisibility as follows:

```md
```text
div-ℤ m n := Σ (k : ℤ), k *ℤ m = n.
```

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Original file line number Diff line number Diff line change
Expand Up @@ -35,7 +35,7 @@ a natural number `k` equipped with an identification `km = n`. Using the
Curry-Howard interpretation of logic into type theory, we express divisibility
as follows:

```md
```text
div-ℕ m n := Σ (k : ℕ), k *ℕ m = n.
```

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Original file line number Diff line number Diff line change
Expand Up @@ -38,7 +38,7 @@ unique natural number `r < d` such that `kd + r = n`.
The following definition produces for each `k : ℕ` a sequence of sequences as
follows:

```md
```text
This is column k
0,…,0,0,0,0,0,0,0,… -- The sequence at row 0 is the constant sequence
Expand Down
Original file line number Diff line number Diff line change
Expand Up @@ -25,7 +25,7 @@ open import foundation.universe-levels
An **initial segment** of the natural numbers is a subtype `P : ℕ → Prop` such
that the implication

```md
```text
P (n + 1) → P n
```

Expand Down
4 changes: 2 additions & 2 deletions src/elementary-number-theory/kolakoski-sequence.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -22,14 +22,14 @@ open import foundation.dependent-pair-types

The Kolakoski sequence

```md
```text
1,2,2,1,1,2,1,2,2,1,2,2,1,1,...
```

is a self-referential sequence of `1`s and `2`s which is the flattening of a
sequence

```md
```text
(1),(2,2),(1,1),(2),(1),(2,2),(1,1)
```

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Original file line number Diff line number Diff line change
Expand Up @@ -19,7 +19,7 @@ The multiset coefficients count the number of multisets of size `k` of elements
of a set of size `n`. In oter words, it counts the number of connected componets
of the type

```md
```text
Σ (A : Fin n → 𝔽), ∥ Fin k ≃ Σ (i : Fin n), A i ∥.
```

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Original file line number Diff line number Diff line change
Expand Up @@ -84,7 +84,7 @@ The delooping of a group homomorphism `f : G → H` is a pointed map
`Bf : BG → BH` equiped with an homotopy witnessing that the following square
commutes :

```md
```text
f
G --------> H
| |
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Original file line number Diff line number Diff line change
Expand Up @@ -19,7 +19,7 @@ open import foundation-core.universe-levels

A commuting 3-simplex of homotopies is a commuting diagram of the form

```md
```text
f ----------> g
| \ ^ |
| \ / |
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/commuting-3-simplices-of-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -19,7 +19,7 @@ open import foundation-core.universe-levels

A commuting 3-simplex is a commuting diagram of the form

```md
```text
A ----------> B
| \ ^ |
| \ / |
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/commuting-cubes-of-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -24,7 +24,7 @@ open import foundation-core.universe-levels
We specify the type of the homotopy witnessing that a cube commutes. Imagine
that the cube is presented as a lattice

```md
```text
*
/ | \
/ | \
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/commuting-squares-of-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -23,7 +23,7 @@ open import foundation-core.universe-levels

A square of maps

```md
```text
A ------> X
| |
| |
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Original file line number Diff line number Diff line change
Expand Up @@ -20,7 +20,7 @@ open import foundation-core.universe-levels

A triangle of homotopies of maps

```md
```text
f ----> g
\ /
\ /
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/commuting-triangles-of-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -19,7 +19,7 @@ open import foundation-core.universe-levels

A triangle of maps

```md
```text
A ----> B
\ /
\ /
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/cones-over-cospans.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -28,7 +28,7 @@ A cone on a cospan `A --f--> X <--g-- B` with vertex `C` is a triple `(p,q,H)`
consisting of a map `p : C → A`, a map `q : C → B`, and a homotopy `H`
witnessing that the square

```md
```text
q
C -----> B
| |
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/equality-fibers-of-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -34,7 +34,7 @@ action on identifications of `f`.
For any map `f : A → B` any `b : B` and any `x y : fib f b`, there is an
equivalence

```md
```text
(x = y) ≃ fib (ap f) ((pr2 x) ∙ (inv (pr2 y)))
```

Expand Down
4 changes: 2 additions & 2 deletions src/foundation-core/equivalences.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -463,7 +463,7 @@ is-equiv-equiv' {f = f} {g} i j H K =

We will assume a commuting square

```md
```text
h
A --------> C
| |
Expand Down Expand Up @@ -562,7 +562,7 @@ module _

Equivalences can be constructed by equational reasoning in the following way:

```md
```text
equivalence-reasoning
X ≃ Y by equiv-1
≃ Z by equiv-2
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/homotopies.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -348,7 +348,7 @@ module _

Homotopies can be constructed by equational reasoning in the following way:

```md
```text
homotopy-reasoning
f ~ g by htpy-1
~ h by htpy-2
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/identity-types.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -469,7 +469,7 @@ ap-binary-concat f refl refl refl refl = refl

Identifications can be constructed by equational reasoning in the following way:

```md
```text
equational-reasoning
x = y by eq-1
= z by eq-2
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/logical-equivalences.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -120,7 +120,7 @@ module _
Logical equivalences can be constructed by equational reasoning in the following
way:

```md
```text
logical-equivalence-reasoning
X ↔ Y by equiv-1
↔ Z by equiv-2
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/morphisms-cospans.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -24,7 +24,7 @@ open import foundation-core.universe-levels

A morphism of cospans is a commuting diagram of the form

```md
```text
A -----> X <----- B
| | |
| | |
Expand Down
2 changes: 1 addition & 1 deletion src/foundation-core/propositions.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -54,7 +54,7 @@ module _
We prove here only that any contractible type is a proposition. The fact that
the empty type and the unit type are propositions can be found in

```md
```text
foundation.empty-types
foundation.unit-type
```
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Original file line number Diff line number Diff line change
Expand Up @@ -33,7 +33,7 @@ open import foundation-core.universe-levels

Let `X` be a type, we have the following equivalence :

```md
```text
Σ ( (U , V , e) : Relaxed-Σ-Decomposition X)
( binary-coproduct-Decomposition U) ≃
Σ ( (A , B , e) : binary-coproduct-Decomposition X)
Expand Down
2 changes: 1 addition & 1 deletion src/foundation/binary-transport.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -21,7 +21,7 @@ open import foundation-core.universe-levels
Given a binary relation `B : A → A → UU` and identifications `p : x = x'` and
`q : y = y'` in `A`, the binary transport of `B` is an operation

```md
```text
binary-tr B p q : B x y → B x' y'
```

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Original file line number Diff line number Diff line change
Expand Up @@ -19,7 +19,7 @@ open import foundation-core.universe-levels

A square of identifications

```md
```text
top
x ------- y
| |
Expand Down
2 changes: 1 addition & 1 deletion src/foundation/commuting-triangles-of-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -22,7 +22,7 @@ open import foundation-core.universe-levels

A triangle of maps

```md
```text
A ----> B
\ /
\ /
Expand Down
2 changes: 1 addition & 1 deletion src/foundation/descent-equivalences.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -24,7 +24,7 @@ open import foundation-core.universe-levels
The descent property of equivalences is a somewhat degenerate form of a descent
property. It asserts that in a commuting diagram of the form

```md
```text
p q
A -----> B -----> C
| | |
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Original file line number Diff line number Diff line change
Expand Up @@ -29,7 +29,7 @@ open import foundation-core.universe-levels
A map `f : A → B` is said to be a **`k`-epimorphism** if the precomposition
function

```md
```text
- ∘ f : (B → X) → (A → X)
```

Expand Down
2 changes: 1 addition & 1 deletion src/foundation/epimorphisms.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -19,7 +19,7 @@ open import foundation-core.universe-levels
A map `f : A → B` is said to be an **epimorphism** if the precomposition
function

```md
```text
- ∘ f : (B → X) → (A → X)
```

Expand Down
4 changes: 2 additions & 2 deletions src/foundation/exponents-set-quotients.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -37,7 +37,7 @@ open import foundation-core.universe-levels
Given a type `A` equipped with an equivalence relation `R` and a type `X`, the
set quotient

```md
```text
(X → A) / ~
```

Expand All @@ -46,7 +46,7 @@ embedding for every `X`, `A`, and `R` if and only if the axiom of choice holds.

Consequently, we get embeddings

```md
```text
((hom-Eq-Rel R S) / ~) ↪ ((A/R) → (B/S))
```

Expand Down
2 changes: 1 addition & 1 deletion src/foundation/fibered-equivalences.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -30,7 +30,7 @@ open import foundation-core.universe-levels

A fibered equivalence is a fibered map

```md
```text
h
A -------> B
| |
Expand Down
2 changes: 1 addition & 1 deletion src/foundation/fibered-involutions.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -24,7 +24,7 @@ open import foundation-core.universe-levels

A fibered involution is a fibered map

```md
```text
h
A -------> A
| |
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2 changes: 1 addition & 1 deletion src/foundation/fibered-maps.lagda.md
Original file line number Diff line number Diff line change
Expand Up @@ -34,7 +34,7 @@ open import foundation-core.universe-levels

Consider a diagram of the form

```md
```text
A B
| |
f| |g
Expand Down
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