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TentativeConvert committed Sep 4, 2024
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Expand Up @@ -338,47 +338,47 @@ Using this theorem, which could be a separate exercise in FunctionSurj, the seco


#### Surj 01
**fun ↦**
**fun ↦**

#### Surj 02
fun ↦
fun ↦

#### Surj 03
**let**
**comp_apply**
**let**
**comp_apply**

#### Surj 04
**if…then…else**
**if_pos** **if_neg**
**if…then…else**
**if_pos** **if_neg**

let
comp_apply
let
comp_apply

#### Surj 05
**funext**
**funext**

#### Surj 06
**congr_arg**
**congr_arg**

#### Surj 07
**congr_fun**
**congr_fun**

#### Surj 08
**RightInverse**
**RightInverse**

congr_fun
congr_fun

#### Surj 09
**LeftInverse**
**LeftInverse**

RightInverse
RightInverse

#### Surj 10
**HasRightInverse**
**HasRightInverse**


#### Surj 11
**Surjective**
**Surjective**

#### Surj 12 SEMIBOSS: function which semiconjugates to successor function is surjective

Expand All @@ -401,61 +401,61 @@ HasRightInverse
Surjective

#### Inj 01
**Injective**
**Injective**

#### Inj 02: bijective ↔ ∃ both inverses
**Bijective** **RightInverse** **Leftinverse**
**Bijective** **RightInverse** **Leftinverse**

**choose**
**choose**

### Inj 03: injective → fibres are singletons
range
Injective
#### Inj 03: injective → fibres are singletons
range
Injective

#### Inj 04: how to use Equiv
**Equiv**
**Equiv**

#### Inj 05: how to construct Equiv
**Equiv**
**Equiv**

RightInverse
RightInverse

#### Inj 04: StrictMono → injective – proof
**StrictMono**
**StrictMono**

Injective
Injective

#### Inj 05: application of StrictMono → application

#### Inj 06: succ has left inverse, using if … then … else
**succ** **if … then … else** **HasLeftInverse**
**succ** **if … then … else** **HasLeftInverse**

#### Inj 07a & b: extend from ℕ to ℤ
**if … then … else**
**if … then … else**

#### Inj 08: image of self vs preimage of LeftInverse
**image**
preimage
LeftInverse
congr_arg
**image**

preimage
LeftInverse
congr_arg

#### Inj 09: HasLeftInverse → Injective

HasLeftInverse
LeftInverse
congr_arg
HasLeftInverse
LeftInverse
congr_arg

#### Inj 10
LeftInverse
RightInverse
Injective
LeftInverse
RightInverse
Injective

#### Inj 11: Injective → HasLeftInverse
range
range

choose
arbitrary
choose
arbitrary

#### Bij 01
**Bijective**
Expand All @@ -471,45 +471,45 @@ choose

#### Bij 04 Equiv (how to use it)

**depends on Bid 02**
**depends on Bij 02**

RightInverse
LeftInverse
Surjective
Injective
congr_arg
…?…
RightInverse
LeftInverse
Surjective
Injective
congr_arg
…?…

#### Bij 05 Equiv (how to construct it)

RightInverse
LeftInverse
Surjective
Injective
RightInverse
LeftInverse
Surjective
Injective


#### Bij 06 Equiv curry/uncurry
**curry_uncurry** **uncurry_curry**
**curry_uncurry** **uncurry_curry**

#### Bij 07 Reading exercise for A → B → C
Surjective
[curry]
Surjective
[curry]

#### Bij 08 Diagonal injective
Injective
congr_fun
[curry]
Fin n
Injective
congr_fun
[curry]
Fin n

#### Bij 09*: Surjective.image_preimage
Surjective
Image
Preimage
Surjective
Image
Preimage

#### Bij 09 BOSS – f surjective ↔ f⁻¹ injective
Surjective
Injective
Image
Preimage
fibre
Surjective
Injective
Image
Preimage
fibre

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