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Equality of conatural numbers #1236

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fredrik-bakke Jan 14, 2025
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Merge branch 'master' into equality-conaturals
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some closure properties of apartness relations
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inequality conaturals
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some basic properties of the type of decreasing binary sequences
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additions booleans
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Decidable posets have decidable equality
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an edit `split-idempotent-maps`
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switch to increasing binary sequences
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work
fredrik-bakke committed Jan 14, 2025
commit 624dff5401c465076c1c40435118c616d05c4e08
6 changes: 3 additions & 3 deletions .vscode/tasks.json
Original file line number Diff line number Diff line change
@@ -59,12 +59,12 @@
"problemMatcher": [],
// "hide": true,
"presentation": {
"reveal": "silent",
"revealProblems": "onProblem",
"close": true
"close": false
},
"runOptions": {
"runOn": "default"
"runOn": "default",
"instanceLimit": 99
}
}
],
2 changes: 1 addition & 1 deletion scripts/remove_unused_imports.py
Original file line number Diff line number Diff line change
@@ -139,5 +139,5 @@ def sort_key(file):
executor.map(lambda file: process_agda_file(
file, agda_options, root, temp_dir), sorted_agda_files)

shutil.rmtree(temp_root)
# shutil.rmtree(temp_root)
sys.exit(status)
1 change: 1 addition & 0 deletions src/elementary-number-theory.lagda.md
Original file line number Diff line number Diff line change
@@ -51,6 +51,7 @@ open import elementary-number-theory.divisibility-integers public
open import elementary-number-theory.divisibility-modular-arithmetic public
open import elementary-number-theory.divisibility-natural-numbers public
open import elementary-number-theory.divisibility-standard-finite-types public
open import elementary-number-theory.equality-conatural-numbers public
open import elementary-number-theory.equality-integers public
open import elementary-number-theory.equality-natural-numbers public
open import elementary-number-theory.euclid-mullin-sequence public
19 changes: 14 additions & 5 deletions src/elementary-number-theory/conatural-numbers.lagda.md
Original file line number Diff line number Diff line change
@@ -18,6 +18,7 @@ open import foundation.maybe
open import foundation.negated-equality
open import foundation.retractions
open import foundation.sections
open import foundation.unit-type
open import foundation.universe-levels

open import foundation-core.identity-types
@@ -41,6 +42,7 @@ initial algebra.
record ℕ∞ : UU lzero
where
coinductive
constructor cons-ℕ∞
field
decons-ℕ∞ : Maybe ℕ∞

@@ -68,12 +70,11 @@ succ-ℕ∞ : ℕ∞ → ℕ∞
decons-ℕ∞ (succ-ℕ∞ x) = unit-Maybe x
```

### The constructor function for conatural numbers
### Alternative definition of the constructor function for conatural numbers

```agda
cons-ℕ∞ : Maybe ℕ∞ → ℕ∞
cons-ℕ∞ (inl x) = succ-ℕ∞ x
cons-ℕ∞ (inr x) = zero-ℕ∞
cons-ℕ∞' : Maybe ℕ∞ → ℕ∞
cons-ℕ∞' = rec-coproduct succ-ℕ∞ (λ _ → zero-ℕ∞)
```

### Alternative definition of the deconstructor function for conatural numbers
@@ -155,13 +156,21 @@ section-decons-ℕ∞' = cons-ℕ∞ , is-section-cons-ℕ∞'

retraction-cons-ℕ∞' : retraction cons-ℕ∞
retraction-cons-ℕ∞' = decons-ℕ∞' , is-section-cons-ℕ∞'

is-section-cons-ℕ∞'' : is-section decons-ℕ∞' cons-ℕ∞'
is-section-cons-ℕ∞'' (inl x) = refl
is-section-cons-ℕ∞'' (inr x) = refl

is-injective-cons-ℕ∞' : is-injective cons-ℕ∞'
is-injective-cons-ℕ∞' =
is-injective-retraction cons-ℕ∞' (decons-ℕ∞' , is-section-cons-ℕ∞'')
```

### The successor function is injective

```agda
is-injective-succ-ℕ∞ : is-injective succ-ℕ∞
is-injective-succ-ℕ∞ p = is-injective-inl (is-injective-cons-ℕ∞ p)
is-injective-succ-ℕ∞ p = is-injective-inl (is-injective-cons-ℕ∞' p)
```

## External links
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