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[ add ] ∸-suc lemma for natural numbers #2757

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15 changes: 15 additions & 0 deletions CHANGELOG.md
Original file line number Diff line number Diff line change
Expand Up @@ -15,6 +15,15 @@ Non-backwards compatible changes
Minor improvements
------------------

* Refactored usages of `+-∸-assoc 1` to `∸-suc` in:
```agda
README.Data.Fin.Relation.Unary.Top
Algebra.Properties.Semiring.Binomial
Data.Fin.Subset.Properties
Data.Nat.Binary.Subtraction
Data.Nat.Combinatorics
```

Deprecated modules
------------------

Expand All @@ -26,3 +35,9 @@ New modules

Additions to existing modules
-----------------------------

* In `Data.Nat.Properties`:
```agda
∸-suc : m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
```

4 changes: 2 additions & 2 deletions doc/README/Data/Fin/Relation/Unary/Top.agda
Original file line number Diff line number Diff line change
Expand Up @@ -17,7 +17,7 @@
module README.Data.Fin.Relation.Unary.Top where

open import Data.Nat.Base using (ℕ; zero; suc; _∸_; _≤_)
open import Data.Nat.Properties using (n∸n≡0; +-∸-assoc; ≤-reflexive)
open import Data.Nat.Properties using (n∸n≡0; ∸-suc; ≤-reflexive)
open import Data.Fin.Base using (Fin; zero; suc; toℕ; fromℕ; inject₁; _>_)
open import Data.Fin.Properties using (toℕ-fromℕ; toℕ<n; toℕ-inject₁)
open import Data.Fin.Induction hiding (>-weakInduction)
Expand Down Expand Up @@ -76,7 +76,7 @@ opposite-prop {suc n} i with view i
... | ‵fromℕ rewrite toℕ-fromℕ n | n∸n≡0 n = refl
... | ‵inject₁ j = begin
suc (toℕ (opposite j)) ≡⟨ cong suc (opposite-prop j) ⟩
suc (n ∸ suc (toℕ j)) ≡⟨ +-∸-assoc 1 (toℕ<n j) ⟨
suc (n ∸ suc (toℕ j)) ≡⟨ ∸-suc (toℕ<n j) ⟨
n ∸ toℕ j ≡⟨ cong (n ∸_) (toℕ-inject₁ j) ⟨
n ∸ toℕ (inject₁ j) ∎ where open ≡-Reasoning

Expand Down
4 changes: 2 additions & 2 deletions src/Algebra/Properties/Semiring/Binomial.agda
Original file line number Diff line number Diff line change
Expand Up @@ -21,7 +21,7 @@ open import Data.Nat.Base as ℕ hiding (_+_; _*_; _^_)
open import Data.Nat.Combinatorics
using (_C_; nCn≡1; nC1≡n; nCk+nC[k+1]≡[n+1]C[k+1])
open import Data.Nat.Properties as ℕ
using (<⇒<ᵇ; n<1+n; n∸n≡0; +-∸-assoc)
using (<⇒<ᵇ; n<1+n; n∸n≡0; ∸-suc)
open import Data.Fin.Base as Fin
using (Fin; zero; suc; toℕ; fromℕ; inject₁)
open import Data.Fin.Patterns using (0F)
Expand Down Expand Up @@ -149,7 +149,7 @@ y*lemma x*y≈y*x {n} j = begin
k≡j = toℕ-inject₁ j

[n-k]≡[n-j] : [n-k] ≡ [n-j]
[n-k]≡[n-j] = ≡.trans (cong (n ∸_) k≡j) (+-∸-assoc 1 (toℕ<n j))
[n-k]≡[n-j] = ≡.trans (cong (n ∸_) k≡j) (∸-suc (toℕ<n j))

------------------------------------------------------------------------
-- Now, a lemma characterising the sum of the term₁ and term₂ expressions
Expand Down
2 changes: 1 addition & 1 deletion src/Data/Fin/Subset/Properties.agda
Original file line number Diff line number Diff line change
Expand Up @@ -367,7 +367,7 @@ p∪∁p≡⊤ (inside ∷ p) = cong (inside ∷_) (p∪∁p≡⊤ p)
∣∁p∣≡n∸∣p∣ (inside ∷ p) = ∣∁p∣≡n∸∣p∣ p
∣∁p∣≡n∸∣p∣ (outside ∷ p) = begin
suc ∣ ∁ p ∣ ≡⟨ cong suc (∣∁p∣≡n∸∣p∣ p) ⟩
suc (_ ∸ ∣ p ∣) ≡⟨ sym (ℕ.+-∸-assoc 1 (∣p∣≤n p)) ⟩
suc (_ ∸ ∣ p ∣) ≡⟨ sym (ℕ.∸-suc (∣p∣≤n p)) ⟩
suc _ ∸ ∣ p ∣ ∎
where open ≡-Reasoning

Expand Down
2 changes: 1 addition & 1 deletion src/Data/Nat/Binary/Subtraction.agda
Original file line number Diff line number Diff line change
Expand Up @@ -88,7 +88,7 @@ toℕ-homo-∸ 2[1+ x ] 1+[2 y ] with x <? y
... | no x≮y = begin
ℕ.suc (2 ℕ.* toℕ (x ∸ y)) ≡⟨ cong (ℕ.suc ∘ (2 ℕ.*_)) (toℕ-homo-∸ x y) ⟩
ℕ.suc (2 ℕ.* (toℕ x ℕ.∸ toℕ y)) ≡⟨ cong ℕ.suc (ℕ.*-distribˡ-∸ 2 (toℕ x) (toℕ y)) ⟩
ℕ.suc (2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y) ≡⟨ sym (ℕ.+-∸-assoc 1 (ℕ.*-monoʳ-≤ 2 (toℕ-mono-≤ (≮⇒≥ x≮y)))) ⟩
ℕ.suc (2 ℕ.* toℕ x ℕ.∸ 2 ℕ.* toℕ y) ≡⟨ sym (ℕ.∸-suc (ℕ.*-monoʳ-≤ 2 (toℕ-mono-≤ (≮⇒≥ x≮y)))) ⟩
ℕ.suc (2 ℕ.* toℕ x) ℕ.∸ 2 ℕ.* toℕ y ≡⟨ sym (cong (ℕ._∸ 2 ℕ.* toℕ y) (ℕ.+-suc (toℕ x) (1 ℕ.* toℕ x))) ⟩
2 ℕ.* (ℕ.suc (toℕ x)) ℕ.∸ ℕ.suc (2 ℕ.* toℕ y) ∎
where open ≡-Reasoning
Expand Down
2 changes: 1 addition & 1 deletion src/Data/Nat/Combinatorics.agda
Original file line number Diff line number Diff line change
Expand Up @@ -122,7 +122,7 @@ module _ {n k} (k<n : k < n) where
[n-k-1]! = [n-k-1] !

[n-k]≡1+[n-k-1] : [n-k] ≡ suc [n-k-1]
[n-k]≡1+[n-k-1] = +-∸-assoc 1 k<n
[n-k]≡1+[n-k-1] = ∸-suc k<n


[n-k]*[n-k-1]!≡[n-k]! : [n-k] * [n-k-1]! ≡ [n-k]!
Expand Down
19 changes: 11 additions & 8 deletions src/Data/Nat/Properties.agda
Original file line number Diff line number Diff line change
Expand Up @@ -1524,6 +1524,10 @@ pred[m∸n]≡m∸[1+n] (suc m) (suc n) = pred[m∸n]≡m∸[1+n] m n
------------------------------------------------------------------------
-- Properties of _∸_ and _≤_/_<_

∸-suc : m ≤ n → suc n ∸ m ≡ suc (n ∸ m)
∸-suc z≤n = refl
∸-suc (s≤s m≤n) = ∸-suc m≤n

m∸n≤m : ∀ m n → m ∸ n ≤ m
m∸n≤m n zero = ≤-refl
m∸n≤m zero (suc n) = ≤-refl
Expand Down Expand Up @@ -1614,12 +1618,11 @@ m≤n⇒n∸m≤n (s≤s m≤n) = m≤n⇒m≤1+n (m≤n⇒n∸m≤n m≤n)
∸-+-assoc (suc m) (suc n) o = ∸-+-assoc m n o

+-∸-assoc : ∀ m {n o} → o ≤ n → (m + n) ∸ o ≡ m + (n ∸ o)
+-∸-assoc m (z≤n {n = n}) = begin-equality m + n ∎
+-∸-assoc m (s≤s {m = o} {n = n} o≤n) = begin-equality
(m + suc n) ∸ suc o ≡⟨ cong (_∸ suc o) (+-suc m n) ⟩
suc (m + n) ∸ suc o ≡⟨⟩
(m + n) ∸ o ≡⟨ +-∸-assoc m o≤n ⟩
m + (n ∸ o) ∎
+-∸-assoc zero {n = n} {o = o} _ = begin-equality n ∸ o ∎
+-∸-assoc (suc m) {n = n} {o = o} o≤n = begin-equality
suc (m + n) ∸ o ≡⟨ ∸-suc (m≤n⇒m≤o+n m o≤n) ⟩
suc ((m + n) ∸ o) ≡⟨ cong suc (+-∸-assoc m o≤n) ⟩
suc (m + (n ∸ o)) ∎

m≤n+o⇒m∸n≤o : ∀ m n {o} → m ≤ n + o → m ∸ n ≤ o
m≤n+o⇒m∸n≤o m zero le = le
Expand All @@ -1634,7 +1637,7 @@ m<n+o⇒m∸n<o (suc m) (suc n) lt = m<n+o⇒m∸n<o m n (s<s⁻¹
m+n≤o⇒m≤o∸n : ∀ m {n o} → m + n ≤ o → m ≤ o ∸ n
m+n≤o⇒m≤o∸n zero le = z≤n
m+n≤o⇒m≤o∸n (suc m) (s≤s le)
rewrite +-∸-assoc 1 (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m le)
rewrite ∸-suc (m+n≤o⇒n≤o m le) = s≤s (m+n≤o⇒m≤o∸n m le)

m≤o∸n⇒m+n≤o : ∀ m {n o} (n≤o : n ≤ o) → m ≤ o ∸ n → m + n ≤ o
m≤o∸n⇒m+n≤o m z≤n le rewrite +-identityʳ m = le
Expand Down Expand Up @@ -1671,7 +1674,7 @@ m∸n+n≡m {m} {n} n≤m = begin-equality
m∸[m∸n]≡n : ∀ {m n} → n ≤ m → m ∸ (m ∸ n) ≡ n
m∸[m∸n]≡n {m} {_} z≤n = n∸n≡0 m
m∸[m∸n]≡n {suc m} {suc n} (s≤s n≤m) = begin-equality
suc m ∸ (m ∸ n) ≡⟨ +-∸-assoc 1 (m∸n≤m m n) ⟩
suc m ∸ (m ∸ n) ≡⟨ ∸-suc (m∸n≤m m n) ⟩
suc (m ∸ (m ∸ n)) ≡⟨ cong suc (m∸[m∸n]≡n n≤m) ⟩
suc n ∎

Expand Down