Lemmas for Positive, Negative, etc. and _+_ and _*_ for rationals

CHANGELOG.md

@@ -316,6 +316,26 @@ Additions to existing modules
   m≤pred[n]⇒suc[m]≤n : .{{NonZero n}} → m ≤ pred n → suc m ≤ n
   ```
 
+* New lemmas in `Data.Rational.Properties`:
+  ```agda
+  nonNeg+nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} q .{{_ : NonNegative q}} → NonNegative (p + q)
+  nonPos+nonPos⇒nonPos : ∀ p .{{_ : NonPositive p}} q .{{_ : NonPositive q}} → NonPositive (p + q)
+  pos+nonNeg⇒pos : ∀ p .{{_ : Positive p}} q .{{_ : NonNegative q}} → Positive (p + q)
+  nonNeg+pos⇒pos : ∀ p .{{_ : NonNegative p}} q .{{_ : Positive q}} → Positive (p + q)
+  pos+pos⇒pos : ∀ p .{{_ : Positive p}} q .{{_ : Positive q}} → Positive (p + q)
+  neg+nonPos⇒neg : ∀ p .{{_ : Negative p}} q .{{_ : NonPositive q}} → Negative (p + q)
+  nonPos+neg⇒neg : ∀ p .{{_ : NonPositive p}} q .{{_ : Negative q}} → Negative (p + q)
+  neg+neg⇒neg : ∀ p .{{_ : Negative p}} q .{{_ : Negative q}} → Negative (p + q)
+  nonNeg*nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} q .{{_ : NonNegative q}} → NonNegative (p * q)
+  nonPos*nonNeg⇒nonPos : ∀ p .{{_ : NonPositive p}} q .{{_ : NonNegative q}} → NonPositive (p * q)
+  nonNeg*nonPos⇒nonPos : ∀ p .{{_ : NonNegative p}} q .{{_ : NonPositive q}} → NonPositive (p * q)
+  nonPos*nonPos⇒nonPos : ∀ p .{{_ : NonPositive p}} q .{{_ : NonPositive q}} → NonNegative (p * q)
+  pos*pos⇒pos : ∀ p .{{_ : Positive p}} q .{{_ : Positive q}} → Positive (p * q)
+  neg*pos⇒neg : ∀ p .{{_ : Negative p}} q .{{_ : Positive q}} → Negative (p * q)
+  pos*neg⇒neg : ∀ p .{{_ : Positive p}} q .{{_ : Negative q}} → Negative (p * q)
+  neg*neg⇒pos : ∀ p .{{_ : Negative p}} q .{{_ : Negative q}} → Positive (p * q)
+  ```
+
 * New lemma in `Data.Vec.Properties`:
   ```agda
   map-concat : map f (concat xss) ≡ concat (map (map f) xss)

src/Data/Rational/Properties.agda

@@ -1012,6 +1012,12 @@ neg-distrib-+ = +-Monomorphism.⁻¹-distrib-∙ ℚᵘ.+-0-isAbelianGroup (ℚ
 +-monoʳ-≤ : ∀ r → (_+_ r) Preserves _≤_ ⟶ _≤_
 +-monoʳ-≤ r p≤q = +-mono-≤ (≤-refl {r}) p≤q
 
+nonNeg+nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} q .{{_ : NonNegative q}} → NonNegative (p + q)
+nonNeg+nonNeg⇒nonNeg p q = nonNegative $ +-mono-≤ (nonNegative⁻¹ p) (nonNegative⁻¹ q)
+
+nonPos+nonPos⇒nonPos : ∀ p .{{_ : NonPositive p}} q .{{_ : NonPositive q}} → NonPositive (p + q)
+nonPos+nonPos⇒nonPos p q = nonPositive $ +-mono-≤ (nonPositive⁻¹ p) (nonPositive⁻¹ q)
+
 ------------------------------------------------------------------------
 -- Properties of _+_ and _<_
 
@@ -1035,6 +1041,24 @@ neg-distrib-+ = +-Monomorphism.⁻¹-distrib-∙ ℚᵘ.+-0-isAbelianGroup (ℚ
 +-monoʳ-< : ∀ r → (_+_ r) Preserves _<_ ⟶ _<_
 +-monoʳ-< r p<q = +-mono-≤-< (≤-refl {r}) p<q
 
+pos+nonNeg⇒pos : ∀ p .{{_ : Positive p}} q .{{_ : NonNegative q}} → Positive (p + q)
+pos+nonNeg⇒pos p q = positive $ +-mono-<-≤ (positive⁻¹ p) (nonNegative⁻¹ q)
+
+nonNeg+pos⇒pos : ∀ p .{{_ : NonNegative p}} q .{{_ : Positive q}} → Positive (p + q)
+nonNeg+pos⇒pos p q = positive $ +-mono-≤-< (nonNegative⁻¹ p) (positive⁻¹ q)
+
+pos+pos⇒pos : ∀ p .{{_ : Positive p}} q .{{_ : Positive q}} → Positive (p + q)
+pos+pos⇒pos p q = positive $ +-mono-< (positive⁻¹ p) (positive⁻¹ q)
+
+neg+nonPos⇒neg : ∀ p .{{_ : Negative p}} q .{{_ : NonPositive q}} → Negative (p + q)
+neg+nonPos⇒neg p q = negative $ +-mono-<-≤ (negative⁻¹ p) (nonPositive⁻¹ q)
+
+nonPos+neg⇒neg : ∀ p .{{_ : NonPositive p}} q .{{_ : Negative q}} → Negative (p + q)
+nonPos+neg⇒neg p q = negative $ +-mono-≤-< (nonPositive⁻¹ p) (negative⁻¹ q)
+
+neg+neg⇒neg : ∀ p .{{_ : Negative p}} q .{{_ : Negative q}} → Negative (p + q)
+neg+neg⇒neg p q = negative $ +-mono-< (negative⁻¹ p) (negative⁻¹ q)
+
 ------------------------------------------------------------------------
 -- Properties of _*_
 ------------------------------------------------------------------------
@@ -1340,6 +1364,34 @@ module _ where
 *-cancelˡ-≤-neg : ∀ r .{{_ : Negative r}} → r * p ≤ r * q → p ≥ q
 *-cancelˡ-≤-neg {p} {q} r rewrite *-comm r p | *-comm r q = *-cancelʳ-≤-neg r
 
+nonNeg*nonNeg⇒nonNeg : ∀ p .{{_ : NonNegative p}} q .{{_ : NonNegative q}} → NonNegative (p * q)
+nonNeg*nonNeg⇒nonNeg p q = nonNegative $ begin
+  0ℚ     ≡⟨ *-zeroʳ p ⟨
+  p * 0ℚ ≤⟨ *-monoˡ-≤-nonNeg p (nonNegative⁻¹ q) ⟩
+  p * q  ∎
+  where open ≤-Reasoning
+
+nonPos*nonNeg⇒nonPos : ∀ p .{{_ : NonPositive p}} q .{{_ : NonNegative q}} → NonPositive (p * q)
+nonPos*nonNeg⇒nonPos p q = nonPositive $ begin
+  p * q  ≤⟨ *-monoˡ-≤-nonPos p (nonNegative⁻¹ q) ⟩
+  p * 0ℚ ≡⟨ *-zeroʳ p ⟩
+  0ℚ     ∎
+  where open ≤-Reasoning
+
+nonNeg*nonPos⇒nonPos : ∀ p .{{_ : NonNegative p}} q .{{_ : NonPositive q}} → NonPositive (p * q)
+nonNeg*nonPos⇒nonPos p q = nonPositive $ begin
+  p * q  ≤⟨ *-monoˡ-≤-nonNeg p (nonPositive⁻¹ q) ⟩
+  p * 0ℚ ≡⟨ *-zeroʳ p ⟩
+  0ℚ     ∎
+  where open ≤-Reasoning
+
+nonPos*nonPos⇒nonPos : ∀ p .{{_ : NonPositive p}} q .{{_ : NonPositive q}} → NonNegative (p * q)
+nonPos*nonPos⇒nonPos p q = nonNegative $ begin
+  0ℚ     ≡⟨ *-zeroʳ p ⟨
+  p * 0ℚ ≤⟨ *-monoˡ-≤-nonPos p (nonPositive⁻¹ q) ⟩
+  p * q  ∎
+  where open ≤-Reasoning
+
 ------------------------------------------------------------------------
 -- Properties of _*_ and _<_
 
@@ -1387,6 +1439,34 @@ module _ where
 *-cancelʳ-<-nonPos : ∀ r .{{_ : NonPositive r}} → p * r < q * r → p > q
 *-cancelʳ-<-nonPos {p} {q} r rewrite *-comm p r | *-comm q r = *-cancelˡ-<-nonPos r
 
+pos*pos⇒pos : ∀ p .{{_ : Positive p}} q .{{_ : Positive q}} → Positive (p * q)
+pos*pos⇒pos p q = positive $ begin-strict
+  0ℚ     ≡⟨ *-zeroʳ p ⟨
+  p * 0ℚ <⟨ *-monoʳ-<-pos p (positive⁻¹ q) ⟩
+  p * q  ∎
+  where open ≤-Reasoning
+
+neg*pos⇒neg : ∀ p .{{_ : Negative p}} q .{{_ : Positive q}} → Negative (p * q)
+neg*pos⇒neg p q = negative $ begin-strict
+  p * q  <⟨ *-monoʳ-<-neg p (positive⁻¹ q) ⟩
+  p * 0ℚ ≡⟨ *-zeroʳ p ⟩
+  0ℚ     ∎
+  where open ≤-Reasoning
+
+pos*neg⇒neg : ∀ p .{{_ : Positive p}} q .{{_ : Negative q}} → Negative (p * q)
+pos*neg⇒neg p q = negative $ begin-strict
+  p * q  <⟨ *-monoʳ-<-pos p (negative⁻¹ q) ⟩
+  p * 0ℚ ≡⟨ *-zeroʳ p ⟩
+  0ℚ     ∎
+  where open ≤-Reasoning
+
+neg*neg⇒pos : ∀ p .{{_ : Negative p}} q .{{_ : Negative q}} → Positive (p * q)
+neg*neg⇒pos p q = positive $ begin-strict
+  0ℚ     ≡⟨ *-zeroʳ p ⟨
+  p * 0ℚ <⟨ *-monoʳ-<-neg p (negative⁻¹ q) ⟩
+  p * q  ∎
+  where open ≤-Reasoning
+
 ------------------------------------------------------------------------
 -- Properties of _⊓_
 ------------------------------------------------------------------------