From 54da0617a511aab4529a12bee360d8df696172bc Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 10 Oct 2025 15:54:25 -0700
Subject: [PATCH 01/22] Reorganize signed arithmetic on rational numbers

---
 src/elementary-number-theory.lagda.md         |   2 +
 ...closed-intervals-rational-numbers.lagda.md |   2 +-
 .../absolute-value-rational-numbers.lagda.md  | 255 ++++++++----------
 ...tion-nonnegative-rational-numbers.lagda.md |  10 +-
 ...ddition-positive-rational-numbers.lagda.md |  14 +-
 .../addition-rational-numbers.lagda.md        |   5 +-
 ...roperty-positive-rational-numbers.lagda.md |   4 +-
 ...imedean-property-rational-numbers.lagda.md |   8 +-
 .../difference-rational-numbers.lagda.md      |   2 +-
 .../distance-rational-numbers.lagda.md        |   2 +-
 ...ive-and-negative-rational-numbers.lagda.md | 106 ++++++++
 .../inequality-rational-numbers.lagda.md      |  11 +-
 ...imum-nonnegative-rational-numbers.lagda.md |   4 +-
 ...inima-and-maxima-rational-numbers.lagda.md |   8 +-
 ...closed-intervals-rational-numbers.lagda.md |   1 +
 ...closed-intervals-rational-numbers.lagda.md |   6 +-
 ...ication-negative-rational-numbers.lagda.md | 159 +++++++++++
 ...tion-nonnegative-rational-numbers.lagda.md |  48 +++-
 ...tion-nonpositive-rational-numbers.lagda.md |  14 +-
 ...ive-and-negative-rational-numbers.lagda.md |   9 +-
 ...ication-positive-rational-numbers.lagda.md |  12 +-
 ...icative-group-of-rational-numbers.lagda.md |   4 +-
 ...closed-intervals-rational-numbers.lagda.md |   6 +-
 .../negative-rational-numbers.lagda.md        | 227 ++++------------
 .../nonnegative-rational-numbers.lagda.md     |  98 +++----
 .../nonpositive-rational-numbers.lagda.md     |  55 ++--
 ...ive-and-negative-rational-numbers.lagda.md | 195 ++++++++++----
 .../positive-rational-numbers.lagda.md        | 104 +++----
 .../rational-numbers.lagda.md                 |  21 +-
 .../squares-rational-numbers.lagda.md         |  19 +-
 ...lity-nonnegative-rational-numbers.lagda.md |  10 -
 ...quality-positive-rational-numbers.lagda.md |   5 +-
 ...trict-inequality-rational-numbers.lagda.md |   7 +-
 ...at-bounded-distance-metric-spaces.lagda.md |   8 +-
 .../located-metric-spaces.lagda.md            |   4 +-
 .../metric-space-of-rational-numbers.lagda.md |   8 +-
 ...onal-sequences-approximating-zero.lagda.md |   1 +
 src/order-theory.lagda.md                     |   1 +
 .../order-preserving-maps-posets.lagda.md     |   6 +-
 ...rder-preserving-maps-total-orders.lagda.md | 193 +++++++++++++
 ...ition-lower-dedekind-real-numbers.lagda.md |  16 +-
 ...ition-upper-dedekind-real-numbers.lagda.md |   4 +-
 ...thmetically-located-dedekind-cuts.lagda.md |   6 +-
 .../multiplication-real-numbers.lagda.md      |   1 +
 ...lower-upper-dedekind-real-numbers.lagda.md |  16 +-
 .../negation-real-numbers.lagda.md            |   2 +-
 .../nonnegative-real-numbers.lagda.md         |   5 +-
 .../positive-real-numbers.lagda.md            |  16 +-
 .../strict-inequality-real-numbers.lagda.md   |   2 +-
 49 files changed, 1052 insertions(+), 670 deletions(-)
 create mode 100644 src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
 create mode 100644 src/order-theory/order-preserving-maps-total-orders.lagda.md

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 662f2615dc5..d6d8fc4df2d 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -93,6 +93,7 @@ open import elementary-number-theory.group-of-integers public
 open import elementary-number-theory.half-integers public
 open import elementary-number-theory.hardy-ramanujan-number public
 open import elementary-number-theory.inclusion-natural-numbers-conatural-numbers public
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers public
 open import elementary-number-theory.inequality-arithmetic-geometric-means-integers public
 open import elementary-number-theory.inequality-arithmetic-geometric-means-rational-numbers public
 open import elementary-number-theory.inequality-conatural-numbers public
@@ -137,6 +138,7 @@ open import elementary-number-theory.multiplication-integers public
 open import elementary-number-theory.multiplication-interior-closed-intervals-rational-numbers public
 open import elementary-number-theory.multiplication-lists-of-natural-numbers public
 open import elementary-number-theory.multiplication-natural-numbers public
+open import elementary-number-theory.multiplication-negative-rational-numbers public
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers public
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers public
 open import elementary-number-theory.multiplication-positive-and-negative-integers public
diff --git a/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md
index 21651d39f03..c0568e3e299 100644
--- a/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-closed-intervals-rational-numbers.lagda.md
@@ -69,7 +69,7 @@ abstract
               ( r≤0))
             ( transitive-leq-ℚ (neg-ℚ r) (neg-ℚ p) (rational-abs-ℚ p)
               ( neg-leq-abs-ℚ p)
-              ( neg-leq-ℚ _ _ p≤r))))
+              ( neg-leq-ℚ p≤r))))
       ( λ 0≤r →
         transitive-leq-ℚ
           ( rational-abs-ℚ r)
diff --git a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
index 93b5e7586ee..fa1c1a6995c 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -12,12 +12,15 @@ module elementary-number-theory.absolute-value-rational-numbers where
 open import elementary-number-theory.addition-nonnegative-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -48,29 +51,27 @@ its negation.
 rational-abs-ℚ : ℚ → ℚ
 rational-abs-ℚ q = max-ℚ q (neg-ℚ q)
 
-opaque
-  unfolding neg-ℚ
-
+abstract
   is-nonnegative-rational-abs-ℚ : (q : ℚ) → is-nonnegative-ℚ (rational-abs-ℚ q)
   is-nonnegative-rational-abs-ℚ q =
     rec-coproduct
-      ( λ 0≤q →
+      ( λ is-nonpos-q →
         inv-tr
           ( is-nonnegative-ℚ)
-          ( right-leq-left-max-ℚ
-            ( q)
-            ( neg-ℚ q)
-            ( transitive-leq-ℚ (neg-ℚ q) zero-ℚ q 0≤q (neg-leq-ℚ zero-ℚ q 0≤q)))
-          ( is-nonnegative-leq-zero-ℚ q 0≤q))
-      ( λ q≤0 →
+          ( left-leq-right-max-ℚ _ _
+            ( leq-nonpositive-nonnegative-ℚ
+              ( q , is-nonpos-q)
+              ( neg-ℚ⁰⁻ (q , is-nonpos-q))))
+          ( is-nonnegative-neg-is-nonpositive-ℚ is-nonpos-q))
+      ( λ is-nonneg-q →
         inv-tr
           ( is-nonnegative-ℚ)
-          ( left-leq-right-max-ℚ
-            ( q)
-            ( neg-ℚ q)
-            ( transitive-leq-ℚ q zero-ℚ (neg-ℚ q) (neg-leq-ℚ q zero-ℚ q≤0) q≤0))
-          ( is-nonnegative-leq-zero-ℚ (neg-ℚ q) (neg-leq-ℚ q zero-ℚ q≤0)))
-      ( linear-leq-ℚ zero-ℚ q)
+          ( right-leq-left-max-ℚ _ _
+            ( leq-nonpositive-nonnegative-ℚ
+              ( neg-ℚ⁰⁺ (q , is-nonneg-q))
+              ( q , is-nonneg-q)))
+          ( is-nonneg-q))
+      ( is-nonpositive-or-nonnegative-ℚ q)
 
 abs-ℚ : ℚ → ℚ⁰⁺
 pr1 (abs-ℚ q) = rational-abs-ℚ q
@@ -92,41 +93,43 @@ abstract
   le-abs-le-le-neg-ℚ p q = le-max-le-both-ℚ q p (neg-ℚ p)
 ```
 
-### The absolute value of a nonnegative rational number is the number itself
+### The absolute value of the negation of `q` is the absolute value of `q`
 
 ```agda
-opaque
-  unfolding neg-ℚ
+abstract
+  abs-neg-ℚ : (q : ℚ) → abs-ℚ (neg-ℚ q) ＝ abs-ℚ q
+  abs-neg-ℚ q =
+    eq-ℚ⁰⁺
+      ( ap (max-ℚ (neg-ℚ q)) (neg-neg-ℚ q) ∙ commutative-max-ℚ _ _)
+
+  rational-abs-neg-ℚ : (q : ℚ) → rational-abs-ℚ (neg-ℚ q) ＝ rational-abs-ℚ q
+  rational-abs-neg-ℚ q = ap rational-ℚ⁰⁺ (abs-neg-ℚ q)
+```
+
+### The absolute value of a nonnegative rational number is the number itself
 
+```agda
+abstract
   abs-rational-ℚ⁰⁺ : (q : ℚ⁰⁺) → abs-ℚ (rational-ℚ⁰⁺ q) ＝ q
-  abs-rational-ℚ⁰⁺ (q , nonneg-q) =
+  abs-rational-ℚ⁰⁺ q =
     eq-ℚ⁰⁺
-      ( right-leq-left-max-ℚ
-        ( q)
-        ( neg-ℚ q)
-        ( transitive-leq-ℚ
-          ( neg-ℚ q)
-          ( zero-ℚ)
-          ( q)
-          ( leq-zero-is-nonnegative-ℚ q nonneg-q)
-          ( neg-leq-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ q nonneg-q))))
+      ( right-leq-left-max-ℚ _ _ (leq-nonpositive-nonnegative-ℚ (neg-ℚ⁰⁺ q) q))
 
   rational-abs-zero-leq-ℚ : (q : ℚ) → leq-ℚ zero-ℚ q → rational-abs-ℚ q ＝ q
   rational-abs-zero-leq-ℚ q 0≤q =
-    ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (q , is-nonnegative-leq-zero-ℚ q 0≤q))
+    ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (q , is-nonnegative-leq-zero-ℚ 0≤q))
 
   rational-abs-leq-zero-ℚ :
     (q : ℚ) → leq-ℚ q zero-ℚ → rational-abs-ℚ q ＝ neg-ℚ q
   rational-abs-leq-zero-ℚ q q≤0 =
-    left-leq-right-max-ℚ
-      ( q)
-      ( neg-ℚ q)
-      ( transitive-leq-ℚ
-        ( q)
-        ( zero-ℚ)
-        ( neg-ℚ q)
-        ( neg-leq-ℚ q zero-ℚ q≤0)
-        ( q≤0))
+    equational-reasoning
+      rational-abs-ℚ q
+      ＝ rational-abs-ℚ (neg-ℚ q)
+        by inv (rational-abs-neg-ℚ q)
+      ＝ neg-ℚ q
+        by
+          ap rational-ℚ⁰⁺
+            ( abs-rational-ℚ⁰⁺ (neg-ℚ⁰⁻ (q , is-nonpositive-leq-zero-ℚ q≤0)))
 ```
 
 ### The absolute value of a positive rational number is the number itself
@@ -139,16 +142,6 @@ abstract
     ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ q))
 ```
 
-### The absolute value of the negation of `q` is the absolute value of `q`
-
-```agda
-abstract
-  abs-neg-ℚ : (q : ℚ) → abs-ℚ (neg-ℚ q) ＝ abs-ℚ q
-  abs-neg-ℚ q =
-    eq-ℚ⁰⁺
-      ( ap (max-ℚ (neg-ℚ q)) (neg-neg-ℚ q) ∙ commutative-max-ℚ _ _)
-```
-
 ### `q` is less than or equal to `abs-ℚ q`
 
 ```agda
@@ -163,36 +156,31 @@ abstract
 ### The absolute value of `q` is zero iff `q` is zero
 
 ```agda
-opaque
-  unfolding neg-ℚ
-
-  eq-zero-eq-abs-zero-ℚ : (q : ℚ) → abs-ℚ q ＝ zero-ℚ⁰⁺ → q ＝ zero-ℚ
-  eq-zero-eq-abs-zero-ℚ q abs=0 =
-    rec-coproduct
-      ( λ 0≤q →
-        antisymmetric-leq-ℚ
-          ( q)
-          ( zero-ℚ)
-          ( tr (leq-ℚ q) (ap rational-ℚ⁰⁺ abs=0) (leq-abs-ℚ q)) 0≤q)
-      ( λ q≤0 →
-        antisymmetric-leq-ℚ
-          ( q)
-          ( zero-ℚ)
-          ( q≤0)
-          ( tr
-            ( leq-ℚ zero-ℚ)
-            ( neg-neg-ℚ q)
-            ( neg-leq-ℚ
-              ( neg-ℚ q)
-              ( zero-ℚ)
-              ( tr
-                ( leq-ℚ (neg-ℚ q))
-                ( ap rational-ℚ⁰⁺ abs=0)
-                ( neg-leq-abs-ℚ q)))))
-      ( linear-leq-ℚ zero-ℚ q)
+abstract
+  eq-zero-eq-abs-zero-ℚ : {q : ℚ} → abs-ℚ q ＝ zero-ℚ⁰⁺ → q ＝ zero-ℚ
+  eq-zero-eq-abs-zero-ℚ {q} abs=0 =
+    antisymmetric-leq-ℚ
+      ( q)
+      ( zero-ℚ)
+      ( transitive-leq-ℚ
+        ( q)
+        ( rational-abs-ℚ q)
+        ( zero-ℚ)
+        ( leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ abs=0))
+        ( leq-abs-ℚ q))
+      ( binary-tr
+        ( leq-ℚ)
+        ( ap (neg-ℚ ∘ rational-ℚ⁰⁺) abs=0 ∙ neg-zero-ℚ)
+        ( neg-neg-ℚ q)
+        ( neg-leq-ℚ (neg-leq-abs-ℚ q)))
 
   abs-zero-ℚ : abs-ℚ zero-ℚ ＝ zero-ℚ⁰⁺
-  abs-zero-ℚ = eq-ℚ⁰⁺ (left-leq-right-max-ℚ _ _ (refl-leq-ℚ zero-ℚ))
+  abs-zero-ℚ =
+    eq-ℚ⁰⁺
+      ( equational-reasoning
+        max-ℚ zero-ℚ (neg-ℚ zero-ℚ)
+        ＝ max-ℚ zero-ℚ zero-ℚ by ap-max-ℚ refl neg-zero-ℚ
+        ＝ zero-ℚ by idempotent-max-ℚ zero-ℚ)
 ```
 
 ### The triangle inequality
@@ -236,78 +224,53 @@ abstract
 ### `|ab| = |a||b|`
 
 ```agda
-opaque
-  unfolding neg-ℚ
-
-  abs-left-mul-nonnegative-ℚ :
-    (q : ℚ) (p : ℚ⁰⁺) → abs-ℚ (rational-ℚ⁰⁺ p *ℚ q) ＝ p *ℚ⁰⁺ abs-ℚ q
-  abs-left-mul-nonnegative-ℚ q p⁰⁺@(p , nonneg-p) with linear-leq-ℚ zero-ℚ q
-  ... | inl 0≤q =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ p *ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-rational-ℚ⁰⁺
-                ( p⁰⁺ *ℚ⁰⁺ (q , is-nonnegative-leq-zero-ℚ q 0≤q)))
-        ＝ p *ℚ rational-abs-ℚ q
-          by ap (p *ℚ_) (inv (rational-abs-zero-leq-ℚ q 0≤q)))
-  ... | inr q≤0 =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ rational-abs-ℚ (neg-ℚ (p *ℚ q))
-          by ap rational-ℚ⁰⁺ (inv (abs-neg-ℚ (p *ℚ q)))
-        ＝ rational-abs-ℚ (p *ℚ neg-ℚ q)
-          by ap rational-abs-ℚ (inv (right-negative-law-mul-ℚ p q))
-        ＝ p *ℚ neg-ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-rational-ℚ⁰⁺
-                ( p⁰⁺ *ℚ⁰⁺
-                  ( neg-ℚ q ,
-                    is-nonnegative-leq-zero-ℚ
-                      ( neg-ℚ q)
-                      ( neg-leq-ℚ q zero-ℚ q≤0))))
-        ＝ p *ℚ rational-abs-ℚ q
-          by ap (p *ℚ_) (inv (rational-abs-leq-zero-ℚ q q≤0)))
+abstract
+  rational-abs-left-mul-nonnegative-ℚ :
+    (q : ℚ) (p : ℚ⁰⁺) →
+    rational-abs-ℚ (rational-ℚ⁰⁺ p *ℚ q) ＝ rational-ℚ⁰⁺ p *ℚ rational-abs-ℚ q
+  rational-abs-left-mul-nonnegative-ℚ q p⁰⁺@(p , _) =
+    equational-reasoning
+      rational-abs-ℚ (p *ℚ q)
+      ＝ max-ℚ (p *ℚ q) (p *ℚ neg-ℚ q)
+        by ap-max-ℚ refl (inv (right-negative-law-mul-ℚ p q))
+      ＝ p *ℚ rational-abs-ℚ q
+        by inv (left-distributive-mul-max-ℚ⁰⁺ p⁰⁺ _ _)
 
   abs-mul-ℚ : (p q : ℚ) → abs-ℚ (p *ℚ q) ＝ abs-ℚ p *ℚ⁰⁺ abs-ℚ q
-  abs-mul-ℚ p q with linear-leq-ℚ zero-ℚ p
-  ... | inl 0≤p =
+  abs-mul-ℚ p q =
     eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ p *ℚ rational-abs-ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-left-mul-nonnegative-ℚ
-                ( q)
-                ( p , is-nonnegative-leq-zero-ℚ p 0≤p))
-        ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
-          by ap (_*ℚ rational-abs-ℚ q) (inv (rational-abs-zero-leq-ℚ p 0≤p)))
-  ... | inr p≤0 =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        rational-abs-ℚ (p *ℚ q)
-        ＝ rational-abs-ℚ (neg-ℚ (p *ℚ q))
-          by ap rational-ℚ⁰⁺ (inv (abs-neg-ℚ (p *ℚ q)))
-        ＝ rational-abs-ℚ (neg-ℚ p *ℚ q)
-          by ap rational-abs-ℚ (inv (left-negative-law-mul-ℚ p q))
-        ＝ neg-ℚ p *ℚ rational-abs-ℚ q
-          by
-            ap
-              ( rational-ℚ⁰⁺)
-              ( abs-left-mul-nonnegative-ℚ
-                ( q)
-                ( neg-ℚ p ,
-                  is-nonnegative-leq-zero-ℚ (neg-ℚ p) (neg-leq-ℚ p zero-ℚ p≤0)))
-        ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
-          by ap (_*ℚ rational-abs-ℚ q) (inv (rational-abs-leq-zero-ℚ p p≤0)))
+      ( rec-coproduct
+        ( λ is-nonpos-p →
+          equational-reasoning
+            rational-abs-ℚ (p *ℚ q)
+            ＝ rational-abs-ℚ (neg-ℚ (p *ℚ q))
+              by inv (rational-abs-neg-ℚ _)
+            ＝ rational-abs-ℚ (neg-ℚ p *ℚ q)
+              by ap rational-abs-ℚ (inv (left-negative-law-mul-ℚ p q))
+            ＝ neg-ℚ p *ℚ rational-abs-ℚ q
+              by
+                rational-abs-left-mul-nonnegative-ℚ
+                  ( q)
+                  ( neg-ℚ⁰⁻ (p , is-nonpos-p))
+            ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
+              by
+                ap-mul-ℚ
+                  ( inv
+                    ( rational-abs-leq-zero-ℚ
+                      ( p)
+                      ( leq-zero-is-nonpositive-ℚ is-nonpos-p)))
+                  ( refl))
+        ( λ is-nonneg-p →
+          equational-reasoning
+            rational-abs-ℚ (p *ℚ q)
+            ＝ p *ℚ rational-abs-ℚ q
+              by rational-abs-left-mul-nonnegative-ℚ q (p , is-nonneg-p)
+            ＝ rational-abs-ℚ p *ℚ rational-abs-ℚ q
+              by
+                ap-mul-ℚ
+                  ( inv (ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ (p , is-nonneg-p))))
+                  ( refl))
+        ( is-nonpositive-or-nonnegative-ℚ p))
 
 abstract
   rational-abs-mul-ℚ :
diff --git a/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md
index cf3e298f8bb..fa24223764b 100644
--- a/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-nonnegative-rational-numbers.lagda.md
@@ -35,12 +35,12 @@ nonnegative.
 
 ```agda
 opaque
-  unfolding add-ℚ
+  unfolding add-ℚ is-nonnegative-ℚ
 
   is-nonnegative-add-ℚ :
-    (p q : ℚ) → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
+    {p q : ℚ} → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
     is-nonnegative-ℚ (p +ℚ q)
-  is-nonnegative-add-ℚ p q nonneg-p nonneg-q =
+  is-nonnegative-add-ℚ {p} {q} nonneg-p nonneg-q =
     is-nonnegative-rational-fraction-ℤ
       ( is-nonnegative-add-fraction-ℤ
         { fraction-ℚ p}
@@ -50,7 +50,7 @@ opaque
 
 add-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
 add-ℚ⁰⁺ (p , nonneg-p) (q , nonneg-q) =
-  ( p +ℚ q , is-nonnegative-add-ℚ p q nonneg-p nonneg-q)
+  ( p +ℚ q , is-nonnegative-add-ℚ nonneg-p nonneg-q)
 
 infixl 35 _+ℚ⁰⁺_
 _+ℚ⁰⁺_ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
@@ -73,7 +73,7 @@ abstract
         ( q)
         ( zero-ℚ)
         ( p)
-        ( leq-zero-is-nonnegative-ℚ p nonneg-p))
+        ( leq-zero-is-nonnegative-ℚ nonneg-p))
 
   is-inflationary-map-right-add-rational-ℚ⁰⁺ :
     (p : ℚ⁰⁺) → is-inflationary-map-Poset ℚ-Poset (_+ℚ rational-ℚ⁰⁺ p)
diff --git a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
index 5b5cc12dc1b..85668704afe 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -57,7 +57,7 @@ positive.
 
 ```agda
 opaque
-  unfolding add-ℚ
+  unfolding add-ℚ is-positive-ℚ
 
   is-positive-add-ℚ :
     {x y : ℚ} → is-positive-ℚ x → is-positive-ℚ y → is-positive-ℚ (x +ℚ y)
@@ -147,9 +147,7 @@ module _
         ( rational-ℚ⁺ x)
         ( zero-ℚ)
         ( rational-ℚ⁺ y)
-        ( le-zero-is-positive-ℚ
-          ( rational-ℚ⁺ y)
-          ( is-positive-rational-ℚ⁺ y)))
+        ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ y)))
 
   le-right-add-ℚ⁺ : le-ℚ⁺ y (x +ℚ⁺ y)
   le-right-add-ℚ⁺ =
@@ -160,9 +158,7 @@ module _
         ( rational-ℚ⁺ y)
         ( zero-ℚ)
         ( rational-ℚ⁺ x)
-        ( le-zero-is-positive-ℚ
-          ( rational-ℚ⁺ x)
-          ( is-positive-rational-ℚ⁺ x)))
+        ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x)))
 ```
 
 ### The positive difference of strictly inequal positive rational numbers
@@ -268,9 +264,7 @@ abstract
         ( x)
         ( zero-ℚ)
         ( rational-ℚ⁺ d)
-        ( le-zero-is-positive-ℚ
-          ( rational-ℚ⁺ d)
-          ( is-positive-rational-ℚ⁺ d)))
+        ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ d)))
 
   le-right-add-rational-ℚ⁺ : (x : ℚ) (d : ℚ⁺) → le-ℚ x (x +ℚ (rational-ℚ⁺ d))
   le-right-add-rational-ℚ⁺ x d =
diff --git a/src/elementary-number-theory/addition-rational-numbers.lagda.md b/src/elementary-number-theory/addition-rational-numbers.lagda.md
index 8d887c25a69..0603a977979 100644
--- a/src/elementary-number-theory/addition-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-rational-numbers.lagda.md
@@ -154,8 +154,7 @@ abstract
 
 ```agda
 opaque
-  unfolding add-ℚ
-  unfolding neg-ℚ
+  unfolding add-ℚ neg-ℚ
 
   left-inverse-law-add-ℚ : (x : ℚ) → (neg-ℚ x) +ℚ x ＝ zero-ℚ
   left-inverse-law-add-ℚ x =
@@ -199,7 +198,7 @@ opaque
   distributive-neg-add-ℚ :
     (x y : ℚ) → neg-ℚ (x +ℚ y) ＝ neg-ℚ x +ℚ neg-ℚ y
   distributive-neg-add-ℚ (x , dxp) (y , dyp) =
-    ( inv (preserves-neg-rational-fraction-ℤ (x +fraction-ℤ y))) ∙
+    ( inv (neg-rational-fraction-ℤ (x +fraction-ℤ y))) ∙
     ( eq-ℚ-sim-fraction-ℤ
       ( neg-fraction-ℤ (x +fraction-ℤ y))
       ( add-fraction-ℤ (neg-fraction-ℤ x) (neg-fraction-ℤ y))
diff --git a/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md b/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md
index 3affd486cfb..458395e0a84 100644
--- a/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/archimedean-property-positive-rational-numbers.lagda.md
@@ -13,7 +13,6 @@ open import elementary-number-theory.archimedean-property-rational-numbers
 open import elementary-number-theory.integers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -22,7 +21,6 @@ open import elementary-number-theory.strict-inequality-positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
-open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
@@ -57,7 +55,7 @@ abstract
         asymmetric-le-ℚ
           ( zero-ℚ)
           ( y)
-          ( le-zero-is-positive-ℚ y pos-y)
+          ( le-zero-is-positive-ℚ pos-y)
           ( tr
             ( le-ℚ y)
             ( equational-reasoning
diff --git a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
index 59ab9240074..f2fdcf6a303 100644
--- a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
@@ -15,7 +15,7 @@ open import elementary-number-theory.integers
 open import elementary-number-theory.multiplication-integer-fractions
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.positive-rational-numbers
+-- open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
@@ -42,7 +42,9 @@ of `ℚ` is that for any two
 an `n : ℕ` such that `y` is less than `n` as a rational number times `x`.
 
 ```agda
-abstract
+opaque
+  unfolding is-positive-ℚ
+
   bound-archimedean-property-ℚ :
     (x y : ℚ) →
     is-positive-ℚ x →
@@ -90,5 +92,5 @@ abstract
   exists-greater-natural-ℚ q =
     map-tot-exists
       ( λ _ → tr (le-ℚ q) (right-unit-law-mul-ℚ _))
-      ( archimedean-property-ℚ one-ℚ q _)
+      ( archimedean-property-ℚ one-ℚ q (is-positive-rational-ℚ⁺ one-ℚ⁺))
 ```
diff --git a/src/elementary-number-theory/difference-rational-numbers.lagda.md b/src/elementary-number-theory/difference-rational-numbers.lagda.md
index 45b52b7ba80..21f068f27b5 100644
--- a/src/elementary-number-theory/difference-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/difference-rational-numbers.lagda.md
@@ -183,7 +183,7 @@ abstract
     equational-reasoning
       rational-ℤ x -ℚ rational-ℤ y
       ＝ rational-ℤ x +ℚ rational-ℤ (neg-ℤ y)
-        by ap (rational-ℤ x +ℚ_) (inv (preserves-neg-rational-ℤ y))
+        by ap (rational-ℤ x +ℚ_) (inv (neg-rational-ℤ y))
       ＝ rational-ℤ (x -ℤ y)
         by add-rational-ℤ x (neg-ℤ y)
 ```
diff --git a/src/elementary-number-theory/distance-rational-numbers.lagda.md b/src/elementary-number-theory/distance-rational-numbers.lagda.md
index 8576576ca69..066898a463e 100644
--- a/src/elementary-number-theory/distance-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-rational-numbers.lagda.md
@@ -141,7 +141,7 @@ abstract
       ( equational-reasoning
         p *ℚ rational-dist-ℚ q r
         ＝ rational-abs-ℚ (p *ℚ (q -ℚ r))
-          by ap rational-ℚ⁰⁺ (inv (abs-left-mul-nonnegative-ℚ _ p⁰⁺))
+          by inv (rational-abs-left-mul-nonnegative-ℚ _ p⁰⁺)
         ＝ rational-abs-ℚ (p *ℚ q +ℚ p *ℚ (neg-ℚ r))
           by ap rational-abs-ℚ (left-distributive-mul-add-ℚ p q (neg-ℚ r))
         ＝ rational-abs-ℚ (p *ℚ q -ℚ p *ℚ r)
diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..3c35c0ff087
--- /dev/null
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -0,0 +1,106 @@
+# Inequalities between positive, negative, nonnegative, and nonpositive rational numbers
+
+```agda
+module elementary-number-theory.inequalities-positive-and-negative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.dependent-pair-types
+```
+
+</details>
+
+## Idea
+
+This page describes
+[inequalities](elementary-number-theory.inequalities-rational-numbers.md) and
+[strict inequalities](elementary-number-theory.strict-inequality-rational-numbers.md)
+between [positive](elementary-number-theory.positive-rational-numbers.md),
+[negative](elementary-number-theory.negative-rational-numbers.md),
+[nonnegative](elementary-number-theory.nonnegative-rational-numbers.md), and
+[nonpositive](elementary-number-theory.nonpositive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md).
+
+## Properties
+
+### Inequalities between rational numbers with different signs
+
+Some inequalities can be deduced directly from the sign of a rational number:
+for example, every negative rational number is less than every nonnegative
+rational number.
+
+#### Any negative rational number is less than any nonnegative rational number
+
+```agda
+abstract
+  le-negative-nonnegative-ℚ :
+    (p : ℚ⁻) (q : ℚ⁰⁺) → le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁰⁺ q)
+  le-negative-nonnegative-ℚ (p , neg-p) (q , nonneg-q) =
+    concatenate-le-leq-ℚ p zero-ℚ q
+      ( le-zero-is-negative-ℚ neg-p)
+      ( leq-zero-is-nonnegative-ℚ nonneg-q)
+
+  leq-negative-nonnegative-ℚ :
+    (p : ℚ⁻) (q : ℚ⁰⁺) → leq-ℚ (rational-ℚ⁻ p) (rational-ℚ⁰⁺ q)
+  leq-negative-nonnegative-ℚ p q = leq-le-ℚ (le-negative-nonnegative-ℚ p q)
+```
+
+#### A nonpositive rational number is less than a positive rational number
+
+```agda
+abstract
+  le-nonpositive-positive-ℚ :
+    (p : ℚ⁰⁻) (q : ℚ⁺) → le-ℚ (rational-ℚ⁰⁻ p) (rational-ℚ⁺ q)
+  le-nonpositive-positive-ℚ (p , nonpos-p) (q , pos-q) =
+    concatenate-leq-le-ℚ p zero-ℚ q
+      ( leq-zero-is-nonpositive-ℚ nonpos-p)
+      ( le-zero-is-positive-ℚ pos-q)
+```
+
+#### A nonpositive rational number is less than or equal to a nonnegative rational number
+
+```agda
+abstract
+  leq-nonpositive-nonnegative-ℚ :
+    (p : ℚ⁰⁻) (q : ℚ⁰⁺) → leq-ℚ (rational-ℚ⁰⁻ p) (rational-ℚ⁰⁺ q)
+  leq-nonpositive-nonnegative-ℚ (p , nonpos-p) (q , nonneg-q) =
+    transitive-leq-ℚ p zero-ℚ q
+      ( leq-zero-is-nonnegative-ℚ nonneg-q)
+      ( leq-zero-is-nonpositive-ℚ nonpos-p)
+```
+
+### Inequalities showing the sign of a rational number
+
+#### If `p < q` and `p` is nonnegative, then `q` is positive
+
+```agda
+abstract
+  is-positive-le-ℚ⁰⁺ :
+    (p : ℚ⁰⁺) (q : ℚ) → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
+  is-positive-le-ℚ⁰⁺ (p , nonneg-p) q p<q =
+    is-positive-le-zero-ℚ
+      ( concatenate-leq-le-ℚ _ _ _ (leq-zero-is-nonnegative-ℚ nonneg-p) p<q)
+```
+
+#### If `p < q` and `q` is nonpositive, then `p` is negative
+
+```agda
+abstract
+  is-negative-le-ℚ⁰⁻ :
+    (q : ℚ⁰⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁰⁻ q) → is-negative-ℚ p
+  is-negative-le-ℚ⁰⁻ (q , nonpos-q) p p<q =
+    is-negative-le-zero-ℚ
+      ( concatenate-le-leq-ℚ p q zero-ℚ
+        ( p<q)
+        ( leq-zero-is-nonpositive-ℚ nonpos-q))
+```
diff --git a/src/elementary-number-theory/inequality-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
index 39552d30824..14d1a9398e4 100644
--- a/src/elementary-number-theory/inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
@@ -340,9 +340,7 @@ module _
   where
 
   opaque
-    unfolding add-ℚ
-    unfolding leq-ℚ-Prop
-    unfolding neg-ℚ
+    unfolding add-ℚ leq-ℚ-Prop neg-ℚ
 
     iff-translate-diff-leq-zero-ℚ : leq-ℚ zero-ℚ (y -ℚ x) ↔ leq-ℚ x y
     iff-translate-diff-leq-zero-ℚ =
@@ -445,11 +443,10 @@ abstract
 
 ```agda
 opaque
-  unfolding leq-ℚ-Prop
-  unfolding neg-ℚ
+  unfolding leq-ℚ-Prop neg-ℚ
 
-  neg-leq-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℚ (neg-ℚ y) (neg-ℚ x)
-  neg-leq-ℚ x y = neg-leq-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
+  neg-leq-ℚ : {x y : ℚ} → leq-ℚ x y → leq-ℚ (neg-ℚ y) (neg-ℚ x)
+  neg-leq-ℚ {x} {y} = neg-leq-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
 ```
 
 ### Transposing additions on inequalities of rational numbers
diff --git a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
index 1bd6c1e4d2f..f7b3619d5e2 100644
--- a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
@@ -36,13 +36,13 @@ abstract
   is-nonnegative-max-ℚ⁰⁺ :
     (p q : ℚ⁰⁺) → is-nonnegative-ℚ (max-ℚ (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q))
   is-nonnegative-max-ℚ⁰⁺ (p , is-nonneg-p) (q , is-nonneg-q) =
-    is-nonnegative-leq-zero-ℚ _
+    is-nonnegative-leq-zero-ℚ
       ( transitive-leq-ℚ
         ( zero-ℚ)
         ( p)
         ( max-ℚ p q)
         ( leq-left-max-ℚ p q)
-        ( leq-zero-is-nonnegative-ℚ p is-nonneg-p))
+        ( leq-zero-is-nonnegative-ℚ is-nonneg-p))
 
 max-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
 max-ℚ⁰⁺ p⁰⁺@(p , _) q⁰⁺@(q , _) = (max-ℚ p q , is-nonnegative-max-ℚ⁰⁺ p⁰⁺ q⁰⁺)
diff --git a/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md b/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md
index 9c62ce02864..87e2c01b901 100644
--- a/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/minima-and-maxima-rational-numbers.lagda.md
@@ -41,10 +41,10 @@ module _
       rec-coproduct
         ( λ p≤q →
           ( ap neg-ℚ (left-leq-right-min-ℚ p q p≤q)) ∙
-          ( inv (right-leq-left-max-ℚ _ _ (neg-leq-ℚ _ _ p≤q))))
+          ( inv (right-leq-left-max-ℚ _ _ (neg-leq-ℚ p≤q))))
         ( λ q≤p →
           ( ap neg-ℚ (right-leq-left-min-ℚ p q q≤p)) ∙
-          ( inv (left-leq-right-max-ℚ _ _ (neg-leq-ℚ _ _ q≤p))))
+          ( inv (left-leq-right-max-ℚ _ _ (neg-leq-ℚ q≤p))))
         ( linear-leq-ℚ p q)
 ```
 
@@ -61,9 +61,9 @@ module _
       rec-coproduct
         ( λ p≤q →
           ( ap neg-ℚ (left-leq-right-max-ℚ _ _ p≤q)) ∙
-          ( inv (right-leq-left-min-ℚ _ _ (neg-leq-ℚ _ _ p≤q))))
+          ( inv (right-leq-left-min-ℚ _ _ (neg-leq-ℚ p≤q))))
         ( λ q≤p →
           ( ap neg-ℚ (right-leq-left-max-ℚ _ _ q≤p)) ∙
-          ( inv (left-leq-right-min-ℚ _ _ (neg-leq-ℚ _ _ q≤p))))
+          ( inv (left-leq-right-min-ℚ _ _ (neg-leq-ℚ q≤p))))
         ( linear-leq-ℚ p q)
 ```
diff --git a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
index 761616c9dcb..a13fd9dd5b3 100644
--- a/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-closed-intervals-rational-numbers.lagda.md
@@ -23,6 +23,7 @@ open import elementary-number-theory.maximum-nonnegative-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minima-and-maxima-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
+open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
diff --git a/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
index b0d806d19f3..b05bb91cc05 100644
--- a/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-interior-closed-intervals-rational-numbers.lagda.md
@@ -9,10 +9,12 @@ module elementary-number-theory.multiplication-interior-closed-intervals-rationa
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.decidable-total-order-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.interior-closed-intervals-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
+open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
@@ -333,7 +335,7 @@ abstract
               inv-tr
                 ( is-nonnegative-ℚ)
                 ( min'=a'c')
-                ( is-nonnegative-mul-ℚ a' c' is-nonneg-a' is-nonneg-c')
+                ( is-nonnegative-mul-ℚ is-nonneg-a' is-nonneg-c')
           in
             rec-coproduct
               ( λ is-neg-a →
@@ -567,7 +569,7 @@ abstract
           ( neg-ℚ)
           ( right-negative-law-lower-bound-mul-closed-interval-ℚ [a,b] [c,d])) ∙
         ( neg-neg-ℚ _))
-      ( neg-le-ℚ _ _
+      ( neg-le-ℚ
         ( le-lower-bound-mul-interior-closed-interval-ℚ
           ( [a,b])
           ( neg-closed-interval-ℚ [c,d])
diff --git a/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..164dcab84e8
--- /dev/null
+++ b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
@@ -0,0 +1,159 @@
+# Multiplication by negative rational numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module elementary-number-theory.multiplication-negative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.identity-types
+open import foundation.transport-along-identifications
+```
+
+</details>
+
+## Idea
+
+The product of two
+[negative rational numbers](elementary-number-theory.negative-rational-numbers.md)
+is their [product](elementary-number-theory.multiplication-rational-numbers.md)
+as [rational numbers](elementary-number-theory.rational-numbers.md), and is
+[positive](elementary-number-theory.positive-rational-numbers.md).
+
+## Properties
+
+### The product of two negative rational numbers is positive
+
+```agda
+opaque
+  unfolding is-negative-ℚ
+
+  is-positive-mul-negative-ℚ :
+    {x y : ℚ} → is-negative-ℚ x → is-negative-ℚ y → is-positive-ℚ (x *ℚ y)
+  is-positive-mul-negative-ℚ {x} {y} P Q =
+    tr
+      ( is-positive-ℚ)
+      ( negative-law-mul-ℚ x y)
+      ( is-positive-mul-ℚ P Q)
+```
+
+### Multiplication by a negative rational number reverses inequality
+
+```agda
+module _
+  (p : ℚ⁻)
+  (q r : ℚ)
+  (H : leq-ℚ q r)
+  where
+
+  abstract
+    reverses-leq-right-mul-ℚ⁻ : leq-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
+    reverses-leq-right-mul-ℚ⁻ =
+      binary-tr
+        ( leq-ℚ)
+        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
+        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
+        ( preserves-leq-right-mul-ℚ⁺
+          ( neg-ℚ⁻ p)
+          ( neg-ℚ r)
+          ( neg-ℚ q)
+          ( neg-leq-ℚ H))
+```
+
+### Multiplication by a negative rational number reverses strict inequality
+
+```agda
+module _
+  (p : ℚ⁻)
+  (q r : ℚ)
+  (H : le-ℚ q r)
+  where
+
+  abstract
+    reverses-le-right-mul-ℚ⁻ : le-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
+    reverses-le-right-mul-ℚ⁻ =
+      binary-tr
+        ( le-ℚ)
+        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
+        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
+        ( preserves-le-right-mul-ℚ⁺
+          ( neg-ℚ⁻ p)
+          ( neg-ℚ r)
+          ( neg-ℚ q)
+          ( neg-le-ℚ H))
+
+    reverses-le-left-mul-ℚ⁻ : le-ℚ (rational-ℚ⁻ p *ℚ r) (rational-ℚ⁻ p *ℚ q)
+    reverses-le-left-mul-ℚ⁻ =
+      binary-tr
+        ( le-ℚ)
+        ( commutative-mul-ℚ _ _)
+        ( commutative-mul-ℚ _ _)
+        ( reverses-le-right-mul-ℚ⁻)
+```
+
+### The negative rational numbers are invertible elements of the multiplicative monoid of rational numbers
+
+```agda
+opaque
+  inv-ℚ⁻ : ℚ⁻ → ℚ⁻
+  inv-ℚ⁻ q = neg-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q))
+
+  left-inverse-law-mul-ℚ⁻ :
+    (q : ℚ⁻) → rational-ℚ⁻ (inv-ℚ⁻ q) *ℚ rational-ℚ⁻ q ＝ one-ℚ
+  left-inverse-law-mul-ℚ⁻ q =
+    inv (swap-neg-mul-ℚ _ _) ∙
+    ap rational-ℚ⁺ (left-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
+
+  right-inverse-law-mul-ℚ⁻ :
+    (q : ℚ⁻) → rational-ℚ⁻ q *ℚ rational-ℚ⁻ (inv-ℚ⁻ q) ＝ one-ℚ
+  right-inverse-law-mul-ℚ⁻ q =
+    swap-neg-mul-ℚ _ _ ∙
+    ap rational-ℚ⁺ (right-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
+```
+
+### Inversion reverses inequality on negative rational numbers
+
+```agda
+opaque
+  unfolding inv-ℚ⁻
+
+  reverses-leq-inv-ℚ⁻ :
+    (p q : ℚ⁻) → leq-ℚ⁻ p q → leq-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
+  reverses-leq-inv-ℚ⁻ p q p≤q =
+    neg-leq-ℚ
+      ( inv-leq-ℚ⁺
+        ( neg-ℚ⁻ q)
+        ( neg-ℚ⁻ p)
+        ( neg-leq-ℚ p≤q))
+```
+
+### Inversion reverses strict inequality on negative rational numbers
+
+```agda
+opaque
+  unfolding inv-ℚ⁻
+
+  reverses-le-inv-ℚ⁻ :
+    (p q : ℚ⁻) → le-ℚ⁻ p q → le-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
+  reverses-le-inv-ℚ⁻ p q p<q =
+    neg-le-ℚ
+      ( inv-le-ℚ⁺
+        ( neg-ℚ⁻ q)
+        ( neg-ℚ⁻ p)
+        ( neg-le-ℚ p<q))
+```
diff --git a/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md
index 203f33961f2..e32b11124b7 100644
--- a/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-nonnegative-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of nonnegative rational numbers
+# Multiplication by nonnegative rational numbers
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -9,9 +9,12 @@ module elementary-number-theory.multiplication-nonnegative-rational-numbers wher
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.decidable-total-order-rational-numbers
 open import elementary-number-theory.inequality-integers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.maximum-rational-numbers
+open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.multiplication-integer-fractions
 open import elementary-number-theory.multiplication-integers
 open import elementary-number-theory.multiplication-positive-and-negative-integers
@@ -23,6 +26,8 @@ open import elementary-number-theory.rational-numbers
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.identity-types
+
+open import order-theory.order-preserving-maps-total-orders
 ```
 
 </details>
@@ -41,12 +46,12 @@ itself nonnegative.
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-nonnegative-ℚ mul-ℚ
 
   is-nonnegative-mul-ℚ :
-    (p q : ℚ) → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
+    {p q : ℚ} → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
     is-nonnegative-ℚ (p *ℚ q)
-  is-nonnegative-mul-ℚ p q nonneg-p nonneg-q =
+  is-nonnegative-mul-ℚ {p} {q} nonneg-p nonneg-q =
     is-nonnegative-rational-fraction-ℤ
       ( is-nonnegative-mul-nonnegative-fraction-ℤ
         { fraction-ℚ p}
@@ -56,7 +61,7 @@ opaque
 
 mul-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
 mul-ℚ⁰⁺ (p , nonneg-p) (q , nonneg-q) =
-  ( p *ℚ q , is-nonnegative-mul-ℚ p q nonneg-p nonneg-q)
+  ( p *ℚ q , is-nonnegative-mul-ℚ nonneg-p nonneg-q)
 
 infixl 35 _*ℚ⁰⁺_
 _*ℚ⁰⁺_ : ℚ⁰⁺ → ℚ⁰⁺ → ℚ⁰⁺
@@ -77,8 +82,7 @@ abstract
 
 ```agda
 opaque
-  unfolding leq-ℚ-Prop
-  unfolding mul-ℚ
+  unfolding is-nonnegative-ℚ leq-ℚ-Prop mul-ℚ
 
   preserves-leq-right-mul-ℚ⁰⁺ :
     (p : ℚ⁰⁺) (q r : ℚ) → leq-ℚ q r →
@@ -123,3 +127,33 @@ abstract
       ( preserves-leq-right-mul-ℚ⁰⁺ s _ _ p≤q)
       ( preserves-leq-left-mul-ℚ⁰⁺ p _ _ r≤s)
 ```
+
+### Multiplication by a nonnegative rational number distributes over the minimum operation
+
+```agda
+abstract
+  left-distributive-mul-min-ℚ⁰⁺ :
+    (p : ℚ⁰⁺) (q r : ℚ) →
+    rational-ℚ⁰⁺ p *ℚ min-ℚ q r ＝
+    min-ℚ (rational-ℚ⁰⁺ p *ℚ q) (rational-ℚ⁰⁺ p *ℚ r)
+  left-distributive-mul-min-ℚ⁰⁺ p⁰⁺@(p , _) =
+    distributive-map-hom-min-Total-Order
+      ( ℚ-Total-Order)
+      ( ℚ-Total-Order)
+      ( p *ℚ_ , preserves-leq-left-mul-ℚ⁰⁺ p⁰⁺)
+```
+
+### Multiplication by a nonnegative rational number distributes over the maximum operation
+
+```agda
+abstract
+  left-distributive-mul-max-ℚ⁰⁺ :
+    (p : ℚ⁰⁺) (q r : ℚ) →
+    rational-ℚ⁰⁺ p *ℚ max-ℚ q r ＝
+    max-ℚ (rational-ℚ⁰⁺ p *ℚ q) (rational-ℚ⁰⁺ p *ℚ r)
+  left-distributive-mul-max-ℚ⁰⁺ p⁰⁺@(p , _) =
+    distributive-map-hom-max-Total-Order
+      ( ℚ-Total-Order)
+      ( ℚ-Total-Order)
+      ( p *ℚ_ , preserves-leq-left-mul-ℚ⁰⁺ p⁰⁺)
+```
diff --git a/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md
index 42b601b5891..f5bb0a14227 100644
--- a/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-nonpositive-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of nonpositive rational numbers
+# Multiplication by nonpositive rational numbers
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -36,7 +36,9 @@ as [rational numbers](elementary-number-theory.rational-numbers.md), and is
 ## Definition
 
 ```agda
-abstract
+opaque
+  unfolding is-nonpositive-ℚ
+
   is-nonnegative-mul-nonpositive-ℚ :
     {x y : ℚ} → is-nonpositive-ℚ x → is-nonpositive-ℚ y →
     is-nonnegative-ℚ (x *ℚ y)
@@ -44,7 +46,7 @@ abstract
     tr
       ( is-nonnegative-ℚ)
       ( negative-law-mul-ℚ x y)
-      ( is-nonnegative-mul-ℚ _ _ nonpos-x nonpos-y)
+      ( is-nonnegative-mul-ℚ nonpos-x nonpos-y)
 
 mul-nonpositive-ℚ : ℚ⁰⁻ → ℚ⁰⁻ → ℚ⁰⁺
 mul-nonpositive-ℚ (p , nonpos-p) (q , nonpos-q) =
@@ -56,7 +58,9 @@ mul-nonpositive-ℚ (p , nonpos-p) (q , nonpos-q) =
 ### Multiplication by a nonpositive rational number reverses inequality
 
 ```agda
-abstract
+opaque
+  unfolding is-nonpositive-ℚ
+
   reverses-leq-right-mul-ℚ⁰⁻ :
     (p : ℚ⁰⁻) (q r : ℚ) → leq-ℚ q r →
     leq-ℚ (r *ℚ rational-ℚ⁰⁻ p) (q *ℚ rational-ℚ⁰⁻ p)
@@ -65,6 +69,6 @@ abstract
       ( leq-ℚ)
       ( ap neg-ℚ (right-negative-law-mul-ℚ r p) ∙ neg-neg-ℚ _)
       ( ap neg-ℚ (right-negative-law-mul-ℚ q p) ∙ neg-neg-ℚ _)
-      ( neg-leq-ℚ _ _
+      ( neg-leq-ℚ
         ( preserves-leq-right-mul-ℚ⁰⁺ (neg-ℚ p , nonpos-p) q r q≤r))
 ```
diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index bf55c8c8e74..63a7876bb54 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of positive, negative, and nonnegative rational numbers
+# Multiplication by positive, negative, and nonnegative rational numbers
 
 ```agda
 module elementary-number-theory.multiplication-positive-and-negative-rational-numbers where
@@ -25,8 +25,9 @@ open import foundation.transport-along-identifications
 ## Idea
 
 When we have information about the sign of the factors of a
-[rational product](elementary-number-theory.multiplication-rational-numbers.md),
-we can deduce the sign of their product too.
+[rational](elementary-number-theory.rational-numbers.md)
+[product](elementary-number-theory.multiplication-rational-numbers.md), we can
+deduce the sign of their product too.
 
 ## Lemmas
 
@@ -34,7 +35,7 @@ we can deduce the sign of their product too.
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ mul-ℚ
 
   is-negative-mul-positive-negative-ℚ :
     {x y : ℚ} → is-positive-ℚ x → is-negative-ℚ y → is-negative-ℚ (x *ℚ y)
diff --git a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
index fd4ad86379d..1c43c7546b9 100644
--- a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
@@ -1,4 +1,4 @@
-# Multiplication of positive rational numbers
+# Multiplication by positive rational numbers
 
 ```agda
 {-# OPTIONS --lossy-unification #-}
@@ -59,7 +59,7 @@ itself positive.
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ mul-ℚ
 
   is-positive-mul-ℚ :
     {x y : ℚ} → is-positive-ℚ x → is-positive-ℚ y → is-positive-ℚ (x *ℚ y)
@@ -147,7 +147,7 @@ module _
   where
 
   opaque
-    unfolding mul-ℚ
+    unfolding is-positive-ℚ mul-ℚ
 
     inv-is-positive-ℚ : ℚ
     pr1 inv-is-positive-ℚ = inv-is-positive-fraction-ℤ (fraction-ℚ x) P
@@ -215,8 +215,7 @@ module _
 
 ```agda
 opaque
-  unfolding le-ℚ-Prop
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ le-ℚ-Prop mul-ℚ
 
   preserves-le-left-mul-ℚ⁺ :
     (p : ℚ⁺) (q r : ℚ) →
@@ -256,8 +255,7 @@ opaque
 
 ```agda
 opaque
-  unfolding leq-ℚ-Prop
-  unfolding mul-ℚ
+  unfolding is-positive-ℚ leq-ℚ-Prop mul-ℚ
 
   preserves-leq-left-mul-ℚ⁺ :
     (p : ℚ⁺) (q r : ℚ) → leq-ℚ q r →
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index 37fa4dc41d4..4db36a011c9 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -138,7 +138,9 @@ module _
       ( H)
       ( inv K)
 
-  abstract
+  opaque
+    unfolding is-negative-ℚ
+
     is-invertible-element-ring-is-nonzero-ℚ :
       is-nonzero-ℚ x → is-invertible-element-Ring ring-ℚ x
     is-invertible-element-ring-is-nonzero-ℚ H =
diff --git a/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md b/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md
index 27b21b5e0b1..36d0cbef5f4 100644
--- a/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negation-closed-intervals-rational-numbers.lagda.md
@@ -32,7 +32,7 @@ of a
 ```agda
 neg-closed-interval-ℚ : closed-interval-ℚ → closed-interval-ℚ
 neg-closed-interval-ℚ ((a , b) , a≤b) =
-  ((neg-ℚ b , neg-ℚ a) , neg-leq-ℚ a b a≤b)
+  ((neg-ℚ b , neg-ℚ a) , neg-leq-ℚ a≤b)
 ```
 
 ## Properties
@@ -48,7 +48,7 @@ abstract
       ( neg-closed-interval-ℚ [a,b])
       ( neg-closed-interval-ℚ [c,d])
   is-interior-neg-closed-interval-ℚ ((a , b) , _) ((c , d) , _) (a<c , d<b) =
-    ( neg-le-ℚ d b d<b , neg-le-ℚ a c a<c)
+    ( neg-le-ℚ d<b , neg-le-ℚ a<c)
 ```
 
 ### Negation preserves nontriviality
@@ -58,5 +58,5 @@ abstract
   is-nontrivial-neg-closed-interval-ℚ :
     ([a,b] : closed-interval-ℚ) → is-proper-closed-interval-ℚ [a,b] →
     is-proper-closed-interval-ℚ (neg-closed-interval-ℚ [a,b])
-  is-nontrivial-neg-closed-interval-ℚ ((a , b) , _) a<b = neg-le-ℚ a b a<b
+  is-nontrivial-neg-closed-interval-ℚ ((a , b) , _) a<b = neg-le-ℚ a<b
 ```
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 0e179958514..be110dba249 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -13,9 +13,6 @@ open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
-open import elementary-number-theory.multiplication-positive-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.negative-integer-fractions
 open import elementary-number-theory.negative-integers
 open import elementary-number-theory.nonzero-rational-numbers
@@ -60,11 +57,12 @@ module _
   (q : ℚ)
   where
 
-  is-negative-ℚ : UU lzero
-  is-negative-ℚ = is-positive-ℚ (neg-ℚ q)
+  opaque
+    is-negative-ℚ : UU lzero
+    is-negative-ℚ = is-positive-ℚ (neg-ℚ q)
 
-  is-prop-is-negative-ℚ : is-prop is-negative-ℚ
-  is-prop-is-negative-ℚ = is-prop-is-positive-ℚ (neg-ℚ q)
+    is-prop-is-negative-ℚ : is-prop is-negative-ℚ
+    is-prop-is-negative-ℚ = is-prop-is-positive-ℚ (neg-ℚ q)
 
   is-negative-prop-ℚ : Prop lzero
   pr1 is-negative-prop-ℚ = is-negative-ℚ
@@ -100,16 +98,14 @@ module _
   is-negative-rational-ℚ⁻ = pr2 x
 
   opaque
-    unfolding neg-ℚ
+    unfolding is-negative-ℚ is-positive-ℚ neg-ℚ
 
     is-negative-fraction-ℚ⁻ : is-negative-fraction-ℤ fraction-ℚ⁻
     is-negative-fraction-ℚ⁻ =
       tr
-        ( is-negative-fraction-ℤ)
-        ( neg-neg-fraction-ℤ fraction-ℚ⁻)
-        ( is-negative-neg-positive-fraction-ℤ
-          ( neg-fraction-ℤ fraction-ℚ⁻)
-          ( is-negative-rational-ℚ⁻))
+        ( is-negative-ℤ)
+        ( neg-neg-ℤ _)
+        ( is-negative-neg-is-positive-ℤ is-negative-rational-ℚ⁻)
 
   is-negative-numerator-ℚ⁻ : is-negative-ℤ numerator-ℚ⁻
   is-negative-numerator-ℚ⁻ = is-negative-fraction-ℚ⁻
@@ -117,12 +113,6 @@ module _
   is-positive-denominator-ℚ⁻ : is-positive-ℤ denominator-ℚ⁻
   is-positive-denominator-ℚ⁻ = is-positive-denominator-ℚ rational-ℚ⁻
 
-  neg-ℚ⁻ : ℚ⁺
-  neg-ℚ⁻ = (neg-ℚ rational-ℚ⁻) , is-negative-rational-ℚ⁻
-
-neg-ℚ⁺ : ℚ⁺ → ℚ⁻
-neg-ℚ⁺ (q , q-is-pos) = neg-ℚ q , inv-tr is-positive-ℚ (neg-neg-ℚ q) q-is-pos
-
 abstract
   eq-ℚ⁻ : {x y : ℚ⁻} → rational-ℚ⁻ x ＝ rational-ℚ⁻ y → x ＝ y
   eq-ℚ⁻ {x} {y} = eq-type-subtype is-negative-prop-ℚ
@@ -141,7 +131,7 @@ is-set-ℚ⁻ = is-set-type-subtype is-negative-prop-ℚ is-set-ℚ
 
 ```agda
 opaque
-  unfolding neg-ℚ
+  unfolding is-negative-ℚ is-positive-ℚ neg-ℚ
 
   is-negative-rational-ℤ :
     (x : ℤ) → is-negative-ℤ x → is-negative-ℚ (rational-ℤ x)
@@ -156,41 +146,49 @@ negative-rational-negative-ℤ (x , x-is-neg) =
 
 ```agda
 opaque
-  unfolding neg-ℚ
-  unfolding rational-fraction-ℤ
+  unfolding is-negative-ℚ
 
   is-negative-rational-fraction-ℤ :
-    {x : fraction-ℤ} (P : is-negative-fraction-ℤ x) →
+    {x : fraction-ℤ} → is-negative-fraction-ℤ x →
     is-negative-ℚ (rational-fraction-ℤ x)
-  is-negative-rational-fraction-ℤ P =
-    is-positive-neg-is-negative-ℤ (is-negative-reduce-fraction-ℤ P)
+  is-negative-rational-fraction-ℤ {x} P =
+    tr
+      ( is-positive-ℚ)
+      ( neg-rational-fraction-ℤ _)
+      ( is-positive-rational-fraction-ℤ
+        ( is-positive-neg-negative-fraction-ℤ x P))
 ```
 
 ### A rational number `x` is negative if and only if `x < 0`
 
 ```agda
-module _
-  (x : ℚ)
-  where
+opaque
+  unfolding is-negative-ℚ
 
-  opaque
-    unfolding neg-ℚ
+  le-zero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → le-ℚ x zero-ℚ
+  le-zero-is-negative-ℚ {x} is-neg-x =
+    binary-tr
+      ( le-ℚ)
+      ( neg-neg-ℚ x)
+      ( neg-zero-ℚ)
+      ( neg-le-ℚ (le-zero-is-positive-ℚ is-neg-x))
+
+  is-negative-le-zero-ℚ : {x : ℚ} → le-ℚ x zero-ℚ → is-negative-ℚ x
+  is-negative-le-zero-ℚ {x} x<0 =
+    is-positive-le-zero-ℚ
+      ( tr
+        ( λ y → le-ℚ y (neg-ℚ x))
+        ( neg-zero-ℚ)
+        ( neg-le-ℚ x<0))
+
+is-negative-iff-le-zero-ℚ : (x : ℚ) → is-negative-ℚ x ↔ le-ℚ x zero-ℚ
+is-negative-iff-le-zero-ℚ x =
+  ( le-zero-is-negative-ℚ ,
+    is-negative-le-zero-ℚ)
 
-    le-zero-is-negative-ℚ : is-negative-ℚ x → le-ℚ x zero-ℚ
-    le-zero-is-negative-ℚ x-is-neg =
-      tr
-        ( λ y → le-ℚ y zero-ℚ)
-        ( neg-neg-ℚ x)
-        ( neg-le-ℚ zero-ℚ (neg-ℚ x) (le-zero-is-positive-ℚ (neg-ℚ x) x-is-neg))
-
-    is-negative-le-zero-ℚ : le-ℚ x zero-ℚ → is-negative-ℚ x
-    is-negative-le-zero-ℚ x-is-neg =
-      is-positive-le-zero-ℚ (neg-ℚ x) (neg-le-ℚ x zero-ℚ x-is-neg)
-
-    is-negative-iff-le-zero-ℚ : is-negative-ℚ x ↔ le-ℚ x zero-ℚ
-    is-negative-iff-le-zero-ℚ =
-      le-zero-is-negative-ℚ ,
-      is-negative-le-zero-ℚ
+abstract
+  leq-zero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → leq-ℚ x zero-ℚ
+  leq-zero-is-negative-ℚ is-neg-x = leq-le-ℚ (le-zero-is-negative-ℚ is-neg-x)
 ```
 
 ### The difference of a rational number with a greater rational number is negative
@@ -200,7 +198,9 @@ module _
   (x y : ℚ) (H : le-ℚ x y)
   where
 
-  abstract
+  opaque
+    unfolding is-negative-ℚ
+
     is-negative-diff-le-ℚ : is-negative-ℚ (x -ℚ y)
     is-negative-diff-le-ℚ =
       inv-tr
@@ -215,7 +215,9 @@ module _
 ### Negative rational numbers are nonzero
 
 ```agda
-abstract
+opaque
+  unfolding is-negative-ℚ
+
   is-nonzero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → is-nonzero-ℚ x
   is-nonzero-is-negative-ℚ {x} H =
     tr
@@ -227,98 +229,6 @@ nonzero-ℚ⁻ : ℚ⁻ → nonzero-ℚ
 nonzero-ℚ⁻ (x , N) = (x , is-nonzero-is-negative-ℚ N)
 ```
 
-### The product of two negative rational numbers is positive
-
-```agda
-opaque
-  unfolding mul-ℚ
-  unfolding rational-fraction-ℤ
-
-  is-positive-mul-negative-ℚ :
-    {x y : ℚ} → is-negative-ℚ x → is-negative-ℚ y → is-positive-ℚ (x *ℚ y)
-  is-positive-mul-negative-ℚ {x} {y} P Q =
-    is-positive-reduce-fraction-ℤ
-      ( is-positive-mul-negative-fraction-ℤ
-        { fraction-ℚ x}
-        { fraction-ℚ y}
-        ( is-negative-fraction-ℚ⁻ (x , P))
-        ( is-negative-fraction-ℚ⁻ (y , Q)))
-```
-
-### Multiplication by a negative rational number reverses inequality
-
-```agda
-module _
-  (p : ℚ⁻)
-  (q r : ℚ)
-  (H : leq-ℚ q r)
-  where
-
-  abstract
-    reverses-leq-right-mul-ℚ⁻ : leq-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
-    reverses-leq-right-mul-ℚ⁻ =
-      binary-tr
-        ( leq-ℚ)
-        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
-        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
-        ( preserves-leq-right-mul-ℚ⁺
-          ( neg-ℚ⁻ p)
-          ( neg-ℚ r)
-          ( neg-ℚ q)
-          ( neg-leq-ℚ q r H))
-```
-
-### Multiplication by a negative rational number reverses strict inequality
-
-```agda
-module _
-  (p : ℚ⁻)
-  (q r : ℚ)
-  (H : le-ℚ q r)
-  where
-
-  abstract
-    reverses-le-right-mul-ℚ⁻ : le-ℚ (r *ℚ rational-ℚ⁻ p) (q *ℚ rational-ℚ⁻ p)
-    reverses-le-right-mul-ℚ⁻ =
-      binary-tr
-        ( le-ℚ)
-        ( negative-law-mul-ℚ r (rational-ℚ⁻ p))
-        ( negative-law-mul-ℚ q (rational-ℚ⁻ p))
-        ( preserves-le-right-mul-ℚ⁺
-          ( neg-ℚ⁻ p)
-          ( neg-ℚ r)
-          ( neg-ℚ q)
-          ( neg-le-ℚ q r H))
-
-    reverses-le-left-mul-ℚ⁻ : le-ℚ (rational-ℚ⁻ p *ℚ r) (rational-ℚ⁻ p *ℚ q)
-    reverses-le-left-mul-ℚ⁻ =
-      binary-tr
-        ( le-ℚ)
-        ( commutative-mul-ℚ _ _)
-        ( commutative-mul-ℚ _ _)
-        ( reverses-le-right-mul-ℚ⁻)
-```
-
-### The negative rational numbers are invertible elements of the multiplicative monoid of rational numbers
-
-```agda
-opaque
-  inv-ℚ⁻ : ℚ⁻ → ℚ⁻
-  inv-ℚ⁻ q = neg-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q))
-
-  left-inverse-law-mul-ℚ⁻ :
-    (q : ℚ⁻) → rational-ℚ⁻ (inv-ℚ⁻ q) *ℚ rational-ℚ⁻ q ＝ one-ℚ
-  left-inverse-law-mul-ℚ⁻ q =
-    inv (swap-neg-mul-ℚ _ _) ∙
-    ap rational-ℚ⁺ (left-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
-
-  right-inverse-law-mul-ℚ⁻ :
-    (q : ℚ⁻) → rational-ℚ⁻ q *ℚ rational-ℚ⁻ (inv-ℚ⁻ q) ＝ one-ℚ
-  right-inverse-law-mul-ℚ⁻ q =
-    swap-neg-mul-ℚ _ _ ∙
-    ap rational-ℚ⁺ (right-inverse-law-mul-ℚ⁺ (neg-ℚ⁻ q))
-```
-
 ### Inequality on negative rational numbers
 
 ```agda
@@ -329,42 +239,6 @@ le-ℚ⁻ : Relation lzero ℚ⁻
 le-ℚ⁻ p q = le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁻ q)
 ```
 
-### Inversion reverses inequality on negative rational numbers
-
-```agda
-opaque
-  unfolding inv-ℚ⁻
-
-  reverses-leq-inv-ℚ⁻ :
-    (p q : ℚ⁻) → leq-ℚ⁻ p q → leq-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
-  reverses-leq-inv-ℚ⁻ p q p≤q =
-    neg-leq-ℚ
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ p)))
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q)))
-      ( inv-leq-ℚ⁺
-        ( neg-ℚ⁻ q)
-        ( neg-ℚ⁻ p)
-        ( neg-leq-ℚ _ _ p≤q))
-```
-
-### Inversion reverses strict inequality on negative rational numbers
-
-```agda
-opaque
-  unfolding inv-ℚ⁻
-
-  reverses-le-inv-ℚ⁻ :
-    (p q : ℚ⁻) → le-ℚ⁻ p q → le-ℚ⁻ (inv-ℚ⁻ q) (inv-ℚ⁻ p)
-  reverses-le-inv-ℚ⁻ p q p<q =
-    neg-le-ℚ
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ p)))
-      ( rational-ℚ⁺ (inv-ℚ⁺ (neg-ℚ⁻ q)))
-      ( inv-le-ℚ⁺
-        ( neg-ℚ⁻ q)
-        ( neg-ℚ⁻ p)
-        ( neg-le-ℚ _ _ p<q))
-```
-
 ### If `p ≤ q` for negative `q`, then `p` is negative
 
 ```agda
@@ -373,8 +247,7 @@ abstract
     (q : ℚ⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
   is-negative-leq-ℚ⁻ (q , neg-q) p p≤q =
     is-negative-le-zero-ℚ
-      ( p)
-      ( concatenate-leq-le-ℚ p q zero-ℚ p≤q (le-zero-is-negative-ℚ q neg-q))
+      ( concatenate-leq-le-ℚ p q zero-ℚ p≤q (le-zero-is-negative-ℚ neg-q))
 
   is-negative-le-ℚ⁻ :
     (q : ℚ⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
diff --git a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
index c7cd39a7481..621a2787146 100644
--- a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
@@ -60,11 +60,12 @@ module _
   (q : ℚ)
   where
 
-  is-nonnegative-ℚ : UU lzero
-  is-nonnegative-ℚ = is-nonnegative-fraction-ℤ (fraction-ℚ q)
+  opaque
+    is-nonnegative-ℚ : UU lzero
+    is-nonnegative-ℚ = is-nonnegative-fraction-ℤ (fraction-ℚ q)
 
-  is-prop-is-nonnegative-ℚ : is-prop is-nonnegative-ℚ
-  is-prop-is-nonnegative-ℚ = is-prop-is-nonnegative-fraction-ℤ (fraction-ℚ q)
+    is-prop-is-nonnegative-ℚ : is-prop is-nonnegative-ℚ
+    is-prop-is-nonnegative-ℚ = is-prop-is-nonnegative-fraction-ℤ (fraction-ℚ q)
 
   is-nonnegative-prop-ℚ : Prop lzero
   pr1 is-nonnegative-prop-ℚ = is-nonnegative-ℚ
@@ -99,20 +100,17 @@ module _
   is-nonnegative-rational-ℚ⁰⁺ : is-nonnegative-ℚ rational-ℚ⁰⁺
   is-nonnegative-rational-ℚ⁰⁺ = pr2 x
 
-  is-nonnegative-fraction-ℚ⁰⁺ : is-nonnegative-fraction-ℤ fraction-ℚ⁰⁺
-  is-nonnegative-fraction-ℚ⁰⁺ = pr2 x
+  opaque
+    unfolding is-nonnegative-ℚ
+
+    is-nonnegative-fraction-ℚ⁰⁺ : is-nonnegative-fraction-ℤ fraction-ℚ⁰⁺
+    is-nonnegative-fraction-ℚ⁰⁺ = pr2 x
 
   is-nonnegative-numerator-ℚ⁰⁺ : is-nonnegative-ℤ numerator-ℚ⁰⁺
   is-nonnegative-numerator-ℚ⁰⁺ = is-nonnegative-fraction-ℚ⁰⁺
 
   is-positive-denominator-ℚ⁰⁺ : is-positive-ℤ denominator-ℚ⁰⁺
   is-positive-denominator-ℚ⁰⁺ = is-positive-denominator-ℚ rational-ℚ⁰⁺
-
-zero-ℚ⁰⁺ : ℚ⁰⁺
-zero-ℚ⁰⁺ = zero-ℚ , _
-
-one-ℚ⁰⁺ : ℚ⁰⁺
-one-ℚ⁰⁺ = one-ℚ , _
 ```
 
 ## Properties
@@ -132,42 +130,49 @@ is-set-ℚ⁰⁺ : is-set ℚ⁰⁺
 is-set-ℚ⁰⁺ = is-set-type-subtype is-nonnegative-prop-ℚ is-set-ℚ
 ```
 
-### All positive rational numbers are nonnegative
-
-```agda
-abstract
-  is-nonnegative-is-positive-ℚ : (q : ℚ) → is-positive-ℚ q → is-nonnegative-ℚ q
-  is-nonnegative-is-positive-ℚ _ = is-nonnegative-is-positive-ℤ
-
-nonnegative-ℚ⁺ : ℚ⁺ → ℚ⁰⁺
-nonnegative-ℚ⁺ (q , H) = (q , is-nonnegative-is-positive-ℚ q H)
-```
-
 ### The rational image of a nonnegative integer is nonnegative
 
 ```agda
-abstract
+opaque
+  unfolding is-nonnegative-ℚ
+
   is-nonnegative-rational-ℤ :
-    (x : ℤ) → is-nonnegative-ℤ x → is-nonnegative-ℚ (rational-ℤ x)
-  is-nonnegative-rational-ℤ _ H = H
+    {x : ℤ} → is-nonnegative-ℤ x → is-nonnegative-ℚ (rational-ℤ x)
+  is-nonnegative-rational-ℤ H = H
 
 nonnegative-rational-nonnegative-ℤ : nonnegative-ℤ → ℚ⁰⁺
 nonnegative-rational-nonnegative-ℤ (x , x-is-neg) =
-  ( rational-ℤ x , is-nonnegative-rational-ℤ x x-is-neg)
+  ( rational-ℤ x , is-nonnegative-rational-ℤ x-is-neg)
 ```
 
 ### The images of natural numbers in the rationals are nonnegative
 
 ```agda
-nonnegative-rational-ℕ : ℕ → ℚ⁰⁺
-nonnegative-rational-ℕ n = (rational-ℕ n , is-nonnegative-int-ℕ n)
+module _
+  (n : ℕ)
+  where
+
+  opaque
+    unfolding is-nonnegative-ℚ
+
+    is-nonnegative-rational-ℕ : is-nonnegative-ℚ (rational-ℕ n)
+    is-nonnegative-rational-ℕ = is-nonnegative-int-ℕ n
+
+  nonnegative-rational-ℕ : ℚ⁰⁺
+  nonnegative-rational-ℕ = (rational-ℕ n , is-nonnegative-rational-ℕ)
+
+zero-ℚ⁰⁺ : ℚ⁰⁺
+zero-ℚ⁰⁺ = nonnegative-rational-ℕ zero-ℕ
+
+one-ℚ⁰⁺ : ℚ⁰⁺
+one-ℚ⁰⁺ = nonnegative-rational-ℕ 1
 ```
 
 ### The rational image of a nonnegative integer fraction is nonnegative
 
 ```agda
 opaque
-  unfolding rational-fraction-ℤ
+  unfolding is-nonnegative-ℚ rational-fraction-ℤ
 
   is-nonnegative-rational-fraction-ℤ :
     {x : fraction-ℤ} (P : is-nonnegative-fraction-ℤ x) →
@@ -182,30 +187,26 @@ opaque
 ### A rational number `x` is nonnegative if and only if `0 ≤ x`
 
 ```agda
-module _
-  (x : ℚ)
-  where
-
-  opaque
-    unfolding leq-ℚ-Prop
+opaque
+  unfolding is-nonnegative-ℚ leq-ℚ-Prop
 
-    leq-zero-is-nonnegative-ℚ : is-nonnegative-ℚ x → leq-ℚ zero-ℚ x
-    leq-zero-is-nonnegative-ℚ =
-      is-nonnegative-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
+  leq-zero-is-nonnegative-ℚ : {x : ℚ} → is-nonnegative-ℚ x → leq-ℚ zero-ℚ x
+  leq-zero-is-nonnegative-ℚ {x} =
+    is-nonnegative-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
 
-    is-nonnegative-leq-zero-ℚ : leq-ℚ zero-ℚ x → is-nonnegative-ℚ x
-    is-nonnegative-leq-zero-ℚ =
-      is-nonnegative-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
+  is-nonnegative-leq-zero-ℚ : {x : ℚ} → leq-ℚ zero-ℚ x → is-nonnegative-ℚ x
+  is-nonnegative-leq-zero-ℚ {x} =
+    is-nonnegative-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
 
-    is-nonnegative-iff-leq-zero-ℚ : is-nonnegative-ℚ x ↔ leq-ℚ zero-ℚ x
-    is-nonnegative-iff-leq-zero-ℚ =
-      ( leq-zero-is-nonnegative-ℚ ,
-        is-nonnegative-leq-zero-ℚ)
+is-nonnegative-iff-leq-zero-ℚ : (x : ℚ) → is-nonnegative-ℚ x ↔ leq-ℚ zero-ℚ x
+is-nonnegative-iff-leq-zero-ℚ x =
+  ( leq-zero-is-nonnegative-ℚ ,
+    is-nonnegative-leq-zero-ℚ)
 
 abstract
   leq-zero-rational-ℚ⁰⁺ : (p : ℚ⁰⁺) → leq-ℚ zero-ℚ (rational-ℚ⁰⁺ p)
   leq-zero-rational-ℚ⁰⁺ (p , is-nonneg-p) =
-    leq-zero-is-nonnegative-ℚ p is-nonneg-p
+    leq-zero-is-nonnegative-ℚ is-nonneg-p
 ```
 
 ### The successor of a nonnegative rational number is positive
@@ -216,12 +217,11 @@ abstract
     (q : ℚ) → is-nonnegative-ℚ q → is-positive-ℚ (succ-ℚ q)
   is-positive-succ-is-nonnegative-ℚ q H =
     is-positive-le-zero-ℚ
-      ( succ-ℚ q)
       ( concatenate-leq-le-ℚ
         ( zero-ℚ)
         ( q)
         ( succ-ℚ q)
-        ( leq-zero-is-nonnegative-ℚ q H)
+        ( leq-zero-is-nonnegative-ℚ H)
         ( le-left-add-rational-ℚ⁺ q one-ℚ⁺))
 
 positive-succ-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁺
diff --git a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
index 1f46950c690..b0506e21798 100644
--- a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
@@ -12,14 +12,12 @@ module elementary-number-theory.nonpositive-rational-numbers where
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
-open import elementary-number-theory.multiplication-nonnegative-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
-open import elementary-number-theory.nonpositive-integers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
@@ -48,10 +46,10 @@ module _
   (q : ℚ)
   where
 
-  is-nonpositive-ℚ : UU lzero
-  is-nonpositive-ℚ = is-nonnegative-ℚ (neg-ℚ q)
+  opaque
+    is-nonpositive-ℚ : UU lzero
+    is-nonpositive-ℚ = is-nonnegative-ℚ (neg-ℚ q)
 
-  abstract
     is-prop-is-nonpositive-ℚ : is-prop is-nonpositive-ℚ
     is-prop-is-nonpositive-ℚ = is-prop-is-nonnegative-ℚ (neg-ℚ q)
 
@@ -98,26 +96,27 @@ abstract
 ### A rational number is nonpositive if and only if it is less than or equal to zero
 
 ```agda
-module _
-  (q : ℚ)
-  where
-
-  opaque
-    unfolding neg-ℚ
-
-    leq-zero-is-nonpositive-ℚ : is-nonpositive-ℚ q → leq-ℚ q zero-ℚ
-    leq-zero-is-nonpositive-ℚ is-nonpos-q =
-      tr
-        ( λ p → leq-ℚ p zero-ℚ)
-        ( neg-neg-ℚ q)
-        ( neg-leq-ℚ _ _ (leq-zero-is-nonnegative-ℚ (neg-ℚ q) is-nonpos-q))
-
-    is-nonpositive-leq-zero-ℚ : leq-ℚ q zero-ℚ → is-nonpositive-ℚ q
-    is-nonpositive-leq-zero-ℚ q≤0 =
-      is-nonnegative-leq-zero-ℚ (neg-ℚ q) (neg-leq-ℚ _ _ q≤0)
-
-    is-nonpositive-iff-leq-zero-ℚ : is-nonpositive-ℚ q ↔ leq-ℚ q zero-ℚ
-    is-nonpositive-iff-leq-zero-ℚ =
-      ( leq-zero-is-nonpositive-ℚ ,
-        is-nonpositive-leq-zero-ℚ)
+opaque
+  unfolding is-nonpositive-ℚ
+
+  leq-zero-is-nonpositive-ℚ : {q : ℚ} → is-nonpositive-ℚ q → leq-ℚ q zero-ℚ
+  leq-zero-is-nonpositive-ℚ {q} is-nonpos-q =
+    binary-tr
+      ( leq-ℚ)
+      ( neg-neg-ℚ q)
+      ( neg-zero-ℚ)
+      ( neg-leq-ℚ (leq-zero-is-nonnegative-ℚ is-nonpos-q))
+
+  is-nonpositive-leq-zero-ℚ : {q : ℚ} → leq-ℚ q zero-ℚ → is-nonpositive-ℚ q
+  is-nonpositive-leq-zero-ℚ {q} q≤0 =
+    is-nonnegative-leq-zero-ℚ
+      ( tr
+        ( λ y → leq-ℚ y (neg-ℚ q))
+        ( neg-zero-ℚ)
+        ( neg-leq-ℚ q≤0))
+
+is-nonpositive-iff-leq-zero-ℚ : (q : ℚ) → is-nonpositive-ℚ q ↔ leq-ℚ q zero-ℚ
+is-nonpositive-iff-leq-zero-ℚ q =
+  ( leq-zero-is-nonpositive-ℚ ,
+    is-nonpositive-leq-zero-ℚ)
 ```
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 836af5c3954..93a69da7873 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -11,14 +11,22 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.nonzero-rational-numbers
+open import elementary-number-theory.positive-and-negative-integers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
+open import foundation.binary-transport
+open import foundation.cartesian-product-types
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.function-types
 open import foundation.functoriality-coproduct-types
 open import foundation.identity-types
+open import foundation.negation
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 ```
 
@@ -35,6 +43,27 @@ In this file, we outline basic relations between
 
 ## Properties
 
+### Trichotomies
+
+#### A rational number is either negative, zero, or positive
+
+```agda
+abstract
+  trichotomy-sign-ℚ :
+    {l : Level} {A : UU l} (x : ℚ) →
+    ( is-negative-ℚ x → A) →
+    ( x ＝ zero-ℚ → A) →
+    ( is-positive-ℚ x → A) →
+    A
+  trichotomy-sign-ℚ x neg zero pos =
+    trichotomy-le-ℚ
+      ( x)
+      ( zero-ℚ)
+      ( λ x<0 → neg (is-negative-le-zero-ℚ x<0))
+      ( zero)
+      ( λ 0<x → pos (is-positive-le-zero-ℚ 0<x))
+```
+
 ### Dichotomies
 
 #### A rational number is either negative or nonnegative
@@ -46,8 +75,8 @@ abstract
     is-negative-ℚ q + is-nonnegative-ℚ q
   decide-is-negative-is-nonnegative-ℚ q =
     map-coproduct
-      ( is-negative-le-zero-ℚ q)
-      ( is-nonnegative-leq-zero-ℚ q)
+      ( is-negative-le-zero-ℚ)
+      ( is-nonnegative-leq-zero-ℚ)
       ( decide-le-leq-ℚ q zero-ℚ)
 ```
 
@@ -60,80 +89,138 @@ abstract
     is-positive-ℚ q + is-nonpositive-ℚ q
   decide-is-positive-is-nonpositive-ℚ q =
     map-coproduct
-      ( is-positive-le-zero-ℚ q)
-      ( is-nonpositive-leq-zero-ℚ q)
+      ( is-positive-le-zero-ℚ)
+      ( is-nonpositive-leq-zero-ℚ)
       ( decide-le-leq-ℚ zero-ℚ q)
 ```
 
-### Trichotomies
-
-#### A rational number is either negative, zero, or positive
+#### A rational number is nonpositive or nonnegative
 
 ```agda
 abstract
-  trichotomy-sign-ℚ :
-    {l : Level} {A : UU l} (x : ℚ) →
-    ( is-negative-ℚ x → A) →
-    ( x ＝ zero-ℚ → A) →
-    ( is-positive-ℚ x → A) →
-    A
-  trichotomy-sign-ℚ x neg zero pos =
-    trichotomy-le-ℚ
-      ( x)
-      ( zero-ℚ)
-      ( λ x<0 → neg (is-negative-le-zero-ℚ x x<0))
-      ( zero)
-      ( λ 0<x → pos (is-positive-le-zero-ℚ x 0<x))
+  is-nonpositive-or-nonnegative-ℚ :
+    (q : ℚ) →
+    is-nonpositive-ℚ q + is-nonnegative-ℚ q
+  is-nonpositive-or-nonnegative-ℚ q =
+    map-coproduct
+      ( is-nonpositive-leq-zero-ℚ)
+      ( is-nonnegative-leq-zero-ℚ)
+      ( linear-leq-ℚ q zero-ℚ)
 ```
 
-### Inequalities
+### If a rational number is nonzero, it is negative or positive
 
-#### A negative rational number is less than a nonnegative rational number
+```agda
+decide-is-negative-is-positive-is-nonzero-ℚ :
+  {q : ℚ} → is-nonzero-ℚ q → is-negative-ℚ q + is-positive-ℚ q
+decide-is-negative-is-positive-is-nonzero-ℚ {q} q≠0 =
+  trichotomy-sign-ℚ q
+    ( inl)
+    ( ex-falso ∘ q≠0)
+    ( inr)
+```
+
+### Implications
+
+#### Any positive rational number is nonnegative
 
 ```agda
-abstract
-  le-negative-nonnegative-ℚ :
-    (p : ℚ⁻) (q : ℚ⁰⁺) → le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁰⁺ q)
-  le-negative-nonnegative-ℚ (p , neg-p) (q , nonneg-q) =
-    concatenate-le-leq-ℚ p zero-ℚ q
-      ( le-zero-is-negative-ℚ p neg-p)
-      ( leq-zero-is-nonnegative-ℚ q nonneg-q)
+opaque
+  unfolding is-nonnegative-ℚ is-positive-ℚ
+
+  is-nonnegative-is-positive-ℚ : {q : ℚ} → is-positive-ℚ q → is-nonnegative-ℚ q
+  is-nonnegative-is-positive-ℚ = is-nonnegative-is-positive-ℤ
+
+nonnegative-ℚ⁺ : ℚ⁺ → ℚ⁰⁺
+nonnegative-ℚ⁺ (q , H) = (q , is-nonnegative-is-positive-ℚ H)
 ```
 
-#### A nonpositive rational number is less than a positive rational number
+### Any negative rational number is nonpositive
 
 ```agda
-abstract
-  le-nonpositive-positive-ℚ :
-    (p : ℚ⁰⁻) (q : ℚ⁺) → le-ℚ (rational-ℚ⁰⁻ p) (rational-ℚ⁺ q)
-  le-nonpositive-positive-ℚ (p , nonpos-p) (q , pos-q) =
-    concatenate-leq-le-ℚ p zero-ℚ q
-      ( leq-zero-is-nonpositive-ℚ p nonpos-p)
-      ( le-zero-is-positive-ℚ q pos-q)
+opaque
+  unfolding is-negative-ℚ is-nonpositive-ℚ
+
+  is-nonpositive-is-negative-ℚ : {q : ℚ} → is-negative-ℚ q → is-nonpositive-ℚ q
+  is-nonpositive-is-negative-ℚ = is-nonnegative-is-positive-ℚ
+
+nonpositive-ℚ⁻ : ℚ⁻ → ℚ⁰⁻
+nonpositive-ℚ⁻ (q , H) = (q , is-nonpositive-is-negative-ℚ H)
 ```
 
-### If `p < q` and `p` is nonnegative, then `q` is positive
+### Operations
+
+#### The negation of a positive rational number is negative
 
 ```agda
-abstract
-  is-positive-le-ℚ⁰⁺ :
-    (p : ℚ⁰⁺) (q : ℚ) → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
-  is-positive-le-ℚ⁰⁺ (p , nonneg-p) q p<q =
-    is-positive-le-zero-ℚ
-      ( q)
-      ( concatenate-leq-le-ℚ _ _ _ (leq-zero-is-nonnegative-ℚ p nonneg-p) p<q)
+opaque
+  unfolding is-negative-ℚ
+
+  is-negative-neg-is-positive-ℚ :
+    {q : ℚ} → is-positive-ℚ q → is-negative-ℚ (neg-ℚ q)
+  is-negative-neg-is-positive-ℚ = inv-tr is-positive-ℚ (neg-neg-ℚ _)
+
+neg-ℚ⁺ : ℚ⁺ → ℚ⁻
+neg-ℚ⁺ (q , is-pos-q) =
+  (neg-ℚ q , is-negative-neg-is-positive-ℚ is-pos-q)
 ```
 
-### If `p < q` and `q` is nonpositive, then `p` is negative
+#### The negation of a negative rational number is positive
+
+```agda
+opaque
+  unfolding is-negative-ℚ
+
+  is-positive-neg-is-negative-ℚ :
+    {q : ℚ} → is-negative-ℚ q → is-positive-ℚ (neg-ℚ q)
+  is-positive-neg-is-negative-ℚ = id
+
+neg-ℚ⁻ : ℚ⁻ → ℚ⁺
+neg-ℚ⁻ (q , is-neg-q) = (neg-ℚ q , is-positive-neg-is-negative-ℚ is-neg-q)
+```
+
+#### The negation of a nonnegative rational number is nonpositive
+
+```agda
+opaque
+  unfolding is-nonpositive-ℚ
+
+  is-nonpositive-neg-is-nonnegative-ℚ :
+    {q : ℚ} → is-nonnegative-ℚ q → is-nonpositive-ℚ (neg-ℚ q)
+  is-nonpositive-neg-is-nonnegative-ℚ = inv-tr is-nonnegative-ℚ (neg-neg-ℚ _)
+
+neg-ℚ⁰⁺ : ℚ⁰⁺ → ℚ⁰⁻
+neg-ℚ⁰⁺ (q , is-nonneg-q) =
+  ( neg-ℚ q , is-nonpositive-neg-is-nonnegative-ℚ is-nonneg-q)
+```
+
+#### The negation of a nonpositive rational number is nonnegative
+
+```agda
+opaque
+  unfolding is-nonpositive-ℚ
+
+  is-nonnegative-neg-is-nonpositive-ℚ :
+    {q : ℚ} → is-nonpositive-ℚ q → is-nonnegative-ℚ (neg-ℚ q)
+  is-nonnegative-neg-is-nonpositive-ℚ = id
+
+neg-ℚ⁰⁻ : ℚ⁰⁻ → ℚ⁰⁺
+neg-ℚ⁰⁻ (q , is-nonpos-q) =
+  ( neg-ℚ q , is-nonnegative-neg-is-nonpositive-ℚ is-nonpos-q)
+```
+
+### Exclusive cases
+
+#### A rational is not both negative and positive
 
 ```agda
 abstract
-  is-negative-le-ℚ⁰⁻ :
-    (q : ℚ⁰⁻) (p : ℚ) → le-ℚ p (rational-ℚ⁰⁻ q) → is-negative-ℚ p
-  is-negative-le-ℚ⁰⁻ (q , nonpos-q) p p<q =
-    is-negative-le-zero-ℚ
-      ( p)
-      ( concatenate-le-leq-ℚ p q zero-ℚ
-        ( p<q)
-        ( leq-zero-is-nonpositive-ℚ q nonpos-q))
+  not-is-negative-is-positive-ℚ :
+    {x : ℚ} → ¬ (is-negative-ℚ x × is-positive-ℚ x)
+  not-is-negative-is-positive-ℚ {x} (N , P) =
+    not-leq-le-ℚ
+      ( zero-ℚ)
+      ( x)
+      ( le-zero-is-positive-ℚ P)
+      ( leq-zero-is-negative-ℚ N)
 ```
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index 33ca6716243..5144d0075c3 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -26,6 +26,7 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.cartesian-product-types
 open import foundation.coproduct-types
+open import foundation.decidable-propositions
 open import foundation.decidable-subtypes
 open import foundation.dependent-pair-types
 open import foundation.function-types
@@ -66,11 +67,16 @@ module _
   (x : ℚ)
   where
 
-  is-positive-ℚ : UU lzero
-  is-positive-ℚ = is-positive-fraction-ℤ (fraction-ℚ x)
+  opaque
+    is-positive-ℚ : UU lzero
+    is-positive-ℚ = is-positive-fraction-ℤ (fraction-ℚ x)
+
+    is-prop-is-positive-ℚ : is-prop is-positive-ℚ
+    is-prop-is-positive-ℚ = is-prop-is-positive-fraction-ℤ (fraction-ℚ x)
 
-  is-prop-is-positive-ℚ : is-prop is-positive-ℚ
-  is-prop-is-positive-ℚ = is-prop-is-positive-fraction-ℤ (fraction-ℚ x)
+    is-decidable-prop-is-positive-ℚ : is-decidable-prop is-positive-ℚ
+    is-decidable-prop-is-positive-ℚ =
+      ( is-prop-is-positive-ℚ , is-decidable-is-positive-ℤ (numerator-ℚ x))
 
   is-positive-prop-ℚ : Prop lzero
   pr1 is-positive-prop-ℚ = is-positive-ℚ
@@ -78,7 +84,7 @@ module _
 
 decidable-subtype-positive-ℚ : decidable-subtype lzero ℚ
 decidable-subtype-positive-ℚ x =
-  decidable-subtype-positive-fraction-ℤ (fraction-ℚ x)
+  ( is-positive-ℚ x , is-decidable-prop-is-positive-ℚ x)
 ```
 
 ### The type of positive rational numbers
@@ -109,11 +115,14 @@ module _
   is-positive-rational-ℚ⁺ : is-positive-ℚ rational-ℚ⁺
   is-positive-rational-ℚ⁺ = pr2 x
 
-  is-positive-fraction-ℚ⁺ : is-positive-fraction-ℤ fraction-ℚ⁺
-  is-positive-fraction-ℚ⁺ = is-positive-rational-ℚ⁺
+  opaque
+    unfolding is-positive-ℚ
+
+    is-positive-fraction-ℚ⁺ : is-positive-fraction-ℤ fraction-ℚ⁺
+    is-positive-fraction-ℚ⁺ = is-positive-rational-ℚ⁺
 
-  is-positive-numerator-ℚ⁺ : is-positive-ℤ numerator-ℚ⁺
-  is-positive-numerator-ℚ⁺ = is-positive-rational-ℚ⁺
+    is-positive-numerator-ℚ⁺ : is-positive-ℤ numerator-ℚ⁺
+    is-positive-numerator-ℚ⁺ = is-positive-rational-ℚ⁺
 
   is-positive-denominator-ℚ⁺ : is-positive-ℤ denominator-ℚ⁺
   is-positive-denominator-ℚ⁺ = is-positive-denominator-ℚ rational-ℚ⁺
@@ -150,19 +159,22 @@ abstract
 ### The rational image of a positive integer is positive
 
 ```agda
-abstract
+opaque
+  unfolding is-positive-ℚ
+
   is-positive-rational-ℤ :
-    (x : ℤ) → is-positive-ℤ x → is-positive-ℚ (rational-ℤ x)
-  is-positive-rational-ℤ x P = P
+    {x : ℤ} → is-positive-ℤ x → is-positive-ℚ (rational-ℤ x)
+  is-positive-rational-ℤ = id
 
 positive-rational-positive-ℤ : positive-ℤ → ℚ⁺
-positive-rational-positive-ℤ (z , pos-z) = rational-ℤ z , pos-z
+positive-rational-positive-ℤ (z , pos-z) =
+    ( rational-ℤ z , is-positive-rational-ℤ pos-z)
 
 positive-rational-ℤ⁺ : ℤ⁺ → ℚ⁺
 positive-rational-ℤ⁺ = positive-rational-positive-ℤ
 
 one-ℚ⁺ : ℚ⁺
-one-ℚ⁺ = (one-ℚ , is-positive-int-positive-ℤ one-positive-ℤ)
+one-ℚ⁺ = positive-rational-ℤ⁺ one-ℤ⁺
 ```
 
 ### The type of positive rational numbers is inhabited
@@ -184,7 +196,7 @@ positive-rational-ℕ⁺ n = positive-rational-positive-ℤ (positive-int-ℕ⁺
 
 ```agda
 opaque
-  unfolding rational-fraction-ℤ
+  unfolding is-positive-ℚ rational-fraction-ℤ
 
   is-positive-rational-fraction-ℤ :
     {x : fraction-ℤ} (P : is-positive-fraction-ℤ x) →
@@ -195,20 +207,16 @@ opaque
 ### A rational number `x` is positive if and only if `0 < x`
 
 ```agda
-module _
-  (x : ℚ)
-  where
-
-  opaque
-    unfolding le-ℚ-Prop
+opaque
+  unfolding is-positive-ℚ le-ℚ-Prop
 
-    le-zero-is-positive-ℚ : is-positive-ℚ x → le-ℚ zero-ℚ x
-    le-zero-is-positive-ℚ =
-      is-positive-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
+  le-zero-is-positive-ℚ : {x : ℚ} → is-positive-ℚ x → le-ℚ zero-ℚ x
+  le-zero-is-positive-ℚ {x} =
+    is-positive-eq-ℤ (inv (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x)))
 
-    is-positive-le-zero-ℚ : le-ℚ zero-ℚ x → is-positive-ℚ x
-    is-positive-le-zero-ℚ =
-      is-positive-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
+  is-positive-le-zero-ℚ : {x : ℚ} → le-ℚ zero-ℚ x → is-positive-ℚ x
+  is-positive-le-zero-ℚ {x} =
+    is-positive-eq-ℤ (cross-mul-diff-zero-fraction-ℤ (fraction-ℚ x))
 ```
 
 ### Zero is not a positive rational number
@@ -217,7 +225,7 @@ module _
 abstract
   is-not-positive-zero-ℚ : ¬ (is-positive-ℚ zero-ℚ)
   is-not-positive-zero-ℚ pos-0 =
-    irreflexive-le-ℚ zero-ℚ (le-zero-is-positive-ℚ zero-ℚ pos-0)
+    irreflexive-le-ℚ zero-ℚ (le-zero-is-positive-ℚ pos-0)
 ```
 
 ### The difference of a rational number with a lesser rational number is positive
@@ -231,7 +239,6 @@ module _
     is-positive-diff-le-ℚ : is-positive-ℚ (y -ℚ x)
     is-positive-diff-le-ℚ =
       is-positive-le-zero-ℚ
-        ( y -ℚ x)
         ( backward-implication
           ( iff-translate-diff-le-zero-ℚ x y)
           ( H))
@@ -252,44 +259,12 @@ module _
     commutative-add-ℚ x (y -ℚ x) ∙ left-law-positive-diff-le-ℚ
 ```
 
-### A nonzero rational number or its negative is positive
-
-```agda
-opaque
-  unfolding neg-ℚ
-
-  decide-is-negative-is-positive-is-nonzero-ℚ :
-    {x : ℚ} → is-nonzero-ℚ x → is-positive-ℚ (neg-ℚ x) + is-positive-ℚ x
-  decide-is-negative-is-positive-is-nonzero-ℚ {x} H =
-    rec-coproduct
-      ( inl ∘ is-positive-neg-is-negative-ℤ)
-      ( inr)
-      ( decide-sign-nonzero-ℤ
-        { numerator-ℚ x}
-        ( is-nonzero-numerator-is-nonzero-ℚ x H))
-```
-
-### A rational and its negative are not both positive
+### Positive rational numbers are nonzero
 
 ```agda
 opaque
-  unfolding neg-ℚ
-
-  not-is-negative-is-positive-ℚ :
-    (x : ℚ) → ¬ (is-positive-ℚ (neg-ℚ x) × is-positive-ℚ x)
-  not-is-negative-is-positive-ℚ x (N , P) =
-    is-not-negative-and-positive-ℤ
-      ( numerator-ℚ x)
-      ( ( is-negative-eq-ℤ
-          (neg-neg-ℤ (numerator-ℚ x))
-          (is-negative-neg-is-positive-ℤ {numerator-ℚ (neg-ℚ x)} N)) ,
-        ( P))
-```
-
-### Positive rational numbers are nonzero
+  unfolding is-positive-ℚ
 
-```agda
-abstract
   is-nonzero-is-positive-ℚ : {x : ℚ} → is-positive-ℚ x → is-nonzero-ℚ x
   is-nonzero-is-positive-ℚ {x} H =
     is-nonzero-is-nonzero-numerator-ℚ x
@@ -309,8 +284,7 @@ abstract
     (p : ℚ⁺) (q : ℚ) → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
   is-positive-leq-ℚ⁺ (p , pos-p) q p≤q =
     is-positive-le-zero-ℚ
-      ( q)
-      ( concatenate-le-leq-ℚ _ _ _ (le-zero-is-positive-ℚ p pos-p) p≤q)
+      ( concatenate-le-leq-ℚ _ _ _ (le-zero-is-positive-ℚ pos-p) p≤q)
 
   is-positive-le-ℚ⁺ :
     (p : ℚ⁺) (q : ℚ) → le-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
diff --git a/src/elementary-number-theory/rational-numbers.lagda.md b/src/elementary-number-theory/rational-numbers.lagda.md
index 2f51198a474..991d52230c5 100644
--- a/src/elementary-number-theory/rational-numbers.lagda.md
+++ b/src/elementary-number-theory/rational-numbers.lagda.md
@@ -149,6 +149,16 @@ opaque
   pr2 (neg-ℚ (x , H)) = is-reduced-neg-fraction-ℤ x H
 ```
 
+### The negation of zero is zero
+
+```agda
+opaque
+  unfolding neg-ℚ
+
+  neg-zero-ℚ : neg-ℚ zero-ℚ ＝ zero-ℚ
+  neg-zero-ℚ = refl
+```
+
 ### The mediant of two rationals
 
 ```agda
@@ -287,9 +297,9 @@ module _
 opaque
   unfolding neg-ℚ
 
-  preserves-neg-rational-ℤ :
+  neg-rational-ℤ :
     (k : ℤ) → rational-ℤ (neg-ℤ k) ＝ neg-ℚ (rational-ℤ k)
-  preserves-neg-rational-ℤ k =
+  neg-rational-ℤ k =
     eq-ℚ (rational-ℤ (neg-ℤ k)) (neg-ℚ (rational-ℤ k)) refl refl
 ```
 
@@ -297,13 +307,12 @@ opaque
 
 ```agda
 opaque
-  unfolding neg-ℚ
-  unfolding rational-fraction-ℤ
+  unfolding neg-ℚ rational-fraction-ℤ
 
-  preserves-neg-rational-fraction-ℤ :
+  neg-rational-fraction-ℤ :
     (x : fraction-ℤ) →
     rational-fraction-ℤ (neg-fraction-ℤ x) ＝ neg-ℚ (rational-fraction-ℤ x)
-  preserves-neg-rational-fraction-ℤ x =
+  neg-rational-fraction-ℤ x =
     ( eq-ℚ-sim-fraction-ℤ
       ( neg-fraction-ℤ x)
       ( fraction-ℚ (neg-ℚ (rational-fraction-ℤ x)))
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index fd41bb63716..154d5e5c068 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -11,7 +11,9 @@ module elementary-number-theory.squares-rational-numbers where
 ```agda
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-nonpositive-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
@@ -52,15 +54,14 @@ square-root-ℚ _ (root , _) = root
 ### Squares in ℚ are nonnegative
 
 ```agda
-is-nonnegative-square-ℚ : (a : ℚ) → is-nonnegative-ℚ (square-ℚ a)
-is-nonnegative-square-ℚ a =
-  rec-coproduct
-    ( λ H →
-      is-nonnegative-is-positive-ℚ
-        ( a *ℚ a)
-        ( is-positive-mul-negative-ℚ {a} {a} H H))
-    ( λ H → is-nonnegative-mul-ℚ a a H H)
-    ( decide-is-negative-is-nonnegative-ℚ a)
+abstract
+  is-nonnegative-square-ℚ : (q : ℚ) → is-nonnegative-ℚ (square-ℚ q)
+  is-nonnegative-square-ℚ q =
+    rec-coproduct
+      ( λ is-nonpos-q →
+        is-nonnegative-mul-nonpositive-ℚ is-nonpos-q is-nonpos-q)
+      ( λ is-nonneg-q → is-nonnegative-mul-ℚ is-nonneg-q is-nonneg-q)
+      ( is-nonpositive-or-nonnegative-ℚ q)
 ```
 
 ### The square of the negation of `x` is the square of `x`
diff --git a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
index a14cf72ef8c..4ef6623999e 100644
--- a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
@@ -37,13 +37,3 @@ le-prop-ℚ⁰⁺ p q = le-ℚ-Prop (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q
 le-ℚ⁰⁺ : (p q : ℚ⁰⁺) → UU lzero
 le-ℚ⁰⁺ p q = type-Prop (le-prop-ℚ⁰⁺ p q)
 ```
-
-## Properties
-
-### Zero is less than positive rational numbers as nonnegative rational numbers
-
-```agda
-abstract
-  le-zero-nonnegative-ℚ⁰⁺ : (q : ℚ⁺) → le-ℚ⁰⁺ zero-ℚ⁰⁺ (nonnegative-ℚ⁺ q)
-  le-zero-nonnegative-ℚ⁰⁺ (q , pos-q) = le-zero-is-positive-ℚ q pos-q
-```
diff --git a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
index c6ca66c15fb..83f68e3629e 100644
--- a/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-positive-rational-numbers.lagda.md
@@ -87,16 +87,15 @@ opaque
   mediant-zero-ℚ⁺ x =
     ( mediant-ℚ zero-ℚ (rational-ℚ⁺ x) ,
       is-positive-le-zero-ℚ
-        ( mediant-ℚ zero-ℚ (rational-ℚ⁺ x))
         ( le-left-mediant-ℚ
           ( zero-ℚ)
           ( rational-ℚ⁺ x)
-          ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))))
+          ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x))))
 
   le-mediant-zero-ℚ⁺ : (x : ℚ⁺) → le-ℚ⁺ (mediant-zero-ℚ⁺ x) x
   le-mediant-zero-ℚ⁺ x =
     le-right-mediant-ℚ
       ( zero-ℚ)
       ( rational-ℚ⁺ x)
-      ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))
+      ( le-zero-is-positive-ℚ (is-positive-rational-ℚ⁺ x))
 ```
diff --git a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
index e3986f030dc..79fe7312b00 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -519,11 +519,10 @@ abstract
 
 ```agda
 opaque
-  unfolding le-ℚ-Prop
-  unfolding neg-ℚ
+  unfolding le-ℚ-Prop neg-ℚ
 
-  neg-le-ℚ : (x y : ℚ) → le-ℚ x y → le-ℚ (neg-ℚ y) (neg-ℚ x)
-  neg-le-ℚ x y = neg-le-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
+  neg-le-ℚ : {x y : ℚ} → le-ℚ x y → le-ℚ (neg-ℚ y) (neg-ℚ x)
+  neg-le-ℚ {x} {y} = neg-le-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)
 ```
 
 ### Transposing additions on strict inequalities of rational numbers
diff --git a/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md b/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md
index b8aa906a321..568cc71d27d 100644
--- a/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md
+++ b/src/metric-spaces/elements-at-bounded-distance-metric-spaces.lagda.md
@@ -259,7 +259,7 @@ module _
           ( ε)
           ( concatenate-leq-le-ℚ q zero-ℚ ε
             ( q≤0)
-            ( le-zero-is-positive-ℚ ε is-pos-ε))
+            ( le-zero-is-positive-ℚ is-pos-ε))
           ( ε<q)
 
     leq-zero-upper-real-dist-Metric-Space :
@@ -267,11 +267,11 @@ module _
     leq-zero-upper-real-dist-Metric-Space q =
       rec-trunc-Prop
         ( le-ℚ-Prop zero-ℚ q)
-        ( λ ((ε , Nεxy) , ε<q) →
+        ( λ (((ε , is-pos-ε) , Nεxy) , ε<q) →
           transitive-le-ℚ
             ( zero-ℚ)
-            ( rational-ℚ⁺ ε)
+            ( ε)
             ( q)
             ( ε<q)
-            ( le-zero-is-positive-ℚ (pr1 ε) (pr2 ε)))
+            ( le-zero-is-positive-ℚ is-pos-ε))
 ```
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 64aa11e62eb..1fef763229a 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -136,12 +136,12 @@ module _
         ( λ 0<p →
           let
             r = mediant-ℚ p q
-            p⁺ = (p , is-positive-le-zero-ℚ p 0<p)
+            p⁺ = (p , is-positive-le-zero-ℚ 0<p)
             p<r = le-left-mediant-ℚ p q p<q
             r<q = le-right-mediant-ℚ p q p<q
             r⁺ =
               ( r ,
-                is-positive-le-zero-ℚ r (transitive-le-ℚ zero-ℚ p r p<r 0<p))
+                is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ p r p<r 0<p))
           in
             map-disjunction
               ( λ ¬Npxy p∈U →
diff --git a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
index 3aae8a5cb1c..5bd39143c27 100644
--- a/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
+++ b/src/metric-spaces/metric-space-of-rational-numbers.lagda.md
@@ -407,8 +407,6 @@ is-cauchy-approximation-rational-ℚ⁺ :
     rational-ℚ⁺
 is-cauchy-approximation-rational-ℚ⁺ ε δ =
   ( leq-le-ℚ
-    { rational-ℚ⁺ δ}
-    { rational-ℚ⁺ ε +ℚ (rational-ℚ⁺ (ε +ℚ⁺ δ))}
     ( transitive-le-ℚ
       ( rational-ℚ⁺ δ)
       ( rational-ℚ⁺ (ε +ℚ⁺ δ))
@@ -418,8 +416,6 @@ is-cauchy-approximation-rational-ℚ⁺ ε δ =
         ( ε +ℚ⁺ δ))
       ( le-right-add-ℚ⁺ ε δ))) ,
   ( leq-le-ℚ
-    { rational-ℚ⁺ ε}
-    { rational-ℚ⁺ δ +ℚ (rational-ℚ⁺ (ε +ℚ⁺ δ))}
     ( transitive-le-ℚ
       ( rational-ℚ⁺ ε)
       ( rational-ℚ⁺ (ε +ℚ⁺ δ))
@@ -444,9 +440,7 @@ is-zero-limit-rational-ℚ⁺ ε δ =
     { zero-ℚ}
     { rational-ℚ⁺ (ε +ℚ⁺ (ε +ℚ⁺ δ))}
     ( le-zero-is-positive-ℚ
-      ( rational-ℚ⁺ (ε +ℚ⁺ (ε +ℚ⁺ δ)))
-      ( is-positive-rational-ℚ⁺
-        (ε +ℚ⁺ (ε +ℚ⁺ δ))))) ,
+      ( is-positive-rational-ℚ⁺ (ε +ℚ⁺ (ε +ℚ⁺ δ))))) ,
   ( leq-le-ℚ
     { rational-ℚ⁺ ε}
     { zero-ℚ +ℚ rational-ℚ⁺ (ε +ℚ⁺ δ)}
diff --git a/src/metric-spaces/rational-sequences-approximating-zero.lagda.md b/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
index dc29336abcd..821404d222f 100644
--- a/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
+++ b/src/metric-spaces/rational-sequences-approximating-zero.lagda.md
@@ -18,6 +18,7 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-natural-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-natural-numbers
diff --git a/src/order-theory.lagda.md b/src/order-theory.lagda.md
index ce2ff9740b2..14e68460e11 100644
--- a/src/order-theory.lagda.md
+++ b/src/order-theory.lagda.md
@@ -111,6 +111,7 @@ open import order-theory.order-preserving-maps-large-posets public
 open import order-theory.order-preserving-maps-large-preorders public
 open import order-theory.order-preserving-maps-posets public
 open import order-theory.order-preserving-maps-preorders public
+open import order-theory.order-preserving-maps-total-orders public
 open import order-theory.ordinals public
 open import order-theory.poset-closed-intervals-lattices public
 open import order-theory.poset-closed-intervals-posets public
diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md
index 1638130acd9..4d127ed3ff8 100644
--- a/src/order-theory/order-preserving-maps-posets.lagda.md
+++ b/src/order-theory/order-preserving-maps-posets.lagda.md
@@ -25,8 +25,10 @@ open import order-theory.posets
 
 ## Idea
 
-A map `f : P → Q` between the underlying types of two posets is said to be
-**order preserving** if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
+A map `f : P → Q` between the underlying types of two
+[posets](order-theory.posets.md) is said to be
+{{#concept "order preserving" Agda=hom-Poset Disambiguation="map between posets"}}
+if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
 
 ## Definition
 
diff --git a/src/order-theory/order-preserving-maps-total-orders.lagda.md b/src/order-theory/order-preserving-maps-total-orders.lagda.md
new file mode 100644
index 00000000000..f3ae69baab9
--- /dev/null
+++ b/src/order-theory/order-preserving-maps-total-orders.lagda.md
@@ -0,0 +1,193 @@
+# Order preserving maps between total orders
+
+```agda
+module order-theory.order-preserving-maps-total-orders where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-functions
+open import foundation.disjunction
+open import foundation.function-types
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import order-theory.order-preserving-maps-posets
+open import order-theory.total-orders
+```
+
+</details>
+
+## Idea
+
+A map `f : P → Q` between the underlying types of two
+[total orders](order-theory.total-orders.md) is said to be
+{{#concept "order preserving" Agda=hom-Total-Order Disambiguation="map between total orders"}}
+if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
+
+## Definition
+
+### Order preserving maps
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (P : Total-Order l1 l2) (Q : Total-Order l3 l4)
+  where
+
+  preserves-order-prop-Total-Order :
+    (type-Total-Order P → type-Total-Order Q) → Prop (l1 ⊔ l2 ⊔ l4)
+  preserves-order-prop-Total-Order =
+    preserves-order-prop-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  preserves-order-Total-Order :
+    (type-Total-Order P → type-Total-Order Q) → UU (l1 ⊔ l2 ⊔ l4)
+  preserves-order-Total-Order =
+    preserves-order-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  is-prop-preserves-order-Total-Order :
+    (f : type-Total-Order P → type-Total-Order Q) →
+    is-prop (preserves-order-Total-Order f)
+  is-prop-preserves-order-Total-Order =
+    is-prop-preserves-order-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  hom-set-Total-Order : Set (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+  hom-set-Total-Order =
+    set-subset
+      ( function-Set (type-Total-Order P) (set-Total-Order Q))
+      ( preserves-order-prop-Total-Order)
+
+  hom-Total-Order : UU (l1 ⊔ l2 ⊔ l3 ⊔ l4)
+  hom-Total-Order = type-Set hom-set-Total-Order
+
+  map-hom-Total-Order :
+    hom-Total-Order → type-Total-Order P → type-Total-Order Q
+  map-hom-Total-Order =
+    map-hom-Poset (poset-Total-Order P) (poset-Total-Order Q)
+
+  preserves-order-hom-Total-Order :
+    (f : hom-Total-Order) → preserves-order-Total-Order (map-hom-Total-Order f)
+  preserves-order-hom-Total-Order =
+    preserves-order-hom-Poset (poset-Total-Order P) (poset-Total-Order Q)
+```
+
+## Properties
+
+### Order-preserving maps distribute over the minimum operation
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (P : Total-Order l1 l2) (Q : Total-Order l3 l4)
+  (f : hom-Total-Order P Q)
+  where
+
+  abstract
+    distributive-map-hom-min-Total-Order :
+      (x y : type-Total-Order P) →
+      map-hom-Total-Order P Q f (min-Total-Order P x y) ＝
+      min-Total-Order Q
+        ( map-hom-Total-Order P Q f x)
+        ( map-hom-Total-Order P Q f y)
+    distributive-map-hom-min-Total-Order x y =
+      elim-disjunction
+        ( Id-Prop
+          ( set-Total-Order Q)
+          ( map-hom-Total-Order P Q f (min-Total-Order P x y))
+          ( min-Total-Order Q
+            ( map-hom-Total-Order P Q f x)
+            ( map-hom-Total-Order P Q f y)))
+        ( λ x≤y →
+          equational-reasoning
+            map-hom-Total-Order P Q f (min-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f x
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( left-leq-right-min-Total-Order P x y x≤y)
+            ＝
+              min-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( left-leq-right-min-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f x y x≤y)))
+        ( λ y≤x →
+          equational-reasoning
+            map-hom-Total-Order P Q f (min-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f y
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( right-leq-left-min-Total-Order P x y y≤x)
+            ＝
+              min-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( right-leq-left-min-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f y x y≤x)))
+        ( is-total-Total-Order P x y)
+```
+
+### Order-preserving maps distribute over the maximum operation
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (P : Total-Order l1 l2) (Q : Total-Order l3 l4)
+  (f : hom-Total-Order P Q)
+  where
+
+  abstract
+    distributive-map-hom-max-Total-Order :
+      (x y : type-Total-Order P) →
+      map-hom-Total-Order P Q f (max-Total-Order P x y) ＝
+      max-Total-Order Q
+        ( map-hom-Total-Order P Q f x)
+        ( map-hom-Total-Order P Q f y)
+    distributive-map-hom-max-Total-Order x y =
+      elim-disjunction
+        ( Id-Prop
+          ( set-Total-Order Q)
+          ( map-hom-Total-Order P Q f (max-Total-Order P x y))
+          ( max-Total-Order Q
+            ( map-hom-Total-Order P Q f x)
+            ( map-hom-Total-Order P Q f y)))
+        ( λ x≤y →
+          equational-reasoning
+            map-hom-Total-Order P Q f (max-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f y
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( left-leq-right-max-Total-Order P x y x≤y)
+            ＝
+              max-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( left-leq-right-max-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f x y x≤y)))
+        ( λ y≤x →
+          equational-reasoning
+            map-hom-Total-Order P Q f (max-Total-Order P x y)
+            ＝ map-hom-Total-Order P Q f x
+              by
+                ap
+                  ( map-hom-Total-Order P Q f)
+                  ( right-leq-left-max-Total-Order P x y y≤x)
+            ＝
+              max-Total-Order Q
+                ( map-hom-Total-Order P Q f x)
+                ( map-hom-Total-Order P Q f y)
+              by
+                inv
+                  ( right-leq-left-max-Total-Order Q _ _
+                    ( preserves-order-hom-Total-Order P Q f y x y≤x)))
+        ( is-total-Total-Order P x y)
+```
diff --git a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
index 6d2c68d957e..d970a20b789 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -12,6 +12,8 @@ module real-numbers.addition-lower-dedekind-real-numbers where
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
@@ -213,9 +215,7 @@ module _
   {l : Level} (x : lower-ℝ l)
   where
 
-  opaque
-    unfolding neg-ℚ
-
+  abstract
     right-unit-law-add-lower-ℝ : add-lower-ℝ x (lower-real-ℚ zero-ℚ) ＝ x
     right-unit-law-add-lower-ℝ =
       eq-sim-cut-lower-ℝ
@@ -231,8 +231,7 @@ module _
                   ( is-in-cut-diff-rational-ℚ⁺-lower-ℝ
                     ( x)
                     ( p)
-                    ( neg-ℚ q ,
-                      is-positive-le-zero-ℚ (neg-ℚ q) (neg-le-ℚ q zero-ℚ q<0))
+                    ( neg-ℚ⁻ (q , is-negative-le-zero-ℚ q<0))
                     ( p<x)))) ,
           (λ p p<x →
             elim-exists
@@ -241,12 +240,11 @@ module _
                 intro-exists
                   ( q , p -ℚ q)
                   ( q<x ,
-                    tr
-                      ( λ r → le-ℚ r zero-ℚ)
+                    binary-tr
+                      ( le-ℚ)
                       ( distributive-neg-diff-ℚ q p)
+                      ( neg-zero-ℚ)
                       ( neg-le-ℚ
-                        ( zero-ℚ)
-                        ( q -ℚ p)
                         ( backward-implication
                           ( iff-translate-diff-le-zero-ℚ p q)
                           ( p<q))) ,
diff --git a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
index adf12e8e83d..44bd7f51aee 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -140,7 +140,7 @@ module _
                   ( q -ℚ px)
                   ( neg-ℚ (q -ℚ p))
                   ( neg-ℚ ε)
-                  ( neg-le-ℚ ε (q -ℚ p) (le-mediant-zero-ℚ⁺ q-p⁺))))
+                  ( neg-le-ℚ (le-mediant-zero-ℚ⁺ q-p⁺))))
               ( y<py) ,
             inv ( is-identity-right-conjugation-add-ℚ (px +ℚ ε) q))
       where open do-syntax-trunc-Prop (cut-add-upper-ℝ q)
@@ -224,7 +224,7 @@ module _
                   ( is-in-cut-add-rational-ℚ⁺-upper-ℝ
                     ( x)
                     ( p)
-                    ( q , is-positive-le-zero-ℚ q 0<q)
+                    ( q , is-positive-le-zero-ℚ 0<q)
                     ( x<p)))) ,
           ( λ q x<q →
             elim-exists
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 3fbc4ba1e4f..c67e8c9b7ac 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -381,7 +381,7 @@ abstract
                     neg-ℚ a
                     ≤ neg-ℚ (b -ℚ ε)
                       by
-                        neg-leq-ℚ _ _
+                        neg-leq-ℚ
                           ( leq-transpose-right-add-ℚ _ _ _ (leq-le-ℚ b<a+ε))
                     ≤ ε -ℚ b
                       by leq-eq-ℚ _ _ (distributive-neg-diff-ℚ _ _)
@@ -390,7 +390,7 @@ abstract
                         preserves-leq-right-add-ℚ ε
                           ( neg-ℚ b)
                           ( neg-ℚ p)
-                          ( neg-leq-ℚ _ _
+                          ( neg-leq-ℚ
                             ( leq-lower-upper-cut-ℝ x p b p<x x<b))
                     ≤ rational-abs-ℚ (ε -ℚ p)
                       by leq-abs-ℚ _
@@ -425,7 +425,7 @@ abstract
                 ( chain-of-inequalities
                     neg-ℚ b
                     ≤ neg-ℚ a
-                      by neg-leq-ℚ _ _ (leq-lower-upper-cut-ℝ x a b a<x x<b)
+                      by neg-leq-ℚ (leq-lower-upper-cut-ℝ x a b a<x x<b)
                     ≤ rational-abs-ℚ a
                       by neg-leq-abs-ℚ a
                     ≤ rational-abs-ℚ a +ℚ ε
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index 7eb3b7277c0..81975525891 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -33,6 +33,7 @@ open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.poset-closed-intervals-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
diff --git a/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
index 5ee824c84c8..59162e2554b 100644
--- a/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
@@ -79,7 +79,7 @@ module _
         (λ p → le-ℚ p q × is-in-cut-neg-lower-ℝ p)
         ( neg-ℚ)
         ( λ p (-q<p , p<x) →
-          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ (neg-ℚ q) p -q<p) ,
+          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ -q<p) ,
           tr (is-in-cut-lower-ℝ x) (inv (neg-neg-ℚ p)) p<x)
         ( forward-implication (is-rounded-cut-lower-ℝ x (neg-ℚ q)) -q<x)
     pr2 (is-rounded-cut-neg-lower-ℝ q) =
@@ -88,7 +88,7 @@ module _
         ( λ p (p<q , -q<x) →
           backward-implication
             ( is-rounded-cut-lower-ℝ x (neg-ℚ q))
-            ( intro-exists (neg-ℚ p) (neg-le-ℚ p q p<q , -q<x)))
+            ( intro-exists (neg-ℚ p) (neg-le-ℚ p<q , -q<x)))
 
   neg-lower-ℝ : upper-ℝ l
   pr1 neg-lower-ℝ = cut-neg-lower-ℝ
@@ -130,7 +130,7 @@ module _
           tr
             ( λ x → le-ℚ x (neg-ℚ p))
             ( neg-neg-ℚ q)
-            ( neg-le-ℚ p (neg-ℚ q) p<-q) ,
+            ( neg-le-ℚ p<-q) ,
           tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)) x<p)
         ( forward-implication (is-rounded-cut-upper-ℝ x (neg-ℚ q)) x<-q)
     pr2 (is-rounded-cut-neg-upper-ℝ q) =
@@ -139,7 +139,7 @@ module _
         ( λ r (q<r , x<-r) →
           backward-implication
             ( is-rounded-cut-upper-ℝ x (neg-ℚ q))
-            ( intro-exists (neg-ℚ r) (neg-le-ℚ q r q<r , x<-r)))
+            ( intro-exists (neg-ℚ r) (neg-le-ℚ q<r , x<-r)))
 
   neg-upper-ℝ : lower-ℝ l
   pr1 neg-upper-ℝ = cut-neg-upper-ℝ
@@ -201,9 +201,9 @@ neg-lower-real-ℚ q =
       ( cut-neg-lower-ℝ (lower-real-ℚ q))
       ( cut-upper-real-ℚ (neg-ℚ q))
       ( (λ p -p<q →
-          tr (le-ℚ (neg-ℚ q)) (neg-neg-ℚ p) (neg-le-ℚ (neg-ℚ p) q -p<q)) ,
+          tr (le-ℚ (neg-ℚ q)) (neg-neg-ℚ p) (neg-le-ℚ -p<q)) ,
         (λ p -q<p →
-          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ (neg-ℚ q) p -q<p))))
+          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ -q<p))))
 ```
 
 ### The negation of a rational projected to an upper real is the projection of its negation as a lower real
@@ -222,10 +222,10 @@ neg-upper-real-ℚ q =
           tr
             ( λ r → le-ℚ r (neg-ℚ q))
             ( neg-neg-ℚ p)
-            ( neg-le-ℚ q (neg-ℚ p) q<-p)) ,
+            ( neg-le-ℚ q<-p)) ,
         (λ p p<-q →
           tr
             ( λ r → le-ℚ r (neg-ℚ p))
             ( neg-neg-ℚ q)
-            ( neg-le-ℚ p (neg-ℚ q) p<-q))))
+            ( neg-le-ℚ p<-q))))
 ```
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 3f84a77108c..a3e57775b74 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -82,7 +82,7 @@ module _
       ( disjunction-Prop (lower-cut-neg-ℝ q) (upper-cut-neg-ℝ r))
       ( inr-disjunction)
       ( inl-disjunction)
-      ( is-located-lower-upper-cut-ℝ x (neg-ℚ r) (neg-ℚ q) (neg-le-ℚ q r q<r))
+      ( is-located-lower-upper-cut-ℝ x (neg-ℚ r) (neg-ℚ q) (neg-le-ℚ q<r))
 
   opaque
     neg-ℝ : ℝ l
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 9f97dc4cc21..7b555aacc04 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -13,6 +13,7 @@ open import elementary-number-theory.addition-nonnegative-rational-numbers
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-nonnegative-rational-numbers
@@ -100,7 +101,7 @@ eq-ℝ⁰⁺ _ _ = eq-type-subtype is-nonnegative-prop-ℝ
 abstract
   is-nonnegative-real-ℚ⁰⁺ : (q : ℚ⁰⁺) → is-nonnegative-ℝ (real-ℚ⁰⁺ q)
   is-nonnegative-real-ℚ⁰⁺ (q , nonneg-q) =
-    preserves-leq-real-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ q nonneg-q)
+    preserves-leq-real-ℚ zero-ℚ q (leq-zero-is-nonnegative-ℚ nonneg-q)
 
 nonnegative-real-ℚ⁰⁺ : ℚ⁰⁺ → ℝ⁰⁺ lzero
 nonnegative-real-ℚ⁰⁺ q = (real-ℚ⁰⁺ q , is-nonnegative-real-ℚ⁰⁺ q)
@@ -270,7 +271,6 @@ module _
         intro-exists
           ( q ,
             is-positive-le-zero-ℚ
-              ( q)
               ( reflects-le-real-ℚ zero-ℚ q
                 ( concatenate-leq-le-ℝ
                   ( zero-ℝ)
@@ -443,7 +443,6 @@ module _
       le-ℝ (real-ℝ⁰⁺ x) (real-ℚ q) → is-positive-ℚ q
     is-positive-le-nonnegative-real-ℚ x<q =
       is-positive-le-zero-ℚ
-        ( q)
         ( reflects-le-real-ℚ _ _
           ( concatenate-leq-le-ℝ _ _ _ (is-nonnegative-real-ℝ⁰⁺ x) x<q))
 ```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 8922e6023bc..4fc3b49bc8b 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -106,7 +106,7 @@ module _
     exists-ℚ⁺-in-lower-cut-is-positive-ℝ =
       elim-exists
         ( ∃ ℚ⁺ (λ p → lower-cut-ℝ x (rational-ℚ⁺ p)))
-        ( λ p (0<p , p<x) → intro-exists (p , is-positive-le-zero-ℚ p 0<p) p<x)
+        ( λ p (0<p , p<x) → intro-exists (p , is-positive-le-zero-ℚ 0<p) p<x)
 
     is-positive-exists-ℚ⁺-in-lower-cut-ℝ :
       exists ℚ⁺ (λ p → lower-cut-ℝ x (rational-ℚ⁺ p)) →
@@ -115,7 +115,7 @@ module _
       elim-exists
         ( is-positive-prop-ℝ x)
         ( λ (p , pos-p) p<x →
-          intro-exists p (le-zero-is-positive-ℚ p pos-p , p<x))
+          intro-exists p (le-zero-is-positive-ℚ pos-p , p<x))
 
   is-positive-iff-exists-ℚ⁺-in-lower-cut-ℝ :
     is-positive-ℝ x ↔ exists ℚ⁺ (λ p → lower-cut-ℝ x (rational-ℚ⁺ p))
@@ -187,13 +187,13 @@ module _
   positive-diff-le-ℝ = (y -ℝ x , is-positive-diff-le-ℝ)
 ```
 
-### Positive rational numbers are positive real numbers
+### The canonical embedding of rational numbers preserves positivity
 
 ```agda
-is-positive-real-positive-ℚ :
+preserves-is-positive-real-ℚ :
   (q : ℚ) → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
-is-positive-real-positive-ℚ q pos-q =
-  preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ q pos-q)
+preserves-is-positive-real-ℚ q pos-q =
+  preserves-le-real-ℚ zero-ℚ q (le-zero-is-positive-ℚ pos-q)
 
 opaque
   unfolding le-ℝ real-ℚ
@@ -204,8 +204,8 @@ opaque
     elim-exists
       ( is-positive-prop-ℚ q)
       ( λ r (0<r , r<q) →
-        is-positive-le-zero-ℚ q (transitive-le-ℚ zero-ℚ r q r<q 0<r))
+        is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ r q r<q 0<r))
 
 positive-real-ℚ⁺ : ℚ⁺ → ℝ⁺ lzero
-positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , is-positive-real-positive-ℚ q pos-q)
+positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ q pos-q)
 ```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index fb1b378d78f..a68d6e8cbd1 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -751,7 +751,7 @@ module _
               ( right-inverse-law-add-ℝ x)
               ( preserves-le-right-add-ℝ (neg-ℝ x) x y x<y))
         intro-exists
-          ( q , is-positive-le-zero-ℚ q (reflects-le-real-ℚ _ _ 0<q))
+          ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ _ _ 0<q))
           ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
 ```
 

From 124c9f689e58dc348553c33bbb7ae304d0254bf5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 10 Oct 2025 16:00:52 -0700
Subject: [PATCH 02/22] make pre-commit

---
 .../archimedean-property-rational-numbers.lagda.md              | 2 +-
 ...inequalities-positive-and-negative-rational-numbers.lagda.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
index f2fdcf6a303..f9f12927fbf 100644
--- a/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/archimedean-property-rational-numbers.lagda.md
@@ -15,7 +15,7 @@ open import elementary-number-theory.integers
 open import elementary-number-theory.multiplication-integer-fractions
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.natural-numbers
--- open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index 3c35c0ff087..ac81b04c088 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -23,7 +23,7 @@ open import foundation.dependent-pair-types
 ## Idea
 
 This page describes
-[inequalities](elementary-number-theory.inequalities-rational-numbers.md) and
+[inequalities](elementary-number-theory.inequality-rational-numbers.md) and
 [strict inequalities](elementary-number-theory.strict-inequality-rational-numbers.md)
 between [positive](elementary-number-theory.positive-rational-numbers.md),
 [negative](elementary-number-theory.negative-rational-numbers.md),

From 9dc460aca8f3d50aede96fcb7222cc61ea54219b Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 11 Oct 2025 21:23:22 -0700
Subject: [PATCH 03/22] Update

---
 ...ive-and-negative-rational-numbers.lagda.md | 15 ++++++++
 .../negative-rational-numbers.lagda.md        |  6 ++--
 .../nonnegative-rational-numbers.lagda.md     | 25 ++++++++++++++
 .../nonpositive-rational-numbers.lagda.md     | 13 +++++++
 ...ive-and-negative-rational-numbers.lagda.md | 20 +++++------
 .../rational-numbers.lagda.md                 |  3 +-
 ...e-roots-positive-rational-numbers.lagda.md |  2 +-
 .../squares-rational-numbers.lagda.md         |  6 ++--
 .../nonnegative-real-numbers.lagda.md         |  7 ++--
 ...re-roots-nonnegative-real-numbers.lagda.md | 34 ++++++++++---------
 10 files changed, 93 insertions(+), 38 deletions(-)

diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index ac81b04c088..414fa5ecdc3 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -11,6 +11,7 @@ open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -55,6 +56,20 @@ abstract
   leq-negative-nonnegative-ℚ p q = leq-le-ℚ (le-negative-nonnegative-ℚ p q)
 ```
 
+#### Any negative rational number is less than any positive rational number
+
+```agda
+abstract
+  le-negative-positive-ℚ :
+    (p : ℚ⁻) (q : ℚ⁺) → le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁺ q)
+  le-negative-positive-ℚ p q =
+    le-negative-nonnegative-ℚ p (nonnegative-ℚ⁺ q)
+
+  leq-negative-positive-ℚ :
+    (p : ℚ⁻) (q : ℚ⁺) → leq-ℚ (rational-ℚ⁻ p) (rational-ℚ⁺ q)
+  leq-negative-positive-ℚ p q = leq-le-ℚ (le-negative-positive-ℚ p q)
+```
+
 #### A nonpositive rational number is less than a positive rational number
 
 ```agda
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 2dc1da8460c..20a8ca5963f 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -264,13 +264,13 @@ opaque
   mediant-zero-ℚ⁻ : ℚ⁻ → ℚ⁻
   mediant-zero-ℚ⁻ (q , is-neg-q) =
     ( mediant-ℚ q zero-ℚ ,
-      is-negative-le-zero-ℚ _
-        ( le-right-mediant-ℚ _ _ (le-zero-is-negative-ℚ q is-neg-q)))
+      is-negative-le-zero-ℚ
+      ( le-right-mediant-ℚ _ _ (le-zero-is-negative-ℚ is-neg-q)))
 
   le-mediant-zero-ℚ⁻ :
     (q : ℚ⁻) → le-ℚ (rational-ℚ⁻ q) (rational-ℚ⁻ (mediant-zero-ℚ⁻ q))
   le-mediant-zero-ℚ⁻ (q , is-neg-q) =
-    le-left-mediant-ℚ _ _ (le-zero-is-negative-ℚ q is-neg-q)
+    le-left-mediant-ℚ _ _ (le-zero-is-negative-ℚ is-neg-q)
 
 rational-mediant-zero-ℚ⁻ : ℚ⁻ → ℚ
 rational-mediant-zero-ℚ⁻ q = rational-ℚ⁻ (mediant-zero-ℚ⁻ q)
diff --git a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
index 621a2787146..b7d23dc65ad 100644
--- a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
@@ -249,3 +249,28 @@ abstract
 nonnegative-diff-leq-ℚ : (x y : ℚ) → leq-ℚ x y → ℚ⁰⁺
 nonnegative-diff-leq-ℚ x y x≤y = (y -ℚ x , is-nonnegative-diff-leq-ℚ x y x≤y)
 ```
+
+### If `x ≤ y` and `x` is nonnegative, then `y` is nonnegative
+
+```agda
+abstract
+  is-nonnegative-leq-ℚ⁰⁺ :
+    (x : ℚ⁰⁺) (y : ℚ) → leq-ℚ (rational-ℚ⁰⁺ x) y →
+    is-nonnegative-ℚ y
+  is-nonnegative-leq-ℚ⁰⁺ (x , is-nonneg-x) y x≤y =
+    is-nonnegative-leq-zero-ℚ
+      ( transitive-leq-ℚ zero-ℚ x y
+        ( x≤y)
+        ( leq-zero-is-nonnegative-ℚ is-nonneg-x))
+```
+
+### If `x < y` and `x` is nonnegative, then `y` is nonnegative
+
+```agda
+abstract
+  is-nonnegative-le-ℚ⁰⁺ :
+    (x : ℚ⁰⁺) (y : ℚ) → le-ℚ (rational-ℚ⁰⁺ x) y →
+    is-nonnegative-ℚ y
+  is-nonnegative-le-ℚ⁰⁺ x y x<y =
+    is-nonnegative-leq-ℚ⁰⁺ x y (leq-le-ℚ x<y)
+```
diff --git a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
index b0506e21798..fa039b16795 100644
--- a/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonpositive-rational-numbers.lagda.md
@@ -120,3 +120,16 @@ is-nonpositive-iff-leq-zero-ℚ q =
   ( leq-zero-is-nonpositive-ℚ ,
     is-nonpositive-leq-zero-ℚ)
 ```
+
+### If `p ≤ q` and `q` is nonpositive, then `p` is nonpositive
+
+```agda
+abstract
+  is-nonpositive-leq-ℚ⁰⁻ :
+    (q : ℚ⁰⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁰⁻ q) → is-nonpositive-ℚ p
+  is-nonpositive-leq-ℚ⁰⁻ (q , nonpos-q) p p≤q =
+    is-nonpositive-leq-zero-ℚ
+      ( transitive-leq-ℚ p q zero-ℚ
+        ( leq-zero-is-nonpositive-ℚ nonpos-q)
+        ( p≤q))
+```
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 8a11f9eb930..a2f18c7d166 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -211,7 +211,7 @@ neg-ℚ⁰⁻ (q , is-nonpos-q) =
 
 ### Exclusive cases
 
-#### A rational is not both negative and positive
+#### A rational number is not both negative and positive
 
 ```agda
 abstract
@@ -225,16 +225,16 @@ abstract
       ( leq-zero-is-negative-ℚ N)
 ```
 
-### If `p ≤ q` and `q` is nonpositive, then `p` is nonpositive
+#### A rational number is not both negative and nonnegative
 
 ```agda
 abstract
-  is-nonpositive-leq-ℚ⁰⁻ :
-    (q : ℚ⁰⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁰⁻ q) → is-nonpositive-ℚ p
-  is-nonpositive-leq-ℚ⁰⁻ (q , nonpos-q) p p≤q =
-    is-nonpositive-leq-zero-ℚ
-      ( p)
-      ( transitive-leq-ℚ p q zero-ℚ
-        ( leq-zero-is-nonpositive-ℚ q nonpos-q)
-        ( p≤q))
+  not-is-negative-is-nonnegative-ℚ :
+    {x : ℚ} → ¬ (is-negative-ℚ x × is-nonnegative-ℚ x)
+  not-is-negative-is-nonnegative-ℚ {x} (N , NN) =
+    not-leq-le-ℚ
+      ( x)
+      ( zero-ℚ)
+      ( le-zero-is-negative-ℚ N)
+      ( leq-zero-is-nonnegative-ℚ NN)
 ```
diff --git a/src/elementary-number-theory/rational-numbers.lagda.md b/src/elementary-number-theory/rational-numbers.lagda.md
index 936003357b2..ed81dcc1dc9 100644
--- a/src/elementary-number-theory/rational-numbers.lagda.md
+++ b/src/elementary-number-theory/rational-numbers.lagda.md
@@ -308,8 +308,7 @@ opaque
 ```agda
 abstract
   eq-neg-one-ℚ : neg-ℚ one-ℚ ＝ neg-one-ℚ
-  eq-neg-one-ℚ =
-    inv (preserves-neg-rational-ℤ one-ℤ)
+  eq-neg-one-ℚ = inv (neg-rational-ℤ one-ℤ)
 ```
 
 ### The reduced fraction of the negative of an integer fraction is the negative of the reduced fraction
diff --git a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
index e9327a4d02f..5864700e784 100644
--- a/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/square-roots-positive-rational-numbers.lagda.md
@@ -260,7 +260,7 @@ abstract
               ( is-identity-right-conjugation-add-ℚ (p *ℚ p) q))
             ( refl)
             ( preserves-le-right-add-ℚ (p *ℚ p) _ _
-              ( neg-le-ℚ _ _
+              ( neg-le-ℚ
                 ( tr
                   ( le-ℚ _)
                   ( ap
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index f40ac8fef9d..16de8f520d9 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -11,6 +11,9 @@ module elementary-number-theory.squares-rational-numbers where
 ```agda
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers
@@ -61,9 +64,8 @@ abstract
     rec-coproduct
       ( λ H →
         is-nonnegative-is-positive-ℚ
-          ( a *ℚ a)
           ( is-positive-mul-negative-ℚ {a} {a} H H))
-      ( λ H → is-nonnegative-mul-ℚ a a H H)
+      ( λ H → is-nonnegative-mul-ℚ H H)
       ( decide-is-negative-is-nonnegative-ℚ a)
 
 nonnegative-square-ℚ : ℚ → ℚ⁰⁺
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index e3f09710870..2dd03cdc110 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -288,7 +288,6 @@ abstract
     is-positive-ℚ q
   is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) q x<q =
     is-positive-le-zero-ℚ
-      ( q)
       ( reflects-le-real-ℚ
         ( zero-ℚ)
         ( q)
@@ -303,7 +302,7 @@ opaque
     {l : Level} → (x : ℝ l) → (upper-cut-ℝ x ⊆ is-positive-prop-ℚ) →
     is-nonnegative-ℝ x
   is-nonnegative-is-positive-upper-cut-ℝ x Uₓ⊆ℚ⁺ =
-    leq-leq'-ℝ zero-ℝ x (λ q q∈Uₓ → le-zero-is-positive-ℚ q (Uₓ⊆ℚ⁺ q q∈Uₓ))
+    leq-leq'-ℝ zero-ℝ x (λ q q∈Uₓ → le-zero-is-positive-ℚ (Uₓ⊆ℚ⁺ q q∈Uₓ))
 ```
 
 ### A real number is nonnegative if and only if every negative rational number is in its lower cut
@@ -316,13 +315,13 @@ opaque
     {l : Level} (x : ℝ l) → (is-negative-prop-ℚ ⊆ lower-cut-ℝ x) →
     is-nonnegative-ℝ x
   is-nonnegative-leq-negative-lower-cut-ℝ x ℚ⁻⊆Lₓ q q<0 =
-    ℚ⁻⊆Lₓ q (is-negative-le-zero-ℚ q q<0)
+    ℚ⁻⊆Lₓ q (is-negative-le-zero-ℚ q<0)
 
   leq-negative-lower-cut-is-nonnegative-ℝ :
     {l : Level} (x : ℝ l) → is-nonnegative-ℝ x →
     (is-negative-prop-ℚ ⊆ lower-cut-ℝ x)
   leq-negative-lower-cut-is-nonnegative-ℝ x 0≤x q is-neg-q =
-    0≤x q (le-zero-is-negative-ℚ q is-neg-q)
+    0≤x q (le-zero-is-negative-ℚ is-neg-q)
 ```
 
 ### Every nonnegative real number has a positive rational number in its upper cut
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index eba2790f9fc..0bcd3acc129 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -9,6 +9,7 @@ module real-numbers.square-roots-nonnegative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minimum-rational-numbers
@@ -88,8 +89,7 @@ module _
         ( λ is-nonneg-neg-one-ℚ →
           ex-falso
             ( not-is-negative-is-nonnegative-ℚ
-              ( is-nonneg-neg-one-ℚ)
-              ( is-negative-rational-ℚ⁻ neg-one-ℚ⁻)))
+              ( is-negative-rational-ℚ⁻ neg-one-ℚ⁻ , is-nonneg-neg-one-ℚ)))
 
     is-inhabited-upper-cut-sqrt-ℝ⁰⁺ : is-inhabited-subtype upper-cut-sqrt-ℝ⁰⁺
     is-inhabited-upper-cut-sqrt-ℝ⁰⁺ =
@@ -122,8 +122,8 @@ module _
                   λ is-nonneg-q' →
                     ex-falso
                       ( not-is-negative-is-nonnegative-ℚ
-                        ( is-nonneg-q')
-                        ( is-negative-rational-ℚ⁻ (mediant-zero-ℚ⁻ q⁻)))))
+                        ( is-negative-rational-ℚ⁻ (mediant-zero-ℚ⁻ q⁻) ,
+                          is-nonneg-q'))))
           ( λ q=0 →
             do
               ( q' , 0<q' , q'<x) ←
@@ -132,9 +132,13 @@ module _
                   ( tr
                     ( is-in-lower-cut-ℝ⁰⁺ x)
                     ( ap-mul-ℚ q=0 q=0 ∙ right-zero-law-mul-ℚ zero-ℚ)
-                    ( q²<x (inv-tr is-nonnegative-ℚ q=0 _)))
+                    ( q²<x
+                      ( inv-tr
+                        ( is-nonnegative-ℚ)
+                        ( q=0)
+                        ( is-nonnegative-rational-ℚ⁰⁺ zero-ℚ⁰⁺))))
               let
-                is-pos-q' = is-positive-le-zero-ℚ q' 0<q'
+                is-pos-q' = is-positive-le-zero-ℚ 0<q'
                 q'⁺ = (q' , is-pos-q')
               ( p⁺@(p , is-pos-p) , p²<q') ← square-le-ℚ⁺ q'⁺
               intro-exists
@@ -142,7 +146,7 @@ module _
                 ( inv-tr
                     ( λ r → le-ℚ r p)
                     ( q=0)
-                    ( le-zero-is-positive-ℚ p is-pos-p) ,
+                    ( le-zero-is-positive-ℚ is-pos-p) ,
                   λ _ →
                     le-lower-cut-ℝ (real-ℝ⁰⁺ x) (p *ℚ p) q' p²<q' q'<x))
           ( λ is-pos-q →
@@ -150,7 +154,7 @@ module _
               (p , q²<p , p<x) ←
                 forward-implication
                   ( is-rounded-lower-cut-ℝ⁰⁺ x (q *ℚ q))
-                  ( q²<x (is-nonnegative-is-positive-ℚ q is-pos-q))
+                  ( q²<x (is-nonnegative-is-positive-ℚ is-pos-q))
               let
                 is-pos-p =
                   is-positive-le-ℚ⁺
@@ -194,7 +198,6 @@ module _
         let
           is-nonneg-r =
             is-nonnegative-is-positive-ℚ
-              ( r)
               ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) r q<r)
         le-lower-cut-ℝ
           ( real-ℝ⁰⁺ x)
@@ -217,8 +220,8 @@ module _
             ( q *ℚ q)
             ( r *ℚ r)
             ( preserves-le-square-ℚ⁰⁺
-              ( q , is-nonnegative-is-positive-ℚ q is-pos-q)
-              ( r , is-nonnegative-is-positive-ℚ r is-pos-r)
+              ( q , is-nonnegative-is-positive-ℚ is-pos-q)
+              ( r , is-nonnegative-is-positive-ℚ is-pos-r)
               q<r)
             ( x<q²))
 
@@ -244,7 +247,7 @@ module _
       is-disjoint-cut-ℝ
         ( real-ℝ⁰⁺ x)
         ( q *ℚ q)
-        ( q<√x (is-nonnegative-is-positive-ℚ q is-pos-q) ,
+        ( q<√x (is-nonnegative-is-positive-ℚ is-pos-q) ,
           x<q²)
 
     is-located-lower-upper-cut-sqrt-ℝ⁰⁺ :
@@ -252,10 +255,11 @@ module _
       type-disjunction-Prop (lower-cut-sqrt-ℝ⁰⁺ p) (upper-cut-sqrt-ℝ⁰⁺ q)
     is-located-lower-upper-cut-sqrt-ℝ⁰⁺ p q p<q =
       rec-coproduct
-        ( λ p<0 →
+        ( λ is-neg-p →
           inl-disjunction
             ( λ is-nonneg-p →
-              ex-falso (not-is-negative-is-nonnegative-ℚ is-nonneg-p p<0)))
+              ex-falso
+                ( not-is-negative-is-nonnegative-ℚ (is-neg-p , is-nonneg-p))))
         ( λ is-nonneg-p →
           map-disjunction
             ( λ p²<x _ → p²<x)
@@ -343,7 +347,6 @@ module _
                   ( λ is-neg-a →
                     ex-falso
                       ( not-is-negative-is-positive-ℚ
-                        ( a *ℚ d)
                         ( is-negative-mul-negative-positive-ℚ
                             ( is-neg-a)
                             ( is-pos-d) ,
@@ -359,7 +362,6 @@ module _
                   ( λ is-neg-c →
                     ex-falso
                       ( not-is-negative-is-positive-ℚ
-                        ( b *ℚ c)
                         ( is-negative-mul-positive-negative-ℚ
                             ( is-pos-b)
                             ( is-neg-c) ,

From e7c64234e700065eaf367b0d1f8a334ad87dad41 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 14:29:14 -0700
Subject: [PATCH 04/22] Progress

---
 ...wers-of-positive-rational-numbers.lagda.md |  0
 ...wers-of-positive-rational-numbers.lagda.md | 49 +++++++++++++++++++
 2 files changed, 49 insertions(+)
 create mode 100644 src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md
new file mode 100644
index 00000000000..e69de29bb2d
diff --git a/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
new file mode 100644
index 00000000000..33cb3b88f16
--- /dev/null
+++ b/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
@@ -0,0 +1,49 @@
+# Powers of positive rational numbers
+
+```agda
+module elementary-number-theory.powers-of-positive-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import group-theory.powers-of-elements-groups
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+on the [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
+power-ℚ⁺ = power-Group group-mul-ℚ⁺
+```
+
+## Properties
+
+### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+
+```agda
+
+```
+
+## See also
+
+- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)

From 9963c472ee1f912b12ac290301d97d71fb48dd33 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 14:49:48 -0700
Subject: [PATCH 05/22] Updates

---
 .../maximum-nonnegative-rational-numbers.lagda.md  |  2 +-
 .../minimum-positive-rational-numbers.lagda.md     |  6 +++---
 ...positive-and-negative-rational-numbers.lagda.md |  6 ++++--
 ...positive-and-negative-rational-numbers.lagda.md | 14 ++++++++++++++
 ...icative-inverses-positive-real-numbers.lagda.md |  5 ++++-
 src/real-numbers/positive-real-numbers.lagda.md    |  1 -
 6 files changed, 26 insertions(+), 8 deletions(-)

diff --git a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
index e58e1036a68..2160af8be1b 100644
--- a/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/maximum-nonnegative-rational-numbers.lagda.md
@@ -37,7 +37,7 @@ abstract
     {p q : ℚ} → is-nonnegative-ℚ p → is-nonnegative-ℚ q →
     is-nonnegative-ℚ (max-ℚ p q)
   is-nonnegative-max-ℚ {p} {q} is-nonneg-p _ =
-    is-nonnegative-leq-zero-ℚ _
+    is-nonnegative-leq-zero-ℚ
       ( transitive-leq-ℚ
         ( zero-ℚ)
         ( p)
diff --git a/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md b/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md
index 47c9745f5f1..b5ccc9867d9 100644
--- a/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/minimum-positive-rational-numbers.lagda.md
@@ -39,10 +39,10 @@ abstract
   is-positive-min-ℚ :
     {p q : ℚ} → is-positive-ℚ p → is-positive-ℚ q → is-positive-ℚ (min-ℚ p q)
   is-positive-min-ℚ {p} {q} is-pos-p is-pos-q =
-    is-positive-le-zero-ℚ _
+    is-positive-le-zero-ℚ
       ( le-min-le-both-ℚ zero-ℚ p q
-        ( le-zero-is-positive-ℚ p is-pos-p)
-        ( le-zero-is-positive-ℚ q is-pos-q))
+        ( le-zero-is-positive-ℚ is-pos-p)
+        ( le-zero-is-positive-ℚ is-pos-q))
 
 min-ℚ⁺ : ℚ⁺ → ℚ⁺ → ℚ⁺
 min-ℚ⁺ (p , is-pos-p) (q , is-pos-q) =
diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index 72a497095a0..8cbfc9610b2 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -71,7 +71,7 @@ mul-negative-positive-ℚ (p , neg-p) (q , pos-q) =
 
 ```agda
 opaque
-  unfolding mul-ℚ
+  unfolding is-nonnegative-ℚ is-positive-ℚ mul-ℚ
 
   is-nonnegative-mul-nonnegative-positive-ℚ :
     {x y : ℚ} → is-nonnegative-ℚ x → is-positive-ℚ y → is-nonnegative-ℚ (x *ℚ y)
@@ -83,7 +83,9 @@ opaque
 ### The product of a nonpositive and a positive rational number is nonpositive
 
 ```agda
-abstract
+opaque
+  unfolding is-nonpositive-ℚ
+
   is-nonpositive-mul-nonpositive-positive-ℚ :
     {x y : ℚ} → is-nonpositive-ℚ x → is-positive-ℚ y → is-nonpositive-ℚ (x *ℚ y)
   is-nonpositive-mul-nonpositive-positive-ℚ is-nonneg-neg-x is-pos-y =
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index a2f18c7d166..af6fead7f8f 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -238,3 +238,17 @@ abstract
       ( le-zero-is-negative-ℚ N)
       ( leq-zero-is-nonnegative-ℚ NN)
 ```
+
+### A rational number is not both positive and nonpositive
+
+```agda
+abstract
+  not-is-positive-is-nonpositive-ℚ :
+    {x : ℚ} → ¬ (is-positive-ℚ x × is-nonpositive-ℚ x)
+  not-is-positive-is-nonpositive-ℚ {x} (P , NP) =
+    not-leq-le-ℚ
+      ( zero-ℚ)
+      ( x)
+      ( le-zero-is-positive-ℚ P)
+      ( leq-zero-is-nonpositive-ℚ NP)
+```
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index b8a957c53fe..384e7af4c8a 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -19,6 +19,7 @@ open import elementary-number-theory.multiplication-closed-intervals-rational-nu
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -443,7 +444,9 @@ module _
                 ( inv-le-ℚ⁺ p⁺ q⁺ p<q)))
         ( λ is-nonpos-p →
           inl-disjunction
-            ( ex-falso ∘ not-is-positive-is-nonpositive-ℚ is-nonpos-p))
+            ( λ is-pos-p →
+              ex-falso
+                ( not-is-positive-is-nonpositive-ℚ (is-pos-p , is-nonpos-p))))
         ( decide-is-positive-is-nonpositive-ℚ p)
 
   opaque
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index a2ce70292e5..9181fcb5459 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -275,7 +275,6 @@ abstract
     {l : Level} (x : ℝ⁺ l) (q : ℚ) → is-in-upper-cut-ℝ⁺ x q → is-positive-ℚ q
   is-positive-is-in-upper-cut-ℝ⁺ x⁺@(x , _) q x<q =
     is-positive-le-zero-ℚ
-      ( q)
       ( le-lower-upper-cut-ℝ x zero-ℚ q (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
 ```
 

From 817f878ff421c7857fb5c279797ee718e95a0300 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 16:39:53 -0700
Subject: [PATCH 06/22] More properties

---
 src/elementary-number-theory.lagda.md         |   2 +
 ...roup-of-positive-rational-numbers.lagda.md |   8 +
 ...wers-of-positive-rational-numbers.lagda.md |  49 ----
 .../powers-positive-rational-numbers.lagda.md | 256 ++++++++++++++++++
 .../powers-of-elements-groups.lagda.md        |  25 ++
 ...ve-inverses-positive-real-numbers.lagda.md |   2 +-
 6 files changed, 292 insertions(+), 50 deletions(-)
 delete mode 100644 src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/powers-positive-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 542cd4fce73..f53d1f7c1a8 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -108,6 +108,7 @@ open import elementary-number-theory.infinitude-of-primes public
 open import elementary-number-theory.initial-segments-natural-numbers public
 open import elementary-number-theory.integer-fractions public
 open import elementary-number-theory.integer-partitions public
+open import elementary-number-theory.integer-powers-of-positive-rational-numbers public
 open import elementary-number-theory.integers public
 open import elementary-number-theory.interior-closed-intervals-rational-numbers public
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers public
@@ -179,6 +180,7 @@ open import elementary-number-theory.positive-integers public
 open import elementary-number-theory.positive-rational-numbers public
 open import elementary-number-theory.powers-integers public
 open import elementary-number-theory.powers-of-two public
+open import elementary-number-theory.powers-positive-rational-numbers public
 open import elementary-number-theory.prime-numbers public
 open import elementary-number-theory.products-of-natural-numbers public
 open import elementary-number-theory.proper-closed-intervals-rational-numbers public
diff --git a/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md
index 9ccaea277e7..3b2ca16a732 100644
--- a/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-positive-rational-numbers.lagda.md
@@ -236,3 +236,11 @@ abstract
       ( is-retraction-left-div-ℚ⁺ p⁺ r)
       ( preserves-le-left-mul-ℚ⁺ (inv-ℚ⁺ p⁺) _ _ pq<pr)
 ```
+
+### The inverse of 1 is 1
+
+```agda
+abstract
+  inv-one-ℚ⁺ : inv-ℚ⁺ one-ℚ⁺ ＝ one-ℚ⁺
+  inv-one-ℚ⁺ = inv-unit-Group group-mul-ℚ⁺
+```
diff --git a/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
deleted file mode 100644
index 33cb3b88f16..00000000000
--- a/src/elementary-number-theory/powers-of-positive-rational-numbers.lagda.md
+++ /dev/null
@@ -1,49 +0,0 @@
-# Powers of positive rational numbers
-
-```agda
-module elementary-number-theory.powers-of-positive-rational-numbers where
-```
-
-<details><summary>Imports</summary>
-
-```agda
-open import elementary-number-theory.multiplication-positive-rational-numbers
-open import elementary-number-theory.rational-numbers
-open import elementary-number-theory.positive-rational-numbers
-open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
-open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
-open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.addition-natural-numbers
-open import elementary-number-theory.multiplication-natural-numbers
-open import group-theory.powers-of-elements-groups
-```
-
-</details>
-
-## Idea
-
-The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
-on the [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
-which is defined by [iteratively](foundation.iterating-functions.md) multiplying
-`x` with itself `n` times.
-
-## Definition
-
-```agda
-power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
-power-ℚ⁺ = power-Group group-mul-ℚ⁺
-```
-
-## Properties
-
-### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
-
-```agda
-
-```
-
-## See also
-
-- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
new file mode 100644
index 00000000000..7815783fb91
--- /dev/null
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -0,0 +1,256 @@
+# Powers of positive rational numbers
+
+```agda
+module elementary-number-theory.powers-positive-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.archimedean-property-rational-numbers
+open import elementary-number-theory.arithmetic-sequences-positive-rational-numbers
+open import elementary-number-theory.bernoullis-inequality-positive-rational-numbers
+open import elementary-number-theory.geometric-sequences-positive-rational-numbers
+open import elementary-number-theory.inequality-positive-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonzero-natural-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-positive-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.existential-quantification
+open import foundation.function-types
+open import foundation.identity-types
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
+
+open import group-theory.multiples-of-elements-abelian-groups
+open import group-theory.powers-of-elements-groups
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+on the [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
+power-ℚ⁺ = power-Group group-mul-ℚ⁺
+```
+
+## Properties
+
+### `1ⁿ = 1`
+
+```agda
+power-one-ℚ⁺ : (n : ℕ) → power-ℚ⁺ n one-ℚ⁺ ＝ one-ℚ⁺
+power-one-ℚ⁺ = power-unit-Group group-mul-ℚ⁺
+```
+
+### `qⁿ⁺¹ = qⁿq`
+
+```agda
+power-succ-ℚ⁺ : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ power-ℚ⁺ n q *ℚ⁺ q
+power-succ-ℚ⁺ = power-succ-Group group-mul-ℚ⁺
+```
+
+### `qⁿ = qqⁿ`
+
+```agda
+power-succ-ℚ⁺' : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ q *ℚ⁺ power-ℚ⁺ n q
+power-succ-ℚ⁺' = power-succ-Group' group-mul-ℚ⁺
+```
+
+### `qᵐ⁺ⁿ = qᵐqⁿ`
+
+```agda
+distributive-power-add-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m +ℕ n) q ＝ power-ℚ⁺ m q *ℚ⁺ power-ℚ⁺ n q
+distributive-power-add-ℚ⁺ m n _ = distributive-power-add-Group group-mul-ℚ⁺ m n
+```
+
+### `(pq)ⁿ=pⁿqⁿ`
+
+```agda
+left-distributive-power-mul-ℚ⁺ :
+  (n : ℕ) (p q : ℚ⁺) → power-ℚ⁺ n (p *ℚ⁺ q) ＝ power-ℚ⁺ n p *ℚ⁺ power-ℚ⁺ n q
+left-distributive-power-mul-ℚ⁺ n p q =
+  left-distributive-multiple-add-Ab abelian-group-mul-ℚ⁺ n
+```
+
+### `pᵐⁿ = (pᵐ)ⁿ`
+
+```agda
+power-mul-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m *ℕ n) q ＝ power-ℚ⁺ n (power-ℚ⁺ m q)
+power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
+```
+
+### If `p` and `q` are positive rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+
+```agda
+abstract
+  preserves-le-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
+    le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
+  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
+  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
+    transitive-le-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-le-right-mul-ℚ⁺ q _ _
+        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
+      ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
+```
+
+### If `p` and `q` are positive rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+
+```agda
+abstract
+  preserves-leq-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → leq-ℚ⁺ p q → leq-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-leq-power-ℚ⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
+  preserves-leq-power-ℚ⁺ 1 p q p≤q = p≤q
+  preserves-leq-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
+    transitive-leq-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-leq-right-mul-ℚ⁺ q _ _ (preserves-leq-power-ℚ⁺ n p q p≤q))
+      ( preserves-leq-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p≤q)
+```
+
+### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  bound-unbounded-power-one-plus-ℚ⁺ ε⁺@(ε , is-pos-ε) b =
+    let
+      (n , b<nε) = bound-archimedean-property-ℚ ε b is-pos-ε
+    in
+      ( n ,
+        transitive-le-ℚ _ _ _
+          ( concatenate-le-leq-ℚ _ _ _
+            ( le-left-add-rational-ℚ⁺ _ one-ℚ⁺)
+            ( binary-tr
+              ( leq-ℚ)
+              ( inv (compute-standard-arithmetic-sequence-ℚ⁺ one-ℚ⁺ ε⁺ n))
+              ( ap
+                ( rational-ℚ⁺)
+                ( equational-reasoning
+                  seq-standard-geometric-sequence-ℚ⁺
+                      ( one-ℚ⁺)
+                      ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                      ( n)
+                  ＝ one-ℚ⁺ *ℚ⁺ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by
+                      inv
+                        ( compute-standard-geometric-sequence-ℚ⁺
+                          ( one-ℚ⁺)
+                          ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                          ( n))
+                  ＝ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by left-unit-law-mul-ℚ⁺ _))
+              ( bernoullis-inequality-ℚ⁺ ε⁺ n)))
+          ( b<nε))
+
+  unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  unbounded-power-one-plus-ℚ⁺ ε b =
+    unit-trunc-Prop (bound-unbounded-power-one-plus-ℚ⁺ ε b)
+```
+
+### If `1 < q`, `qⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
+    let
+      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
+      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+    in
+      ( n ,
+        tr
+          ( le-ℚ b)
+          ( ap
+            ( rational-ℚ⁺ ∘ power-ℚ⁺ n)
+            ( eq-ℚ⁺ (is-identity-right-conjugation-add-ℚ one-ℚ q)))
+          ( b<⟨1+q-1⟩ⁿ))
+
+  unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  unbounded-power-greater-than-one-ℚ⁺ q b 1<q =
+    unit-trunc-Prop (bound-unbounded-power-greater-than-one-ℚ⁺ q b 1<q)
+```
+
+### If `ε` is a positive rational number less than one, `εⁿ` becomes arbitrarily small
+
+```agda
+abstract
+  bound-arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    Σ ℕ (λ n → le-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1 =
+    let
+      1/δ = inv-ℚ⁺ δ
+      1/ε = inv-ℚ⁺ ε
+      1<1/ε =
+        tr
+          ( λ q → le-ℚ⁺ q 1/ε)
+          ( inv-one-ℚ⁺)
+          ( inv-le-ℚ⁺ ε one-ℚ⁺ ε<1)
+      (n , 1/δ<⟨1/ε⟩ⁿ) =
+        bound-unbounded-power-greater-than-one-ℚ⁺ 1/ε (rational-ℚ⁺ 1/δ) 1<1/ε
+    in
+      ( n ,
+        binary-tr
+          ( le-ℚ⁺)
+          ( equational-reasoning
+            inv-ℚ⁺ (power-ℚ⁺ n 1/ε)
+            ＝ inv-ℚ⁺ (inv-ℚ⁺ (power-ℚ⁺ n ε))
+              by ap inv-ℚ⁺ (power-inv-Group group-mul-ℚ⁺ n ε)
+            ＝ power-ℚ⁺ n ε
+              by inv-inv-ℚ⁺ (power-ℚ⁺ n ε))
+          ( inv-inv-ℚ⁺ δ)
+          ( inv-le-ℚ⁺ 1/δ (power-ℚ⁺ n 1/ε) 1/δ<⟨1/ε⟩ⁿ))
+
+  arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    exists ℕ (λ n → le-prop-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε =
+    unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
+```
+
+## See also
+
+- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
diff --git a/src/group-theory/powers-of-elements-groups.lagda.md b/src/group-theory/powers-of-elements-groups.lagda.md
index a9e28db03ab..f2e9511b258 100644
--- a/src/group-theory/powers-of-elements-groups.lagda.md
+++ b/src/group-theory/powers-of-elements-groups.lagda.md
@@ -11,6 +11,7 @@ open import elementary-number-theory.addition-natural-numbers
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.natural-numbers
 
+open import foundation.action-on-identifications-functions
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
@@ -175,6 +176,30 @@ module _
   power-mul-Group = power-mul-Monoid (monoid-Group G)
 ```
 
+### The inverse of `x`, raised to the power `n`, is the inverse of `x` raised to the power `n`
+
+```agda
+module _
+  {l : Level} (G : Group l)
+  where
+
+  abstract
+    power-inv-Group :
+      (n : ℕ) (x : type-Group G) →
+      power-Group G n (inv-Group G x) ＝ inv-Group G (power-Group G n x)
+    power-inv-Group 0 _ = inv (inv-unit-Group G)
+    power-inv-Group 1 _ = refl
+    power-inv-Group (succ-ℕ n@(succ-ℕ _)) x =
+      equational-reasoning
+        mul-Group G (power-Group G n (inv-Group G x)) (inv-Group G x)
+        ＝ mul-Group G (inv-Group G (power-Group G n x)) (inv-Group G x)
+          by ap-mul-Group G (power-inv-Group n x) refl
+        ＝ inv-Group G (mul-Group G x (power-Group G n x))
+          by inv (distributive-inv-mul-Group G)
+        ＝ inv-Group G (power-Group G (succ-ℕ n) x)
+          by ap (inv-Group G) (inv (power-succ-Group' G n x))
+```
+
 ### Group homomorphisms preserve powers
 
 ```agda
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 384e7af4c8a..06b96659065 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -10,6 +10,7 @@ module real-numbers.multiplicative-inverses-positive-real-numbers where
 
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-positive-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
@@ -19,7 +20,6 @@ open import elementary-number-theory.multiplication-closed-intervals-rational-nu
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers

From 984d8d40f546f7aef53a05614035a3d2615a048c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 16:43:12 -0700
Subject: [PATCH 07/22] Back out integer powers

---
 .../integer-powers-of-positive-rational-numbers.lagda.md          | 0
 1 file changed, 0 insertions(+), 0 deletions(-)
 delete mode 100644 src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md b/src/elementary-number-theory/integer-powers-of-positive-rational-numbers.lagda.md
deleted file mode 100644
index e69de29bb2d..00000000000

From a69f76695b653a7178ad6277aa99b9713adfd65f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 14 Oct 2025 20:45:33 -0700
Subject: [PATCH 08/22] Fixes

---
 src/elementary-number-theory.lagda.md                           | 1 -
 .../powers-positive-rational-numbers.lagda.md                   | 2 +-
 2 files changed, 1 insertion(+), 2 deletions(-)

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index f53d1f7c1a8..be3b1e0ad01 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -108,7 +108,6 @@ open import elementary-number-theory.infinitude-of-primes public
 open import elementary-number-theory.initial-segments-natural-numbers public
 open import elementary-number-theory.integer-fractions public
 open import elementary-number-theory.integer-partitions public
-open import elementary-number-theory.integer-powers-of-positive-rational-numbers public
 open import elementary-number-theory.integers public
 open import elementary-number-theory.interior-closed-intervals-rational-numbers public
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers public
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 7815783fb91..811235c5420 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -45,7 +45,7 @@ open import group-theory.powers-of-elements-groups
 ## Idea
 
 The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
 on the [positive](elementary-number-theory.positive-rational-numbers.md)
 [rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
 which is defined by [iteratively](foundation.iterating-functions.md) multiplying

From ea3fa12c81c28d11c10da66a04d9e42d6a7baf98 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 15 Oct 2025 09:15:11 -0700
Subject: [PATCH 09/22] Fix merge

---
 ...ion-positive-and-negative-rational-numbers.lagda.md |  6 ++----
 .../positive-and-negative-rational-numbers.lagda.md    | 10 ----------
 ...ltiplicative-inverses-nonzero-real-numbers.lagda.md |  4 +---
 src/real-numbers/negative-real-numbers.lagda.md        |  4 ++--
 .../square-roots-nonnegative-real-numbers.lagda.md     | 10 ++--------
 5 files changed, 7 insertions(+), 27 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index e8b894794eb..eda0123b5e7 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -148,8 +148,7 @@ abstract
             ( y≠0)
             ( λ is-pos-y →
               ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( x *ℚ y)
+                ( not-is-negative-is-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-x is-pos-y ,
                     is-pos-xy))))
         ( λ x=0 →
@@ -164,8 +163,7 @@ abstract
             ( y)
             ( λ is-neg-y →
               ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( x *ℚ y)
+                ( not-is-negative-is-positive-ℚ
                   ( is-negative-mul-positive-negative-ℚ is-pos-x is-neg-y ,
                     is-pos-xy)))
             ( y≠0)
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index ed64fcf3fdf..00e1063ad38 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -239,16 +239,6 @@ abstract
       ( leq-zero-is-nonnegative-ℚ NN)
 ```
 
-### A positive rational number is not negative
-
-```agda
-abstract
-  not-is-negative-is-positive-ℚ :
-    {q : ℚ} → is-positive-ℚ q → ¬ (is-negative-ℚ q)
-  not-is-negative-is-positive-ℚ {q} is-pos-q =
-    not-is-negative-is-nonnegative-ℚ (is-nonnegative-is-positive-ℚ q is-pos-q)
-```
-
 ### If `p < q` and `q` is nonpositive, then `p` is negative
 
 ```agda
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index 79a1985bb71..43c55227e30 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -151,7 +151,6 @@ opaque
       let
         is-positive-mul p q {r} lb1 lb2 =
           is-positive-le-zero-ℚ
-            ( p *ℚ q)
             ( concatenate-le-leq-ℚ
               ( zero-ℚ)
               ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -173,8 +172,7 @@ opaque
                   ( intro-exists (b , is-neg-b) x<b)))
             ( λ (is-pos-b , is-pos-d) →
               ex-falso
-                ( is-not-negative-and-positive-ℚ
-                  ( a *ℚ d)
+                ( not-is-negative-is-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-a is-pos-d ,
                     is-positive-mul a d
                       ( leq-right-min-ℚ _ _)
diff --git a/src/real-numbers/negative-real-numbers.lagda.md b/src/real-numbers/negative-real-numbers.lagda.md
index f33b4ee1af5..fd87ef7b8c7 100644
--- a/src/real-numbers/negative-real-numbers.lagda.md
+++ b/src/real-numbers/negative-real-numbers.lagda.md
@@ -77,7 +77,7 @@ module _
     exists-ℚ⁻-in-upper-cut-is-negative-ℝ =
       elim-exists
         ( ∃ ℚ⁻ (λ p → upper-cut-ℝ x (rational-ℚ⁻ p)))
-        ( λ p (x<p , p<0) → intro-exists (p , is-negative-le-zero-ℚ p p<0) x<p)
+        ( λ p (x<p , p<0) → intro-exists (p , is-negative-le-zero-ℚ p<0) x<p)
 
     is-negative-exists-ℚ⁻-in-upper-cut-ℝ :
       exists ℚ⁻ (λ p → upper-cut-ℝ x (rational-ℚ⁻ p)) → is-negative-ℝ x
@@ -85,7 +85,7 @@ module _
       elim-exists
         ( is-negative-prop-ℝ x)
         ( λ (p , is-neg-p) x<p →
-          intro-exists p (x<p , le-zero-is-negative-ℚ p is-neg-p))
+          intro-exists p (x<p , le-zero-is-negative-ℚ is-neg-p))
 
     is-negative-iff-exists-ℚ⁻-in-upper-cut-ℝ :
       is-negative-ℝ x ↔ exists ℚ⁻ (λ p → upper-cut-ℝ x (rational-ℚ⁻ p))
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index a10b9ce4c7d..0bcd3acc129 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -354,10 +354,7 @@ module _
                             ( a)
                             ( d)
                             ( leq-right-min-ℚ _ _)
-                            ( leq-left-min-ℚ _ _))
-                        ( is-negative-mul-negative-positive-ℚ
-                            ( is-neg-a)
-                            ( is-pos-d))))
+                            ( leq-left-min-ℚ _ _))))
                   ( id)
                   ( decide-is-negative-is-nonnegative-ℚ a)
               is-nonneg-c =
@@ -372,10 +369,7 @@ module _
                             ( b)
                             ( c)
                             ( leq-left-min-ℚ _ _)
-                            ( leq-right-min-ℚ _ _))
-                        ( is-negative-mul-positive-negative-ℚ
-                            ( is-pos-b)
-                            ( is-neg-c))))
+                            ( leq-right-min-ℚ _ _))))
                   ( id)
                   ( decide-is-negative-is-nonnegative-ℚ c)
               a' = max-ℚ a c

From c7176d6136c1caff408391a579cde7cf849712ea Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 15 Oct 2025 09:29:06 -0700
Subject: [PATCH 10/22] make pre-commit

---
 .../multiplicative-inverses-positive-real-numbers.lagda.md      | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 384e7af4c8a..06b96659065 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -10,6 +10,7 @@ module real-numbers.multiplicative-inverses-positive-real-numbers where
 
 ```agda
 open import elementary-number-theory.closed-intervals-rational-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.maximum-positive-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
@@ -19,7 +20,6 @@ open import elementary-number-theory.multiplication-closed-intervals-rational-nu
 open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.multiplicative-group-of-positive-rational-numbers
 open import elementary-number-theory.nonpositive-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers

From df308e2425fb8b1863565200a6a735d7e5afc7cd Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 15 Oct 2025 13:58:21 -0700
Subject: [PATCH 11/22] Fix merge

---
 ...ication-negative-rational-numbers.lagda.md | 19 +++++++++----------
 .../squares-real-numbers.lagda.md             |  4 ++--
 2 files changed, 11 insertions(+), 12 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
index 5a41de6f305..bd53f0e9c03 100644
--- a/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-negative-rational-numbers.lagda.md
@@ -21,18 +21,9 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
-open import elementary-number-theory.multiplication-positive-rational-numbers
-open import elementary-number-theory.multiplication-rational-numbers
-open import elementary-number-theory.negative-integer-fractions
-open import elementary-number-theory.negative-rational-numbers
-open import elementary-number-theory.positive-and-negative-rational-numbers
-open import elementary-number-theory.positive-integer-fractions
-open import elementary-number-theory.positive-rational-numbers
-open import elementary-number-theory.rational-numbers
-
-open import foundation.dependent-pair-types
 ```
 
 </details>
@@ -60,6 +51,14 @@ opaque
       ( is-positive-ℚ)
       ( negative-law-mul-ℚ x y)
       ( is-positive-mul-ℚ P Q)
+
+mul-ℚ⁻ : ℚ⁻ → ℚ⁻ → ℚ⁺
+mul-ℚ⁻ (p , is-neg-p) (q , is-neg-q) =
+  (p *ℚ q , is-positive-mul-negative-ℚ is-neg-p is-neg-q)
+
+infixl 40 _*ℚ⁻_
+_*ℚ⁻_ : ℚ⁻ → ℚ⁻ → ℚ⁺
+_*ℚ⁻_ = mul-ℚ⁻
 ```
 
 ### Multiplication by a negative rational number reverses inequality
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 9420a081e95..c6fedb34990 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -98,7 +98,7 @@ opaque
             ( is-positive-prop-ℚ q)
             ( λ a'<0 →
               let
-                a'⁻ = (a' , is-negative-le-zero-ℚ a' a'<0)
+                a'⁻ = (a' , is-negative-le-zero-ℚ a'<0)
               in
                 is-positive-le-ℚ⁺
                   ( a'⁻ *ℚ⁻ a'⁻)
@@ -113,7 +113,7 @@ opaque
                     ( [a',b'][a',b']<q)))
             ( λ 0<b' →
               let
-                b'⁺ = (b' , is-positive-le-zero-ℚ b' 0<b')
+                b'⁺ = (b' , is-positive-le-zero-ℚ 0<b')
               in
                 is-positive-le-ℚ⁺
                   ( b'⁺ *ℚ⁺ b'⁺)

From 607eba5ec1a6559aeea74938da2646f7f2ce780e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 09:39:18 -0700
Subject: [PATCH 12/22] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 ...lities-positive-and-negative-rational-numbers.lagda.md | 8 ++++----
 ...cation-positive-and-negative-rational-numbers.lagda.md | 3 +--
 2 files changed, 5 insertions(+), 6 deletions(-)

diff --git a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
index 414fa5ecdc3..88ac145a32a 100644
--- a/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequalities-positive-and-negative-rational-numbers.lagda.md
@@ -36,8 +36,8 @@ between [positive](elementary-number-theory.positive-rational-numbers.md),
 
 ### Inequalities between rational numbers with different signs
 
-Some inequalities can be deduced directly from the sign of a rational number:
-for example, every negative rational number is less than every nonnegative
+Some inequalities can be deduced directly from the sign of a rational number.
+For example, every negative rational number is less than every nonnegative
 rational number.
 
 #### Any negative rational number is less than any nonnegative rational number
@@ -70,7 +70,7 @@ abstract
   leq-negative-positive-ℚ p q = leq-le-ℚ (le-negative-positive-ℚ p q)
 ```
 
-#### A nonpositive rational number is less than a positive rational number
+#### A nonpositive rational number is less than any positive rational number
 
 ```agda
 abstract
@@ -82,7 +82,7 @@ abstract
       ( le-zero-is-positive-ℚ pos-q)
 ```
 
-#### A nonpositive rational number is less than or equal to a nonnegative rational number
+#### A nonpositive rational number is less than or equal to any nonnegative rational number
 
 ```agda
 abstract
diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index eda0123b5e7..39db46c6512 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -30,8 +30,7 @@ open import foundation.transport-along-identifications
 ## Idea
 
 When we have information about the sign of the factors of a
-[rational](elementary-number-theory.rational-numbers.md)
-[product](elementary-number-theory.multiplication-rational-numbers.md), we can
+[product](elementary-number-theory.multiplication-rational-numbers.md) of [rational numbers](elementary-number-theory.rational-numbers.md), we can
 deduce the sign of their product too.
 
 ## Lemmas

From daa9fc084e696aa8775564973cdcc2804bd7e3aa Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 09:43:42 -0700
Subject: [PATCH 13/22] Respond to review comments

---
 ...positive-and-negative-rational-numbers.lagda.md |  9 +++++----
 ...positive-and-negative-rational-numbers.lagda.md | 14 +++++++-------
 .../order-preserving-maps-posets.lagda.md          |  2 +-
 .../order-preserving-maps-total-orders.lagda.md    |  2 +-
 ...licative-inverses-nonzero-real-numbers.lagda.md |  2 +-
 ...icative-inverses-positive-real-numbers.lagda.md |  2 +-
 .../square-roots-nonnegative-real-numbers.lagda.md | 10 +++++-----
 7 files changed, 21 insertions(+), 20 deletions(-)

diff --git a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
index 39db46c6512..4cbe0a91112 100644
--- a/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-and-negative-rational-numbers.lagda.md
@@ -30,8 +30,9 @@ open import foundation.transport-along-identifications
 ## Idea
 
 When we have information about the sign of the factors of a
-[product](elementary-number-theory.multiplication-rational-numbers.md) of [rational numbers](elementary-number-theory.rational-numbers.md), we can
-deduce the sign of their product too.
+[product](elementary-number-theory.multiplication-rational-numbers.md) of
+[rational numbers](elementary-number-theory.rational-numbers.md), we can deduce
+the sign of their product too.
 
 ## Lemmas
 
@@ -147,7 +148,7 @@ abstract
             ( y≠0)
             ( λ is-pos-y →
               ex-falso
-                ( not-is-negative-is-positive-ℚ
+                ( is-not-negative-and-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-x is-pos-y ,
                     is-pos-xy))))
         ( λ x=0 →
@@ -162,7 +163,7 @@ abstract
             ( y)
             ( λ is-neg-y →
               ex-falso
-                ( not-is-negative-is-positive-ℚ
+                ( is-not-negative-and-positive-ℚ
                   ( is-negative-mul-positive-negative-ℚ is-pos-x is-neg-y ,
                     is-pos-xy)))
             ( y≠0)
diff --git a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index 00e1063ad38..e0447a64bbc 100644
--- a/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
@@ -215,9 +215,9 @@ neg-ℚ⁰⁻ (q , is-nonpos-q) =
 
 ```agda
 abstract
-  not-is-negative-is-positive-ℚ :
+  is-not-negative-and-positive-ℚ :
     {x : ℚ} → ¬ (is-negative-ℚ x × is-positive-ℚ x)
-  not-is-negative-is-positive-ℚ {x} (N , P) =
+  is-not-negative-and-positive-ℚ {x} (N , P) =
     not-leq-le-ℚ
       ( zero-ℚ)
       ( x)
@@ -229,9 +229,9 @@ abstract
 
 ```agda
 abstract
-  not-is-negative-is-nonnegative-ℚ :
+  is-not-negative-and-nonnegativeℚ :
     {x : ℚ} → ¬ (is-negative-ℚ x × is-nonnegative-ℚ x)
-  not-is-negative-is-nonnegative-ℚ {x} (N , NN) =
+  is-not-negative-and-nonnegativeℚ {x} (N , NN) =
     not-leq-le-ℚ
       ( x)
       ( zero-ℚ)
@@ -239,13 +239,13 @@ abstract
       ( leq-zero-is-nonnegative-ℚ NN)
 ```
 
-### If `p < q` and `q` is nonpositive, then `p` is negative
+### A rational number is not both positive and nonpositive
 
 ```agda
 abstract
-  not-is-positive-is-nonpositive-ℚ :
+  is-not-positive-and-nonpositive-ℚ :
     {x : ℚ} → ¬ (is-positive-ℚ x × is-nonpositive-ℚ x)
-  not-is-positive-is-nonpositive-ℚ {x} (P , NP) =
+  is-not-positive-and-nonpositive-ℚ {x} (P , NP) =
     not-leq-le-ℚ
       ( zero-ℚ)
       ( x)
diff --git a/src/order-theory/order-preserving-maps-posets.lagda.md b/src/order-theory/order-preserving-maps-posets.lagda.md
index 4d127ed3ff8..bcd5ba75286 100644
--- a/src/order-theory/order-preserving-maps-posets.lagda.md
+++ b/src/order-theory/order-preserving-maps-posets.lagda.md
@@ -27,7 +27,7 @@ open import order-theory.posets
 
 A map `f : P → Q` between the underlying types of two
 [posets](order-theory.posets.md) is said to be
-{{#concept "order preserving" Agda=hom-Poset Disambiguation="map between posets"}}
+{{#concept "order preserving" WD="increasing function" WDID=Q3075182 Agda=hom-Poset Disambiguation="map between posets"}}
 if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
 
 ## Definition
diff --git a/src/order-theory/order-preserving-maps-total-orders.lagda.md b/src/order-theory/order-preserving-maps-total-orders.lagda.md
index f3ae69baab9..161a0dafd86 100644
--- a/src/order-theory/order-preserving-maps-total-orders.lagda.md
+++ b/src/order-theory/order-preserving-maps-total-orders.lagda.md
@@ -26,7 +26,7 @@ open import order-theory.total-orders
 
 A map `f : P → Q` between the underlying types of two
 [total orders](order-theory.total-orders.md) is said to be
-{{#concept "order preserving" Agda=hom-Total-Order Disambiguation="map between total orders"}}
+{{#concept "order preserving" WD="increasing function" WDID=Q3075182 Agda=hom-Total-Order Disambiguation="map between total orders"}}
 if `x ≤ y` in `P` implies `f x ≤ f y` in `Q`.
 
 ## Definition
diff --git a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
index 43c55227e30..f91c1d2bfcf 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -172,7 +172,7 @@ opaque
                   ( intro-exists (b , is-neg-b) x<b)))
             ( λ (is-pos-b , is-pos-d) →
               ex-falso
-                ( not-is-negative-is-positive-ℚ
+                ( is-not-negative-and-positive-ℚ
                   ( is-negative-mul-negative-positive-ℚ is-neg-a is-pos-d ,
                     is-positive-mul a d
                       ( leq-right-min-ℚ _ _)
diff --git a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
index 06b96659065..73cd01dce63 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -446,7 +446,7 @@ module _
           inl-disjunction
             ( λ is-pos-p →
               ex-falso
-                ( not-is-positive-is-nonpositive-ℚ (is-pos-p , is-nonpos-p))))
+                ( is-not-positive-and-nonpositive-ℚ (is-pos-p , is-nonpos-p))))
         ( decide-is-positive-is-nonpositive-ℚ p)
 
   opaque
diff --git a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
index aa52cf66ce6..a9576b242d4 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -92,7 +92,7 @@ module _
         ( neg-one-ℚ)
         ( λ is-nonneg-neg-one-ℚ →
           ex-falso
-            ( not-is-negative-is-nonnegative-ℚ
+            ( is-not-negative-and-nonnegativeℚ
               ( is-negative-rational-ℚ⁻ neg-one-ℚ⁻ , is-nonneg-neg-one-ℚ)))
 
     is-inhabited-upper-cut-sqrt-ℝ⁰⁺ : is-inhabited-subtype upper-cut-sqrt-ℝ⁰⁺
@@ -125,7 +125,7 @@ module _
                 ( le-mediant-zero-ℚ⁻ q⁻ ,
                   λ is-nonneg-q' →
                     ex-falso
-                      ( not-is-negative-is-nonnegative-ℚ
+                      ( is-not-negative-and-nonnegativeℚ
                         ( is-negative-rational-ℚ⁻ (mediant-zero-ℚ⁻ q⁻) ,
                           is-nonneg-q'))))
           ( λ q=0 →
@@ -263,7 +263,7 @@ module _
           inl-disjunction
             ( λ is-nonneg-p →
               ex-falso
-                ( not-is-negative-is-nonnegative-ℚ (is-neg-p , is-nonneg-p))))
+                ( is-not-negative-and-nonnegativeℚ (is-neg-p , is-nonneg-p))))
         ( λ is-nonneg-p →
           map-disjunction
             ( λ p²<x _ → p²<x)
@@ -350,7 +350,7 @@ module _
                 rec-coproduct
                   ( λ is-neg-a →
                     ex-falso
-                      ( not-is-negative-is-positive-ℚ
+                      ( is-not-negative-and-positive-ℚ
                         ( is-negative-mul-negative-positive-ℚ
                             ( is-neg-a)
                             ( is-pos-d) ,
@@ -365,7 +365,7 @@ module _
                 rec-coproduct
                   ( λ is-neg-c →
                     ex-falso
-                      ( not-is-negative-is-positive-ℚ
+                      ( is-not-negative-and-positive-ℚ
                         ( is-negative-mul-positive-negative-ℚ
                             ( is-pos-b)
                             ( is-neg-c) ,

From 0905e0fd91186800a05f80be2911e727c1562d80 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:07:16 -0700
Subject: [PATCH 14/22] Progress

---
 .../powers-rational-numbers.lagda.md          | 228 ++++++++++++++++++
 1 file changed, 228 insertions(+)
 create mode 100644 src/elementary-number-theory/powers-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
new file mode 100644
index 00000000000..ee922d0fad0
--- /dev/null
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -0,0 +1,228 @@
+# Powers of rational numbers
+
+```agda
+module elementary-number-theory.powers-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
+open import group-theory.powers-of-elements-monoids
+open import foundation.identity-types
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
+on the [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
+which is defined by [iteratively](foundation.iterating-functions.md) multiplying
+`x` with itself `n` times.
+
+## Definition
+
+```agda
+power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
+power-ℚ⁺ = power-Group group-mul-ℚ⁺
+```
+
+## Properties
+
+### `1ⁿ = 1`
+
+```agda
+power-one-ℚ⁺ : (n : ℕ) → power-ℚ⁺ n one-ℚ⁺ ＝ one-ℚ⁺
+power-one-ℚ⁺ = power-unit-Group group-mul-ℚ⁺
+```
+
+### `qⁿ⁺¹ = qⁿq`
+
+```agda
+power-succ-ℚ⁺ : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ power-ℚ⁺ n q *ℚ⁺ q
+power-succ-ℚ⁺ = power-succ-Group group-mul-ℚ⁺
+```
+
+### `qⁿ = qqⁿ`
+
+```agda
+power-succ-ℚ⁺' : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ q *ℚ⁺ power-ℚ⁺ n q
+power-succ-ℚ⁺' = power-succ-Group' group-mul-ℚ⁺
+```
+
+### `qᵐ⁺ⁿ = qᵐqⁿ`
+
+```agda
+distributive-power-add-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m +ℕ n) q ＝ power-ℚ⁺ m q *ℚ⁺ power-ℚ⁺ n q
+distributive-power-add-ℚ⁺ m n _ = distributive-power-add-Group group-mul-ℚ⁺ m n
+```
+
+### `(pq)ⁿ=pⁿqⁿ`
+
+```agda
+left-distributive-power-mul-ℚ⁺ :
+  (n : ℕ) (p q : ℚ⁺) → power-ℚ⁺ n (p *ℚ⁺ q) ＝ power-ℚ⁺ n p *ℚ⁺ power-ℚ⁺ n q
+left-distributive-power-mul-ℚ⁺ n p q =
+  left-distributive-multiple-add-Ab abelian-group-mul-ℚ⁺ n
+```
+
+### `pᵐⁿ = (pᵐ)ⁿ`
+
+```agda
+power-mul-ℚ⁺ :
+  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m *ℕ n) q ＝ power-ℚ⁺ n (power-ℚ⁺ m q)
+power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
+```
+
+### If `p` and `q` are positive rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+
+```agda
+abstract
+  preserves-le-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
+    le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
+  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
+  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
+    transitive-le-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-le-right-mul-ℚ⁺ q _ _
+        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
+      ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
+```
+
+### If `p` and `q` are positive rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+
+```agda
+abstract
+  preserves-leq-power-ℚ⁺ :
+    (n : ℕ) (p q : ℚ⁺) → leq-ℚ⁺ p q → leq-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
+  preserves-leq-power-ℚ⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
+  preserves-leq-power-ℚ⁺ 1 p q p≤q = p≤q
+  preserves-leq-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
+    transitive-leq-ℚ⁺
+      ( power-ℚ⁺ (succ-ℕ n) p)
+      ( power-ℚ⁺ n p *ℚ⁺ q)
+      ( power-ℚ⁺ (succ-ℕ n) q)
+      ( preserves-leq-right-mul-ℚ⁺ q _ _ (preserves-leq-power-ℚ⁺ n p q p≤q))
+      ( preserves-leq-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p≤q)
+```
+
+### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  bound-unbounded-power-one-plus-ℚ⁺ ε⁺@(ε , is-pos-ε) b =
+    let
+      (n , b<nε) = bound-archimedean-property-ℚ ε b is-pos-ε
+    in
+      ( n ,
+        transitive-le-ℚ _ _ _
+          ( concatenate-le-leq-ℚ _ _ _
+            ( le-left-add-rational-ℚ⁺ _ one-ℚ⁺)
+            ( binary-tr
+              ( leq-ℚ)
+              ( inv (compute-standard-arithmetic-sequence-ℚ⁺ one-ℚ⁺ ε⁺ n))
+              ( ap
+                ( rational-ℚ⁺)
+                ( equational-reasoning
+                  seq-standard-geometric-sequence-ℚ⁺
+                      ( one-ℚ⁺)
+                      ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                      ( n)
+                  ＝ one-ℚ⁺ *ℚ⁺ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by
+                      inv
+                        ( compute-standard-geometric-sequence-ℚ⁺
+                          ( one-ℚ⁺)
+                          ( one-ℚ⁺ +ℚ⁺ ε⁺)
+                          ( n))
+                  ＝ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
+                    by left-unit-law-mul-ℚ⁺ _))
+              ( bernoullis-inequality-ℚ⁺ ε⁺ n)))
+          ( b<nε))
+
+  unbounded-power-one-plus-ℚ⁺ :
+    (ε : ℚ⁺) (b : ℚ) →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
+  unbounded-power-one-plus-ℚ⁺ ε b =
+    unit-trunc-Prop (bound-unbounded-power-one-plus-ℚ⁺ ε b)
+```
+
+### If `1 < q`, `qⁿ` grows without bound
+
+```agda
+abstract
+  bound-unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
+    let
+      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
+      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+    in
+      ( n ,
+        tr
+          ( le-ℚ b)
+          ( ap
+            ( rational-ℚ⁺ ∘ power-ℚ⁺ n)
+            ( eq-ℚ⁺ (is-identity-right-conjugation-add-ℚ one-ℚ q)))
+          ( b<⟨1+q-1⟩ⁿ))
+
+  unbounded-power-greater-than-one-ℚ⁺ :
+    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
+    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n q)))
+  unbounded-power-greater-than-one-ℚ⁺ q b 1<q =
+    unit-trunc-Prop (bound-unbounded-power-greater-than-one-ℚ⁺ q b 1<q)
+```
+
+### If `ε` is a positive rational number less than one, `εⁿ` becomes arbitrarily small
+
+```agda
+abstract
+  bound-arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    Σ ℕ (λ n → le-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1 =
+    let
+      1/δ = inv-ℚ⁺ δ
+      1/ε = inv-ℚ⁺ ε
+      1<1/ε =
+        tr
+          ( λ q → le-ℚ⁺ q 1/ε)
+          ( inv-one-ℚ⁺)
+          ( inv-le-ℚ⁺ ε one-ℚ⁺ ε<1)
+      (n , 1/δ<⟨1/ε⟩ⁿ) =
+        bound-unbounded-power-greater-than-one-ℚ⁺ 1/ε (rational-ℚ⁺ 1/δ) 1<1/ε
+    in
+      ( n ,
+        binary-tr
+          ( le-ℚ⁺)
+          ( equational-reasoning
+            inv-ℚ⁺ (power-ℚ⁺ n 1/ε)
+            ＝ inv-ℚ⁺ (inv-ℚ⁺ (power-ℚ⁺ n ε))
+              by ap inv-ℚ⁺ (power-inv-Group group-mul-ℚ⁺ n ε)
+            ＝ power-ℚ⁺ n ε
+              by inv-inv-ℚ⁺ (power-ℚ⁺ n ε))
+          ( inv-inv-ℚ⁺ δ)
+          ( inv-le-ℚ⁺ 1/δ (power-ℚ⁺ n 1/ε) 1/δ<⟨1/ε⟩ⁿ))
+
+  arbitrarily-small-power-le-one-ℚ⁺ :
+    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
+    exists ℕ (λ n → le-prop-ℚ⁺ (power-ℚ⁺ n ε) δ)
+  arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε =
+    unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
+```
+
+## See also
+
+- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)

From 06518412e816c73dff5818342930a5783b9ce9f4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:07:41 -0700
Subject: [PATCH 15/22] Clarify concept disambiguation

---
 .../powers-positive-rational-numbers.lagda.md                   | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 811235c5420..789446dfa67 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -45,7 +45,7 @@ open import group-theory.powers-of-elements-groups
 ## Idea
 
 The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
+{{#concept "power operation" Disambiguation="a positive rational number to a natural number power" Agda=power-ℚ⁺}}
 on the [positive](elementary-number-theory.positive-rational-numbers.md)
 [rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
 which is defined by [iteratively](foundation.iterating-functions.md) multiplying

From e0a24b825f0e63177d5c9fe62c33c3a092b47e2d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:08:06 -0700
Subject: [PATCH 16/22] Progress

---
 .../powers-rational-numbers.lagda.md                     | 9 ++++-----
 1 file changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index ee922d0fad0..2f5c88228e5 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -17,11 +17,10 @@ open import foundation.identity-types
 ## Idea
 
 The
-{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ⁺}}
-on the [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
-which is defined by [iteratively](foundation.iterating-functions.md) multiplying
-`x` with itself `n` times.
+{{#concept "power operation" Disambiguation="raising a rational number to a natural number power" Agda=power-ℚ}}
+on the [rational numbers](elementary-number-theory.rational-numbers.md)
+`n x ↦ xⁿ`, is defined by [iteratively](foundation.iterating-functions.md)
+multiplying `x` with itself `n` times.
 
 ## Definition
 

From c5b3be8a204d11ef7e5b5439126af5ecc24b5cd4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:08:45 -0700
Subject: [PATCH 17/22] Further clarify doc

---
 .../powers-positive-rational-numbers.lagda.md             | 8 +++++---
 1 file changed, 5 insertions(+), 3 deletions(-)

diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 789446dfa67..61a906395d3 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -47,9 +47,11 @@ open import group-theory.powers-of-elements-groups
 The
 {{#concept "power operation" Disambiguation="a positive rational number to a natural number power" Agda=power-ℚ⁺}}
 on the [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational numbers](elementary-number-theory.rational-numbers.md) `n x ↦ xⁿ`,
-which is defined by [iteratively](foundation.iterating-functions.md) multiplying
-`x` with itself `n` times.
+[rational numbers](elementary-number-theory.rational-numbers.md) is the
+operation `n x ↦ xⁿ`, which is defined by
+[iteratively](foundation.iterating-functions.md) multiplying `x` with itself `n`
+times. This file covers the case where `n` is a
+[natural number](elementary-number-theory.natural-numbers.md).
 
 ## Definition
 

From a8c35cd424f51abc92b27cfe1e0f2330b3df1025 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 10:55:55 -0700
Subject: [PATCH 18/22] Progress

---
 src/elementary-number-theory.lagda.md         |   1 +
 .../parity-natural-numbers.lagda.md           |  10 +-
 .../powers-rational-numbers.lagda.md          | 271 +++++++++---------
 .../squares-rational-numbers.lagda.md         |  24 ++
 ...s-of-elements-commutative-monoids.lagda.md |   8 +-
 5 files changed, 168 insertions(+), 146 deletions(-)

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index be3b1e0ad01..0e518ce0112 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -180,6 +180,7 @@ open import elementary-number-theory.positive-rational-numbers public
 open import elementary-number-theory.powers-integers public
 open import elementary-number-theory.powers-of-two public
 open import elementary-number-theory.powers-positive-rational-numbers public
+open import elementary-number-theory.powers-rational-numbers public
 open import elementary-number-theory.prime-numbers public
 open import elementary-number-theory.products-of-natural-numbers public
 open import elementary-number-theory.proper-closed-intervals-rational-numbers public
diff --git a/src/elementary-number-theory/parity-natural-numbers.lagda.md b/src/elementary-number-theory/parity-natural-numbers.lagda.md
index 503b5245425..9fa410e7f87 100644
--- a/src/elementary-number-theory/parity-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/parity-natural-numbers.lagda.md
@@ -29,9 +29,13 @@ open import foundation.universe-levels
 
 ## Idea
 
-Parity partitions the natural numbers into two classes: the even and the odd
-natural numbers. Even natural numbers are those that are divisible by two, and
-odd natural numbers are those that aren't.
+{{#concept "Parity" WDID=Q230967 WD="parity"}} partitions the
+[natural numbers](elementary-number-theory.natural-numbers.md) into two classes:
+the {{#concept "even" WDID=Q13366104 WD="even number" Agda=is-even-ℕ}} and the
+{{#concept "odd" WDID=Q13366129 WD="odd number" Agda=is-odd-ℕ}} natural numbers.
+Even natural numbers are those that are
+[divisible](elementary-number-theory.divisibility-natural-numbers.md) by two,
+and odd natural numbers are those that [aren't](foundation.negation.md).
 
 ## Definition
 
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 2f5c88228e5..83f95cea235 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -7,9 +7,32 @@ module elementary-number-theory.powers-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.absolute-value-rational-numbers
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-positive-and-negative-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
-open import group-theory.powers-of-elements-monoids
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.negative-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.parity-natural-numbers
+open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.squares-rational-numbers
+
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
 open import foundation.identity-types
+open import foundation.transport-along-identifications
+
+open import group-theory.powers-of-elements-commutative-monoids
+open import group-theory.powers-of-elements-monoids
 ```
 
 </details>
@@ -25,8 +48,8 @@ multiplying `x` with itself `n` times.
 ## Definition
 
 ```agda
-power-ℚ⁺ : ℕ → ℚ⁺ → ℚ⁺
-power-ℚ⁺ = power-Group group-mul-ℚ⁺
+power-ℚ : ℕ → ℚ → ℚ
+power-ℚ = power-Monoid monoid-mul-ℚ
 ```
 
 ## Properties
@@ -34,194 +57,162 @@ power-ℚ⁺ = power-Group group-mul-ℚ⁺
 ### `1ⁿ = 1`
 
 ```agda
-power-one-ℚ⁺ : (n : ℕ) → power-ℚ⁺ n one-ℚ⁺ ＝ one-ℚ⁺
-power-one-ℚ⁺ = power-unit-Group group-mul-ℚ⁺
+abstract
+  power-one-ℚ : (n : ℕ) → power-ℚ n one-ℚ ＝ one-ℚ
+  power-one-ℚ = power-unit-Monoid monoid-mul-ℚ
 ```
 
-### `qⁿ⁺¹ = qⁿq`
+### `qⁿ¹ = qⁿq`
 
 ```agda
-power-succ-ℚ⁺ : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ power-ℚ⁺ n q *ℚ⁺ q
-power-succ-ℚ⁺ = power-succ-Group group-mul-ℚ⁺
+abstract
+  power-succ-ℚ : (n : ℕ) (q : ℚ) → power-ℚ (succ-ℕ n) q ＝ power-ℚ n q *ℚ q
+  power-succ-ℚ = power-succ-Monoid monoid-mul-ℚ
 ```
 
 ### `qⁿ = qqⁿ`
 
 ```agda
-power-succ-ℚ⁺' : (n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (succ-ℕ n) q ＝ q *ℚ⁺ power-ℚ⁺ n q
-power-succ-ℚ⁺' = power-succ-Group' group-mul-ℚ⁺
+abstract
+  power-succ-ℚ' : (n : ℕ) (q : ℚ) → power-ℚ (succ-ℕ n) q ＝ q *ℚ power-ℚ n q
+  power-succ-ℚ' = power-succ-Monoid' monoid-mul-ℚ
 ```
 
-### `qᵐ⁺ⁿ = qᵐqⁿ`
+### `qᵐⁿ = qᵐqⁿ`
 
 ```agda
-distributive-power-add-ℚ⁺ :
-  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m +ℕ n) q ＝ power-ℚ⁺ m q *ℚ⁺ power-ℚ⁺ n q
-distributive-power-add-ℚ⁺ m n _ = distributive-power-add-Group group-mul-ℚ⁺ m n
+abstract
+  distributive-power-add-ℚ :
+    (m n : ℕ) (q : ℚ) → power-ℚ (m +ℕ n) q ＝ power-ℚ m q *ℚ power-ℚ n q
+  distributive-power-add-ℚ m n _ =
+    distributive-power-add-Monoid monoid-mul-ℚ m n
 ```
 
 ### `(pq)ⁿ=pⁿqⁿ`
 
 ```agda
-left-distributive-power-mul-ℚ⁺ :
-  (n : ℕ) (p q : ℚ⁺) → power-ℚ⁺ n (p *ℚ⁺ q) ＝ power-ℚ⁺ n p *ℚ⁺ power-ℚ⁺ n q
-left-distributive-power-mul-ℚ⁺ n p q =
-  left-distributive-multiple-add-Ab abelian-group-mul-ℚ⁺ n
+abstract
+  left-distributive-power-mul-ℚ :
+    (n : ℕ) (p q : ℚ) → power-ℚ n (p *ℚ q) ＝ power-ℚ n p *ℚ power-ℚ n q
+  left-distributive-power-mul-ℚ n p q =
+    distributive-power-mul-Commutative-Monoid commutative-monoid-mul-ℚ n
 ```
 
 ### `pᵐⁿ = (pᵐ)ⁿ`
 
 ```agda
-power-mul-ℚ⁺ :
-  (m n : ℕ) (q : ℚ⁺) → power-ℚ⁺ (m *ℕ n) q ＝ power-ℚ⁺ n (power-ℚ⁺ m q)
-power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
+abstract
+  power-mul-ℚ :
+    (m n : ℕ) (q : ℚ) → power-ℚ (m *ℕ n) q ＝ power-ℚ n (power-ℚ m q)
+  power-mul-ℚ m n q = power-mul-Monoid monoid-mul-ℚ m n
 ```
 
-### If `p` and `q` are positive rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+### Even powers of rational numbers are nonnegative
 
 ```agda
 abstract
-  preserves-le-power-ℚ⁺ :
-    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
-    le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
-  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
-  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
-  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
-    transitive-le-ℚ⁺
-      ( power-ℚ⁺ (succ-ℕ n) p)
-      ( power-ℚ⁺ n p *ℚ⁺ q)
-      ( power-ℚ⁺ (succ-ℕ n) q)
-      ( preserves-le-right-mul-ℚ⁺ q _ _
-        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
-      ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
+  is-nonnegative-even-power-ℚ :
+    (n : ℕ) (q : ℚ) → is-even-ℕ n → is-nonnegative-ℚ (power-ℚ n q)
+  is-nonnegative-even-power-ℚ _ q (k , refl) =
+    inv-tr
+      ( is-nonnegative-ℚ)
+      ( power-mul-ℚ k 2 q)
+      ( is-nonnegative-square-ℚ (power-ℚ k q))
+
+nonnegative-even-power-ℚ :
+  (n : ℕ) (q : ℚ) → is-even-ℕ n → ℚ⁰⁺
+nonnegative-even-power-ℚ n q even-n =
+  ( power-ℚ n q , is-nonnegative-even-power-ℚ n q even-n)
 ```
 
-### If `p` and `q` are positive rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+### Powers of positive rational numbers are positive
 
 ```agda
 abstract
-  preserves-leq-power-ℚ⁺ :
-    (n : ℕ) (p q : ℚ⁺) → leq-ℚ⁺ p q → leq-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
-  preserves-leq-power-ℚ⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
-  preserves-leq-power-ℚ⁺ 1 p q p≤q = p≤q
-  preserves-leq-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
-    transitive-leq-ℚ⁺
-      ( power-ℚ⁺ (succ-ℕ n) p)
-      ( power-ℚ⁺ n p *ℚ⁺ q)
-      ( power-ℚ⁺ (succ-ℕ n) q)
-      ( preserves-leq-right-mul-ℚ⁺ q _ _ (preserves-leq-power-ℚ⁺ n p q p≤q))
-      ( preserves-leq-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p≤q)
+  is-positive-power-ℚ⁺ :
+    (n : ℕ) (q : ℚ⁺) → is-positive-ℚ (power-ℚ n (rational-ℚ⁺ q))
+  is-positive-power-ℚ⁺ 0 q = is-positive-rational-ℚ⁺ one-ℚ⁺
+  is-positive-power-ℚ⁺ (succ-ℕ n) q⁺@(q , is-pos-q) =
+    inv-tr
+      ( is-positive-ℚ)
+      ( power-succ-ℚ n q)
+      ( is-positive-mul-ℚ (is-positive-power-ℚ⁺ n q⁺) is-pos-q)
 ```
 
-### For any positive rational `ε`, `(1 + ε)ⁿ` grows without bound
+### Even powers of negative rational numbers are positive
 
 ```agda
 abstract
-  bound-unbounded-power-one-plus-ℚ⁺ :
-    (ε : ℚ⁺) (b : ℚ) →
-    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
-  bound-unbounded-power-one-plus-ℚ⁺ ε⁺@(ε , is-pos-ε) b =
-    let
-      (n , b<nε) = bound-archimedean-property-ℚ ε b is-pos-ε
-    in
-      ( n ,
-        transitive-le-ℚ _ _ _
-          ( concatenate-le-leq-ℚ _ _ _
-            ( le-left-add-rational-ℚ⁺ _ one-ℚ⁺)
-            ( binary-tr
-              ( leq-ℚ)
-              ( inv (compute-standard-arithmetic-sequence-ℚ⁺ one-ℚ⁺ ε⁺ n))
-              ( ap
-                ( rational-ℚ⁺)
-                ( equational-reasoning
-                  seq-standard-geometric-sequence-ℚ⁺
-                      ( one-ℚ⁺)
-                      ( one-ℚ⁺ +ℚ⁺ ε⁺)
-                      ( n)
-                  ＝ one-ℚ⁺ *ℚ⁺ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
-                    by
-                      inv
-                        ( compute-standard-geometric-sequence-ℚ⁺
-                          ( one-ℚ⁺)
-                          ( one-ℚ⁺ +ℚ⁺ ε⁺)
-                          ( n))
-                  ＝ power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε⁺)
-                    by left-unit-law-mul-ℚ⁺ _))
-              ( bernoullis-inequality-ℚ⁺ ε⁺ n)))
-          ( b<nε))
-
-  unbounded-power-one-plus-ℚ⁺ :
-    (ε : ℚ⁺) (b : ℚ) →
-    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n (one-ℚ⁺ +ℚ⁺ ε))))
-  unbounded-power-one-plus-ℚ⁺ ε b =
-    unit-trunc-Prop (bound-unbounded-power-one-plus-ℚ⁺ ε b)
+  is-positive-even-power-ℚ⁻ :
+    (n : ℕ) (q : ℚ⁻) → is-even-ℕ n → is-positive-ℚ (power-ℚ n (rational-ℚ⁻ q))
+  is-positive-even-power-ℚ⁻ _ q⁻@(q , is-neg-q) (k , refl) =
+    inv-tr
+      ( is-positive-ℚ)
+      ( equational-reasoning
+        power-ℚ (k *ℕ 2) q
+        ＝ power-ℚ (2 *ℕ k) q
+          by ap (λ m → power-ℚ m q) (commutative-mul-ℕ k 2)
+        ＝ power-ℚ k (square-ℚ q)
+          by power-mul-ℚ 2 k q)
+      ( is-positive-power-ℚ⁺ k (square-ℚ⁻ q⁻))
 ```
 
-### If `1 < q`, `qⁿ` grows without bound
+### Odd powers of negative rational numbers are negative
 
 ```agda
 abstract
-  bound-unbounded-power-greater-than-one-ℚ⁺ :
-    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
-    Σ ℕ (λ n → le-ℚ b (rational-ℚ⁺ (power-ℚ⁺ n q)))
-  bound-unbounded-power-greater-than-one-ℚ⁺ q⁺@(q , _) b 1<q =
+  is-negative-odd-power-ℚ⁻ :
+    (n : ℕ) (q : ℚ⁻) → is-odd-ℕ n → is-negative-ℚ (power-ℚ n (rational-ℚ⁻ q))
+  is-negative-odd-power-ℚ⁻ n q⁻@(q , is-neg-q) odd-n =
     let
-      q-1⁺ = positive-diff-le-ℚ one-ℚ q 1<q
-      (n , b<⟨1+q-1⟩ⁿ) = bound-unbounded-power-one-plus-ℚ⁺ q-1⁺ b
+      (k , k2+1=n) = has-odd-expansion-is-odd n odd-n
     in
-      ( n ,
-        tr
-          ( le-ℚ b)
-          ( ap
-            ( rational-ℚ⁺ ∘ power-ℚ⁺ n)
-            ( eq-ℚ⁺ (is-identity-right-conjugation-add-ℚ one-ℚ q)))
-          ( b<⟨1+q-1⟩ⁿ))
-
-  unbounded-power-greater-than-one-ℚ⁺ :
-    (q : ℚ⁺) (b : ℚ) → le-ℚ⁺ one-ℚ⁺ q →
-    exists ℕ (λ n → le-ℚ-Prop b (rational-ℚ⁺ (power-ℚ⁺ n q)))
-  unbounded-power-greater-than-one-ℚ⁺ q b 1<q =
-    unit-trunc-Prop (bound-unbounded-power-greater-than-one-ℚ⁺ q b 1<q)
+      tr
+        ( is-negative-ℚ)
+        ( equational-reasoning
+          power-ℚ (k *ℕ 2) q *ℚ q
+          ＝ power-ℚ (succ-ℕ (k *ℕ 2)) q
+            by inv (power-succ-ℚ (k *ℕ 2) q)
+          ＝ power-ℚ n q
+            by ap (λ m → power-ℚ m q) k2+1=n)
+        ( is-negative-mul-positive-negative-ℚ
+          ( is-positive-even-power-ℚ⁻ (k *ℕ 2) q⁻ (k , refl))
+          ( is-neg-q))
 ```
 
-### If `ε` is a positive rational number less than one, `εⁿ` becomes arbitrarily small
+### If `|p| ≤ |q|`, `|pⁿ| ≤ |qⁿ|`
 
 ```agda
 abstract
-  bound-arbitrarily-small-power-le-one-ℚ⁺ :
-    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
-    Σ ℕ (λ n → le-ℚ⁺ (power-ℚ⁺ n ε) δ)
-  bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ ε<1 =
-    let
-      1/δ = inv-ℚ⁺ δ
-      1/ε = inv-ℚ⁺ ε
-      1<1/ε =
-        tr
-          ( λ q → le-ℚ⁺ q 1/ε)
-          ( inv-one-ℚ⁺)
-          ( inv-le-ℚ⁺ ε one-ℚ⁺ ε<1)
-      (n , 1/δ<⟨1/ε⟩ⁿ) =
-        bound-unbounded-power-greater-than-one-ℚ⁺ 1/ε (rational-ℚ⁺ 1/δ) 1<1/ε
-    in
-      ( n ,
-        binary-tr
-          ( le-ℚ⁺)
-          ( equational-reasoning
-            inv-ℚ⁺ (power-ℚ⁺ n 1/ε)
-            ＝ inv-ℚ⁺ (inv-ℚ⁺ (power-ℚ⁺ n ε))
-              by ap inv-ℚ⁺ (power-inv-Group group-mul-ℚ⁺ n ε)
-            ＝ power-ℚ⁺ n ε
-              by inv-inv-ℚ⁺ (power-ℚ⁺ n ε))
-          ( inv-inv-ℚ⁺ δ)
-          ( inv-le-ℚ⁺ 1/δ (power-ℚ⁺ n 1/ε) 1/δ<⟨1/ε⟩ⁿ))
-
-  arbitrarily-small-power-le-one-ℚ⁺ :
-    (ε δ : ℚ⁺) → le-ℚ⁺ ε one-ℚ⁺ →
-    exists ℕ (λ n → le-prop-ℚ⁺ (power-ℚ⁺ n ε) δ)
-  arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε =
-    unit-trunc-Prop (bound-arbitrarily-small-power-le-one-ℚ⁺ ε δ 1<ε)
+  preserves-leq-abs-power-ℚ :
+    (n : ℕ) (p q : ℚ) → leq-ℚ⁰⁺ (abs-ℚ p) (abs-ℚ q) →
+    leq-ℚ⁰⁺ (abs-ℚ (power-ℚ n p)) (abs-ℚ (power-ℚ n q))
+  preserves-leq-abs-power-ℚ 0 p q _ = refl-leq-ℚ _
+  preserves-leq-abs-power-ℚ (succ-ℕ n) p q |p|≤|q| =
+    binary-tr
+      ( leq-ℚ)
+      ( equational-reasoning
+        rational-abs-ℚ (power-ℚ n p) *ℚ rational-abs-ℚ p
+        ＝ rational-abs-ℚ (power-ℚ n p *ℚ p)
+          by inv (rational-abs-mul-ℚ (power-ℚ n p) p)
+        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) p)
+          by ap rational-abs-ℚ (inv (power-succ-ℚ n p)))
+      ( equational-reasoning
+        rational-abs-ℚ (power-ℚ n q) *ℚ rational-abs-ℚ q
+        ＝ rational-abs-ℚ (power-ℚ n q *ℚ q)
+          by inv (rational-abs-mul-ℚ (power-ℚ n q) q)
+        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) q)
+          by ap rational-abs-ℚ (inv (power-succ-ℚ n q)))
+      ( preserves-leq-mul-ℚ⁰⁺
+        ( abs-ℚ (power-ℚ n p))
+        ( abs-ℚ (power-ℚ n q))
+        ( abs-ℚ p)
+        ( abs-ℚ q)
+        ( preserves-leq-abs-power-ℚ n p q |p|≤|q|)
+        ( |p|≤|q|))
 ```
 
 ## See also
 
-- [Powers of elements of a group](group-theory.powers-of-elements-groups.md)
+- [Powers of elements of a monoid](group-theory.powers-of-elements-monoids.md)
+- [Powers of elements of a commutative monoid](group-theory.powers-of-elements-commutative-monoids.md)
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index 16de8f520d9..d9c561a0178 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -9,11 +9,13 @@ module elementary-number-theory.squares-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.multiplication-negative-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
 open import elementary-number-theory.multiplication-nonpositive-rational-numbers
@@ -171,6 +173,10 @@ abstract
     (x : ℚ) → is-negative-ℚ x → is-positive-ℚ (square-ℚ x)
   is-positive-square-negative-ℚ x neg-x =
     is-positive-mul-negative-ℚ {x} {x} neg-x neg-x
+
+square-ℚ⁻ : ℚ⁻ → ℚ⁺
+square-ℚ⁻ (q , is-neg-q) =
+  (square-ℚ q , is-positive-square-negative-ℚ q is-neg-q)
 ```
 
 ### If the square of a rational number is 0, it is zero
@@ -274,3 +280,21 @@ abstract
             ( preserves-leq-square-ℚ⁰⁺ q⁰⁺ p⁰⁺ q≤p)))
       ( decide-le-leq-ℚ p q)
 ```
+
+### `|p|² = p²`
+
+```agda
+abstract
+  square-abs-ℚ : (q : ℚ) → square-ℚ (rational-abs-ℚ q) ＝ square-ℚ q
+  square-abs-ℚ q =
+    rec-coproduct
+      ( λ q≤-q →
+        equational-reasoning
+          square-ℚ (rational-abs-ℚ q)
+          ＝ square-ℚ (neg-ℚ q)
+            by ap square-ℚ (left-leq-right-max-ℚ _ _ q≤-q)
+          ＝ square-ℚ q
+            by square-neg-ℚ q)
+      ( λ -q≤q → ap square-ℚ (right-leq-left-max-ℚ _ _ -q≤q))
+      ( linear-leq-ℚ q (neg-ℚ q))
+```
diff --git a/src/group-theory/powers-of-elements-commutative-monoids.lagda.md b/src/group-theory/powers-of-elements-commutative-monoids.lagda.md
index 344471dcb03..54c92815ab0 100644
--- a/src/group-theory/powers-of-elements-commutative-monoids.lagda.md
+++ b/src/group-theory/powers-of-elements-commutative-monoids.lagda.md
@@ -144,13 +144,15 @@ module _
 
   distributive-power-mul-Commutative-Monoid :
     (n : ℕ) {x y : type-Commutative-Monoid M} →
-    (H : mul-Commutative-Monoid M x y ＝ mul-Commutative-Monoid M y x) →
     power-Commutative-Monoid M n (mul-Commutative-Monoid M x y) ＝
     mul-Commutative-Monoid M
       ( power-Commutative-Monoid M n x)
       ( power-Commutative-Monoid M n y)
-  distributive-power-mul-Commutative-Monoid =
-    distributive-power-mul-Monoid (monoid-Commutative-Monoid M)
+  distributive-power-mul-Commutative-Monoid n =
+    distributive-power-mul-Monoid
+      ( monoid-Commutative-Monoid M)
+      ( n)
+      ( commutative-mul-Commutative-Monoid M _ _)
 ```
 
 ### Powers by products of natural numbers are iterated powers

From 906c3fb059f557700d7b67fb336fef916c093875 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 12:21:35 -0700
Subject: [PATCH 19/22] Progress

---
 src/elementary-number-theory.lagda.md         |   2 +
 .../absolute-value-rational-numbers.lagda.md  |  14 +-
 ...lity-nonnegative-rational-numbers.lagda.md |   8 +
 ...d-of-nonnegative-rational-numbers.lagda.md |  55 ++++
 ...cative-monoid-of-rational-numbers.lagda.md |   4 +-
 .../nonnegative-rational-numbers.lagda.md     |   3 +
 .../parity-natural-numbers.lagda.md           |   9 +
 ...wers-nonnegative-rational-numbers.lagda.md | 146 ++++++++++
 .../powers-positive-rational-numbers.lagda.md |  10 +-
 .../powers-rational-numbers.lagda.md          | 258 +++++++++++++++---
 ...lity-nonnegative-rational-numbers.lagda.md |  22 ++
 .../submonoids-commutative-monoids.lagda.md   |   5 +-
 src/group-theory/submonoids.lagda.md          |   6 +-
 13 files changed, 495 insertions(+), 47 deletions(-)
 create mode 100644 src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md
 create mode 100644 src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md

diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 0e518ce0112..6655218d7ca 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -148,6 +148,7 @@ open import elementary-number-theory.multiplicative-group-of-positive-rational-n
 open import elementary-number-theory.multiplicative-group-of-rational-numbers public
 open import elementary-number-theory.multiplicative-inverses-positive-integer-fractions public
 open import elementary-number-theory.multiplicative-monoid-of-natural-numbers public
+open import elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers public
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers public
 open import elementary-number-theory.multiplicative-units-integers public
 open import elementary-number-theory.multiplicative-units-standard-cyclic-rings public
@@ -178,6 +179,7 @@ open import elementary-number-theory.positive-integer-fractions public
 open import elementary-number-theory.positive-integers public
 open import elementary-number-theory.positive-rational-numbers public
 open import elementary-number-theory.powers-integers public
+open import elementary-number-theory.powers-nonnegative-rational-numbers public
 open import elementary-number-theory.powers-of-two public
 open import elementary-number-theory.powers-positive-rational-numbers public
 open import elementary-number-theory.powers-rational-numbers public
diff --git a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
index fa1c1a6995c..4b0ad01d885 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -174,13 +174,15 @@ abstract
         ( neg-neg-ℚ q)
         ( neg-leq-ℚ (neg-leq-abs-ℚ q)))
 
+  rational-abs-zero-ℚ : rational-abs-ℚ zero-ℚ ＝ zero-ℚ
+  rational-abs-zero-ℚ =
+    equational-reasoning
+      max-ℚ zero-ℚ (neg-ℚ zero-ℚ)
+      ＝ max-ℚ zero-ℚ zero-ℚ by ap-max-ℚ refl neg-zero-ℚ
+      ＝ zero-ℚ by idempotent-max-ℚ zero-ℚ
+
   abs-zero-ℚ : abs-ℚ zero-ℚ ＝ zero-ℚ⁰⁺
-  abs-zero-ℚ =
-    eq-ℚ⁰⁺
-      ( equational-reasoning
-        max-ℚ zero-ℚ (neg-ℚ zero-ℚ)
-        ＝ max-ℚ zero-ℚ zero-ℚ by ap-max-ℚ refl neg-zero-ℚ
-        ＝ zero-ℚ by idempotent-max-ℚ zero-ℚ)
+  abs-zero-ℚ = eq-ℚ⁰⁺ rational-abs-zero-ℚ
 ```
 
 ### The triangle inequality
diff --git a/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md
index 7ec428e6cf1..a78e7658d87 100644
--- a/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-nonnegative-rational-numbers.lagda.md
@@ -51,3 +51,11 @@ decidable-total-order-ℚ⁰⁺ =
     ℚ-Decidable-Total-Order
     is-nonnegative-prop-ℚ
 ```
+
+### Inequality on the nonnegative rational numbers is transitive
+
+```agda
+abstract
+  transitive-leq-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → leq-ℚ⁰⁺ q r → leq-ℚ⁰⁺ p q → leq-ℚ⁰⁺ p r
+  transitive-leq-ℚ⁰⁺ (p , _) (q , _) (r , _) = transitive-leq-ℚ p q r
+```
diff --git a/src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..da8d59e5b14
--- /dev/null
+++ b/src/elementary-number-theory/multiplicative-monoid-of-nonnegative-rational-numbers.lagda.md
@@ -0,0 +1,55 @@
+# The multiplicative monoid of nonnegative rational numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import group-theory.commutative-monoids
+open import group-theory.monoids
+open import group-theory.submonoids-commutative-monoids
+```
+
+</details>
+
+## Idea
+
+The type of
+[nonnegative](elementary-number-theory.nonnegative-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) equipped with
+[multiplication](elementary-number-theory.multiplication-nonnegative-rational-numbers.md)
+is a [commutative monoid](group-theory.commutative-monoids.md) with unit
+`one-ℚ⁰⁺`.
+
+## Definitions
+
+### The multiplicative commutative monoid of nonnegative rational numbers
+
+```agda
+nonnegative-submonoid-commutative-monoid-mul-ℚ :
+  Commutative-Submonoid lzero commutative-monoid-mul-ℚ
+nonnegative-submonoid-commutative-monoid-mul-ℚ =
+  ( is-nonnegative-prop-ℚ ,
+    is-nonnegative-one-ℚ ,
+    λ _ _ → is-nonnegative-mul-ℚ)
+
+commutative-monoid-mul-ℚ⁰⁺ : Commutative-Monoid lzero
+commutative-monoid-mul-ℚ⁰⁺ =
+  commutative-monoid-Commutative-Submonoid
+    ( commutative-monoid-mul-ℚ)
+    ( nonnegative-submonoid-commutative-monoid-mul-ℚ)
+
+monoid-mul-ℚ⁰⁺ : Monoid lzero
+monoid-mul-ℚ⁰⁺ = monoid-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
diff --git a/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md
index 781ac3df4c6..a3ac5b301d7 100644
--- a/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-monoid-of-rational-numbers.lagda.md
@@ -27,8 +27,8 @@ open import group-theory.semigroups
 
 The type of [rational numbers](elementary-number-theory.rational-numbers.md)
 equipped with
-[multiplication](elementary-number-theory.addition-rational-numbers.md) is a
-[commutative monoid](group-theory.commutative-monoids.md) with unit `one-ℚ`.
+[multiplication](elementary-number-theory.multiplication-rational-numbers.md) is
+a [commutative monoid](group-theory.commutative-monoids.md) with unit `one-ℚ`.
 
 ## Definitions
 
diff --git a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
index b7d23dc65ad..f16ddf889fb 100644
--- a/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/nonnegative-rational-numbers.lagda.md
@@ -166,6 +166,9 @@ zero-ℚ⁰⁺ = nonnegative-rational-ℕ zero-ℕ
 
 one-ℚ⁰⁺ : ℚ⁰⁺
 one-ℚ⁰⁺ = nonnegative-rational-ℕ 1
+
+is-nonnegative-one-ℚ : is-nonnegative-ℚ one-ℚ
+is-nonnegative-one-ℚ = is-nonnegative-rational-ℕ 1
 ```
 
 ### The rational image of a nonnegative integer fraction is nonnegative
diff --git a/src/elementary-number-theory/parity-natural-numbers.lagda.md b/src/elementary-number-theory/parity-natural-numbers.lagda.md
index 9fa410e7f87..5a7c82123c1 100644
--- a/src/elementary-number-theory/parity-natural-numbers.lagda.md
+++ b/src/elementary-number-theory/parity-natural-numbers.lagda.md
@@ -12,6 +12,7 @@ open import elementary-number-theory.equality-natural-numbers
 open import elementary-number-theory.modular-arithmetic-standard-finite-types
 open import elementary-number-theory.multiplication-natural-numbers
 open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonzero-natural-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.decidable-types
@@ -69,6 +70,14 @@ is-odd-one-ℕ : is-odd-ℕ 1
 is-odd-one-ℕ H = Eq-eq-ℕ (is-one-div-one-ℕ 2 H)
 ```
 
+### An odd natural number is nonzero
+
+```agda
+abstract
+  is-nonzero-is-odd-ℕ : {n : ℕ} → is-odd-ℕ n → is-nonzero-ℕ n
+  is-nonzero-is-odd-ℕ odd-n refl = odd-n is-even-zero-ℕ
+```
+
 ### A natural number `x` is even if and only if `x + 2` is even
 
 ```agda
diff --git a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
new file mode 100644
index 00000000000..9ad7fa97c80
--- /dev/null
+++ b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
@@ -0,0 +1,146 @@
+# Powers of nonnegative rational numbers
+
+```agda
+module elementary-number-theory.powers-nonnegative-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
+open import elementary-number-theory.inequality-rational-numbers
+open import elementary-number-theory.multiplication-natural-numbers
+open import elementary-number-theory.multiplication-nonnegative-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.multiplicative-monoid-of-nonnegative-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-nonnegative-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.identity-types
+
+open import group-theory.powers-of-elements-commutative-monoids
+open import group-theory.powers-of-elements-monoids
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "power operation" Disambiguation="a nonnegative rational number to a natural number power" Agda=power-ℚ⁰⁺}}
+on the [nonnegative](elementary-number-theory.nonnegative-rational-numbers.md)
+[rational numbers](elementary-number-theory.rational-numbers.md) is the
+operation `n x ↦ xⁿ`, which is defined by
+[iteratively](foundation.iterating-functions.md) multiplying `x` with itself `n`
+times. This file covers the case where `n` is a
+[natural number](elementary-number-theory.natural-numbers.md).
+
+## Definition
+
+```agda
+power-ℚ⁰⁺ : ℕ → ℚ⁰⁺ → ℚ⁰⁺
+power-ℚ⁰⁺ = power-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
+
+## Properties
+
+### `1ⁿ = 1`
+
+```agda
+power-one-ℚ⁰⁺ : (n : ℕ) → power-ℚ⁰⁺ n one-ℚ⁰⁺ ＝ one-ℚ⁰⁺
+power-one-ℚ⁰⁺ = power-unit-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
+
+### `qⁿ⁺¹ = qⁿq`
+
+```agda
+power-succ-ℚ⁰⁺ :
+  (n : ℕ) (q : ℚ⁰⁺) → power-ℚ⁰⁺ (succ-ℕ n) q ＝ power-ℚ⁰⁺ n q *ℚ⁰⁺ q
+power-succ-ℚ⁰⁺ = power-succ-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺
+```
+
+### `qⁿ = qqⁿ`
+
+```agda
+power-succ-ℚ⁰⁺' :
+  (n : ℕ) (q : ℚ⁰⁺) → power-ℚ⁰⁺ (succ-ℕ n) q ＝ q *ℚ⁰⁺ power-ℚ⁰⁺ n q
+power-succ-ℚ⁰⁺' = power-succ-Commutative-Monoid' commutative-monoid-mul-ℚ⁰⁺
+```
+
+### `qᵐ⁺ⁿ = qᵐqⁿ`
+
+```agda
+abstract
+  distributive-power-add-ℚ⁰⁺ :
+    (m n : ℕ) (q : ℚ⁰⁺) →
+    power-ℚ⁰⁺ (m +ℕ n) q ＝ power-ℚ⁰⁺ m q *ℚ⁰⁺ power-ℚ⁰⁺ n q
+  distributive-power-add-ℚ⁰⁺ m n _ =
+    distributive-power-add-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺ m n
+```
+
+### `(pq)ⁿ=pⁿqⁿ`
+
+```agda
+abstract
+  left-distributive-power-mul-ℚ⁰⁺ :
+    (n : ℕ) (p q : ℚ⁰⁺) →
+    power-ℚ⁰⁺ n (p *ℚ⁰⁺ q) ＝ power-ℚ⁰⁺ n p *ℚ⁰⁺ power-ℚ⁰⁺ n q
+  left-distributive-power-mul-ℚ⁰⁺ n p q =
+    distributive-power-mul-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺ n
+```
+
+### `pᵐⁿ = (pᵐ)ⁿ`
+
+```agda
+power-mul-ℚ⁰⁺ :
+  (m n : ℕ) (q : ℚ⁰⁺) → power-ℚ⁰⁺ (m *ℕ n) q ＝ power-ℚ⁰⁺ n (power-ℚ⁰⁺ m q)
+power-mul-ℚ⁰⁺ m n q =
+  power-mul-Commutative-Monoid commutative-monoid-mul-ℚ⁰⁺ m n
+```
+
+### If `p` and `q` are nonnegative rational numbers with `p < q` and `n` is nonzero, `pⁿ < qⁿ`
+
+```agda
+abstract
+  preserves-le-power-ℚ⁰⁺ :
+    (n : ℕ) → is-nonzero-ℕ n → (p q : ℚ⁰⁺) → le-ℚ⁰⁺ p q →
+    le-ℚ⁰⁺ (power-ℚ⁰⁺ n p) (power-ℚ⁰⁺ n q)
+  preserves-le-power-ℚ⁰⁺ 0 H p q p<q = ex-falso (H refl)
+  preserves-le-power-ℚ⁰⁺ 1 _ p q p<q = p<q
+  preserves-le-power-ℚ⁰⁺ (succ-ℕ n@(succ-ℕ _)) _ p q p<q =
+    concatenate-leq-le-ℚ⁰⁺
+      ( power-ℚ⁰⁺ (succ-ℕ n) p)
+      ( power-ℚ⁰⁺ n p *ℚ⁰⁺ q)
+      ( power-ℚ⁰⁺ (succ-ℕ n) q)
+      ( preserves-leq-left-mul-ℚ⁰⁺ (power-ℚ⁰⁺ n p) _ _ (leq-le-ℚ p<q))
+      ( preserves-le-right-mul-ℚ⁺
+        ( rational-ℚ⁰⁺ q , is-positive-le-ℚ⁰⁺ p _ p<q)
+        ( rational-ℚ⁰⁺ (power-ℚ⁰⁺ n p))
+        ( rational-ℚ⁰⁺ (power-ℚ⁰⁺ n q))
+        ( preserves-le-power-ℚ⁰⁺ n (is-nonzero-succ-ℕ _) p q p<q))
+```
+
+### If `p` and `q` are nonnegative rational numbers with `p ≤ q`, `pⁿ ≤ qⁿ`
+
+```agda
+abstract
+  preserves-leq-power-ℚ⁰⁺ :
+    (n : ℕ) (p q : ℚ⁰⁺) → leq-ℚ⁰⁺ p q → leq-ℚ⁰⁺ (power-ℚ⁰⁺ n p) (power-ℚ⁰⁺ n q)
+  preserves-leq-power-ℚ⁰⁺ 0 _ _ _ = refl-leq-ℚ one-ℚ
+  preserves-leq-power-ℚ⁰⁺ 1 p q p≤q = p≤q
+  preserves-leq-power-ℚ⁰⁺ (succ-ℕ n@(succ-ℕ _)) p q p≤q =
+    transitive-leq-ℚ⁰⁺
+      ( power-ℚ⁰⁺ (succ-ℕ n) p)
+      ( power-ℚ⁰⁺ n p *ℚ⁰⁺ q)
+      ( power-ℚ⁰⁺ (succ-ℕ n) q)
+      ( preserves-leq-right-mul-ℚ⁰⁺ q _ _ (preserves-leq-power-ℚ⁰⁺ n p q p≤q))
+      ( preserves-leq-left-mul-ℚ⁰⁺ (power-ℚ⁰⁺ n p) _ _ p≤q)
+```
diff --git a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
index 61a906395d3..3ca56e989fa 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -113,17 +113,17 @@ power-mul-ℚ⁺ m n q = power-mul-Group group-mul-ℚ⁺ m n
 ```agda
 abstract
   preserves-le-power-ℚ⁺ :
-    (n : ℕ) (p q : ℚ⁺) → le-ℚ⁺ p q → is-nonzero-ℕ n →
+    (n : ℕ) → is-nonzero-ℕ n → (p q : ℚ⁺) → le-ℚ⁺ p q →
     le-ℚ⁺ (power-ℚ⁺ n p) (power-ℚ⁺ n q)
-  preserves-le-power-ℚ⁺ 0 p q p<q H = ex-falso (H refl)
-  preserves-le-power-ℚ⁺ 1 p q p<q H = p<q
-  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) p q p<q H =
+  preserves-le-power-ℚ⁺ 0 H p q p<q = ex-falso (H refl)
+  preserves-le-power-ℚ⁺ 1 _ p q p<q = p<q
+  preserves-le-power-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) _ p q p<q =
     transitive-le-ℚ⁺
       ( power-ℚ⁺ (succ-ℕ n) p)
       ( power-ℚ⁺ n p *ℚ⁺ q)
       ( power-ℚ⁺ (succ-ℕ n) q)
       ( preserves-le-right-mul-ℚ⁺ q _ _
-        ( preserves-le-power-ℚ⁺ n p q p<q (is-nonzero-succ-ℕ _)))
+        ( preserves-le-power-ℚ⁺ n (is-nonzero-succ-ℕ _) p q p<q))
       ( preserves-le-left-mul-ℚ⁺ (power-ℚ⁺ n p) _ _ p<q)
 ```
 
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 83f95cea235..1374740c371 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -9,6 +9,7 @@ module elementary-number-theory.powers-rational-numbers where
 ```agda
 open import elementary-number-theory.absolute-value-rational-numbers
 open import elementary-number-theory.addition-natural-numbers
+open import elementary-number-theory.inequalities-positive-and-negative-rational-numbers
 open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.multiplication-natural-numbers
@@ -21,12 +22,17 @@ open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.parity-natural-numbers
+open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
+open import elementary-number-theory.powers-nonnegative-rational-numbers
+open import elementary-number-theory.powers-positive-rational-numbers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.squares-rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
+open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.transport-along-identifications
@@ -54,6 +60,31 @@ power-ℚ = power-Monoid monoid-mul-ℚ
 
 ## Properties
 
+### The power operation on rational numbers agrees with the power operation on positive rational numbers
+
+```agda
+abstract
+  power-rational-ℚ⁺ :
+    (n : ℕ) (q : ℚ⁺) → power-ℚ n (rational-ℚ⁺ q) ＝ rational-ℚ⁺ (power-ℚ⁺ n q)
+  power-rational-ℚ⁺ 0 _ = refl
+  power-rational-ℚ⁺ 1 _ = refl
+  power-rational-ℚ⁺ (succ-ℕ n@(succ-ℕ _)) q =
+    ap-mul-ℚ (power-rational-ℚ⁺ n q) refl
+```
+
+### The power operation on rational numbers agrees with the power operation on nonnegative rational numbers
+
+```agda
+abstract
+  power-rational-ℚ⁰⁺ :
+    (n : ℕ) (q : ℚ⁰⁺) →
+    power-ℚ n (rational-ℚ⁰⁺ q) ＝ rational-ℚ⁰⁺ (power-ℚ⁰⁺ n q)
+  power-rational-ℚ⁰⁺ 0 _ = refl
+  power-rational-ℚ⁰⁺ 1 _ = refl
+  power-rational-ℚ⁰⁺ (succ-ℕ n@(succ-ℕ _)) q =
+    ap-mul-ℚ (power-rational-ℚ⁰⁺ n q) refl
+```
+
 ### `1ⁿ = 1`
 
 ```agda
@@ -105,6 +136,11 @@ abstract
   power-mul-ℚ :
     (m n : ℕ) (q : ℚ) → power-ℚ (m *ℕ n) q ＝ power-ℚ n (power-ℚ m q)
   power-mul-ℚ m n q = power-mul-Monoid monoid-mul-ℚ m n
+
+  power-mul-ℚ' :
+    (m n : ℕ) (q : ℚ) → power-ℚ (m *ℕ n) q ＝ power-ℚ m (power-ℚ n q)
+  power-mul-ℚ' m n q =
+    ap (λ k → power-ℚ k q) (commutative-mul-ℕ m n) ∙ power-mul-ℚ n m q
 ```
 
 ### Even powers of rational numbers are nonnegative
@@ -131,12 +167,11 @@ nonnegative-even-power-ℚ n q even-n =
 abstract
   is-positive-power-ℚ⁺ :
     (n : ℕ) (q : ℚ⁺) → is-positive-ℚ (power-ℚ n (rational-ℚ⁺ q))
-  is-positive-power-ℚ⁺ 0 q = is-positive-rational-ℚ⁺ one-ℚ⁺
-  is-positive-power-ℚ⁺ (succ-ℕ n) q⁺@(q , is-pos-q) =
+  is-positive-power-ℚ⁺ n q =
     inv-tr
       ( is-positive-ℚ)
-      ( power-succ-ℚ n q)
-      ( is-positive-mul-ℚ (is-positive-power-ℚ⁺ n q⁺) is-pos-q)
+      ( power-rational-ℚ⁺ n q)
+      ( is-positive-rational-ℚ⁺ (power-ℚ⁺ n q))
 ```
 
 ### Even powers of negative rational numbers are positive
@@ -148,12 +183,7 @@ abstract
   is-positive-even-power-ℚ⁻ _ q⁻@(q , is-neg-q) (k , refl) =
     inv-tr
       ( is-positive-ℚ)
-      ( equational-reasoning
-        power-ℚ (k *ℕ 2) q
-        ＝ power-ℚ (2 *ℕ k) q
-          by ap (λ m → power-ℚ m q) (commutative-mul-ℕ k 2)
-        ＝ power-ℚ k (square-ℚ q)
-          by power-mul-ℚ 2 k q)
+      ( power-mul-ℚ' k 2 q)
       ( is-positive-power-ℚ⁺ k (square-ℚ⁻ q⁻))
 ```
 
@@ -180,6 +210,66 @@ abstract
           ( is-neg-q))
 ```
 
+### For even `n`, `(-q)ⁿ = qⁿ`
+
+```agda
+abstract
+  even-power-neg-ℚ :
+    (n : ℕ) (q : ℚ) → is-even-ℕ n → power-ℚ n (neg-ℚ q) ＝ power-ℚ n q
+  even-power-neg-ℚ _ q (k , refl) =
+    equational-reasoning
+      power-ℚ (k *ℕ 2) (neg-ℚ q)
+      ＝ power-ℚ k (square-ℚ (neg-ℚ q))
+        by power-mul-ℚ' k 2 (neg-ℚ q)
+      ＝ power-ℚ k (square-ℚ q)
+        by ap (power-ℚ k) (square-neg-ℚ q)
+      ＝ power-ℚ (k *ℕ 2) q
+        by inv (power-mul-ℚ' k 2 q)
+```
+
+### For odd `n`, `(-q)ⁿ = -(qⁿ)`
+
+```agda
+abstract
+  odd-power-neg-ℚ :
+    (n : ℕ) (q : ℚ) → is-odd-ℕ n → power-ℚ n (neg-ℚ q) ＝ neg-ℚ (power-ℚ n q)
+  odd-power-neg-ℚ n q odd-n =
+    let (k , k2+1=n) = has-odd-expansion-is-odd n odd-n
+    in
+      equational-reasoning
+        power-ℚ n (neg-ℚ q)
+        ＝ power-ℚ (succ-ℕ (k *ℕ 2)) (neg-ℚ q)
+          by ap (λ m → power-ℚ m (neg-ℚ q)) (inv k2+1=n)
+        ＝ power-ℚ (k *ℕ 2) (neg-ℚ q) *ℚ neg-ℚ q
+          by power-succ-ℚ (k *ℕ 2) (neg-ℚ q)
+        ＝ power-ℚ (k *ℕ 2) q *ℚ neg-ℚ q
+          by ap-mul-ℚ (even-power-neg-ℚ _ q (k , refl)) refl
+        ＝ neg-ℚ (power-ℚ (k *ℕ 2) q *ℚ q)
+          by right-negative-law-mul-ℚ _ _
+        ＝ neg-ℚ (power-ℚ (succ-ℕ (k *ℕ 2)) q)
+          by ap neg-ℚ (inv (power-succ-ℚ (k *ℕ 2) q))
+        ＝ neg-ℚ (power-ℚ n q)
+          by ap (λ m → neg-ℚ (power-ℚ m q)) k2+1=n
+```
+
+### `|q|ⁿ=|qⁿ|`
+
+```agda
+abstract
+  power-rational-abs-ℚ :
+    (n : ℕ) (q : ℚ) →
+    power-ℚ n (rational-abs-ℚ q) ＝ rational-abs-ℚ (power-ℚ n q)
+  power-rational-abs-ℚ 0 q = inv (ap rational-ℚ⁰⁺ (abs-rational-ℚ⁰⁺ one-ℚ⁰⁺))
+  power-rational-abs-ℚ 1 q = refl
+  power-rational-abs-ℚ (succ-ℕ n@(succ-ℕ _)) q =
+    equational-reasoning
+      power-ℚ n (rational-abs-ℚ q) *ℚ rational-abs-ℚ q
+      ＝ rational-abs-ℚ (power-ℚ n q) *ℚ rational-abs-ℚ q
+        by ap-mul-ℚ (power-rational-abs-ℚ n q) refl
+      ＝ rational-abs-ℚ (power-ℚ n q *ℚ q)
+        by inv (rational-abs-mul-ℚ _ _)
+```
+
 ### If `|p| ≤ |q|`, `|pⁿ| ≤ |qⁿ|`
 
 ```agda
@@ -187,29 +277,135 @@ abstract
   preserves-leq-abs-power-ℚ :
     (n : ℕ) (p q : ℚ) → leq-ℚ⁰⁺ (abs-ℚ p) (abs-ℚ q) →
     leq-ℚ⁰⁺ (abs-ℚ (power-ℚ n p)) (abs-ℚ (power-ℚ n q))
-  preserves-leq-abs-power-ℚ 0 p q _ = refl-leq-ℚ _
-  preserves-leq-abs-power-ℚ (succ-ℕ n) p q |p|≤|q| =
+  preserves-leq-abs-power-ℚ n p q |p|≤|q| =
     binary-tr
       ( leq-ℚ)
-      ( equational-reasoning
-        rational-abs-ℚ (power-ℚ n p) *ℚ rational-abs-ℚ p
-        ＝ rational-abs-ℚ (power-ℚ n p *ℚ p)
-          by inv (rational-abs-mul-ℚ (power-ℚ n p) p)
-        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) p)
-          by ap rational-abs-ℚ (inv (power-succ-ℚ n p)))
-      ( equational-reasoning
-        rational-abs-ℚ (power-ℚ n q) *ℚ rational-abs-ℚ q
-        ＝ rational-abs-ℚ (power-ℚ n q *ℚ q)
-          by inv (rational-abs-mul-ℚ (power-ℚ n q) q)
-        ＝ rational-abs-ℚ (power-ℚ (succ-ℕ n) q)
-          by ap rational-abs-ℚ (inv (power-succ-ℚ n q)))
-      ( preserves-leq-mul-ℚ⁰⁺
-        ( abs-ℚ (power-ℚ n p))
-        ( abs-ℚ (power-ℚ n q))
-        ( abs-ℚ p)
-        ( abs-ℚ q)
-        ( preserves-leq-abs-power-ℚ n p q |p|≤|q|)
-        ( |p|≤|q|))
+      ( inv (power-rational-ℚ⁰⁺ n (abs-ℚ p)) ∙ power-rational-abs-ℚ n p)
+      ( inv (power-rational-ℚ⁰⁺ n (abs-ℚ q)) ∙ power-rational-abs-ℚ n q)
+      ( preserves-leq-power-ℚ⁰⁺ n (abs-ℚ p) (abs-ℚ q) |p|≤|q|)
+```
+
+### Odd powers of rational numbers preserve inequality
+
+```agda
+abstract
+  preserves-leq-odd-power-ℚ :
+    (n : ℕ) (p q : ℚ) → is-odd-ℕ n → leq-ℚ p q →
+    leq-ℚ (power-ℚ n p) (power-ℚ n q)
+  preserves-leq-odd-power-ℚ n p q odd-n p≤q =
+    rec-coproduct
+      ( λ is-neg-p →
+        rec-coproduct
+          ( λ is-neg-q →
+            let
+              p⁻ = (p , is-neg-p)
+              q⁻ = (q , is-neg-q)
+            in
+              binary-tr
+                ( leq-ℚ)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ p))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n p))
+                    by ap neg-ℚ (odd-power-neg-ℚ n p odd-n)
+                  ＝ power-ℚ n p
+                    by neg-neg-ℚ _)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ q))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n q))
+                    by ap neg-ℚ (odd-power-neg-ℚ n q odd-n)
+                  ＝ power-ℚ n q
+                    by neg-neg-ℚ _)
+                ( neg-leq-ℚ
+                  ( preserves-leq-power-ℚ⁺
+                    ( n)
+                    ( neg-ℚ⁻ q⁻)
+                    ( neg-ℚ⁻ p⁻)
+                    ( neg-leq-ℚ p≤q))))
+          ( λ is-nonneg-q →
+            inv-tr
+              ( leq-ℚ (power-ℚ n p))
+              ( power-rational-ℚ⁰⁺ n (q , is-nonneg-q))
+              ( leq-negative-nonnegative-ℚ
+                ( power-ℚ n p , is-negative-odd-power-ℚ⁻ n (p , is-neg-p) odd-n)
+                ( power-ℚ⁰⁺ n (q , is-nonneg-q))))
+          ( decide-is-negative-is-nonnegative-ℚ q))
+      ( λ is-nonneg-p →
+        let
+          p⁰⁺ = (p , is-nonneg-p)
+          q⁰⁺ = (q , is-nonnegative-leq-ℚ⁰⁺ p⁰⁺ q p≤q)
+        in
+          binary-tr
+            ( leq-ℚ)
+            ( inv (power-rational-ℚ⁰⁺ n p⁰⁺))
+            ( inv (power-rational-ℚ⁰⁺ n q⁰⁺))
+            ( preserves-leq-power-ℚ⁰⁺ n p⁰⁺ q⁰⁺ p≤q))
+      ( decide-is-negative-is-nonnegative-ℚ p)
+```
+
+### Odd powers of rational numbers preserve strict inequality
+
+```agda
+abstract
+  preserves-le-odd-power-ℚ :
+    (n : ℕ) (p q : ℚ) → is-odd-ℕ n → le-ℚ p q →
+    le-ℚ (power-ℚ n p) (power-ℚ n q)
+  preserves-le-odd-power-ℚ n p q odd-n p<q =
+    rec-coproduct
+      ( λ is-neg-p →
+        rec-coproduct
+          ( λ is-neg-q →
+            let
+              p⁻ = (p , is-neg-p)
+              q⁻ = (q , is-neg-q)
+            in
+              binary-tr
+                ( le-ℚ)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ p))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ p⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n p))
+                    by ap neg-ℚ (odd-power-neg-ℚ n p odd-n)
+                  ＝ power-ℚ n p
+                    by neg-neg-ℚ _)
+                ( equational-reasoning
+                  neg-ℚ (rational-ℚ⁺ (power-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (power-ℚ n (neg-ℚ q))
+                    by ap neg-ℚ (inv (power-rational-ℚ⁺ n (neg-ℚ⁻ q⁻)))
+                  ＝ neg-ℚ (neg-ℚ (power-ℚ n q))
+                    by ap neg-ℚ (odd-power-neg-ℚ n q odd-n)
+                  ＝ power-ℚ n q
+                    by neg-neg-ℚ _)
+                ( neg-le-ℚ
+                  ( preserves-le-power-ℚ⁺
+                    ( n)
+                    ( is-nonzero-is-odd-ℕ odd-n)
+                    ( neg-ℚ⁻ q⁻)
+                    ( neg-ℚ⁻ p⁻)
+                    ( neg-le-ℚ p<q))))
+          ( λ is-nonneg-q →
+            inv-tr
+              ( le-ℚ (power-ℚ n p))
+              ( power-rational-ℚ⁰⁺ n (q , is-nonneg-q))
+              ( le-negative-nonnegative-ℚ
+                ( power-ℚ n p , is-negative-odd-power-ℚ⁻ n (p , is-neg-p) odd-n)
+                ( power-ℚ⁰⁺ n (q , is-nonneg-q))))
+          ( decide-is-negative-is-nonnegative-ℚ q))
+      ( λ is-nonneg-p →
+        let
+          p⁰⁺ = (p , is-nonneg-p)
+          q⁰⁺ = (q , is-nonnegative-le-ℚ⁰⁺ p⁰⁺ q p<q)
+        in
+          binary-tr
+            ( le-ℚ)
+            ( inv (power-rational-ℚ⁰⁺ n p⁰⁺))
+            ( inv (power-rational-ℚ⁰⁺ n q⁰⁺))
+            ( preserves-le-power-ℚ⁰⁺ n (is-nonzero-is-odd-ℕ odd-n) p⁰⁺ q⁰⁺ p<q))
+      ( decide-is-negative-is-nonnegative-ℚ p)
 ```
 
 ## See also
diff --git a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
index 4ef6623999e..c82fd46745e 100644
--- a/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-nonnegative-rational-numbers.lagda.md
@@ -7,6 +7,7 @@ module elementary-number-theory.strict-inequality-nonnegative-rational-numbers w
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequality-nonnegative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
@@ -37,3 +38,24 @@ le-prop-ℚ⁰⁺ p q = le-ℚ-Prop (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q
 le-ℚ⁰⁺ : (p q : ℚ⁰⁺) → UU lzero
 le-ℚ⁰⁺ p q = type-Prop (le-prop-ℚ⁰⁺ p q)
 ```
+
+## Properties
+
+### Strict inequality on the nonnegative rational numbers is transitive
+
+```agda
+abstract
+  transitive-le-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → le-ℚ⁰⁺ q r → le-ℚ⁰⁺ p q → le-ℚ⁰⁺ p r
+  transitive-le-ℚ⁰⁺ (p , _) (q , _) (r , _) = transitive-le-ℚ p q r
+```
+
+### Concatenation laws with inequality
+
+```agda
+abstract
+  concatenate-le-leq-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → le-ℚ⁰⁺ p q → leq-ℚ⁰⁺ q r → le-ℚ⁰⁺ p r
+  concatenate-le-leq-ℚ⁰⁺ (p , _) (q , _) (r , _) = concatenate-le-leq-ℚ p q r
+
+  concatenate-leq-le-ℚ⁰⁺ : (p q r : ℚ⁰⁺) → leq-ℚ⁰⁺ p q → le-ℚ⁰⁺ q r → le-ℚ⁰⁺ p r
+  concatenate-leq-le-ℚ⁰⁺ (p , _) (q , _) (r , _) = concatenate-leq-le-ℚ p q r
+```
diff --git a/src/group-theory/submonoids-commutative-monoids.lagda.md b/src/group-theory/submonoids-commutative-monoids.lagda.md
index 1b93a31af20..6734dc7ba91 100644
--- a/src/group-theory/submonoids-commutative-monoids.lagda.md
+++ b/src/group-theory/submonoids-commutative-monoids.lagda.md
@@ -27,7 +27,10 @@ open import group-theory.subsets-commutative-monoids
 
 ## Idea
 
-A submonoid of a commutative monoid `M` is a subset of `M` that contains the
+A
+{{#concept "submonoid" Disambiguation="of a commutative monoid" Agda=Commutative-Submonoid}}
+of a [commutative monoid](group-theory.commutative-monoids.md) `M` is a
+[subset](group-theory.subsets-commutative-monoids.md) of `M` that contains the
 unit of `M` and is closed under multiplication.
 
 ## Definitions
diff --git a/src/group-theory/submonoids.lagda.md b/src/group-theory/submonoids.lagda.md
index 4245b22e102..28dacdedee4 100644
--- a/src/group-theory/submonoids.lagda.md
+++ b/src/group-theory/submonoids.lagda.md
@@ -27,8 +27,10 @@ open import group-theory.subsets-monoids
 
 ## Idea
 
-A submonoid of a monoid `M` is a subset of `M` that contains the unit of `M` and
-is closed under multiplication.
+A {{#concept "submonoid" WDID=Q121499459 WD="submonoid" Agda=Submonoid}} of a
+[monoid](group-theory.monoids.md) `M` is a
+[subset](group-theory.subsets-monoids.md) of `M` that contains the unit of `M`
+and is closed under multiplication.
 
 ## Definitions
 

From 92b42a5ad47796ab1923535a41077a5cdc97a22c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 12:37:31 -0700
Subject: [PATCH 20/22] Correct superscripts

---
 src/elementary-number-theory/powers-rational-numbers.lagda.md | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index 1374740c371..392ea1998cf 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -93,7 +93,7 @@ abstract
   power-one-ℚ = power-unit-Monoid monoid-mul-ℚ
 ```
 
-### `qⁿ¹ = qⁿq`
+### `qⁿ⁺¹ = qⁿq`
 
 ```agda
 abstract
@@ -101,7 +101,7 @@ abstract
   power-succ-ℚ = power-succ-Monoid monoid-mul-ℚ
 ```
 
-### `qⁿ = qqⁿ`
+### `qⁿ⁺¹ = qqⁿ`
 
 ```agda
 abstract

From 978b5045b61f96dfd1b8dc09145956ea5da15ce8 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 15:47:48 -0700
Subject: [PATCH 21/22] Progress

---
 ...of-elements-commutative-semirings.lagda.md |  15 ++
 ...icative-group-of-rational-numbers.lagda.md |  21 ++
 .../arithmetic-sequences-semigroups.lagda.md  |  27 +++
 src/lists/sequences.lagda.md                  |   9 +
 src/ring-theory.lagda.md                      |   1 +
 .../geometric-sequences-rings.lagda.md        | 216 ++++++++++++++++++
 .../geometric-sequences-semirings.lagda.md    |  36 ++-
 7 files changed, 323 insertions(+), 2 deletions(-)
 create mode 100644 src/ring-theory/geometric-sequences-rings.lagda.md

diff --git a/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md b/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md
index 0bda772d1a1..a5727d03a8e 100644
--- a/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md
+++ b/src/commutative-algebra/powers-of-elements-commutative-semirings.lagda.md
@@ -51,6 +51,21 @@ module _
     power-succ-Semiring (semiring-Commutative-Semiring A)
 ```
 
+### `xⁿ⁺¹ = xxⁿ`
+
+```agda
+module _
+  {l : Level} (A : Commutative-Semiring l)
+  where
+
+  power-succ-Commutative-Semiring' :
+    (n : ℕ) (x : type-Commutative-Semiring A) →
+    power-Commutative-Semiring A (succ-ℕ n) x ＝
+    mul-Commutative-Semiring A x (power-Commutative-Semiring A n x)
+  power-succ-Commutative-Semiring' =
+    power-succ-Semiring' (semiring-Commutative-Semiring A)
+```
+
 ### Powers by sums of natural numbers are products of powers
 
 ```agda
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index 4db36a011c9..a35af131e80 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -10,8 +10,10 @@ module elementary-number-theory.multiplicative-group-of-rational-numbers where
 
 ```agda
 open import elementary-number-theory.multiplication-positive-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -25,6 +27,7 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.function-types
+open import foundation.negated-equality
 open import foundation.identity-types
 open import foundation.subtypes
 open import foundation.transport-along-identifications
@@ -86,6 +89,9 @@ _*ℚˣ_ = mul-ℚˣ
 
 inv-ℚˣ : ℚˣ → ℚˣ
 inv-ℚˣ = inv-group-of-units-Ring ring-ℚ
+
+rational-inv-ℚˣ : ℚˣ → ℚ
+rational-inv-ℚˣ q = rational-ℚˣ (inv-ℚˣ q)
 ```
 
 ### Inverse laws in the multiplicative group of rational numbers
@@ -169,3 +175,18 @@ invertible-ℚ⁺ = invertible-nonzero-ℚ ∘ nonzero-ℚ⁺
 invertible-ℚ⁻ : ℚ⁻ → ℚˣ
 invertible-ℚ⁻ = invertible-nonzero-ℚ ∘ nonzero-ℚ⁻
 ```
+
+### If `a ≠ b`, `b - a` is invertible
+
+```agda
+abstract
+  is-invertible-diff-neq-ℚ :
+    (a b : ℚ) → a ≠ b → is-invertible-element-Ring ring-ℚ (b -ℚ a)
+  is-invertible-diff-neq-ℚ a b a≠b =
+    is-invertible-element-ring-is-nonzero-ℚ
+      ( b -ℚ a)
+      ( λ b-a=0 → a≠b (inv (eq-is-unit-right-div-Group group-add-ℚ b-a=0)))
+
+invertible-diff-neq-ℚ : (a b : ℚ) → a ≠ b → ℚˣ
+invertible-diff-neq-ℚ a b a≠b = (b -ℚ a , is-invertible-diff-neq-ℚ a b a≠b)
+```
diff --git a/src/group-theory/arithmetic-sequences-semigroups.lagda.md b/src/group-theory/arithmetic-sequences-semigroups.lagda.md
index f58aa8571c4..0238c5bd311 100644
--- a/src/group-theory/arithmetic-sequences-semigroups.lagda.md
+++ b/src/group-theory/arithmetic-sequences-semigroups.lagda.md
@@ -13,6 +13,7 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
+open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
@@ -188,6 +189,32 @@ module _
       ( refl)
 ```
 
+### The tail of a standard arithmetic sequence is another standard arithmetic sequence
+
+```agda
+module _
+  {l : Level} (G : Semigroup l) (a d : type-Semigroup G)
+  where
+
+  abstract
+    htpy-tail-standard-arithmetric-sequence-Semigroup :
+      tail-sequence
+        ( seq-standard-arithmetic-sequence-Semigroup G a d) ~
+      seq-standard-arithmetic-sequence-Semigroup G (mul-Semigroup G a d) d
+    htpy-tail-standard-arithmetric-sequence-Semigroup 0 = refl
+    htpy-tail-standard-arithmetric-sequence-Semigroup (succ-ℕ n) =
+      ap-mul-Semigroup G
+        ( htpy-tail-standard-arithmetric-sequence-Semigroup n)
+        ( refl)
+
+    eq-tail-standard-arithmetric-sequence-Semigroup :
+      tail-sequence
+        ( seq-standard-arithmetic-sequence-Semigroup G a d) ＝
+      seq-standard-arithmetic-sequence-Semigroup G (mul-Semigroup G a d) d
+    eq-tail-standard-arithmetric-sequence-Semigroup =
+      eq-htpy htpy-tail-standard-arithmetric-sequence-Semigroup
+```
+
 ## External links
 
 - [Arithmetic progressions](https://en.wikipedia.org/wiki/Arithmetic_progression)
diff --git a/src/lists/sequences.lagda.md b/src/lists/sequences.lagda.md
index 2549a666e11..a09c39a608b 100644
--- a/src/lists/sequences.lagda.md
+++ b/src/lists/sequences.lagda.md
@@ -7,6 +7,8 @@ module lists.sequences where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.natural-numbers
+
 open import foundation.universe-levels
 
 open import foundation-core.function-types
@@ -36,6 +38,13 @@ sequence : {l : Level} → UU l → UU l
 sequence A = dependent-sequence (λ _ → A)
 ```
 
+### The tail of a sequence
+
+```agda
+tail-sequence : {l : Level} {A : UU l} → sequence A → sequence A
+tail-sequence u n = u (succ-ℕ n)
+```
+
 ### Functorial action on maps of sequences
 
 ```agda
diff --git a/src/ring-theory.lagda.md b/src/ring-theory.lagda.md
index a16c402dafa..7322fd40478 100644
--- a/src/ring-theory.lagda.md
+++ b/src/ring-theory.lagda.md
@@ -29,6 +29,7 @@ open import ring-theory.full-ideals-rings public
 open import ring-theory.function-rings public
 open import ring-theory.function-semirings public
 open import ring-theory.generating-elements-rings public
+open import ring-theory.geometric-sequences-rings public
 open import ring-theory.geometric-sequences-semirings public
 open import ring-theory.groups-of-units-rings public
 open import ring-theory.homomorphisms-cyclic-rings public
diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
new file mode 100644
index 00000000000..90497c4db21
--- /dev/null
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -0,0 +1,216 @@
+# Geometric sequences in rings
+
+```agda
+module ring-theory.geometric-sequences-rings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.homotopies
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.universe-levels
+
+open import group-theory.arithmetic-sequences-semigroups
+
+open import lists.sequences
+
+open import ring-theory.geometric-sequences-semirings
+open import ring-theory.powers-of-elements-rings
+open import ring-theory.rings
+```
+
+</details>
+
+## Ideas
+
+A
+{{#concept "geometric sequence" Disambiguation="in a ring" Agda=geometric-sequence-Ring}}
+in a [ring](ring-theory.semirings.md) is an
+[arithmetic sequence](ring-theory.geometric-sequence-semirings.md) in the
+underlying [semiring](ring-theory.semirings.md).
+
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
+semiring.
+
+## Definitions
+
+### Geometric sequences in rings
+
+```agda
+module _
+  {l : Level} (R : Ring l)
+  where
+
+  geometric-sequence-Ring : UU l
+  geometric-sequence-Ring =
+    geometric-sequence-Semiring (semiring-Ring R)
+
+module _
+  {l : Level} (R : Ring l)
+  (u : geometric-sequence-Ring R)
+  where
+
+  seq-geometric-sequence-Ring : ℕ → type-Ring R
+  seq-geometric-sequence-Ring =
+    seq-geometric-sequence-Semiring (semiring-Ring R) u
+
+  common-ratio-geometric-sequence-Ring : type-Ring R
+  common-ratio-geometric-sequence-Ring =
+    common-ratio-geometric-sequence-Semiring (semiring-Ring R) u
+
+  is-common-ratio-geometric-sequence-Ring :
+    ( n : ℕ) →
+    ( seq-geometric-sequence-Ring (succ-ℕ n)) ＝
+    ( mul-Ring
+      ( R)
+      ( seq-geometric-sequence-Ring n)
+      ( common-ratio-geometric-sequence-Ring))
+  is-common-ratio-geometric-sequence-Ring =
+    is-common-difference-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( u)
+
+  initial-term-geometric-sequence-Ring : type-Ring R
+  initial-term-geometric-sequence-Ring =
+    initial-term-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( u)
+```
+
+### The standard geometric sequences in a semiring
+
+The standard geometric sequence with initial term `a` and common factor `r` is
+the sequence `u` defined by:
+
+- `u₀ = a`
+- `uₙ₊₁ = uₙ * r`
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a r : type-Ring R)
+  where
+
+  standard-geometric-sequence-Ring : geometric-sequence-Ring R
+  standard-geometric-sequence-Ring =
+    standard-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( a)
+      ( r)
+
+  seq-standard-geometric-sequence-Ring : ℕ → type-Ring R
+  seq-standard-geometric-sequence-Ring =
+    seq-geometric-sequence-Ring R
+      standard-geometric-sequence-Ring
+```
+
+### The geometric sequences `n ↦ a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a r : type-Ring R)
+  where
+
+  mul-pow-nat-Ring : sequence (type-Ring R)
+  mul-pow-nat-Ring = mul-pow-nat-Semiring (semiring-Ring R) a r
+
+  geometric-mul-pow-nat-Ring : geometric-sequence-Ring R
+  geometric-mul-pow-nat-Ring =
+    geometric-mul-pow-nat-Semiring (semiring-Ring R) a r
+
+  initial-term-mul-pow-nat-Ring : type-Ring R
+  initial-term-mul-pow-nat-Ring =
+    initial-term-mul-pow-nat-Semiring (semiring-Ring R) a r
+
+  eq-initial-term-mul-pow-nat-Ring :
+    initial-term-mul-pow-nat-Ring ＝ a
+  eq-initial-term-mul-pow-nat-Ring =
+    eq-initial-term-mul-pow-nat-Semiring (semiring-Ring R) a r
+```
+
+## Properties
+
+### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Ring l)
+  (u : geometric-sequence-Ring R)
+  where
+
+  htpy-seq-standard-geometric-sequence-Ring :
+    ( seq-geometric-sequence-Ring R
+      ( standard-geometric-sequence-Ring R
+        ( initial-term-geometric-sequence-Ring R u)
+        ( common-ratio-geometric-sequence-Ring R u))) ~
+    ( seq-geometric-sequence-Ring R u)
+  htpy-seq-standard-geometric-sequence-Ring =
+    htpy-seq-standard-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( u)
+```
+
+### The nth term of an geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a r : type-Ring R)
+  where
+
+  htpy-mul-pow-standard-geometric-sequence-Ring :
+    mul-pow-nat-Ring R a r ~ seq-standard-geometric-sequence-Ring R a r
+  htpy-mul-pow-standard-geometric-sequence-Ring =
+    htpy-mul-pow-standard-geometric-sequence-Semiring (semiring-Ring R) a r
+```
+
+```agda
+module _
+  {l : Level} (R : Ring l) (u : geometric-sequence-Ring R)
+  where
+
+  htpy-mul-pow-geometric-sequence-Ring :
+    mul-pow-nat-Ring
+      ( R)
+      ( initial-term-geometric-sequence-Ring R u)
+      ( common-ratio-geometric-sequence-Ring R u) ~
+    seq-geometric-sequence-Ring R u
+  htpy-mul-pow-geometric-sequence-Ring =
+    htpy-mul-pow-geometric-sequence-Semiring (semiring-Ring R) u
+```
+
+### Constant sequences are geometric with common ratio one
+
+```agda
+module _
+  {l : Level} (R : Ring l) (a : type-Ring R)
+  where
+
+  one-is-common-ratio-const-sequence-Ring :
+    is-common-difference-sequence-Semigroup
+      ( multiplicative-semigroup-Ring R)
+      ( λ _ → a)
+      ( one-Ring R)
+  one-is-common-ratio-const-sequence-Ring n =
+    inv (right-unit-law-mul-Ring R a)
+
+  geometric-const-sequence-Ring : geometric-sequence-Ring R
+  geometric-const-sequence-Ring =
+    geometric-const-sequence-Semiring (semiring-Ring R) a
+```
+
+## See also
+
+- [Geometric sequences in a semiring](ring-theory.geometric-sequences-semirings.md)
+
+## External links
+
+- [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
+  at Wikipedia
diff --git a/src/ring-theory/geometric-sequences-semirings.lagda.md b/src/ring-theory/geometric-sequences-semirings.lagda.md
index 2b0107a2dbd..3d7dff3e3ac 100644
--- a/src/ring-theory/geometric-sequences-semirings.lagda.md
+++ b/src/ring-theory/geometric-sequences-semirings.lagda.md
@@ -13,6 +13,7 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
+open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
 open import foundation.propositions
@@ -54,6 +55,11 @@ module _
     arithmetic-sequence-Semigroup
       ( multiplicative-semigroup-Semiring R)
 
+  is-geometric-sequence-Semiring : sequence (type-Semiring R) → UU l
+  is-geometric-sequence-Semiring =
+    is-arithmetic-sequence-Semigroup
+      ( multiplicative-semigroup-Semiring R)
+
 module _
   {l : Level} (R : Semiring l)
   (u : geometric-sequence-Semiring R)
@@ -66,8 +72,8 @@ module _
       ( u)
 
   is-geometric-seq-geometric-sequence-Semiring :
-    is-arithmetic-sequence-Semigroup
-      ( multiplicative-semigroup-Semiring R)
+    is-geometric-sequence-Semiring
+      ( R)
       ( seq-geometric-sequence-Semiring)
   is-geometric-seq-geometric-sequence-Semiring =
     is-arithmetic-seq-arithmetic-sequence-Semigroup
@@ -267,6 +273,32 @@ module _
     ( one-Semiring R , one-is-common-ratio-const-sequence-Semiring)
 ```
 
+### The tail of a standard geometric sequence is another standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Semiring l) (a r : type-Semiring R)
+  where
+
+  abstract
+    htpy-tail-standard-geometric-sequence-Semiring :
+      tail-sequence
+        ( seq-standard-geometric-sequence-Semiring R a r) ~
+      seq-standard-geometric-sequence-Semiring R (mul-Semiring R a r) r
+    htpy-tail-standard-geometric-sequence-Semiring =
+      htpy-tail-standard-arithmetric-sequence-Semigroup
+        ( multiplicative-semigroup-Semiring R)
+        ( a)
+        ( r)
+
+    eq-tail-standard-geometric-sequence-Semiring :
+      tail-sequence
+        ( seq-standard-geometric-sequence-Semiring R a r) ＝
+      seq-standard-geometric-sequence-Semiring R (mul-Semiring R a r) r
+    eq-tail-standard-geometric-sequence-Semiring =
+      eq-htpy htpy-tail-standard-geometric-sequence-Semiring
+```
+
 ## External links
 
 - [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)

From e332e1993006ccbeed9f7b35f0e8499bc429155c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 17 Oct 2025 16:26:05 -0700
Subject: [PATCH 22/22] Sums of geometric series for rational numbers

---
 src/commutative-algebra.lagda.md              |   2 +
 ...etric-sequences-commutative-rings.lagda.md | 311 +++++++++++++
 ...c-sequences-commutative-semirings.lagda.md | 439 ++++++++++++++++++
 src/elementary-number-theory.lagda.md         |   1 +
 ...metric-sequences-rational-numbers.lagda.md | 203 ++++++++
 ...icative-group-of-rational-numbers.lagda.md |  23 +-
 .../geometric-sequences-rings.lagda.md        |   2 +-
 .../geometric-sequences-semirings.lagda.md    |   2 +-
 8 files changed, 978 insertions(+), 5 deletions(-)
 create mode 100644 src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
 create mode 100644 src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
 create mode 100644 src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md

diff --git a/src/commutative-algebra.lagda.md b/src/commutative-algebra.lagda.md
index 0ca53e325c4..801d517bd88 100644
--- a/src/commutative-algebra.lagda.md
+++ b/src/commutative-algebra.lagda.md
@@ -22,6 +22,8 @@ open import commutative-algebra.formal-power-series-commutative-semirings public
 open import commutative-algebra.full-ideals-commutative-rings public
 open import commutative-algebra.function-commutative-rings public
 open import commutative-algebra.function-commutative-semirings public
+open import commutative-algebra.geometric-sequences-commutative-rings public
+open import commutative-algebra.geometric-sequences-commutative-semirings public
 open import commutative-algebra.groups-of-units-commutative-rings public
 open import commutative-algebra.homomorphisms-commutative-rings public
 open import commutative-algebra.homomorphisms-commutative-semirings public
diff --git a/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
new file mode 100644
index 00000000000..ee6b868c9e9
--- /dev/null
+++ b/src/commutative-algebra/geometric-sequences-commutative-rings.lagda.md
@@ -0,0 +1,311 @@
+# Geometric sequences in commutative rings
+
+```agda
+module commutative-algebra.geometric-sequences-commutative-rings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.commutative-rings
+open import commutative-algebra.commutative-semirings
+open import commutative-algebra.geometric-sequences-commutative-semirings
+open import commutative-algebra.powers-of-elements-commutative-rings
+
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.function-types
+open import foundation.homotopies
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.universe-levels
+
+open import group-theory.arithmetic-sequences-semigroups
+
+open import lists.finite-sequences
+open import lists.sequences
+```
+
+</details>
+
+## Ideas
+
+A
+{{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Ring}}
+in a [semiring](ring-theory.semirings.md) is an
+[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
+semiring multiplicative [semigroup](group-theory.semigroups.md).
+
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
+semiring.
+
+## Definitions
+
+### Geometric sequences in semirings
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  where
+
+  geometric-sequence-Commutative-Ring : UU l
+  geometric-sequence-Commutative-Ring =
+    geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+
+  is-geometric-sequence-Commutative-Ring :
+    sequence (type-Commutative-Ring R) → UU l
+  is-geometric-sequence-Commutative-Ring =
+    is-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+
+module _
+  {l : Level} (R : Commutative-Ring l)
+  (u : geometric-sequence-Commutative-Ring R)
+  where
+
+  seq-geometric-sequence-Commutative-Ring : ℕ → type-Commutative-Ring R
+  seq-geometric-sequence-Commutative-Ring =
+    seq-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  is-geometric-seq-geometric-sequence-Commutative-Ring :
+    is-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( seq-geometric-sequence-Commutative-Ring)
+  is-geometric-seq-geometric-sequence-Commutative-Ring =
+    is-geometric-seq-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  common-ratio-geometric-sequence-Commutative-Ring :
+    type-Commutative-Ring R
+  common-ratio-geometric-sequence-Commutative-Ring =
+    common-ratio-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  is-common-ratio-geometric-sequence-Commutative-Ring :
+    ( n : ℕ) →
+    ( seq-geometric-sequence-Commutative-Ring (succ-ℕ n)) ＝
+    ( mul-Commutative-Ring
+      ( R)
+      ( seq-geometric-sequence-Commutative-Ring n)
+      ( common-ratio-geometric-sequence-Commutative-Ring))
+  is-common-ratio-geometric-sequence-Commutative-Ring =
+    is-common-ratio-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+
+  initial-term-geometric-sequence-Commutative-Ring :
+    type-Commutative-Ring R
+  initial-term-geometric-sequence-Commutative-Ring =
+    initial-term-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+```
+
+### The standard geometric sequences in a semiring
+
+The standard geometric sequence with initial term `a` and common factor `r` is
+the sequence `u` defined by:
+
+- `u₀ = a`
+- `uₙ₊₁ = uₙ * r`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  standard-geometric-sequence-Commutative-Ring :
+    geometric-sequence-Commutative-Ring R
+  standard-geometric-sequence-Commutative-Ring =
+    standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+
+  seq-standard-geometric-sequence-Commutative-Ring :
+    ℕ → type-Commutative-Ring R
+  seq-standard-geometric-sequence-Commutative-Ring =
+    seq-geometric-sequence-Commutative-Ring R
+      standard-geometric-sequence-Commutative-Ring
+
+  is-geometric-standard-geometric-sequence-Commutative-Ring :
+    is-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( seq-standard-geometric-sequence-Commutative-Ring)
+  is-geometric-standard-geometric-sequence-Commutative-Ring =
+    is-geometric-seq-geometric-sequence-Commutative-Ring R
+      standard-geometric-sequence-Commutative-Ring
+```
+
+### The geometric sequences `n ↦ a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  mul-pow-nat-Commutative-Ring : ℕ → type-Commutative-Ring R
+  mul-pow-nat-Commutative-Ring n =
+    mul-Commutative-Ring R a (power-Commutative-Ring R n r)
+
+  geometric-mul-pow-nat-Commutative-Ring : geometric-sequence-Commutative-Ring R
+  geometric-mul-pow-nat-Commutative-Ring =
+    geometric-mul-pow-nat-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+
+  initial-term-mul-pow-nat-Commutative-Ring : type-Commutative-Ring R
+  initial-term-mul-pow-nat-Commutative-Ring =
+    initial-term-geometric-sequence-Commutative-Ring
+      ( R)
+      ( geometric-mul-pow-nat-Commutative-Ring)
+
+  eq-initial-term-mul-pow-nat-Commutative-Ring :
+    initial-term-mul-pow-nat-Commutative-Ring ＝ a
+  eq-initial-term-mul-pow-nat-Commutative-Ring =
+    right-unit-law-mul-Commutative-Ring R a
+```
+
+## Properties
+
+### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  (u : geometric-sequence-Commutative-Ring R)
+  where
+
+  htpy-seq-standard-geometric-sequence-Commutative-Ring :
+    ( seq-geometric-sequence-Commutative-Ring R
+      ( standard-geometric-sequence-Commutative-Ring R
+        ( initial-term-geometric-sequence-Commutative-Ring R u)
+        ( common-ratio-geometric-sequence-Commutative-Ring R u))) ~
+    ( seq-geometric-sequence-Commutative-Ring R u)
+  htpy-seq-standard-geometric-sequence-Commutative-Ring =
+    htpy-seq-standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+```
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Ring :
+    mul-pow-nat-Commutative-Ring R a r ~
+    seq-standard-geometric-sequence-Commutative-Ring R a r
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Ring =
+    htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+
+  initial-term-standard-geometric-sequence-Commutative-Ring :
+    seq-standard-geometric-sequence-Commutative-Ring R a r 0 ＝ a
+  initial-term-standard-geometric-sequence-Commutative-Ring =
+    ( inv (htpy-mul-pow-standard-geometric-sequence-Commutative-Ring 0)) ∙
+    ( eq-initial-term-mul-pow-nat-Commutative-Ring R a r)
+```
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l)
+  (u : geometric-sequence-Commutative-Ring R)
+  where
+
+  htpy-mul-pow-geometric-sequence-Commutative-Ring :
+    mul-pow-nat-Commutative-Ring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Ring R u)
+      ( common-ratio-geometric-sequence-Commutative-Ring R u) ~
+    seq-geometric-sequence-Commutative-Ring R u
+  htpy-mul-pow-geometric-sequence-Commutative-Ring n =
+    ( htpy-mul-pow-standard-geometric-sequence-Commutative-Ring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Ring R u)
+      ( common-ratio-geometric-sequence-Commutative-Ring R u)
+      ( n)) ∙
+    ( htpy-seq-standard-geometric-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( u)
+      ( n))
+```
+
+### Constant sequences are geometric with common ratio one
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a : type-Commutative-Ring R)
+  where
+
+  geometric-const-sequence-Commutative-Ring :
+    geometric-sequence-Commutative-Ring R
+  geometric-const-sequence-Commutative-Ring =
+    geometric-const-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+```
+
+### Finite geometric sequences
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  standard-geometric-fin-sequence-Commutative-Ring :
+    (n : ℕ) → fin-sequence (type-Commutative-Ring R) n
+  standard-geometric-fin-sequence-Commutative-Ring =
+    fin-sequence-sequence
+      ( seq-standard-geometric-sequence-Commutative-Ring R a r)
+
+  sum-standard-geometric-fin-sequence-Commutative-Ring :
+    (n : ℕ) → type-Commutative-Ring R
+  sum-standard-geometric-fin-sequence-Commutative-Ring =
+    sum-standard-geometric-fin-sequence-Commutative-Semiring
+      ( commutative-semiring-Commutative-Ring R)
+      ( a)
+      ( r)
+```
+
+### The sum of a finite geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Ring l) (a r : type-Commutative-Ring R)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Ring :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-Commutative-Ring R
+        ( a)
+        ( r)
+        ( succ-ℕ n) ＝
+      add-Commutative-Ring R
+        ( a)
+        ( mul-Commutative-Ring R
+          ( r)
+          ( sum-standard-geometric-fin-sequence-Commutative-Ring R a r n))
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Ring =
+      compute-sum-standard-geometric-fin-sequence-succ-Commutative-Semiring
+        ( commutative-semiring-Commutative-Ring R)
+        ( a)
+        ( r)
+```
diff --git a/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
new file mode 100644
index 00000000000..821a30d43c3
--- /dev/null
+++ b/src/commutative-algebra/geometric-sequences-commutative-semirings.lagda.md
@@ -0,0 +1,439 @@
+# Geometric sequences in commutative semirings
+
+```agda
+module commutative-algebra.geometric-sequences-commutative-semirings where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.commutative-semirings
+open import commutative-algebra.powers-of-elements-commutative-semirings
+open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-semirings
+
+open import elementary-number-theory.natural-numbers
+
+open import foundation.action-on-identifications-binary-functions
+open import foundation.action-on-identifications-functions
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-extensionality
+open import foundation.function-types
+open import foundation.homotopies
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.universe-levels
+
+open import group-theory.arithmetic-sequences-semigroups
+
+open import lists.finite-sequences
+open import lists.sequences
+
+open import ring-theory.geometric-sequences-semirings
+open import ring-theory.powers-of-elements-semirings
+open import ring-theory.semirings
+
+open import univalent-combinatorics.standard-finite-types
+```
+
+</details>
+
+## Ideas
+
+A
+{{#concept "geometric sequence" Disambiguation="in a commutative semiring" Agda=geometric-sequence-Commutative-Semiring}}
+in a [semiring](ring-theory.semirings.md) is an
+[geometric sequence](ring-theory.geometric-sequences-semirings.md) in the
+semiring multiplicative [semigroup](group-theory.semigroups.md).
+
+These are sequences of the form `n ↦ a * rⁿ`, for elements `a`, `r` in the
+semiring.
+
+## Definitions
+
+### Geometric sequences in semirings
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  where
+
+  geometric-sequence-Commutative-Semiring : UU l
+  geometric-sequence-Commutative-Semiring =
+    geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+
+  is-geometric-sequence-Commutative-Semiring :
+    sequence (type-Commutative-Semiring R) → UU l
+  is-geometric-sequence-Commutative-Semiring =
+    is-geometric-sequence-Semiring (semiring-Commutative-Semiring R)
+
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  (u : geometric-sequence-Commutative-Semiring R)
+  where
+
+  seq-geometric-sequence-Commutative-Semiring : ℕ → type-Commutative-Semiring R
+  seq-geometric-sequence-Commutative-Semiring =
+    seq-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  is-geometric-seq-geometric-sequence-Commutative-Semiring :
+    is-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( seq-geometric-sequence-Commutative-Semiring)
+  is-geometric-seq-geometric-sequence-Commutative-Semiring =
+    is-geometric-seq-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  common-ratio-geometric-sequence-Commutative-Semiring :
+    type-Commutative-Semiring R
+  common-ratio-geometric-sequence-Commutative-Semiring =
+    common-ratio-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  is-common-ratio-geometric-sequence-Commutative-Semiring :
+    ( n : ℕ) →
+    ( seq-geometric-sequence-Commutative-Semiring (succ-ℕ n)) ＝
+    ( mul-Commutative-Semiring
+      ( R)
+      ( seq-geometric-sequence-Commutative-Semiring n)
+      ( common-ratio-geometric-sequence-Commutative-Semiring))
+  is-common-ratio-geometric-sequence-Commutative-Semiring =
+    is-common-ratio-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+
+  initial-term-geometric-sequence-Commutative-Semiring :
+    type-Commutative-Semiring R
+  initial-term-geometric-sequence-Commutative-Semiring =
+    initial-term-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+```
+
+### The standard geometric sequences in a semiring
+
+The standard geometric sequence with initial term `a` and common factor `r` is
+the sequence `u` defined by:
+
+- `u₀ = a`
+- `uₙ₊₁ = uₙ * r`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  standard-geometric-sequence-Commutative-Semiring :
+    geometric-sequence-Commutative-Semiring R
+  standard-geometric-sequence-Commutative-Semiring =
+    standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( a)
+      ( r)
+
+  seq-standard-geometric-sequence-Commutative-Semiring :
+    ℕ → type-Commutative-Semiring R
+  seq-standard-geometric-sequence-Commutative-Semiring =
+    seq-geometric-sequence-Commutative-Semiring R
+      standard-geometric-sequence-Commutative-Semiring
+
+  is-geometric-standard-geometric-sequence-Commutative-Semiring :
+    is-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( seq-standard-geometric-sequence-Commutative-Semiring)
+  is-geometric-standard-geometric-sequence-Commutative-Semiring =
+    is-geometric-seq-geometric-sequence-Commutative-Semiring R
+      standard-geometric-sequence-Commutative-Semiring
+```
+
+### The geometric sequences `n ↦ a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  mul-pow-nat-Commutative-Semiring : ℕ → type-Commutative-Semiring R
+  mul-pow-nat-Commutative-Semiring n =
+    mul-Commutative-Semiring R a (power-Commutative-Semiring R n r)
+
+  geometric-mul-pow-nat-Commutative-Semiring :
+    geometric-sequence-Commutative-Semiring R
+  geometric-mul-pow-nat-Commutative-Semiring =
+    geometric-mul-pow-nat-Semiring (semiring-Commutative-Semiring R) a r
+
+  initial-term-mul-pow-nat-Commutative-Semiring : type-Commutative-Semiring R
+  initial-term-mul-pow-nat-Commutative-Semiring =
+    initial-term-geometric-sequence-Commutative-Semiring
+      ( R)
+      ( geometric-mul-pow-nat-Commutative-Semiring)
+
+  eq-initial-term-mul-pow-nat-Commutative-Semiring :
+    initial-term-mul-pow-nat-Commutative-Semiring ＝ a
+  eq-initial-term-mul-pow-nat-Commutative-Semiring =
+    right-unit-law-mul-Commutative-Semiring R a
+```
+
+## Properties
+
+### Any geometric sequence in a semiring is homotopic to a standard geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  (u : geometric-sequence-Commutative-Semiring R)
+  where
+
+  htpy-seq-standard-geometric-sequence-Commutative-Semiring :
+    ( seq-geometric-sequence-Commutative-Semiring R
+      ( standard-geometric-sequence-Commutative-Semiring R
+        ( initial-term-geometric-sequence-Commutative-Semiring R u)
+        ( common-ratio-geometric-sequence-Commutative-Semiring R u))) ~
+    ( seq-geometric-sequence-Commutative-Semiring R u)
+  htpy-seq-standard-geometric-sequence-Commutative-Semiring =
+    htpy-seq-standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+```
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring :
+    mul-pow-nat-Commutative-Semiring R a r ~
+    seq-standard-geometric-sequence-Commutative-Semiring R a r
+  htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring =
+    htpy-mul-pow-standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( a)
+      ( r)
+
+  initial-term-standard-geometric-sequence-Commutative-Semiring :
+    seq-standard-geometric-sequence-Commutative-Semiring R a r 0 ＝ a
+  initial-term-standard-geometric-sequence-Commutative-Semiring =
+    ( inv (htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring 0)) ∙
+    ( eq-initial-term-mul-pow-nat-Commutative-Semiring R a r)
+```
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l)
+  (u : geometric-sequence-Commutative-Semiring R)
+  where
+
+  htpy-mul-pow-geometric-sequence-Commutative-Semiring :
+    mul-pow-nat-Commutative-Semiring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Semiring R u)
+      ( common-ratio-geometric-sequence-Commutative-Semiring R u) ~
+    seq-geometric-sequence-Commutative-Semiring R u
+  htpy-mul-pow-geometric-sequence-Commutative-Semiring n =
+    ( htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+      ( R)
+      ( initial-term-geometric-sequence-Commutative-Semiring R u)
+      ( common-ratio-geometric-sequence-Commutative-Semiring R u)
+      ( n)) ∙
+    ( htpy-seq-standard-geometric-sequence-Semiring
+      ( semiring-Commutative-Semiring R)
+      ( u)
+      ( n))
+```
+
+### Constant sequences are geometric with common ratio one
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a : type-Commutative-Semiring R)
+  where
+
+  geometric-const-sequence-Commutative-Semiring :
+    geometric-sequence-Commutative-Semiring R
+  geometric-const-sequence-Commutative-Semiring =
+    geometric-const-sequence-Semiring (semiring-Commutative-Semiring R) a
+```
+
+### Finite geometric sequences
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  standard-geometric-fin-sequence-Commutative-Semiring :
+    (n : ℕ) → fin-sequence (type-Commutative-Semiring R) n
+  standard-geometric-fin-sequence-Commutative-Semiring =
+    fin-sequence-sequence
+      ( seq-standard-geometric-sequence-Commutative-Semiring R a r)
+
+  sum-standard-geometric-fin-sequence-Commutative-Semiring :
+    (n : ℕ) → type-Commutative-Semiring R
+  sum-standard-geometric-fin-sequence-Commutative-Semiring n =
+    sum-fin-sequence-type-Commutative-Semiring R n
+      ( standard-geometric-fin-sequence-Commutative-Semiring n)
+```
+
+### The sum of a finite geometric sequence
+
+```agda
+module _
+  {l : Level} (R : Commutative-Semiring l) (a r : type-Commutative-Semiring R)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Semiring :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-Commutative-Semiring R
+        ( a)
+        ( r)
+        ( succ-ℕ n) ＝
+      add-Commutative-Semiring R
+        ( a)
+        ( mul-Commutative-Semiring R
+          ( r)
+          ( sum-standard-geometric-fin-sequence-Commutative-Semiring R a r n))
+    compute-sum-standard-geometric-fin-sequence-succ-Commutative-Semiring n =
+      equational-reasoning
+        sum-standard-geometric-fin-sequence-Commutative-Semiring R
+          ( a)
+          ( r)
+          ( succ-ℕ n)
+        ＝
+          add-Commutative-Semiring R
+            ( term-Fin a r (succ-ℕ n) (zero-Fin n))
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i → term-Fin a r (succ-ℕ n) (inr-Fin n i)))
+          by
+            snoc-sum-fin-sequence-type-Commutative-Semiring R n
+              ( standard-geometric-fin-sequence-Commutative-Semiring R
+                ( a)
+                ( r)
+                ( succ-ℕ n))
+              ( refl)
+        ＝
+          add-Commutative-Semiring R
+            ( term a r 0)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i → term a r (succ-ℕ (nat-Fin n i))))
+          by
+            ap-add-Commutative-Semiring R
+              ( ap
+                ( seq-standard-geometric-sequence-Commutative-Semiring R a r)
+                ( is-zero-nat-zero-Fin {n}))
+              ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                ( λ i → ap (term a r) (nat-inr-Fin n i)))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i →
+                mul-Commutative-Semiring R
+                  ( a)
+                  ( power-Commutative-Semiring R (succ-ℕ (nat-Fin n i)) r)))
+            by
+              ap-add-Commutative-Semiring R
+                ( initial-term-standard-geometric-sequence-Commutative-Semiring
+                  ( R)
+                  ( a)
+                  ( r))
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( λ i →
+                    inv
+                      ( htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+                        ( R)
+                        ( a)
+                        ( r)
+                        ( succ-ℕ (nat-Fin n i)))))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i →
+                mul-Commutative-Semiring R
+                  ( a)
+                  ( mul-Commutative-Semiring R
+                    ( r)
+                    ( power-Commutative-Semiring R (nat-Fin n i) r))))
+            by
+              ap-add-Commutative-Semiring R
+                ( refl)
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( λ i →
+                    ap-mul-Commutative-Semiring R
+                      ( refl)
+                      ( power-succ-Commutative-Semiring' R (nat-Fin n i) r)))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( sum-fin-sequence-type-Commutative-Semiring R n
+              ( λ i →
+                mul-Commutative-Semiring R
+                  ( r)
+                  ( mul-Commutative-Semiring R
+                    ( a)
+                    ( power-Commutative-Semiring R (nat-Fin n i) r))))
+            by
+              ap-add-Commutative-Semiring R
+                ( refl)
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( λ i → left-swap-mul-Commutative-Semiring R _ _ _))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( mul-Commutative-Semiring R
+              ( r)
+              ( sum-fin-sequence-type-Commutative-Semiring R n
+                ( λ i →
+                  mul-Commutative-Semiring R
+                    ( a)
+                    ( power-Commutative-Semiring R (nat-Fin n i) r))))
+          by
+            ap-add-Commutative-Semiring R
+              ( refl)
+              ( inv
+                ( left-distributive-mul-sum-fin-sequence-type-Commutative-Semiring
+                  ( R)
+                  ( n)
+                  ( r)
+                  ( _)))
+        ＝
+          add-Commutative-Semiring R
+            ( a)
+            ( mul-Commutative-Semiring R
+              ( r)
+              ( sum-standard-geometric-fin-sequence-Commutative-Semiring R
+                ( a)
+                ( r)
+                ( n)))
+          by
+            ap-add-Commutative-Semiring R
+              ( refl)
+              ( ap-mul-Commutative-Semiring R
+                ( refl)
+                ( htpy-sum-fin-sequence-type-Commutative-Semiring R n
+                  ( htpy-mul-pow-standard-geometric-sequence-Commutative-Semiring
+                      ( R)
+                      ( a)
+                      ( r) ∘
+                    nat-Fin n)))
+      where
+        term :
+          type-Commutative-Semiring R → type-Commutative-Semiring R →
+          sequence (type-Commutative-Semiring R)
+        term a' r' =
+          seq-standard-geometric-sequence-Commutative-Semiring R a' r'
+        term-Fin :
+          type-Commutative-Semiring R → type-Commutative-Semiring R →
+          (n : ℕ) → fin-sequence (type-Commutative-Semiring R) n
+        term-Fin a' r' =
+          standard-geometric-fin-sequence-Commutative-Semiring R a' r'
+```
diff --git a/src/elementary-number-theory.lagda.md b/src/elementary-number-theory.lagda.md
index 6655218d7ca..51a5fb4b31e 100644
--- a/src/elementary-number-theory.lagda.md
+++ b/src/elementary-number-theory.lagda.md
@@ -85,6 +85,7 @@ open import elementary-number-theory.finitary-natural-numbers public
 open import elementary-number-theory.finitely-cyclic-maps public
 open import elementary-number-theory.fundamental-theorem-of-arithmetic public
 open import elementary-number-theory.geometric-sequences-positive-rational-numbers public
+open import elementary-number-theory.geometric-sequences-rational-numbers public
 open import elementary-number-theory.goldbach-conjecture public
 open import elementary-number-theory.greatest-common-divisor-integers public
 open import elementary-number-theory.greatest-common-divisor-natural-numbers public
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
new file mode 100644
index 00000000000..747150c04bb
--- /dev/null
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -0,0 +1,203 @@
+# Geometric sequences of rational numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module elementary-number-theory.geometric-sequences-rational-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import commutative-algebra.geometric-sequences-commutative-rings
+open import commutative-algebra.sums-of-finite-sequences-of-elements-commutative-rings
+
+open import elementary-number-theory.addition-rational-numbers
+open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.multiplication-rational-numbers
+open import elementary-number-theory.multiplicative-group-of-rational-numbers
+open import elementary-number-theory.natural-numbers
+open import elementary-number-theory.powers-rational-numbers
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.ring-of-rational-numbers
+
+open import foundation.identity-types
+open import foundation.negated-equality
+open import foundation.universe-levels
+
+open import group-theory.groups
+
+open import lists.sequences
+
+open import order-theory.strictly-increasing-sequences-strictly-preordered-sets
+```
+
+</details>
+
+## Idea
+
+A
+{{#concept "geometric sequence" Disambiguation="of rational numbers" Agda=geometric-sequence-ℚ}}
+of [rational numbers](elementary-number-theory.positive-rational-numbers.md) is
+a
+[geometric sequence](commutative-algebra.geometric-sequences-commutative-rings.md)
+in the
+[commutative ring of rational numbers](elementary-number-theory.ring-of-rational-numbers.md).
+
+## Definitions
+
+```agda
+geometric-sequence-ℚ : UU lzero
+geometric-sequence-ℚ = geometric-sequence-Commutative-Ring commutative-ring-ℚ
+
+seq-geometric-sequence-ℚ : geometric-sequence-ℚ → sequence ℚ
+seq-geometric-sequence-ℚ =
+  seq-geometric-sequence-Commutative-Ring commutative-ring-ℚ
+
+standard-geometric-sequence-ℚ : ℚ → ℚ → geometric-sequence-ℚ
+standard-geometric-sequence-ℚ =
+  standard-geometric-sequence-Commutative-Ring commutative-ring-ℚ
+
+seq-standard-geometric-sequence-ℚ : ℚ → ℚ → sequence ℚ
+seq-standard-geometric-sequence-ℚ =
+  seq-standard-geometric-sequence-Commutative-Ring commutative-ring-ℚ
+```
+
+## Properties
+
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+
+```agda
+module _
+  (a r : ℚ)
+  where
+
+  compute-standard-geometric-sequence-ℚ :
+    (n : ℕ) → seq-standard-geometric-sequence-ℚ a r n ＝ a *ℚ power-ℚ n r
+  compute-standard-geometric-sequence-ℚ n =
+    inv
+      ( htpy-mul-pow-standard-geometric-sequence-Commutative-Ring
+        ( commutative-ring-ℚ)
+        ( a)
+        ( r)
+        ( n))
+```
+
+### If `r ≠ 1`, the sum of the first `n` elements of the standard geometric sequence `n ↦ arⁿ` is `a(1-rⁿ)/(1-r)`
+
+```agda
+sum-standard-geometric-fin-sequence-ℚ : ℚ → ℚ → ℕ → ℚ
+sum-standard-geometric-fin-sequence-ℚ =
+  sum-standard-geometric-fin-sequence-Commutative-Ring commutative-ring-ℚ
+
+module _
+  (a r : ℚ) (r≠1 : r ≠ one-ℚ)
+  where
+
+  abstract
+    compute-sum-standard-geometric-fin-sequence-ℚ :
+      (n : ℕ) →
+      sum-standard-geometric-fin-sequence-ℚ a r n ＝
+      ( a *ℚ
+        ( (one-ℚ -ℚ power-ℚ n r) *ℚ
+          rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)))
+    compute-sum-standard-geometric-fin-sequence-ℚ 0 =
+      inv
+        ( equational-reasoning
+          a *ℚ ((one-ℚ -ℚ one-ℚ) *ℚ _)
+          ＝ a *ℚ (zero-ℚ *ℚ _)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ (right-inverse-law-add-ℚ one-ℚ) refl)
+          ＝ a *ℚ zero-ℚ
+            by ap-mul-ℚ refl (left-zero-law-mul-ℚ _)
+          ＝ zero-ℚ
+            by right-zero-law-mul-ℚ a)
+    compute-sum-standard-geometric-fin-sequence-ℚ (succ-ℕ n) =
+      let
+        1/1-r = rational-inv-ℚˣ (invertible-diff-neq-ℚ r one-ℚ r≠1)
+      in
+        equational-reasoning
+          sum-standard-geometric-fin-sequence-ℚ a r (succ-ℕ n)
+          ＝
+            sum-standard-geometric-fin-sequence-ℚ a r n +ℚ
+            seq-standard-geometric-sequence-ℚ a r n
+            by
+              cons-sum-fin-sequence-type-Commutative-Ring
+                ( commutative-ring-ℚ)
+                ( n)
+                ( _)
+                ( refl)
+          ＝
+            ( a *ℚ
+              ( (one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r)) +ℚ
+            ( a *ℚ power-ℚ n r)
+            by
+              ap-add-ℚ
+                ( compute-sum-standard-geometric-fin-sequence-ℚ n)
+                ( compute-standard-geometric-sequence-ℚ a r n)
+          ＝
+            a *ℚ
+            (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r) +ℚ power-ℚ n r)
+            by inv (left-distributive-mul-add-ℚ a _ _)
+          ＝
+            a *ℚ
+            ( (((one-ℚ -ℚ power-ℚ n r) *ℚ 1/1-r) +ℚ
+              (power-ℚ n r *ℚ (one-ℚ -ℚ r)) *ℚ 1/1-r))
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-add-ℚ
+                  ( refl)
+                  ( inv
+                    ( cancel-right-mul-div-ℚˣ _
+                      ( invertible-diff-neq-ℚ r one-ℚ r≠1))))
+          ＝
+            a *ℚ
+            ( ( (one-ℚ -ℚ power-ℚ n r) +ℚ (power-ℚ n r *ℚ (one-ℚ -ℚ r))) *ℚ
+              1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( inv (right-distributive-mul-add-ℚ _ _ 1/1-r))
+          ＝
+            a *ℚ
+            ( ( one-ℚ -ℚ power-ℚ n r +ℚ
+                ((power-ℚ n r *ℚ one-ℚ) -ℚ (power-ℚ n r *ℚ r))) *ℚ
+              1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ
+                  ( ap-add-ℚ refl (left-distributive-mul-diff-ℚ _ _ _))
+                  ( refl))
+          ＝
+            a *ℚ
+            ( ( one-ℚ -ℚ power-ℚ n r +ℚ
+                ((power-ℚ n r -ℚ power-ℚ (succ-ℕ n) r))) *ℚ
+              1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ
+                  ( ap-add-ℚ
+                    ( refl)
+                    ( ap-diff-ℚ
+                      ( right-unit-law-mul-ℚ _)
+                      ( inv (power-succ-ℚ n r))))
+                  ( refl))
+          ＝ a *ℚ ((one-ℚ -ℚ power-ℚ (succ-ℕ n) r) *ℚ 1/1-r)
+            by
+              ap-mul-ℚ
+                ( refl)
+                ( ap-mul-ℚ
+                  ( mul-right-div-Group group-add-ℚ _ _ _)
+                  ( refl))
+```
+
+## External links
+
+- [Geometric progressions](https://en.wikipedia.org/wiki/Geometric_progression)
+  at Wikipedia
diff --git a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
index a35af131e80..3df03c15ddf 100644
--- a/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplicative-group-of-rational-numbers.lagda.md
@@ -9,11 +9,11 @@ module elementary-number-theory.multiplicative-group-of-rational-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
+open import elementary-number-theory.difference-rational-numbers
+open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.multiplicative-monoid-of-rational-numbers
-open import elementary-number-theory.difference-rational-numbers
 open import elementary-number-theory.negative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
@@ -27,8 +27,8 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.function-types
-open import foundation.negated-equality
 open import foundation.identity-types
+open import foundation.negated-equality
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -190,3 +190,20 @@ abstract
 invertible-diff-neq-ℚ : (a b : ℚ) → a ≠ b → ℚˣ
 invertible-diff-neq-ℚ a b a≠b = (b -ℚ a , is-invertible-diff-neq-ℚ a b a≠b)
 ```
+
+### Cancellation laws
+
+```agda
+abstract
+  cancel-right-mul-div-ℚˣ :
+    (p : ℚ) (q : ℚˣ) → (p *ℚ rational-ℚˣ q) *ℚ rational-inv-ℚˣ q ＝ p
+  cancel-right-mul-div-ℚˣ p q =
+    equational-reasoning
+      p *ℚ rational-ℚˣ q *ℚ rational-inv-ℚˣ q
+      ＝ p *ℚ (rational-ℚˣ q *ℚ rational-inv-ℚˣ q)
+        by associative-mul-ℚ _ _ _
+      ＝ p *ℚ rational-ℚˣ one-ℚˣ
+        by ap-mul-ℚ refl (ap rational-ℚˣ (right-inverse-law-mul-ℚˣ q))
+      ＝ p
+        by right-unit-law-mul-ℚ p
+```
diff --git a/src/ring-theory/geometric-sequences-rings.lagda.md b/src/ring-theory/geometric-sequences-rings.lagda.md
index 90497c4db21..9458cf5a5fa 100644
--- a/src/ring-theory/geometric-sequences-rings.lagda.md
+++ b/src/ring-theory/geometric-sequences-rings.lagda.md
@@ -158,7 +158,7 @@ module _
       ( u)
 ```
 
-### The nth term of an geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
 
 ```agda
 module _
diff --git a/src/ring-theory/geometric-sequences-semirings.lagda.md b/src/ring-theory/geometric-sequences-semirings.lagda.md
index 3d7dff3e3ac..1e560ceabd0 100644
--- a/src/ring-theory/geometric-sequences-semirings.lagda.md
+++ b/src/ring-theory/geometric-sequences-semirings.lagda.md
@@ -211,7 +211,7 @@ module _
       ( u)
 ```
 
-### The nth term of an geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
+### The nth term of a geometric sequence with initial term `a` and common ratio `r` is `a * rⁿ`
 
 ```agda
 module _