From 380382bfd9c357f6017d92126b8533f7b2065606 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 14:41:01 -0800
Subject: [PATCH 01/21] Progress

---
 .../absolute-value-rational-numbers.lagda.md  |   2 +-
 ...ddition-positive-rational-numbers.lagda.md |  13 +-
 .../distance-rational-numbers.lagda.md        |   5 +-
 .../inequality-rational-numbers.lagda.md      |  15 +-
 ...closed-intervals-rational-numbers.lagda.md |  16 +-
 ...ication-positive-rational-numbers.lagda.md |   2 +-
 .../negative-rational-numbers.lagda.md        |   4 +-
 .../positive-rational-numbers.lagda.md        |   2 +-
 ...e-roots-positive-rational-numbers.lagda.md |  22 +-
 ...trict-inequality-rational-numbers.lagda.md |  33 +-
 .../metrics-of-metric-spaces.lagda.md         |   8 +-
 src/metric-spaces/metrics.lagda.md            |  12 +-
 src/real-numbers.lagda.md                     |   7 +
 ...lue-closed-intervals-real-numbers.lagda.md |   2 +-
 .../absolute-value-real-numbers.lagda.md      |  26 +-
 ...ition-lower-dedekind-real-numbers.lagda.md |   2 +-
 ...addition-nonnegative-real-numbers.lagda.md | 136 ++++++++
 ...ition-upper-dedekind-real-numbers.lagda.md |   2 +-
 ...thmetically-located-dedekind-cuts.lagda.md |  20 +-
 .../binary-maximum-real-numbers.lagda.md      |  23 +-
 .../binary-minimum-real-numbers.lagda.md      |  13 +-
 ...ompleteness-dedekind-real-numbers.lagda.md |  13 +-
 .../closed-intervals-real-numbers.lagda.md    |   6 +-
 .../dedekind-real-numbers.lagda.md            |   8 +-
 .../distance-real-numbers.lagda.md            |   6 +-
 ...nequalities-addition-real-numbers.lagda.md | 234 +++++++++++++
 ...equality-nonnegative-real-numbers.lagda.md | 159 +++++++++
 .../inequality-real-numbers.lagda.md          | 267 ++-------------
 .../infima-families-real-numbers.lagda.md     |  17 +-
 ...ally-bounded-subsets-real-numbers.lagda.md |  11 +-
 .../isometry-addition-real-numbers.lagda.md   |   1 +
 .../isometry-negation-real-numbers.lagda.md   |  29 ++
 ...nuity-multiplication-real-numbers.lagda.md |  27 +-
 ...imum-finite-families-real-numbers.lagda.md |   1 +
 ...space-of-nonnegative-real-numbers.lagda.md |  34 ++
 .../metric-space-of-real-numbers.lagda.md     |  21 +-
 ...tiplication-negative-real-numbers.lagda.md |   4 +-
 ...lication-nonnegative-real-numbers.lagda.md |   5 +-
 .../multiplication-real-numbers.lagda.md      |  12 +-
 ...ve-inverses-positive-real-numbers.lagda.md |   3 +-
 .../negation-real-numbers.lagda.md            |   2 +-
 .../nonnegative-real-numbers.lagda.md         | 319 +----------------
 ...ositive-and-negative-real-numbers.lagda.md |   6 +-
 .../positive-real-numbers.lagda.md            |  39 +--
 ...sing-universe-levels-real-numbers.lagda.md |   2 +-
 .../rational-real-numbers.lagda.md            |  14 +-
 ...equality-nonnegative-real-numbers.lagda.md |  71 ++++
 ...aturation-inequality-real-numbers.lagda.md |  29 +-
 ...milarity-nonnegative-real-numbers.lagda.md |  70 ++++
 ...re-roots-nonnegative-real-numbers.lagda.md |   9 +-
 .../squares-real-numbers.lagda.md             |  10 +-
 ...nequalities-addition-real-numbers.lagda.md | 289 ++++++++++++++++
 ...equality-nonnegative-real-numbers.lagda.md |  42 ++-
 .../strict-inequality-real-numbers.lagda.md   | 323 ++----------------
 .../suprema-families-real-numbers.lagda.md    |   5 +-
 ...action-cuts-dedekind-real-numbers.lagda.md |  13 +-
 56 files changed, 1340 insertions(+), 1126 deletions(-)
 create mode 100644 src/real-numbers/addition-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/inequalities-addition-real-numbers.lagda.md
 create mode 100644 src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
 create mode 100644 src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md

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 4b0ad01d88..031a23160f 100644
--- a/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/absolute-value-rational-numbers.lagda.md
@@ -166,7 +166,7 @@ abstract
         ( q)
         ( rational-abs-ℚ q)
         ( zero-ℚ)
-        ( leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ abs=0))
+        ( leq-eq-ℚ (ap rational-ℚ⁰⁺ abs=0))
         ( leq-abs-ℚ q))
       ( binary-tr
         ( leq-ℚ)
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 85668704af..7e1cf13e5b 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -169,7 +169,7 @@ module _
   where
 
   le-diff-ℚ⁺ : ℚ⁺
-  le-diff-ℚ⁺ = positive-diff-le-ℚ (rational-ℚ⁺ x) (rational-ℚ⁺ y) H
+  le-diff-ℚ⁺ = positive-diff-le-ℚ H
 
   left-diff-law-add-ℚ⁺ : le-diff-ℚ⁺ +ℚ⁺ x ＝ y
   left-diff-law-add-ℚ⁺ =
@@ -342,15 +342,8 @@ module _
             ( le-right-mediant-ℚ y x I)
             ( tr
               ( leq-ℚ x)
-              ( right-law-positive-diff-le-ℚ
-                ( y)
-                ( mediant-ℚ y x)
-                ( le-left-mediant-ℚ y x I))
-              ( H
-                ( positive-diff-le-ℚ
-                  ( y)
-                  ( mediant-ℚ y x)
-                  ( le-left-mediant-ℚ y x I))))))
+              ( right-law-positive-diff-le-ℚ (le-left-mediant-ℚ y x I))
+              ( H (positive-diff-le-ℚ (le-left-mediant-ℚ y x I))))))
       ( id)
       ( decide-le-leq-ℚ y x)
 
diff --git a/src/elementary-number-theory/distance-rational-numbers.lagda.md b/src/elementary-number-theory/distance-rational-numbers.lagda.md
index 6a9bcc3d5d..3926d248dc 100644
--- a/src/elementary-number-theory/distance-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-rational-numbers.lagda.md
@@ -191,10 +191,7 @@ abstract
       ( rational-dist-ℚ p q)
       ( (rational-abs-ℚ p) +ℚ (rational-abs-ℚ (neg-ℚ q)))
       ( rational-abs-ℚ p +ℚ rational-abs-ℚ q)
-      ( leq-eq-ℚ
-        ( (rational-abs-ℚ p) +ℚ (rational-abs-ℚ (neg-ℚ q)))
-        ( rational-abs-ℚ p +ℚ rational-abs-ℚ q)
-        ( ap (add-ℚ (rational-abs-ℚ p) ∘ rational-ℚ⁰⁺) (abs-neg-ℚ q)))
+      ( leq-eq-ℚ (ap (add-ℚ (rational-abs-ℚ p) ∘ rational-ℚ⁰⁺) (abs-neg-ℚ q)))
       ( triangle-inequality-abs-ℚ p (neg-ℚ q))
 ```
 
diff --git a/src/elementary-number-theory/inequality-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
index 14d1a9398e..29d6fb57a9 100644
--- a/src/elementary-number-theory/inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
@@ -126,8 +126,8 @@ opaque
     refl-leq-ℤ (numerator-ℚ x *ℤ denominator-ℚ x)
 
 abstract
-  leq-eq-ℚ : (x y : ℚ) → x ＝ y → leq-ℚ x y
-  leq-eq-ℚ x y x=y = tr (leq-ℚ x) x=y (refl-leq-ℚ x)
+  leq-eq-ℚ : {x y : ℚ} → x ＝ y → leq-ℚ x y
+  leq-eq-ℚ {x} refl = refl-leq-ℚ x
 ```
 
 ### Inequality on the rational numbers is antisymmetric
@@ -294,8 +294,8 @@ opaque
   unfolding leq-ℚ-Prop
 
   preserves-leq-rational-ℤ :
-    (x y : ℤ) → leq-ℤ x y → leq-ℚ (rational-ℤ x) (rational-ℤ y)
-  preserves-leq-rational-ℤ x y =
+    {x y : ℤ} → leq-ℤ x y → leq-ℚ (rational-ℤ x) (rational-ℤ y)
+  preserves-leq-rational-ℤ {x} {y} =
     binary-tr leq-ℤ
       ( inv (right-unit-law-mul-ℤ x))
       ( inv (right-unit-law-mul-ℤ y))
@@ -309,7 +309,7 @@ opaque
 
   iff-leq-rational-ℤ :
     (x y : ℤ) → leq-ℤ x y ↔ leq-ℚ (rational-ℤ x) (rational-ℤ y)
-  pr1 (iff-leq-rational-ℤ x y) = preserves-leq-rational-ℤ x y
+  pr1 (iff-leq-rational-ℤ x y) = preserves-leq-rational-ℤ
   pr2 (iff-leq-rational-ℤ x y) = reflects-leq-rational-ℤ x y
 ```
 
@@ -324,8 +324,9 @@ abstract
     iff-leq-int-ℕ x y
 
   preserves-leq-rational-ℕ :
-    (x y : ℕ) → leq-ℕ x y → leq-ℚ (rational-ℕ x) (rational-ℕ y)
-  preserves-leq-rational-ℕ x y = forward-implication (iff-leq-rational-ℕ x y)
+    {x y : ℕ} → leq-ℕ x y → leq-ℚ (rational-ℕ x) (rational-ℕ y)
+  preserves-leq-rational-ℕ {x} {y} =
+    forward-implication (iff-leq-rational-ℕ x y)
 
   reflects-leq-rational-ℕ :
     (x y : ℕ) → leq-ℚ (rational-ℕ x) (rational-ℕ y) → leq-ℕ x y
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 8ec9ae546c..698a8adb82 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
@@ -924,7 +924,7 @@ abstract
           rational-dist-ℚ (a *ℚ q) (b *ℚ q)
           ≤ rational-dist-ℚ a b *ℚ rational-abs-ℚ q
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ( ap
                     ( rational-abs-ℚ)
                     ( inv (right-distributive-mul-diff-ℚ _ _ _))) ∙
@@ -939,7 +939,7 @@ abstract
           ≤ width-closed-interval-ℚ [a,b] *ℚ
             rational-max-abs-closed-interval-ℚ [c,d]
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap-mul-ℚ
                   ( eq-width-dist-lower-upper-bounds-closed-interval-ℚ [a,b])
                   ( refl))
@@ -951,7 +951,7 @@ abstract
           rational-dist-ℚ (p *ℚ c) (p *ℚ d)
           ≤ rational-dist-ℚ c d *ℚ rational-abs-ℚ p
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ( ap
                     ( rational-abs-ℚ)
                     ( inv (left-distributive-mul-diff-ℚ _ _ _))) ∙
@@ -966,7 +966,7 @@ abstract
                 ( leq-max-abs-is-in-closed-interval-ℚ [a,b] p p∈[a,b])
           ≤ _
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap-mul-ℚ
                   ( eq-width-dist-lower-upper-bounds-closed-interval-ℚ [c,d])
                   ( refl))
@@ -983,7 +983,7 @@ abstract
           rational-dist-ℚ (a *ℚ c) (b *ℚ d)
           ≤ rational-abs-ℚ ((a *ℚ c -ℚ b *ℚ c) +ℚ (b *ℚ c -ℚ b *ℚ d))
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap
                   ( rational-abs-ℚ)
                   ( inv ( mul-right-div-Group group-add-ℚ _ _ _)))
@@ -996,7 +996,7 @@ abstract
           rational-dist-ℚ (a *ℚ d) (b *ℚ c)
           ≤ rational-abs-ℚ ((a *ℚ d -ℚ b *ℚ d) +ℚ (b *ℚ d -ℚ b *ℚ c))
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap
                   ( rational-abs-ℚ)
                   ( inv ( mul-right-div-Group group-add-ℚ _ _ _)))
@@ -1006,7 +1006,7 @@ abstract
           ≤ rational-dist-ℚ (a *ℚ d) (b *ℚ d) +ℚ
             rational-dist-ℚ (b *ℚ c) (b *ℚ d)
             by
-              leq-eq-ℚ _ _
+              leq-eq-ℚ
                 ( ap-add-ℚ refl (ap rational-ℚ⁰⁺ (commutative-dist-ℚ _ _)))
           ≤ <b-a><max|c||d|> +ℚ <d-c><max|a||b|>
             by preserves-leq-add-ℚ |ad-bd|≤<b-a>max|c||d| |bc-bd|≤<d-c>max|a||b|
@@ -1042,7 +1042,7 @@ abstract
         chain-of-inequalities
           rational-dist-ℚ q q
           ≤ zero-ℚ
-            by leq-eq-ℚ _ _ (rational-dist-self-ℚ q)
+            by leq-eq-ℚ (rational-dist-self-ℚ q)
           ≤ <b-a><max|c||d|> +ℚ <d-c><max|a||b|>
             by
               leq-zero-rational-ℚ⁰⁺ (<b-a><max|c||d|>⁰⁺ +ℚ⁰⁺ <d-c><max|a||b|>⁰⁺)
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 e0fb4cbbe4..492703832c 100644
--- a/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/multiplication-positive-rational-numbers.lagda.md
@@ -357,6 +357,6 @@ abstract
   leq-left-mul-leq-one-ℚ⁺ p⁺@(p , _) p≤1 q⁺@(q , _) =
     trichotomy-le-ℚ p one-ℚ
       ( λ p<1 → leq-le-ℚ (le-left-mul-less-than-one-ℚ⁺ p⁺ p<1 q⁺))
-      ( λ p=1 → leq-eq-ℚ _ _ (ap-mul-ℚ p=1 refl ∙ left-unit-law-mul-ℚ q))
+      ( λ p=1 → leq-eq-ℚ (ap-mul-ℚ p=1 refl ∙ left-unit-law-mul-ℚ q))
       ( λ 1<p → ex-falso (not-leq-le-ℚ one-ℚ p 1<p p≤1))
 ```
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 20a8ca5963..f5754c5d86 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -198,7 +198,7 @@ abstract
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (H : le-ℚ x y)
   where
 
   opaque
@@ -209,7 +209,7 @@ module _
       inv-tr
         ( is-positive-ℚ)
         ( distributive-neg-diff-ℚ x y)
-        ( is-positive-diff-le-ℚ x y H)
+        ( is-positive-diff-le-ℚ H)
 
   negative-diff-le-ℚ : ℚ⁻
   negative-diff-le-ℚ = x -ℚ y , is-negative-diff-le-ℚ
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index 4eed261513..dec1ae2814 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -235,7 +235,7 @@ abstract
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (H : le-ℚ x y)
   where
 
   abstract
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 efd3983670..7687c6d3d4 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
@@ -102,7 +102,7 @@ module _
               ( le-ℚ)
               ( inv q=1)
               ( inv (mul-rational-ℕ 2 2))
-              ( preserves-le-rational-ℕ 1 4 _)))
+              ( preserves-le-rational-ℕ {1} {4} _)))
         ( λ 1<q →
           ( q , le-left-mul-greater-than-one-ℚ⁺ q 1<q q))
 
@@ -120,7 +120,7 @@ abstract
   bound-rounded-below-square-le-ℚ⁺ p⁺@(p , _) q⁺@(q , _) p²<q =
     let
       (δ₁⁺@(δ₁ , _) , δ₁+δ₁<⟨q-p²⟩/p) =
-        bound-double-le-ℚ⁺ (positive-diff-le-ℚ (p *ℚ p) q p²<q *ℚ⁺ inv-ℚ⁺ p⁺)
+        bound-double-le-ℚ⁺ (positive-diff-le-ℚ p²<q *ℚ⁺ inv-ℚ⁺ p⁺)
       (δ₂⁺@(δ₂ , _) , δ₂+δ₂<δ₁) = bound-double-le-ℚ⁺ δ₁⁺
       δ₃⁺@(δ₃ , _) = min-ℚ⁺ δ₂⁺ p⁺
       δ₂<δ₁ : le-ℚ δ₂ δ₁
@@ -141,22 +141,22 @@ abstract
           ( chain-of-inequalities
             (p +ℚ δ₃) *ℚ (p +ℚ δ₃)
             ≤ (p +ℚ δ₃) *ℚ p +ℚ (p +ℚ δ₃) *ℚ δ₃
-              by leq-eq-ℚ _ _ (left-distributive-mul-add-ℚ (p +ℚ δ₃) p δ₃)
+              by leq-eq-ℚ (left-distributive-mul-add-ℚ (p +ℚ δ₃) p δ₃)
             ≤ (p +ℚ δ₃) *ℚ p +ℚ ((p *ℚ δ₃) +ℚ (δ₃ *ℚ δ₃))
-              by leq-eq-ℚ _ _
+              by leq-eq-ℚ
                 ( ap-add-ℚ refl (right-distributive-mul-add-ℚ _ _ _))
             ≤ (p +ℚ δ₃) *ℚ p +ℚ ((δ₃ *ℚ p) +ℚ (δ₃ *ℚ p))
               by
                 preserves-leq-right-add-ℚ _ _ _
                   ( preserves-leq-add-ℚ
-                    ( leq-eq-ℚ _ _ (commutative-mul-ℚ _ _))
+                    ( leq-eq-ℚ (commutative-mul-ℚ _ _))
                     ( preserves-leq-left-mul-ℚ⁺
                       ( δ₃⁺)
                       ( δ₃)
                       ( p)
                       ( leq-right-min-ℚ⁺ _ _)))
             ≤ (p +ℚ (δ₃ +ℚ δ₃ +ℚ δ₃)) *ℚ p
-              by leq-eq-ℚ _ _
+              by leq-eq-ℚ
                 ( equational-reasoning
                   (p +ℚ δ₃) *ℚ p +ℚ (δ₃ *ℚ p +ℚ δ₃ *ℚ p)
                   ＝ (p +ℚ δ₃) *ℚ p +ℚ (δ₃ +ℚ δ₃) *ℚ p
@@ -185,7 +185,7 @@ abstract
                       ( leq-le-ℚ δ₂+δ₂<δ₁)
                       ( leq-le-ℚ δ₂<δ₁)))
             ≤ p *ℚ p +ℚ (δ₁ +ℚ δ₁) *ℚ p
-              by leq-eq-ℚ _ _ (right-distributive-mul-add-ℚ _ _ _))
+              by leq-eq-ℚ (right-distributive-mul-add-ℚ _ _ _))
           ( tr
             ( le-ℚ (p *ℚ p +ℚ (δ₁ +ℚ δ₁) *ℚ p))
             ( equational-reasoning
@@ -200,7 +200,7 @@ abstract
                       ( is-section-right-div-Group
                         ( group-mul-ℚ⁺)
                         ( p⁺)
-                        ( positive-diff-le-ℚ _ _ p²<q)))
+                        ( positive-diff-le-ℚ p²<q)))
               ＝ q
                 by is-identity-right-conjugation-add-ℚ (p *ℚ p) q)
             ( preserves-le-right-add-ℚ (p *ℚ p) _ _
@@ -223,7 +223,7 @@ abstract
   bound-rounded-above-square-le-ℚ⁺ p⁺@(p , _) q⁺@(q , _) q<p² =
     let
       (δ⁺@(δ , _) , δ+δ<⟨p²-q⟩/p) =
-        bound-double-le-ℚ⁺ ( positive-diff-le-ℚ _ _ q<p² *ℚ⁺ inv-ℚ⁺ p⁺)
+        bound-double-le-ℚ⁺ ( positive-diff-le-ℚ q<p² *ℚ⁺ inv-ℚ⁺ p⁺)
       δ<p =
         transitive-le-ℚ δ (δ +ℚ δ) p
           ( transitive-le-ℚ
@@ -238,7 +238,7 @@ abstract
             ( δ+δ<⟨p²-q⟩/p))
           ( le-right-add-rational-ℚ⁺ δ δ⁺)
     in
-      ( positive-diff-le-ℚ _ _ δ<p ,
+      ( positive-diff-le-ℚ δ<p ,
         le-diff-rational-ℚ⁺ p δ⁺ ,
         transitive-le-ℚ
           ( q)
@@ -271,7 +271,7 @@ abstract
                     ( is-section-right-div-Group
                       ( group-mul-ℚ⁺)
                       ( p⁺)
-                      ( positive-diff-le-ℚ _ _ q<p²)))
+                      ( positive-diff-le-ℚ q<p²)))
                   ( preserves-le-right-mul-ℚ⁺ p⁺ _ _ δ+δ<⟨p²-q⟩/p))))))
 
   rounded-above-square-le-ℚ⁺ :
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 fad1efcdb1..db95edd8b7 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -148,19 +148,15 @@ irreflexive-le-ℚ =
 ### Strict inequality on the rationals is transitive
 
 ```agda
-module _
-  (x y z : ℚ)
-  where
-
-  opaque
-    unfolding le-ℚ-Prop
+abstract opaque
+  unfolding le-ℚ-Prop
 
-    transitive-le-ℚ : le-ℚ y z → le-ℚ x y → le-ℚ x z
-    transitive-le-ℚ =
-      transitive-le-fraction-ℤ
-        ( fraction-ℚ x)
-        ( fraction-ℚ y)
-        ( fraction-ℚ z)
+  transitive-le-ℚ : (x y z : ℚ) → le-ℚ y z → le-ℚ x y → le-ℚ x z
+  transitive-le-ℚ x y z =
+    transitive-le-fraction-ℤ
+      ( fraction-ℚ x)
+      ( fraction-ℚ y)
+      ( fraction-ℚ z)
 ```
 
 ### Concatenation rules for inequality and strict inequality on the rational numbers
@@ -297,7 +293,7 @@ module _
 
 ```agda
 module _
-  (x y : ℤ)
+  {x y : ℤ}
   where
 
   opaque
@@ -322,12 +318,12 @@ module _
 
 ```agda
 module _
-  (m n : ℕ)
+  {m n : ℕ}
   where
 
   abstract
     iff-le-rational-ℕ : le-ℕ m n ↔ le-ℚ (rational-ℕ m) (rational-ℕ n)
-    iff-le-rational-ℕ = iff-le-rational-ℤ _ _ ∘iff iff-le-int-ℕ m n
+    iff-le-rational-ℕ = iff-le-rational-ℤ ∘iff iff-le-int-ℕ m n
 
     preserves-le-rational-ℕ : le-ℕ m n → le-ℚ (rational-ℕ m) (rational-ℕ n)
     preserves-le-rational-ℕ = forward-implication iff-le-rational-ℕ
@@ -344,9 +340,7 @@ module _
   where
 
   opaque
-    unfolding add-ℚ
-    unfolding le-ℚ-Prop
-    unfolding neg-ℚ
+    unfolding add-ℚ le-ℚ-Prop neg-ℚ
 
     iff-translate-diff-le-zero-ℚ : le-ℚ zero-ℚ (y -ℚ x) ↔ le-ℚ x y
     iff-translate-diff-le-zero-ℚ =
@@ -478,8 +472,7 @@ module _
 
 ```agda
 opaque
-  unfolding le-ℚ-Prop
-  unfolding leq-ℚ-Prop
+  unfolding le-ℚ-Prop leq-ℚ-Prop
 
   decide-le-leq-ℚ : (x y : ℚ) → le-ℚ x y + leq-ℚ y x
   decide-le-leq-ℚ x y =
diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index 641b00f2ba..7a8cea0f77 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -23,7 +23,11 @@ open import metric-spaces.equality-of-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.metrics
 
+open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.saturation-inequality-nonnegative-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -154,10 +158,10 @@ module _
                     ( r⁺)
                     ( backward-implication
                       ( is-metric-M-ρ r⁺ y z)
-                      ( leq-le-ℝ _ _ ρyz<r))
+                      ( leq-le-ℝ ρyz<r))
                     ( backward-implication
                       ( is-metric-M-ρ q⁺ x y)
-                      ( leq-le-ℝ _ _ ρxy<q)))))
+                      ( leq-le-ℝ ρxy<q)))))
               ( d'<d))
 
     is-extensional-is-metric-of-Metric-Space :
diff --git a/src/metric-spaces/metrics.lagda.md b/src/metric-spaces/metrics.lagda.md
index 66155d05b2..6ed22735b3 100644
--- a/src/metric-spaces/metrics.lagda.md
+++ b/src/metric-spaces/metrics.lagda.md
@@ -1,6 +1,8 @@
 # Metrics
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module metric-spaces.metrics where
 ```
 
@@ -36,9 +38,13 @@ open import metric-spaces.saturated-rational-neighborhood-relations
 open import metric-spaces.symmetric-rational-neighborhood-relations
 open import metric-spaces.triangular-rational-neighborhood-relations
 
+open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.saturation-inequality-nonnegative-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
@@ -338,9 +344,7 @@ module _
         ( not-leq-le-ℝ
           ( real-ℚ⁺ δ)
           ( real-ℝ⁰⁺ (dist-Metric X μ x y)))
-        ( leq-le-ℝ
-          ( real-ℝ⁰⁺ (dist-Metric X μ x y))
-          ( real-ℚ⁺ ε))
+        ( leq-le-ℝ)
         ( is-located-le-ℝ
           ( real-ℝ⁰⁺ (dist-Metric X μ x y))
           ( rational-ℚ⁺ δ)
@@ -365,6 +369,6 @@ module _
       bounded-dist-Metric-Space (metric-space-Metric X μ) x y
     bounded-dist-metric-space-Metric x y =
       map-tot-exists
-        ( λ ε → leq-le-ℝ (real-ℝ⁰⁺ (dist-Metric X μ x y)) (real-ℚ⁺ ε))
+        ( λ ε → leq-le-ℝ)
         ( le-some-positive-rational-ℝ⁰⁺ (dist-Metric X μ x y))
 ```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index dec3db12f1..1a10077ffc 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -8,6 +8,7 @@ module real-numbers where
 open import real-numbers.absolute-value-closed-intervals-real-numbers public
 open import real-numbers.absolute-value-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
+open import real-numbers.addition-nonnegative-real-numbers public
 open import real-numbers.addition-real-numbers public
 open import real-numbers.addition-upper-dedekind-real-numbers public
 open import real-numbers.apartness-real-numbers public
@@ -22,7 +23,9 @@ open import real-numbers.difference-real-numbers public
 open import real-numbers.distance-real-numbers public
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers public
 open import real-numbers.finitely-enumerable-subsets-real-numbers public
+open import real-numbers.inequalities-addition-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
+open import real-numbers.inequality-nonnegative-real-numbers public
 open import real-numbers.inequality-positive-real-numbers public
 open import real-numbers.inequality-real-numbers public
 open import real-numbers.inequality-upper-dedekind-real-numbers public
@@ -42,6 +45,7 @@ open import real-numbers.maximum-finite-families-real-numbers public
 open import real-numbers.maximum-inhabited-finitely-enumerable-subsets-real-numbers public
 open import real-numbers.maximum-lower-dedekind-real-numbers public
 open import real-numbers.maximum-upper-dedekind-real-numbers public
+open import real-numbers.metric-space-of-nonnegative-real-numbers public
 open import real-numbers.metric-space-of-real-numbers public
 open import real-numbers.minimum-finite-families-real-numbers public
 open import real-numbers.minimum-inhabited-finitely-enumerable-subsets-real-numbers public
@@ -67,10 +71,13 @@ open import real-numbers.rational-real-numbers public
 open import real-numbers.rational-upper-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
+open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
+open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
 open import real-numbers.squares-real-numbers public
+open import real-numbers.strict-inequalities-addition-real-numbers public
 open import real-numbers.strict-inequality-nonnegative-real-numbers public
 open import real-numbers.strict-inequality-positive-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
diff --git a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
index 194cae093f..f040689b7c 100644
--- a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
@@ -52,7 +52,7 @@ abstract
       ( z)
       ( max-ℝ (neg-ℝ x) y)
       ( transitive-leq-ℝ _ _ _ (leq-right-max-ℝ _ _) z≤y)
-      ( transitive-leq-ℝ _ _ _ (leq-left-max-ℝ _ _) (neg-leq-ℝ _ _ x≤z))
+      ( transitive-leq-ℝ _ _ _ (leq-left-max-ℝ _ _) (neg-leq-ℝ x≤z))
 ```
 
 ### The maximum absolute value of an interval is nonnegative
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 8695d96e15..d31b66d8b8 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -28,7 +28,9 @@ open import metric-spaces.short-functions-metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.isometry-negation-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
@@ -96,7 +98,7 @@ opaque
       ( lower-cut-ℝ (abs-ℝ x) q)
       ( id)
       ( λ (x<0 , 0<x) → ex-falso (is-disjoint-cut-ℝ x zero-ℚ (0<x , x<0)))
-      ( is-located-lower-upper-cut-ℝ (abs-ℝ x) q zero-ℚ q<0)
+      ( is-located-lower-upper-cut-ℝ (abs-ℝ x) q<0)
 
 nonnegative-abs-ℝ : {l : Level} → ℝ l → ℝ⁰⁺ l
 nonnegative-abs-ℝ x = (abs-ℝ x , is-nonnegative-abs-ℝ x)
@@ -134,10 +136,10 @@ module _
       tr
         ( leq-ℝ (neg-ℝ (abs-ℝ x)))
         ( neg-neg-ℝ x)
-        ( neg-leq-ℝ (neg-ℝ x) (abs-ℝ x) neg-leq-abs-ℝ)
+        ( neg-leq-ℝ neg-leq-abs-ℝ)
 
     neg-leq-neg-abs-ℝ : leq-ℝ (neg-ℝ (abs-ℝ x)) (neg-ℝ x)
-    neg-leq-neg-abs-ℝ = neg-leq-ℝ x (abs-ℝ x) leq-abs-ℝ
+    neg-leq-neg-abs-ℝ = neg-leq-ℝ leq-abs-ℝ
 ```
 
 ### If `|x| ＝ 0` then `x ＝ 0`
@@ -231,23 +233,11 @@ module _
       leq-abs-leq-leq-neg-ℝ
         ( x +ℝ y)
         ( abs-ℝ x +ℝ abs-ℝ y)
-        ( preserves-leq-add-ℝ
-          ( x)
-          ( abs-ℝ x)
-          ( y)
-          ( abs-ℝ y)
-          ( leq-abs-ℝ x)
-          ( leq-abs-ℝ y))
+        ( preserves-leq-add-ℝ (leq-abs-ℝ x) (leq-abs-ℝ y))
         ( inv-tr
           ( λ z → leq-ℝ z (abs-ℝ x +ℝ abs-ℝ y))
           ( distributive-neg-add-ℝ x y)
-          ( preserves-leq-add-ℝ
-            ( neg-ℝ x)
-            ( abs-ℝ x)
-            ( neg-ℝ y)
-            ( abs-ℝ y)
-            ( neg-leq-abs-ℝ x)
-            ( neg-leq-abs-ℝ y)))
+          ( preserves-leq-add-ℝ (neg-leq-abs-ℝ x) (neg-leq-abs-ℝ y)))
 ```
 
 ### The absolute value is a short function
@@ -354,7 +344,7 @@ module _
     leq-abs-sqrt-square-ℝ' :
       leq-ℝ' (real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x)) (abs-ℝ x)
     leq-abs-sqrt-square-ℝ' q |x|<q@(x<q , -x<q) =
-      ( is-positive-is-in-upper-cut-ℝ⁰⁺ (nonnegative-abs-ℝ x) q (x<q , -x<q) ,
+      ( is-positive-is-in-upper-cut-ℝ⁰⁺ (nonnegative-abs-ℝ x) (x<q , -x<q) ,
         is-in-upper-cut-square-pos-neg-ℝ x q x<q -x<q)
 
     eq-abs-sqrt-square-ℝ : abs-ℝ x ＝ real-sqrt-ℝ⁰⁺ (nonnegative-square-ℝ x)
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 d970a20b78..bd1e58abfe 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -114,7 +114,7 @@ module _
         (r , q<r , r<x+y) ← ∃r
         ((rx , ry) , (rx<x , ry<y , r=rx+ry)) ← r<x+y
         let
-          r-q⁺ = positive-diff-le-ℚ q r q<r
+          r-q⁺ = positive-diff-le-ℚ q<r
           ε⁺@(ε , _) = mediant-zero-ℚ⁺ r-q⁺
         intro-exists
           ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
new file mode 100644
index 0000000000..0735a72392
--- /dev/null
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,136 @@
+# Addition of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.addition-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.addition-nonnegative-rational-numbers
+open import elementary-number-theory.addition-positive-rational-numbers
+open import elementary-number-theory.nonnegative-rational-numbers
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.identity-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequality-nonnegative-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+The [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are closed under
+[addition](real-numbers.addition-real-numbers.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  real-add-ℝ⁰⁺ : ℝ (l1 ⊔ l2)
+  real-add-ℝ⁰⁺ = real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y
+
+  abstract
+    is-nonnegative-real-add-ℝ⁰⁺ : is-nonnegative-ℝ real-add-ℝ⁰⁺
+    is-nonnegative-real-add-ℝ⁰⁺ =
+      tr
+        ( λ z → leq-ℝ z (real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y))
+        ( left-unit-law-add-ℝ zero-ℝ)
+        ( preserves-leq-add-ℝ
+          ( is-nonnegative-real-ℝ⁰⁺ x)
+          ( is-nonnegative-real-ℝ⁰⁺ y))
+
+  add-ℝ⁰⁺ : ℝ⁰⁺ (l1 ⊔ l2)
+  add-ℝ⁰⁺ = (real-add-ℝ⁰⁺ , is-nonnegative-real-add-ℝ⁰⁺)
+
+infixl 35 _+ℝ⁰⁺_
+
+_+ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → ℝ⁰⁺ (l1 ⊔ l2)
+_+ℝ⁰⁺_ = add-ℝ⁰⁺
+```
+
+## Properties
+
+### Unit laws for addition
+
+```agda
+module _
+  {l : Level} (x : ℝ⁰⁺ l)
+  where
+
+  abstract
+    left-unit-law-add-ℝ⁰⁺ : zero-ℝ⁰⁺ +ℝ⁰⁺ x ＝ x
+    left-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (left-unit-law-add-ℝ _)
+
+    right-unit-law-add-ℝ⁰⁺ : x +ℝ⁰⁺ zero-ℝ⁰⁺ ＝ x
+    right-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (right-unit-law-add-ℝ _)
+```
+
+### Addition preserves inequality
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
+  where
+
+  abstract
+    preserves-leq-add-ℝ⁰⁺ :
+      leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ z w → leq-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
+    preserves-leq-add-ℝ⁰⁺ = preserves-leq-add-ℝ
+```
+
+### The canonical embedding of nonnegative rational numbers to nonnegative real numbers preserves addition
+
+```agda
+abstract
+  add-nonnegative-real-ℚ⁰⁺ :
+    (p q : ℚ⁰⁺) →
+    nonnegative-real-ℚ⁰⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁰⁺ q ＝
+    nonnegative-real-ℚ⁰⁺ (p +ℚ⁰⁺ q)
+  add-nonnegative-real-ℚ⁰⁺ p q =
+    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q))
+```
+
+### The canonical embedding of positive rational numbers to nonnegative real numbers preserves addition
+
+```agda
+abstract
+  add-nonnegative-real-ℚ⁺ :
+    (p q : ℚ⁺) →
+    nonnegative-real-ℚ⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁺ q ＝
+    nonnegative-real-ℚ⁺ (p +ℚ⁺ q)
+  add-nonnegative-real-ℚ⁺ p q =
+    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁺ p) (rational-ℚ⁺ q))
+```
+
+### Addition preserves strict inequality
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
+  where
+
+  abstract
+    preserves-le-add-ℝ⁰⁺ :
+      le-ℝ⁰⁺ x y → le-ℝ⁰⁺ z w → le-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
+    preserves-le-add-ℝ⁰⁺ = preserves-le-add-ℝ
+```
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 44bd7f51ae..616124b951 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -111,7 +111,7 @@ module _
         (p , p<q , x+y<p) ← ∃p
         ((px , py) , (x<px , y<py , p=px+py)) ← x+y<p
         let
-          q-p⁺ = positive-diff-le-ℚ p q p<q
+          q-p⁺ = positive-diff-le-ℚ p<q
           ε⁺@(ε , _) = mediant-zero-ℚ⁺ q-p⁺
         intro-exists
           ( px +ℚ ε , q -ℚ (px +ℚ ε))
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index af40140487..ffca8483f9 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -204,7 +204,7 @@ module _
                       ( preserves-leq-left-add-ℚ (q -ℚ p) p' p p'≤p)))
                   ( q'∈U)))
             ( decide-le-leq-ℚ p p'))
-        ( arithmetically-located (positive-diff-le-ℚ p q p<q))
+        ( arithmetically-located (positive-diff-le-ℚ p<q))
 
     is-located-is-weakly-arithmetically-located-lower-upper-ℝ :
       is-weakly-arithmetically-located-lower-upper-ℝ x y →
@@ -270,11 +270,7 @@ module _
                 ( is-in-upper-cut-ℝ x)
                 ( associative-add-ℚ p ε ε)
                 ( x<p+2ε)))
-        ( is-located-lower-upper-cut-ℝ
-          ( x)
-          ( p +ℚ ε)
-          ( (p +ℚ ε) +ℚ ε)
-          ( le-right-add-rational-ℚ⁺ (p +ℚ ε) ε⁺))
+        ( is-located-lower-upper-cut-ℝ x (le-right-add-rational-ℚ⁺ (p +ℚ ε) ε⁺))
 
     is-arithmetically-located-ℝ :
       is-arithmetically-located-lower-upper-ℝ (lower-real-ℝ x) (upper-real-ℝ x)
@@ -376,7 +372,7 @@ abstract
                     ≤ rational-ℕ m +ℚ rational-ℕ n
                       by preserves-leq-right-add-ℚ _ _ _ (leq-le-ℚ max|p||q|<n)
                     ≤ rational-ℕ (m +ℕ n)
-                      by leq-eq-ℚ _ _ (add-rational-ℕ _ _))
+                      by leq-eq-ℚ (add-rational-ℕ _ _))
                 ( chain-of-inequalities
                     neg-ℚ a
                     ≤ neg-ℚ (b -ℚ ε)
@@ -384,7 +380,7 @@ abstract
                         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 _ _)
+                      by leq-eq-ℚ (distributive-neg-diff-ℚ _ _)
                     ≤ ε -ℚ p
                       by
                         preserves-leq-right-add-ℚ ε
@@ -399,7 +395,7 @@ abstract
                     ≤ ε +ℚ max-ℚ (rational-abs-ℚ p) (rational-abs-ℚ q)
                       by
                         preserves-leq-add-ℚ
-                          ( leq-eq-ℚ _ _ (rational-abs-rational-ℚ⁺ ε⁺))
+                          ( leq-eq-ℚ (rational-abs-rational-ℚ⁺ ε⁺))
                           ( leq-left-max-ℚ _ _)
                     ≤ rational-ℕ m +ℚ rational-ℕ n
                       by
@@ -407,7 +403,7 @@ abstract
                           ( leq-le-ℚ ε<m)
                           ( leq-le-ℚ max|p||q|<n)
                     ≤ rational-ℕ (m +ℕ n)
-                      by leq-eq-ℚ _ _ (add-rational-ℕ _ _))
+                      by leq-eq-ℚ (add-rational-ℕ _ _))
             |b|≤|a|+ε =
               leq-abs-leq-leq-neg-ℚ b (rational-abs-ℚ a +ℚ ε)
                 ( chain-of-inequalities
@@ -419,9 +415,7 @@ abstract
                     ≤ rational-abs-ℚ a +ℚ rational-abs-ℚ ε
                       by triangle-inequality-abs-ℚ _ _
                     ≤ rational-abs-ℚ a +ℚ ε
-                      by
-                        leq-eq-ℚ _ _
-                          ( ap-add-ℚ refl (rational-abs-rational-ℚ⁺ ε⁺)))
+                      by leq-eq-ℚ (ap-add-ℚ refl (rational-abs-rational-ℚ⁺ ε⁺)))
                 ( chain-of-inequalities
                     neg-ℚ b
                     ≤ neg-ℚ a
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 7d521f85c1..2482c2f855 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -32,6 +32,7 @@ open import order-theory.least-upper-bounds-large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
@@ -96,8 +97,8 @@ module _
           map-disjunction
             ( λ p<y → inr-disjunction p<y)
             ( x<q ,_)
-            ( is-located-lower-upper-cut-ℝ y p q p<q))
-        ( is-located-lower-upper-cut-ℝ x p q p<q)
+            ( is-located-lower-upper-cut-ℝ y p<q))
+        ( is-located-lower-upper-cut-ℝ x p<q)
       where
         claim : Prop (l1 ⊔ l2)
         claim =
@@ -331,8 +332,6 @@ module _
             ( max-ℝ x z)
             ( max-ℝ x z +ℝ real-ℚ⁺ d)
             ( leq-le-ℝ
-              ( max-ℝ x z)
-              ( max-ℝ x z +ℝ real-ℚ⁺ d)
               ( le-left-add-real-ℝ⁺
                 ( max-ℝ x z)
                 ( positive-real-ℚ⁺ d)))
@@ -453,14 +452,14 @@ module _
             ( max-ℝ x y)
             ( le-diff-real-ℝ⁺ (max-ℝ x y) (positive-real-ℚ⁺ ε⁺))
         (r , q-<ℝ-r , r<max) ← dense-rational-le-ℝ (real-ℚ q) (max-ℝ x y) q<max
-        let q<r = reflects-le-real-ℚ q r q-<ℝ-r
+        let q<r = reflects-le-real-ℚ q-<ℝ-r
         map-disjunction
           ( λ q<x →
             transitive-le-ℝ
               ( max-ℝ x y -ℝ real-ℚ ε)
               ( real-ℚ q)
               ( x)
-              ( le-real-is-in-lower-cut-ℚ q x q<x)
+              ( le-real-is-in-lower-cut-ℚ x q<x)
               ( max-ε<q))
           ( λ x<r →
             elim-disjunction
@@ -470,7 +469,7 @@ module _
                   ( max-ℝ x y -ℝ real-ℚ ε)
                   ( real-ℚ q)
                   ( y)
-                  ( le-real-is-in-lower-cut-ℚ q y q<y)
+                  ( le-real-is-in-lower-cut-ℚ y q<y)
                   ( max-ε<q))
               ( λ y<r →
                 ex-falso
@@ -478,11 +477,9 @@ module _
                     ( max-ℝ x y)
                     ( concatenate-leq-le-ℝ (max-ℝ x y) (real-ℚ r) (max-ℝ x y)
                       ( leq-max-leq-leq-ℝ x y (real-ℚ r)
-                        ( leq-le-ℝ x (real-ℚ r)
-                          ( le-real-is-in-upper-cut-ℚ r x x<r))
-                        ( leq-le-ℝ y (real-ℚ r)
-                          ( le-real-is-in-upper-cut-ℚ r y y<r)))
+                        ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ x x<r))
+                        ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ y y<r)))
                       ( r<max))))
-              ( is-located-lower-upper-cut-ℝ y q r q<r))
-          ( is-located-lower-upper-cut-ℝ x q r q<r)
+              ( is-located-lower-upper-cut-ℝ y q<r))
+          ( is-located-lower-upper-cut-ℝ x q<r)
 ```
diff --git a/src/real-numbers/binary-minimum-real-numbers.lagda.md b/src/real-numbers/binary-minimum-real-numbers.lagda.md
index f156bb4674..121bf5d42b 100644
--- a/src/real-numbers/binary-minimum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-minimum-real-numbers.lagda.md
@@ -79,10 +79,8 @@ module _
       tr
         ( λ w → leq-ℝ w (min-ℝ x y))
         ( neg-neg-ℝ z)
-        ( neg-leq-ℝ (max-ℝ (neg-ℝ x) (neg-ℝ y)) (neg-ℝ z)
-          ( leq-max-leq-leq-ℝ _ _ (neg-ℝ z)
-            ( neg-leq-ℝ z x z≤x)
-            ( neg-leq-ℝ z y z≤y)))
+        ( neg-leq-ℝ
+          ( leq-max-leq-leq-ℝ _ _ (neg-ℝ z) (neg-leq-ℝ z≤x) (neg-leq-ℝ z≤y)))
     pr2 (is-greatest-binary-lower-bound-min-ℝ z) z≤min =
       let
         case :
@@ -93,7 +91,7 @@ module _
             ( tr
               ( leq-ℝ (min-ℝ x y))
               ( neg-neg-ℝ _)
-              ( neg-leq-ℝ (neg-ℝ w) (max-ℝ (neg-ℝ x) (neg-ℝ y)) -w≤max))
+              ( neg-leq-ℝ -w≤max))
             ( z≤min)
       in (case x (leq-left-max-ℝ _ _) , case y (leq-right-max-ℝ _ _))
 
@@ -172,10 +170,11 @@ module _
           le-ℝ (max-ℝ (neg-ℝ x) (neg-ℝ y) -ℝ real-ℚ⁺ ε) (neg-ℝ w) →
           le-ℝ w (min-ℝ x y +ℝ real-ℚ⁺ ε)
         case w max-ε<-w =
-          binary-tr le-ℝ
+          binary-tr
+            ( le-ℝ)
             ( neg-neg-ℝ w)
             ( distributive-neg-diff-ℝ _ _ ∙ commutative-add-ℝ _ _)
-            ( neg-le-ℝ _ (neg-ℝ w) max-ε<-w)
+            ( neg-le-ℝ max-ε<-w)
       in
         elim-disjunction
           ( (le-prop-ℝ x (min-ℝ x y +ℝ real-ℚ⁺ ε)) ∨
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index c80918effe..2a3e529451 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -45,6 +45,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.distance-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.metric-space-of-real-numbers
@@ -369,10 +370,8 @@ module _
         ( upper-real-lim-cauchy-approximation-ℝ)
     is-located-lower-upper-cut-lim-cauchy-approximation-ℝ p q p<q =
       let
-        ε'⁺@(ε' , _) , 2ε'⁺<q-p =
-          bound-double-le-ℚ⁺ (positive-diff-le-ℚ p q p<q)
-        ε⁺@(ε , _) , 2ε⁺<ε'⁺ =
-          bound-double-le-ℚ⁺ ε'⁺
+        ε'⁺@(ε' , _) , 2ε'⁺<q-p = bound-double-le-ℚ⁺ (positive-diff-le-ℚ p<q)
+        ε⁺@(ε , _) , 2ε⁺<ε'⁺ = bound-double-le-ℚ⁺ ε'⁺
         2ε' = ε' +ℚ ε'
         2ε = ε +ℚ ε
         4ε = 2ε +ℚ 2ε
@@ -383,8 +382,6 @@ module _
           ( intro-exists (ε⁺ , ε⁺))
           ( is-located-lower-upper-cut-ℝ
             ( map-cauchy-approximation-ℝ x ε⁺)
-            ( p +ℚ 2ε)
-            ( q -ℚ 2ε)
             ( le-transpose-left-add-ℚ
               ( p +ℚ 2ε)
               ( 2ε)
@@ -479,8 +476,6 @@ module _
               ( ε+θ)
               ( lim)
               ( leq-le-ℝ
-                ( xε -ℝ ε+θ)
-                ( lim)
                 ( intro-exists
                   ( q)
                   ( xε-ε-θ<q ,
@@ -500,8 +495,6 @@ module _
                 ( xε)
                 ( ε+θ)
                 ( leq-le-ℝ
-                  ( lim)
-                  ( xε +ℝ ε+θ)
                   ( intro-exists
                     ( r)
                     ( intro-exists
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index 2369f9cf54..846087aa48 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -84,7 +84,7 @@ upper-bound-closed-interval-ℝ =
 ```agda
 unit-closed-interval-ℝ : closed-interval-ℝ lzero lzero
 unit-closed-interval-ℝ =
-  ((zero-ℝ , one-ℝ) , preserves-leq-real-ℚ zero-ℚ one-ℚ leq-zero-one-ℚ)
+  ((zero-ℝ , one-ℝ) , preserves-leq-real-ℚ leq-zero-one-ℚ)
 ```
 
 ### Closed intervals in the real numbers are closed in the metric space of real numbers
@@ -103,7 +103,7 @@ opaque
         let open do-syntax-trunc-Prop (lower-cut-ℝ x q)
         in do
           (r , q<r , r<a) ← forward-implication (is-rounded-lower-cut-ℝ a q) q<a
-          let ε⁺@(ε , _) = positive-diff-le-ℚ q r q<r
+          let ε⁺@(ε , _) = positive-diff-le-ℚ q<r
           (y , Nεxy , a≤y , _) ← H ε⁺
           pr1 Nεxy
             ( q)
@@ -117,7 +117,7 @@ opaque
         let open do-syntax-trunc-Prop (lower-cut-ℝ b q)
         in do
           (r , q<r , r<x) ← forward-implication (is-rounded-lower-cut-ℝ x q) q<x
-          let ε⁺@(ε , _) = positive-diff-le-ℚ q r q<r
+          let ε⁺@(ε , _) = positive-diff-le-ℚ q<r
           (y , Nεxy , _ , y≤b) ← H ε⁺
           y≤b
             ( q)
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 3ce56e8f4b..a6183b2bc4 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -179,9 +179,9 @@ module _
   is-disjoint-cut-ℝ = pr1 (pr2 (pr2 x))
 
   is-located-lower-upper-cut-ℝ :
-    (q r : ℚ) → le-ℚ q r →
+    {q r : ℚ} → le-ℚ q r →
     type-disjunction-Prop (lower-cut-ℝ q) (upper-cut-ℝ r)
-  is-located-lower-upper-cut-ℝ = pr2 (pr2 (pr2 x))
+  is-located-lower-upper-cut-ℝ {q} {r} = pr2 (pr2 (pr2 x)) q r
 
   cut-ℝ : subtype l ℚ
   cut-ℝ q =
@@ -315,7 +315,7 @@ module _
           ( lower-cut-ℝ x p)
           ( upper-cut-ℝ x q)
           ( pr2 I)
-          ( is-located-lower-upper-cut-ℝ x p q ( pr1 I)))
+          ( is-located-lower-upper-cut-ℝ x ( pr1 I)))
 
   subset-lower-complement-upper-cut-lower-cut-ℝ :
     lower-cut-ℝ x ⊆ lower-complement-upper-cut-ℝ x
@@ -382,7 +382,7 @@ module _
           ( lower-cut-ℝ x p)
           ( upper-cut-ℝ x q)
           ( pr2 I)
-          ( is-located-lower-upper-cut-ℝ x p q (pr1 I)))
+          ( is-located-lower-upper-cut-ℝ x (pr1 I)))
 
   subset-upper-complement-lower-cut-upper-cut-ℝ :
     upper-cut-ℝ x ⊆ upper-complement-lower-cut-ℝ x
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 446e7c220a..df3a005ee3 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -28,6 +28,7 @@ open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-real-numbers
@@ -138,14 +139,12 @@ abstract
     lower-neighborhood-ℝ d y x
   lower-neighborhood-diff-ℝ d⁺@(d , _) x y x-y≤d q q+d<x =
     is-in-lower-cut-le-real-ℚ
-      ( q)
       ( y)
       ( concatenate-le-leq-ℝ
         ( real-ℚ q)
         ( x -ℝ real-ℚ d)
         ( y)
         ( le-real-is-in-lower-cut-ℚ
-          ( q)
           ( x -ℝ real-ℚ d)
           ( transpose-add-is-in-lower-cut-ℝ x q d q+d<x))
         ( swap-right-diff-leq-ℝ x y (real-ℚ d) x-y≤d))
@@ -213,9 +212,6 @@ abstract
     dist-ℝ x z ≤-ℝ dist-ℝ x y +ℝ dist-ℝ y z
   triangle-inequality-dist-ℝ x y z =
     preserves-leq-left-sim-ℝ
-      ( dist-ℝ x y +ℝ dist-ℝ y z)
-      ( abs-ℝ (x -ℝ y +ℝ y -ℝ z))
-      ( abs-ℝ (x -ℝ z))
       ( preserves-sim-abs-ℝ
         ( similarity-reasoning-ℝ
           x -ℝ y +ℝ y -ℝ z
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
new file mode 100644
index 0000000000..654f922bc1
--- /dev/null
+++ b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
@@ -0,0 +1,234 @@
+# Inequalities about addition and subtraction on the real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.inequalities-addition-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.function-types
+open import foundation.functoriality-cartesian-product-types
+open import foundation.logical-equivalences
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import logic.functoriality-existential-quantification
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
+```
+
+</details>
+
+## Idea
+
+This file describes lemmas about
+[inequalities](real-numbers.inequality-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md) related to
+[addition](real-numbers.addition-real-numbers.md) and
+[subtraction](real-numbers.difference-real-numbers.md).
+
+## Lemmas
+
+### Inequality on the real numbers is translation invariant
+
+```agda
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  opaque
+    unfolding add-ℝ leq-ℝ
+
+    preserves-leq-right-add-ℝ : leq-ℝ x y → leq-ℝ (x +ℝ z) (y +ℝ z)
+    preserves-leq-right-add-ℝ x≤y _ =
+      map-tot-exists (λ (qx , _) → map-product (x≤y qx) id)
+
+    preserves-leq-left-add-ℝ : leq-ℝ x y → leq-ℝ (z +ℝ x) (z +ℝ y)
+    preserves-leq-left-add-ℝ x≤y _ =
+      map-tot-exists (λ (_ , qx) → map-product id (map-product (x≤y qx) id))
+
+abstract
+  preserves-leq-diff-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    leq-ℝ x y → leq-ℝ (x -ℝ z) (y -ℝ z)
+  preserves-leq-diff-ℝ z = preserves-leq-right-add-ℝ (neg-ℝ z)
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  abstract
+    reflects-leq-right-add-ℝ : leq-ℝ (x +ℝ z) (y +ℝ z) → leq-ℝ x y
+    reflects-leq-right-add-ℝ x+z≤y+z =
+      preserves-leq-sim-ℝ
+        ( (x +ℝ z) -ℝ z)
+        ( x)
+        ( (y +ℝ z) -ℝ z)
+        ( y)
+        ( cancel-right-add-diff-ℝ x z)
+        ( cancel-right-add-diff-ℝ y z)
+        ( preserves-leq-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z≤y+z)
+
+    reflects-leq-left-add-ℝ : leq-ℝ (z +ℝ x) (z +ℝ y) → leq-ℝ x y
+    reflects-leq-left-add-ℝ z+x≤z+y =
+      reflects-leq-right-add-ℝ
+        ( binary-tr
+          ( leq-ℝ)
+          ( commutative-add-ℝ z x)
+          ( commutative-add-ℝ z y)
+          ( z+x≤z+y))
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  iff-translate-right-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (x +ℝ z) (y +ℝ z)
+  pr1 iff-translate-right-leq-ℝ = preserves-leq-right-add-ℝ z x y
+  pr2 iff-translate-right-leq-ℝ = reflects-leq-right-add-ℝ z x y
+
+  iff-translate-left-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (z +ℝ x) (z +ℝ y)
+  pr1 iff-translate-left-leq-ℝ = preserves-leq-left-add-ℝ z x y
+  pr2 iff-translate-left-leq-ℝ = reflects-leq-left-add-ℝ z x y
+
+abstract
+  preserves-leq-add-ℝ :
+    {l1 l2 l3 l4 : Level} {a : ℝ l1} {b : ℝ l2} {c : ℝ l3} {d : ℝ l4} →
+    leq-ℝ a b → leq-ℝ c d → leq-ℝ (a +ℝ c) (b +ℝ d)
+  preserves-leq-add-ℝ {a = a} {b = b} {c = c} {d = d} a≤b c≤d =
+    transitive-leq-ℝ
+      ( a +ℝ c)
+      ( a +ℝ d)
+      ( b +ℝ d)
+      ( preserves-leq-right-add-ℝ d a b a≤b)
+      ( preserves-leq-left-add-ℝ a c d c≤d)
+```
+
+### Transposition laws
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    leq-transpose-left-diff-ℝ : leq-ℝ (x -ℝ y) z → leq-ℝ x (z +ℝ y)
+    leq-transpose-left-diff-ℝ x-y≤z =
+      preserves-leq-left-sim-ℝ
+        ( cancel-right-diff-add-ℝ x y)
+        ( preserves-leq-right-add-ℝ y (x -ℝ y) z x-y≤z)
+
+    leq-transpose-left-add-ℝ : leq-ℝ (x +ℝ y) z → leq-ℝ x (z -ℝ y)
+    leq-transpose-left-add-ℝ x+y≤z =
+      preserves-leq-left-sim-ℝ
+        ( cancel-right-add-diff-ℝ x y)
+        ( preserves-leq-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y≤z)
+
+    leq-transpose-right-add-ℝ : leq-ℝ x (y +ℝ z) → leq-ℝ (x -ℝ z) y
+    leq-transpose-right-add-ℝ x≤y+z =
+      preserves-leq-right-sim-ℝ
+        ( cancel-right-add-diff-ℝ y z)
+        ( preserves-leq-right-add-ℝ (neg-ℝ z) x (y +ℝ z) x≤y+z)
+
+    leq-transpose-right-diff-ℝ : leq-ℝ x (y -ℝ z) → leq-ℝ (x +ℝ z) y
+    leq-transpose-right-diff-ℝ x≤y-z =
+      preserves-leq-right-sim-ℝ
+        ( cancel-right-diff-add-ℝ y z)
+        ( preserves-leq-right-add-ℝ z x (y -ℝ z) x≤y-z)
+```
+
+### Swapping laws
+
+```agda
+abstract
+  swap-right-diff-leq-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    leq-ℝ (x -ℝ y) z → leq-ℝ (x -ℝ z) y
+  swap-right-diff-leq-ℝ x y z x-y≤z =
+    leq-transpose-right-add-ℝ
+      ( x)
+      ( y)
+      ( z)
+      ( tr
+        ( leq-ℝ x)
+        ( commutative-add-ℝ _ _)
+        ( leq-transpose-left-diff-ℝ x y z x-y≤z))
+```
+
+### Addition of real numbers preserves lower neighborhoods
+
+```agda
+module _
+  {l1 l2 l3 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    preserves-lower-neighborhood-leq-left-add-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d =
+      inv-tr
+        ( leq-ℝ (x +ℝ y))
+        ( associative-add-ℝ x z (real-ℚ⁺ d))
+        ( preserves-leq-left-add-ℝ
+          ( x)
+          ( y)
+          ( z +ℝ real-ℚ⁺ d)
+          ( z≤y+d))
+
+    preserves-lower-neighborhood-leq-right-add-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-right-add-ℝ z≤y+d =
+      binary-tr
+        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
+        ( commutative-add-ℝ x y)
+        ( commutative-add-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d)
+```
+
+### Addition of real numbers reflects lower neighborhoods
+
+```agda
+module _
+  {l1 l2 l3 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    reflects-lower-neighborhood-leq-left-add-ℝ :
+      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d) →
+      leq-ℝ y (z +ℝ real-ℚ⁺ d)
+    reflects-lower-neighborhood-leq-left-add-ℝ x+y≤x+z+d =
+      reflects-leq-left-add-ℝ
+        ( x)
+        ( y)
+        ( z +ℝ real-ℚ⁺ d)
+        ( tr
+          ( leq-ℝ (x +ℝ y))
+          ( associative-add-ℝ x z (real-ℚ⁺ d))
+          ( x+y≤x+z+d))
+
+    reflects-lower-neighborhood-leq-right-add-ℝ :
+      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d) →
+      leq-ℝ y (z +ℝ real-ℚ⁺ d)
+    reflects-lower-neighborhood-leq-right-add-ℝ y+x≤z+y+d =
+      reflects-lower-neighborhood-leq-left-add-ℝ
+        ( binary-tr
+          ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
+          ( commutative-add-ℝ y x)
+          ( commutative-add-ℝ z x)
+          ( y+x≤z+y+d))
+```
diff --git a/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
new file mode 100644
index 0000000000..fa71aea44a
--- /dev/null
+++ b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,159 @@
+# Inequality of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.inequality-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+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.coproduct-types
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.function-types
+open import foundation.logical-equivalences
+open import foundation.propositional-truncations
+open import foundation.propositions
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+The
+{{#concept "standard ordering" Disambiguation="on the nonnegative real numbers" Agda=leq-ℝ⁰⁺}}
+on the [nonnegative real numbers](real-numbers.nonnegative-real-numbers.md) is
+inherited from the [standard ordering](real-numbers.inequality-real-numbers.md)
+on [real numbers](real-numbers.dedekind-real-numbers.md).
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  leq-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
+  leq-prop-ℝ⁰⁺ = leq-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+
+  leq-ℝ⁰⁺ : UU (l1 ⊔ l2)
+  leq-ℝ⁰⁺ = type-Prop leq-prop-ℝ⁰⁺
+```
+
+## Properties
+
+### Zero is less than or equal to every nonnegative real number
+
+```agda
+leq-zero-ℝ⁰⁺ : {l : Level} (x : ℝ⁰⁺ l) → leq-ℝ⁰⁺ zero-ℝ⁰⁺ x
+leq-zero-ℝ⁰⁺ = is-nonnegative-real-ℝ⁰⁺
+```
+
+### Similarity preserves inequality
+
+```agda
+module _
+  {l1 l2 l3 : Level} (z : ℝ⁰⁺ l1) (x : ℝ⁰⁺ l2) (y : ℝ⁰⁺ l3) (x~y : sim-ℝ⁰⁺ x y)
+  where
+
+  abstract
+    preserves-leq-left-sim-ℝ⁰⁺ : leq-ℝ⁰⁺ x z → leq-ℝ⁰⁺ y z
+    preserves-leq-left-sim-ℝ⁰⁺ = preserves-leq-left-sim-ℝ x~y
+```
+
+### Inequality is transitive
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3)
+  where
+
+  transitive-leq-ℝ⁰⁺ : leq-ℝ⁰⁺ y z → leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ x z
+  transitive-leq-ℝ⁰⁺ = transitive-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z)
+```
+
+### If `x` is less than all the positive rational numbers `y` is less than, then `x ≤ y`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    leq-le-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) → le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
+        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
+      leq-ℝ⁰⁺ x y
+    leq-le-positive-rational-ℝ⁰⁺ H =
+      leq-le-rational-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+        ( λ q y<q →
+          rec-coproduct
+            ( λ 0<q →
+              let open do-syntax-trunc-Prop (le-prop-ℝ (real-ℝ⁰⁺ x) (real-ℚ q))
+              in do
+                (p , y<p , p<q) ← dense-rational-le-ℝ _ _ y<q
+                transitive-le-ℝ _ (real-ℚ p) _
+                  ( p<q)
+                  ( H
+                    ( p , is-positive-le-nonnegative-real-ℚ y p y<p)
+                    ( y<p)))
+            ( λ q≤0 → ex-falso (not-le-leq-zero-rational-ℝ⁰⁺ y q q≤0 y<q))
+            ( decide-le-leq-ℚ zero-ℚ q))
+```
+
+### If `x` is less than or equal to all the positive rational numbers `y` is less than or equal to, then `x ≤ y`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    leq-leq-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) → leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
+        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
+      leq-ℝ⁰⁺ x y
+    leq-leq-positive-rational-ℝ⁰⁺ H =
+      leq-le-positive-rational-ℝ⁰⁺ x y
+        ( λ q y<q →
+          let open do-syntax-trunc-Prop (le-prop-ℝ _ _)
+          in do
+            (r , y<r , r<q) ← dense-rational-le-ℝ _ _ y<q
+            concatenate-leq-le-ℝ _ _ _
+              ( H
+                ( r , is-positive-le-nonnegative-real-ℚ y r y<r)
+                ( leq-le-ℝ y<r))
+              ( r<q))
+```
+
+### If `x` is less than or equal to the same positive rational numbers `y` is less than or equal to, then `x` and `y` are similar
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    sim-leq-same-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) →
+        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
+      sim-ℝ⁰⁺ x y
+    sim-leq-same-positive-rational-ℝ⁰⁺ H =
+      sim-sim-leq-ℝ
+        ( leq-leq-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
+          leq-leq-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
+```
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index d441631cb9..d9b18435f1 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -193,14 +193,14 @@ opaque
   refl-leq-ℝ : {l : Level} (x : ℝ l) → leq-ℝ x x
   refl-leq-ℝ x = refl-leq-Large-Preorder lower-ℝ-Large-Preorder (lower-real-ℝ x)
 
-  leq-eq-ℝ : {l : Level} (x y : ℝ l) → x ＝ y → leq-ℝ x y
-  leq-eq-ℝ x y x=y = tr (leq-ℝ x) x=y (refl-leq-ℝ x)
+  leq-eq-ℝ : {l : Level} {x y : ℝ l} → x ＝ y → leq-ℝ x y
+  leq-eq-ℝ {x = x} refl = refl-leq-ℝ x
 
 opaque
   unfolding leq-ℝ sim-ℝ
 
-  leq-sim-ℝ : {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ x y → leq-ℝ x y
-  leq-sim-ℝ _ _ = pr1
+  leq-sim-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → sim-ℝ x y → leq-ℝ x y
+  leq-sim-ℝ = pr1
 ```
 
 ### Inequality on the real numbers is antisymmetric
@@ -288,24 +288,28 @@ opaque
 ### The canonical map from rational numbers to the reals preserves and reflects inequality
 
 ```agda
-opaque
-  unfolding leq-ℝ real-ℚ
+module _
+  {x y : ℚ}
+  where
+
+  opaque
+    unfolding leq-ℝ real-ℚ
 
-  preserves-leq-real-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
-  preserves-leq-real-ℚ = preserves-leq-lower-real-ℚ
+    preserves-leq-real-ℚ : leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
+    preserves-leq-real-ℚ = preserves-leq-lower-real-ℚ x y
 
-  reflects-leq-real-ℚ : (x y : ℚ) → leq-ℝ (real-ℚ x) (real-ℚ y) → leq-ℚ x y
-  reflects-leq-real-ℚ = reflects-leq-lower-real-ℚ
+    reflects-leq-real-ℚ : leq-ℝ (real-ℚ x) (real-ℚ y) → leq-ℚ x y
+    reflects-leq-real-ℚ = reflects-leq-lower-real-ℚ x y
 
-  iff-leq-real-ℚ : (x y : ℚ) → leq-ℚ x y ↔ leq-ℝ (real-ℚ x) (real-ℚ y)
-  iff-leq-real-ℚ = iff-leq-lower-real-ℚ
+    iff-leq-real-ℚ : leq-ℚ x y ↔ leq-ℝ (real-ℚ x) (real-ℚ y)
+    iff-leq-real-ℚ = iff-leq-lower-real-ℚ x y
 ```
 
 ### Negation reverses inequality on the real numbers
 
 ```agda
 module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
   opaque
@@ -319,7 +323,7 @@ module _
 
 ```agda
 module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) (x~y : sim-ℝ x y)
+  {l1 l2 l3 : Level} {z : ℝ l1} {x : ℝ l2} {y : ℝ l3} (x~y : sim-ℝ x y)
   where
 
   opaque
@@ -340,241 +344,8 @@ module _
   preserves-leq-sim-ℝ : leq-ℝ x1 y1 → leq-ℝ x2 y2
   preserves-leq-sim-ℝ x1≤y1 =
     preserves-leq-left-sim-ℝ
-      ( y2)
-      ( x1)
-      ( x2)
       ( x1~x2)
-      ( preserves-leq-right-sim-ℝ x1 y1 y2 y1~y2 x1≤y1)
-```
-
-### Inequality on the real numbers is translation invariant
-
-```agda
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  opaque
-    unfolding add-ℝ leq-ℝ
-
-    preserves-leq-right-add-ℝ : leq-ℝ x y → leq-ℝ (x +ℝ z) (y +ℝ z)
-    preserves-leq-right-add-ℝ x≤y _ =
-      map-tot-exists (λ (qx , _) → map-product (x≤y qx) id)
-
-    preserves-leq-left-add-ℝ : leq-ℝ x y → leq-ℝ (z +ℝ x) (z +ℝ y)
-    preserves-leq-left-add-ℝ x≤y _ =
-      map-tot-exists (λ (_ , qx) → map-product id (map-product (x≤y qx) id))
-
-abstract
-  preserves-leq-diff-ℝ :
-    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
-    leq-ℝ x y → leq-ℝ (x -ℝ z) (y -ℝ z)
-  preserves-leq-diff-ℝ z = preserves-leq-right-add-ℝ (neg-ℝ z)
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  abstract
-    reflects-leq-right-add-ℝ : leq-ℝ (x +ℝ z) (y +ℝ z) → leq-ℝ x y
-    reflects-leq-right-add-ℝ x+z≤y+z =
-      preserves-leq-sim-ℝ
-        ( (x +ℝ z) -ℝ z)
-        ( x)
-        ( (y +ℝ z) -ℝ z)
-        ( y)
-        ( cancel-right-add-diff-ℝ x z)
-        ( cancel-right-add-diff-ℝ y z)
-        ( preserves-leq-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z≤y+z)
-
-    reflects-leq-left-add-ℝ : leq-ℝ (z +ℝ x) (z +ℝ y) → leq-ℝ x y
-    reflects-leq-left-add-ℝ z+x≤z+y =
-      reflects-leq-right-add-ℝ
-        ( binary-tr
-          ( leq-ℝ)
-          ( commutative-add-ℝ z x)
-          ( commutative-add-ℝ z y)
-          ( z+x≤z+y))
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  iff-translate-right-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (x +ℝ z) (y +ℝ z)
-  pr1 iff-translate-right-leq-ℝ = preserves-leq-right-add-ℝ z x y
-  pr2 iff-translate-right-leq-ℝ = reflects-leq-right-add-ℝ z x y
-
-  iff-translate-left-leq-ℝ : leq-ℝ x y ↔ leq-ℝ (z +ℝ x) (z +ℝ y)
-  pr1 iff-translate-left-leq-ℝ = preserves-leq-left-add-ℝ z x y
-  pr2 iff-translate-left-leq-ℝ = reflects-leq-left-add-ℝ z x y
-
-abstract
-  preserves-leq-add-ℝ :
-    {l1 l2 l3 l4 : Level} (a : ℝ l1) (b : ℝ l2) (c : ℝ l3) (d : ℝ l4) →
-    leq-ℝ a b → leq-ℝ c d → leq-ℝ (a +ℝ c) (b +ℝ d)
-  preserves-leq-add-ℝ a b c d a≤b c≤d =
-    transitive-leq-ℝ
-      ( a +ℝ c)
-      ( a +ℝ d)
-      ( b +ℝ d)
-      ( preserves-leq-right-add-ℝ d a b a≤b)
-      ( preserves-leq-left-add-ℝ a c d c≤d)
-```
-
-### Transposition laws
-
-```agda
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    leq-transpose-left-diff-ℝ : leq-ℝ (x -ℝ y) z → leq-ℝ x (z +ℝ y)
-    leq-transpose-left-diff-ℝ x-y≤z =
-      preserves-leq-left-sim-ℝ
-        ( z +ℝ y)
-        ( (x -ℝ y) +ℝ y)
-        ( x)
-        ( cancel-right-diff-add-ℝ x y)
-        ( preserves-leq-right-add-ℝ y (x -ℝ y) z x-y≤z)
-
-    leq-transpose-left-add-ℝ : leq-ℝ (x +ℝ y) z → leq-ℝ x (z -ℝ y)
-    leq-transpose-left-add-ℝ x+y≤z =
-      preserves-leq-left-sim-ℝ
-        (z -ℝ y)
-        ( (x +ℝ y) -ℝ y)
-        ( x)
-        ( cancel-right-add-diff-ℝ x y)
-        ( preserves-leq-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y≤z)
-
-    leq-transpose-right-add-ℝ : leq-ℝ x (y +ℝ z) → leq-ℝ (x -ℝ z) y
-    leq-transpose-right-add-ℝ x≤y+z =
-      preserves-leq-right-sim-ℝ
-        ( x -ℝ z)
-        ( (y +ℝ z) -ℝ z)
-        ( y)
-        ( cancel-right-add-diff-ℝ y z)
-        ( preserves-leq-right-add-ℝ (neg-ℝ z) x (y +ℝ z) x≤y+z)
-
-    leq-transpose-right-diff-ℝ : leq-ℝ x (y -ℝ z) → leq-ℝ (x +ℝ z) y
-    leq-transpose-right-diff-ℝ x≤y-z =
-      preserves-leq-right-sim-ℝ
-        ( x +ℝ z)
-        ( (y -ℝ z) +ℝ z)
-        ( y)
-        ( cancel-right-diff-add-ℝ y z)
-        ( preserves-leq-right-add-ℝ z x (y -ℝ z) x≤y-z)
-```
-
-### Swapping laws
-
-```agda
-abstract
-  swap-right-diff-leq-ℝ :
-    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
-    leq-ℝ (x -ℝ y) z → leq-ℝ (x -ℝ z) y
-  swap-right-diff-leq-ℝ x y z x-y≤z =
-    leq-transpose-right-add-ℝ
-      ( x)
-      ( y)
-      ( z)
-      ( tr
-        ( leq-ℝ x)
-        ( commutative-add-ℝ _ _)
-        ( leq-transpose-left-diff-ℝ x y z x-y≤z))
-```
-
-### Addition of real numbers preserves lower neighborhoods
-
-```agda
-module _
-  {l1 l2 l3 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    preserves-lower-neighborhood-leq-left-add-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d =
-      inv-tr
-        ( leq-ℝ (x +ℝ y))
-        ( associative-add-ℝ x z (real-ℚ⁺ d))
-        ( preserves-leq-left-add-ℝ
-          ( x)
-          ( y)
-          ( z +ℝ real-ℚ⁺ d)
-          ( z≤y+d))
-
-    preserves-lower-neighborhood-leq-right-add-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-right-add-ℝ z≤y+d =
-      binary-tr
-        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
-        ( commutative-add-ℝ x y)
-        ( commutative-add-ℝ x z)
-        ( preserves-lower-neighborhood-leq-left-add-ℝ z≤y+d)
-```
-
-### Addition of real numbers reflects lower neighborhoods
-
-```agda
-module _
-  {l1 l2 l3 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    reflects-lower-neighborhood-leq-left-add-ℝ :
-      leq-ℝ (x +ℝ y) ((x +ℝ z) +ℝ real-ℚ⁺ d) →
-      leq-ℝ y (z +ℝ real-ℚ⁺ d)
-    reflects-lower-neighborhood-leq-left-add-ℝ x+y≤x+z+d =
-      reflects-leq-left-add-ℝ
-        ( x)
-        ( y)
-        ( z +ℝ real-ℚ⁺ d)
-        ( tr
-          ( leq-ℝ (x +ℝ y))
-          ( associative-add-ℝ x z (real-ℚ⁺ d))
-          ( x+y≤x+z+d))
-
-    reflects-lower-neighborhood-leq-right-add-ℝ :
-      leq-ℝ (y +ℝ x) ((z +ℝ x) +ℝ real-ℚ⁺ d) →
-      leq-ℝ y (z +ℝ real-ℚ⁺ d)
-    reflects-lower-neighborhood-leq-right-add-ℝ y+x≤z+y+d =
-      reflects-lower-neighborhood-leq-left-add-ℝ
-        ( binary-tr
-          ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
-          ( commutative-add-ℝ y x)
-          ( commutative-add-ℝ z x)
-          ( y+x≤z+y+d))
-```
-
-### Negation of real numbers reverses lower neighborhoods
-
-```agda
-module _
-  {l1 l2 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2)
-  where
-
-  reverses-lower-neighborhood-neg-ℝ :
-    leq-ℝ x (y +ℝ real-ℚ⁺ d) →
-    leq-ℝ (neg-ℝ y) (neg-ℝ x +ℝ real-ℚ⁺ d)
-  reverses-lower-neighborhood-neg-ℝ x≤y+d =
-    tr
-      ( leq-ℝ (neg-ℝ y))
-      ( ( distributive-neg-add-ℝ x ((neg-ℝ ∘ real-ℚ ∘ rational-ℚ⁺) d)) ∙
-        ( ap (add-ℝ (neg-ℝ x)) (neg-neg-ℝ (real-ℚ⁺ d))))
-      ( neg-leq-ℝ
-        ( x -ℝ real-ℚ⁺ d)
-        ( y)
-        ( leq-transpose-right-add-ℝ
-          ( x)
-          ( y)
-          ( real-ℚ⁺ d)
-          ( x≤y+d)))
+      ( preserves-leq-right-sim-ℝ y1~y2 x1≤y1)
 ```
 
 ### `x ≤ q` for a rational `q` if and only if `q ∉ lower-cut-ℝ x`
diff --git a/src/real-numbers/infima-families-real-numbers.lagda.md b/src/real-numbers/infima-families-real-numbers.lagda.md
index 6b3be4283a..add144950d 100644
--- a/src/real-numbers/infima-families-real-numbers.lagda.md
+++ b/src/real-numbers/infima-families-real-numbers.lagda.md
@@ -41,6 +41,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
@@ -149,7 +150,7 @@ module _
             not-leq-le-ℝ (y i) z
               ( transitive-le-ℝ (y i) (x +ℝ real-ℚ ε) z
                 ( le-transpose-right-diff-ℝ' _ z _
-                  ( le-real-is-in-lower-cut-ℚ ε (z -ℝ x) ε<z-x))
+                  ( le-real-is-in-lower-cut-ℚ (z -ℝ x) ε<z-x))
                 ( yᵢ<x+ε))
               ( z≤yᵢ i))
     pr2 (is-greatest-lower-bound-is-infimum-family-ℝ z) z≤x i =
@@ -257,7 +258,7 @@ module _
         ( i)
         ( transitive-le-ℝ (y i) (x +ℝ real-ℚ ε) z
           ( le-transpose-right-diff-ℝ' _ z _
-            ( le-real-is-in-lower-cut-ℚ ε (z -ℝ x) ε<z-x))
+            ( le-real-is-in-lower-cut-ℚ (z -ℝ x) ε<z-x))
           ( yᵢ<x+ε))
 
   le-infimum-iff-le-element-family-ℝ :
@@ -284,7 +285,7 @@ module _
       tr
         ( leq-ℝ (neg-ℝ x))
         ( neg-neg-ℝ (y i))
-        ( neg-leq-ℝ _ _
+        ( neg-leq-ℝ
           ( is-upper-bound-is-supremum-family-ℝ
             ( neg-ℝ ∘ y)
             ( x)
@@ -299,7 +300,7 @@ module _
           binary-tr le-ℝ
             ( neg-neg-ℝ _)
             ( distributive-neg-diff-ℝ _ _ ∙ commutative-add-ℝ _ _)
-            ( neg-le-ℝ (x -ℝ real-ℚ⁺ ε) (neg-ℝ (y i)) x-ε<-yᵢ))
+            ( neg-le-ℝ x-ε<-yᵢ))
         ( is-approximated-below-is-supremum-family-ℝ
           ( neg-ℝ ∘ y)
           ( x)
@@ -327,7 +328,7 @@ module _
       tr
         ( λ w → leq-ℝ w (neg-ℝ x))
         ( neg-neg-ℝ (y i))
-        ( neg-leq-ℝ _ _
+        ( neg-leq-ℝ
           ( is-lower-bound-is-infimum-family-ℝ
             ( neg-ℝ ∘ y)
             ( x)
@@ -342,7 +343,7 @@ module _
           binary-tr le-ℝ
             ( distributive-neg-add-ℝ _ _)
             ( neg-neg-ℝ (y i))
-            ( neg-le-ℝ (neg-ℝ (y i)) (x +ℝ real-ℚ⁺ ε) -yᵢ<x+ε))
+            ( neg-le-ℝ -yᵢ<x+ε))
         ( is-approximated-above-is-infimum-family-ℝ
           ( neg-ℝ ∘ y)
           ( x)
@@ -373,7 +374,7 @@ module _
       tr
         ( leq-ℝ (neg-ℝ x))
         ( neg-neg-ℝ s)
-        ( neg-leq-ℝ _ _
+        ( neg-leq-ℝ
           ( is-upper-bound-is-supremum-family-ℝ
             ( inclusion-subset-ℝ (neg-subset-ℝ S))
             ( x)
@@ -397,7 +398,7 @@ module _
             ( le-ℝ)
             ( refl)
             ( distributive-neg-diff-ℝ x (real-ℚ⁺ ε) ∙ commutative-add-ℝ _ _)
-            ( neg-le-ℝ _ _ x-ε<s))
+            ( neg-le-ℝ x-ε<s))
 
     is-infimum-neg-supremum-neg-subset-ℝ : is-infimum-subset-ℝ S (neg-ℝ x)
     is-infimum-neg-supremum-neg-subset-ℝ =
diff --git a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
index 0549a21316..21124540b7 100644
--- a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
@@ -44,6 +44,8 @@ open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.closed-intervals-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.infima-and-suprema-families-real-numbers
 open import real-numbers.infima-families-real-numbers
@@ -53,6 +55,7 @@ open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
@@ -220,14 +223,14 @@ module _
                 ( max-net η +ℝ real-ℚ⁺ η)
                 ( max-net η +ℝ real-ℚ⁺ (ε +ℚ⁺ η))
                 ( preserves-leq-left-add-ℝ (max-net η) _ _
-                  ( preserves-leq-real-ℚ _ _ (leq-left-add-rational-ℚ⁺ _ ε)))
+                  ( preserves-leq-real-ℚ (leq-left-add-rational-ℚ⁺ _ ε)))
                 ( bound ε η))
               ( transitive-leq-ℝ
                 ( max-net η)
                 ( max-net ε +ℝ real-ℚ⁺ ε)
                 ( max-net ε +ℝ real-ℚ⁺ (ε +ℚ⁺ η))
                 ( preserves-leq-left-add-ℝ (max-net ε) _ _
-                  ( preserves-leq-real-ℚ _ _ (leq-right-add-rational-ℚ⁺ _ η)))
+                  ( preserves-leq-real-ℚ (leq-right-add-rational-ℚ⁺ _ η)))
                 ( bound η ε)))
 
     sup-modulated-totally-bounded-subset-ℝ : ℝ l2
@@ -281,7 +284,7 @@ module _
                               ( pr1 ∘ pr1)
                               ( (y , y∈S) , y∈net-ε'))))))
                     ( preserves-le-left-add-ℝ sup _ _
-                      ( preserves-le-real-ℚ _ _ ε'+ε'<ε))))))
+                      ( preserves-le-real-ℚ ε'+ε'<ε))))))
 
     is-approximated-below-sup-modulated-totally-bounded-subset-ℝ :
       is-approximated-below-family-ℝ
@@ -315,7 +318,7 @@ module _
               ( sup -ℝ real-ℚ⁺ ε)
               ( sup -ℝ real-ℚ⁺ (ε' +ℚ⁺ ε'))
               ( max-net ε' -ℝ real-ℚ⁺ ε')
-              ( reverses-le-diff-ℝ sup _ _ (preserves-le-real-ℚ _ _ ε'+ε'<ε))
+              ( reverses-le-diff-ℝ sup _ _ (preserves-le-real-ℚ ε'+ε'<ε))
               ( tr
                 ( λ y → leq-ℝ y (max-net ε' -ℝ real-ℚ⁺ ε'))
                 ( associative-add-ℝ _ _ _ ∙
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index e46f33719e..a7d3686d03 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -24,6 +24,7 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index 8854080f8e..7e5f7562a2 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -11,19 +11,24 @@ module real-numbers.isometry-negation-real-numbers where
 ```agda
 open import elementary-number-theory.positive-rational-numbers
 
+open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.function-types
 open import foundation.identity-types
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-spaces
 
+open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
 ```
 
 </details>
@@ -36,6 +41,30 @@ open import real-numbers.negation-real-numbers
 
 ## Definitions
 
+### Negation of real numbers reverses lower neighborhoods
+
+```agda
+module _
+  {l1 l2 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2)
+  where
+
+  reverses-lower-neighborhood-neg-ℝ :
+    leq-ℝ x (y +ℝ real-ℚ⁺ d) →
+    leq-ℝ (neg-ℝ y) (neg-ℝ x +ℝ real-ℚ⁺ d)
+  reverses-lower-neighborhood-neg-ℝ x≤y+d =
+    tr
+      ( leq-ℝ (neg-ℝ y))
+      ( ( distributive-neg-add-ℝ x ((neg-ℝ ∘ real-ℚ ∘ rational-ℚ⁺) d)) ∙
+        ( ap (add-ℝ (neg-ℝ x)) (neg-neg-ℝ (real-ℚ⁺ d))))
+      ( neg-leq-ℝ
+        ( leq-transpose-right-add-ℝ
+          ( x)
+          ( y)
+          ( real-ℚ⁺ d)
+          ( x≤y+d)))
+```
+
 ### Negation of a real number preserves neighborhoods
 
 ```agda
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index 9e1a1f4786..7b3d8f8c66 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -30,11 +30,13 @@ open import order-theory.large-posets
 
 open import real-numbers.absolute-value-closed-intervals-real-numbers
 open import real-numbers.absolute-value-real-numbers
+open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers
 open import real-numbers.metric-space-of-real-numbers
@@ -90,7 +92,7 @@ module _
               ( chain-of-inequalities
                 dist-ℝ (c *ℝ x) (c *ℝ y)
                 ≤ abs-ℝ c *ℝ dist-ℝ x y
-                  by leq-eq-ℝ _ _ (inv (left-distributive-abs-mul-dist-ℝ _ _ _))
+                  by leq-eq-ℝ (inv (left-distributive-abs-mul-dist-ℝ _ _ _))
                 ≤ real-ℚ⁺ q *ℝ real-ℚ⁺ ε
                   by
                     preserves-leq-mul-ℝ⁰⁺
@@ -98,14 +100,10 @@ module _
                       ( nonnegative-real-ℚ⁺ q)
                       ( nonnegative-dist-ℝ x y)
                       ( nonnegative-real-ℚ⁺ ε)
-                      ( leq-le-ℝ _ _
-                        ( le-real-is-in-upper-cut-ℚ
-                          ( rational-ℚ⁺ q)
-                          ( abs-ℝ c)
-                          ( |c|<q)))
+                      ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ (abs-ℝ c) |c|<q))
                       ( leq-dist-neighborhood-ℝ ε x y Nεxy)
                 ≤ real-ℚ⁺ (q *ℚ⁺ ε)
-                  by leq-eq-ℝ _ _ (mul-real-ℚ _ _)))
+                  by leq-eq-ℝ (mul-real-ℚ _ _)))
 
     is-lipschitz-left-mul-ℝ :
       is-lipschitz-function-Metric-Space
@@ -228,14 +226,14 @@ module _
                   by triangle-inequality-dist-ℝ _ _ _
                 ≤ dist-ℝ x₁ x₂ *ℝ abs-ℝ y₁ +ℝ abs-ℝ x₂ *ℝ dist-ℝ y₁ y₂
                   by
-                    leq-eq-ℝ _ _
+                    leq-eq-ℝ
                       ( inv
                         ( ap-add-ℝ
                           ( right-distributive-abs-mul-dist-ℝ x₁ x₂ y₁)
                           ( left-distributive-abs-mul-dist-ℝ x₂ y₁ y₂)))
                 ≤ real-ℚ⁺ ε *ℝ my +ℝ mx *ℝ real-ℚ⁺ ε
                   by
-                    preserves-leq-add-ℝ _ _ _ _
+                    preserves-leq-add-ℝ
                       ( preserves-leq-mul-ℝ⁰⁺
                         ( nonnegative-dist-ℝ x₁ x₂)
                         ( nonnegative-real-ℚ⁺ ε)
@@ -251,12 +249,10 @@ module _
                         ( is-max-mx (x₂ , x₂∈X))
                         ( leq-dist-neighborhood-ℝ ε y₁ y₂ Nεy₁y₂))
                 ≤ my *ℝ real-ℚ⁺ ε +ℝ mx *ℝ real-ℚ⁺ ε
-                  by
-                    leq-eq-ℝ _ _
-                      ( ap-add-ℝ (commutative-mul-ℝ _ _) refl)
+                  by leq-eq-ℝ (ap-add-ℝ (commutative-mul-ℝ _ _) refl)
                 ≤ (my +ℝ mx) *ℝ real-ℚ⁺ ε
                   by
-                    leq-eq-ℝ _ _
+                    leq-eq-ℝ
                       ( inv (right-distributive-mul-add-ℝ my mx (real-ℚ⁺ ε)))
                 ≤ real-ℚ q *ℝ real-ℚ⁺ ε
                   by
@@ -264,10 +260,9 @@ module _
                       ( nonnegative-real-ℚ⁺ ε)
                       ( my +ℝ mx)
                       ( real-ℚ q)
-                      ( leq-le-ℝ _ _
-                        ( le-real-is-in-upper-cut-ℚ q (my +ℝ mx) my+mx<q))
+                      ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ (my +ℝ mx) my+mx<q))
                 ≤ real-ℚ⁺ (q⁺ *ℚ⁺ ε)
-                  by leq-eq-ℝ _ _ (mul-real-ℚ q (rational-ℚ⁺ ε))))
+                  by leq-eq-ℝ (mul-real-ℚ q (rational-ℚ⁺ ε))))
 
   lipschitz-mul-inhabited-totally-bounded-subset-ℝ :
     lipschitz-function-Metric-Space
diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 9486dd6024..5e757a72e7 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -43,6 +43,7 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.suprema-families-real-numbers
 
diff --git a/src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md b/src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md
new file mode 100644
index 0000000000..6628c69c5c
--- /dev/null
+++ b/src/real-numbers/metric-space-of-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,34 @@
+# The metric space of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.metric-space-of-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+
+open import metric-spaces.metric-spaces
+
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.subsets-real-numbers
+```
+
+</details>
+
+## Idea
+
+The [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) form a
+[subspace](metric-spaces.subspaces-metric-spaces.md) of the
+[metric space of real numbers](real-numbers.metric-space-of-real-numbers.md).
+
+## Definition
+
+```agda
+metric-space-ℝ⁰⁺ : (l : Level) → Metric-Space (lsuc l) l
+metric-space-ℝ⁰⁺ l = metric-space-subset-ℝ (is-nonnegative-prop-ℝ {l})
+```
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index 9c80701de0..5afe6b7b56 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -44,10 +44,12 @@ open import metric-spaces.triangular-rational-neighborhood-relations
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers
 ```
@@ -170,15 +172,11 @@ module _
         elim-exists
           ( lower-cut-ℝ x r)
           ( λ r' (K , I') →
-            H ( positive-diff-le-ℚ (r +ℚ rational-ℚ⁺ ε) r' K)
+            H ( positive-diff-le-ℚ K)
               ( r)
               ( tr
                 ( is-in-lower-cut-ℝ y)
-                ( ( inv
-                    ( right-law-positive-diff-le-ℚ
-                      ( r +ℚ rational-ℚ⁺ ε)
-                      ( r')
-                      ( K))) ∙
+                ( ( inv (right-law-positive-diff-le-ℚ K)) ∙
                   ( associative-add-ℚ
                     ( r)
                     ( rational-ℚ⁺ ε)
@@ -221,11 +219,11 @@ module _
               ( lower-cut-ℝ y r)
               ( λ s (r<s , Lxs) →
                 pr2
-                  ( H (positive-diff-le-ℚ r s r<s))
+                  ( H (positive-diff-le-ℚ r<s))
                   ( r)
                   ( inv-tr
                     ( λ u → is-in-lower-cut-ℝ x u)
-                    ( right-law-positive-diff-le-ℚ r s r<s)
+                    ( right-law-positive-diff-le-ℚ r<s)
                     ( Lxs)))
               ( forward-implication (is-rounded-lower-cut-ℝ x r) Lxr))
           ( λ r Lyr →
@@ -233,11 +231,11 @@ module _
               ( lower-cut-ℝ x r)
               ( λ s (r<s , Lys) →
                 pr1
-                  ( H (positive-diff-le-ℚ r s r<s))
+                  ( H (positive-diff-le-ℚ r<s))
                   ( r)
                   ( inv-tr
                     ( λ u → is-in-lower-cut-ℝ y u)
-                    ( right-law-positive-diff-le-ℚ r s r<s)
+                    ( right-law-positive-diff-le-ℚ r<s)
                     ( Lys)))
               ( forward-implication (is-rounded-lower-cut-ℝ y r) Lyr)))
 
@@ -284,7 +282,6 @@ module _
       lower-neighborhood-ℝ d x y
     lower-neighborhood-real-bound-leq-ℝ (d , _) x y y≤x+d q q+d<y =
       is-in-lower-cut-le-real-ℚ
-        ( q)
         ( x)
         ( concatenate-le-leq-ℝ
           ( real-ℚ q)
@@ -297,7 +294,7 @@ module _
             ( inv-tr
               ( λ z → le-ℝ z y)
               ( add-real-ℚ q d)
-              ( le-real-is-in-lower-cut-ℚ (q +ℚ d) y q+d<y)))
+              ( le-real-is-in-lower-cut-ℚ y q+d<y)))
           ( leq-transpose-right-add-ℝ y x (real-ℚ d) y≤x+d))
 
   opaque
diff --git a/src/real-numbers/multiplication-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
index af8ae9cbfa..e13cdd5cc2 100644
--- a/src/real-numbers/multiplication-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
@@ -76,7 +76,7 @@ abstract
       ( le-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-le-ℝ _ _ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-le-ℝ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
 ```
 
 ### Multiplication by a negative real number reverses inequality
@@ -91,5 +91,5 @@ abstract
       ( leq-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-leq-ℝ _ _ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-leq-ℝ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
 ```
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index 706360afa7..dd18d3c8e5 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -27,6 +27,7 @@ open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
@@ -62,8 +63,8 @@ opaque
               ([c,d]@((c , d) , _) , c<y , y<d)) ,
             [a,b][c,d]<q) ← xy<q
           let
-            b⁺ = (b , is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) b x<b)
-            d⁺ = (d , is-positive-is-in-upper-cut-ℝ⁰⁺ (y , 0≤y) d y<d)
+            b⁺ = (b , is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) x<b)
+            d⁺ = (d , is-positive-is-in-upper-cut-ℝ⁰⁺ (y , 0≤y) y<d)
           is-positive-le-ℚ⁺
             ( b⁺ *ℚ⁺ d⁺)
             ( q)
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index ea78969785..e26e0fb218 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -477,9 +477,9 @@ module _
                         r<y ,
                         y<s))
               ≤ rational-ℕ N
-                by preserves-leq-rational-ℕ _ _ (right-leq-max-ℕ _ _)
+                by preserves-leq-rational-ℕ (right-leq-max-ℕ _ _)
               ≤ rational-ℕ (succ-ℕ N)
-                by preserves-leq-rational-ℕ _ _ (succ-leq-ℕ N)
+                by preserves-leq-rational-ℕ (succ-leq-ℕ N)
           max|p||q|≤sN =
             chain-of-inequalities
               max-ℚ (rational-abs-ℚ p) (rational-abs-ℚ q)
@@ -495,9 +495,9 @@ module _
                         p<x ,
                         x<q))
               ≤ rational-ℕ N
-                by preserves-leq-rational-ℕ _ _ (left-leq-max-ℕ _ _)
+                by preserves-leq-rational-ℕ (left-leq-max-ℕ _ _)
               ≤ rational-ℕ (succ-ℕ N)
-                by preserves-leq-rational-ℕ _ _ (succ-leq-ℕ N)
+                by preserves-leq-rational-ℕ (succ-leq-ℕ N)
           [p,q] = ((p , q) , p≤q)
           [r,s] = ((r , s) , r≤s)
           a = lower-bound-mul-closed-interval-ℚ [p,q] [r,s]
@@ -531,7 +531,7 @@ module _
               ≤ ( rational-ℚ⁺ εx +ℚ rational-ℚ⁺ εy) *ℚ
                 rational-ℕ (succ-ℕ N)
                 by
-                  leq-eq-ℚ _ _
+                  leq-eq-ℚ
                     ( inv (right-distributive-mul-add-ℚ _ _ _))
               ≤ rational-ℚ⁺
                   ( ( εx₀ *ℚ⁺ positive-reciprocal-rational-succ-ℕ N) +ℚ⁺
@@ -551,7 +551,7 @@ module _
                         ( εy₀ *ℚ⁺ positive-reciprocal-rational-succ-ℕ N)))
               ≤ rational-ℚ⁺ ε
                 by
-                  leq-eq-ℚ _ _
+                  leq-eq-ℚ
                     ( ap-mul-ℚ
                       ( inv (right-distributive-mul-add-ℚ _ _ _))
                       ( refl) ∙
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 73cd01dce6..d17a359116 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -58,6 +58,7 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-positive-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-positive-real-numbers
 open import real-numbers.strict-inequality-real-numbers
@@ -439,8 +440,6 @@ module _
                         ( x<p⁻¹))))
               ( is-located-lower-upper-cut-ℝ
                 ( real-ℝ⁺ x)
-                ( rational-ℚ⁺ (inv-ℚ⁺ q⁺))
-                ( rational-ℚ⁺ (inv-ℚ⁺ p⁺))
                 ( inv-le-ℚ⁺ p⁺ q⁺ p<q)))
         ( λ is-nonpos-p →
           inl-disjunction
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 832384eb28..183ddbcd09 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -83,7 +83,7 @@ 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))
+      ( is-located-lower-upper-cut-ℝ x (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 bf51c3d32b..6b5a11da37 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -40,6 +40,7 @@ open import metric-spaces.metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.saturation-inequality-real-numbers
@@ -125,12 +126,14 @@ eq-ℝ⁰⁺ _ _ = eq-type-subtype is-nonnegative-prop-ℝ
 
 ```agda
 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 nonneg-q)
+  preserves-is-nonnegative-real-ℚ :
+    {q : ℚ} → is-nonnegative-ℚ q → is-nonnegative-ℝ (real-ℚ q)
+  preserves-is-nonnegative-real-ℚ is-nonneg-q =
+    preserves-leq-real-ℚ (leq-zero-is-nonnegative-ℚ is-nonneg-q)
 
 nonnegative-real-ℚ⁰⁺ : ℚ⁰⁺ → ℝ⁰⁺ lzero
-nonnegative-real-ℚ⁰⁺ q = (real-ℚ⁰⁺ q , is-nonnegative-real-ℚ⁰⁺ q)
+nonnegative-real-ℚ⁰⁺ (q , is-nonneg-q) =
+  ( real-ℚ q , preserves-is-nonnegative-real-ℚ is-nonneg-q)
 
 nonnegative-real-ℚ⁺ : ℚ⁺ → ℝ⁰⁺ lzero
 nonnegative-real-ℚ⁺ q = nonnegative-real-ℚ⁰⁺ (nonnegative-ℚ⁺ q)
@@ -146,102 +149,19 @@ one-ℝ⁰⁺ : ℝ⁰⁺ lzero
 one-ℝ⁰⁺ = nonnegative-real-ℚ⁰⁺ one-ℚ⁰⁺
 ```
 
-### Similarity of nonnegative real numbers
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  sim-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
-  sim-prop-ℝ⁰⁺ = sim-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-
-  sim-ℝ⁰⁺ : UU (l1 ⊔ l2)
-  sim-ℝ⁰⁺ = sim-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-
-infix 6 _~ℝ⁰⁺_
-_~ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → UU (l1 ⊔ l2)
-_~ℝ⁰⁺_ = sim-ℝ⁰⁺
-
-sim-zero-prop-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → Prop l
-sim-zero-prop-ℝ⁰⁺ = sim-prop-ℝ⁰⁺ zero-ℝ⁰⁺
-
-sim-zero-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → UU l
-sim-zero-ℝ⁰⁺ = sim-ℝ⁰⁺ zero-ℝ⁰⁺
-
-eq-sim-ℝ⁰⁺ : {l : Level} (x y : ℝ⁰⁺ l) → sim-ℝ⁰⁺ x y → x ＝ y
-eq-sim-ℝ⁰⁺ x y x~y = eq-ℝ⁰⁺ x y (eq-sim-ℝ {x = real-ℝ⁰⁺ x} {y = real-ℝ⁰⁺ y} x~y)
-```
-
-#### Similarity is symmetric
-
-```agda
-abstract
-  symmetric-sim-ℝ⁰⁺ :
-    {l1 l2 : Level} → (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) → x ~ℝ⁰⁺ y → y ~ℝ⁰⁺ x
-  symmetric-sim-ℝ⁰⁺ _ _ = symmetric-sim-ℝ
-```
-
-### Inequality on nonnegative real numbers
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  leq-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
-  leq-prop-ℝ⁰⁺ = leq-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-
-  leq-ℝ⁰⁺ : UU (l1 ⊔ l2)
-  leq-ℝ⁰⁺ = type-Prop leq-prop-ℝ⁰⁺
-```
-
-### Zero is less than or equal to every nonnegative real number
-
-```agda
-leq-zero-ℝ⁰⁺ : {l : Level} (x : ℝ⁰⁺ l) → leq-ℝ⁰⁺ zero-ℝ⁰⁺ x
-leq-zero-ℝ⁰⁺ = is-nonnegative-real-ℝ⁰⁺
-```
-
-### Similarity preserves inequality
-
-```agda
-module _
-  {l1 l2 l3 : Level} (z : ℝ⁰⁺ l1) (x : ℝ⁰⁺ l2) (y : ℝ⁰⁺ l3) (x~y : sim-ℝ⁰⁺ x y)
-  where
-
-  abstract
-    preserves-leq-left-sim-ℝ⁰⁺ : leq-ℝ⁰⁺ x z → leq-ℝ⁰⁺ y z
-    preserves-leq-left-sim-ℝ⁰⁺ =
-      preserves-leq-left-sim-ℝ (real-ℝ⁰⁺ z) _ _ x~y
-```
-
-### Inequality is transitive
-
-```agda
-module _
-  {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3)
-  where
-
-  transitive-leq-ℝ⁰⁺ : leq-ℝ⁰⁺ y z → leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ x z
-  transitive-leq-ℝ⁰⁺ = transitive-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z)
-```
-
 ### A real number is nonnegative if and only if every element of its upper cut is positive
 
 ```agda
 abstract
   is-positive-is-in-upper-cut-ℝ⁰⁺ :
-    {l : Level} → (x : ℝ⁰⁺ l) (q : ℚ) → is-in-upper-cut-ℝ⁰⁺ x q →
+    {l : Level} → (x : ℝ⁰⁺ l) {q : ℚ} → is-in-upper-cut-ℝ⁰⁺ x q →
     is-positive-ℚ q
-  is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) q x<q =
+  is-positive-is-in-upper-cut-ℝ⁰⁺ (x , 0≤x) x<q =
     is-positive-le-zero-ℚ
       ( reflects-le-real-ℚ
-        ( zero-ℚ)
-        ( q)
         ( concatenate-leq-le-ℝ zero-ℝ x _
           ( 0≤x)
-          ( le-real-is-in-upper-cut-ℚ q x x<q)))
+          ( le-real-is-in-upper-cut-ℚ x x<q)))
 
 opaque
   unfolding leq-ℝ leq-ℝ' real-ℚ
@@ -283,123 +203,7 @@ abstract
     let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → upper-cut-ℝ⁰⁺ x (rational-ℚ⁺ q)))
     in do
       (q , x<q) ← is-inhabited-upper-cut-ℝ (real-ℝ⁰⁺ x)
-      intro-exists (q , is-positive-is-in-upper-cut-ℝ⁰⁺ x q x<q) x<q
-```
-
-### Addition on nonnegative real numbers
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  real-add-ℝ⁰⁺ : ℝ (l1 ⊔ l2)
-  real-add-ℝ⁰⁺ = real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y
-
-  abstract
-    is-nonnegative-real-add-ℝ⁰⁺ : is-nonnegative-ℝ real-add-ℝ⁰⁺
-    is-nonnegative-real-add-ℝ⁰⁺ =
-      tr
-        ( λ z → leq-ℝ z (real-ℝ⁰⁺ x +ℝ real-ℝ⁰⁺ y))
-        ( left-unit-law-add-ℝ zero-ℝ)
-        ( preserves-leq-add-ℝ
-          ( zero-ℝ)
-          ( real-ℝ⁰⁺ x)
-          ( zero-ℝ)
-          ( real-ℝ⁰⁺ y)
-          ( is-nonnegative-real-ℝ⁰⁺ x)
-          ( is-nonnegative-real-ℝ⁰⁺ y))
-
-  add-ℝ⁰⁺ : ℝ⁰⁺ (l1 ⊔ l2)
-  add-ℝ⁰⁺ = (real-add-ℝ⁰⁺ , is-nonnegative-real-add-ℝ⁰⁺)
-
-infixl 35 _+ℝ⁰⁺_
-
-_+ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → ℝ⁰⁺ (l1 ⊔ l2)
-_+ℝ⁰⁺_ = add-ℝ⁰⁺
-```
-
-### Unit laws for addition
-
-```agda
-module _
-  {l : Level} (x : ℝ⁰⁺ l)
-  where
-
-  abstract
-    left-unit-law-add-ℝ⁰⁺ : zero-ℝ⁰⁺ +ℝ⁰⁺ x ＝ x
-    left-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (left-unit-law-add-ℝ _)
-
-    right-unit-law-add-ℝ⁰⁺ : x +ℝ⁰⁺ zero-ℝ⁰⁺ ＝ x
-    right-unit-law-add-ℝ⁰⁺ = eq-ℝ⁰⁺ _ _ (right-unit-law-add-ℝ _)
-```
-
-### Addition preserves inequality
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
-  where
-
-  abstract
-    preserves-leq-add-ℝ⁰⁺ :
-      leq-ℝ⁰⁺ x y → leq-ℝ⁰⁺ z w → leq-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
-    preserves-leq-add-ℝ⁰⁺ =
-      preserves-leq-add-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z) (real-ℝ⁰⁺ w)
-```
-
-### If `x ≤ y + ε` for every `ε : ℚ⁺`, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    saturated-leq-ℝ⁰⁺ :
-      ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (y +ℝ⁰⁺ nonnegative-real-ℚ⁺ ε)) →
-      leq-ℝ⁰⁺ x y
-    saturated-leq-ℝ⁰⁺ = saturated-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-```
-
-### If a nonnegative real number is less than or equal to all positive rational numbers, it is similar to zero
-
-```agda
-sim-zero-le-positive-rational-ℝ⁰⁺ :
-  {l : Level} (x : ℝ⁰⁺ l) →
-  ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (nonnegative-real-ℚ⁺ ε)) →
-  sim-zero-ℝ⁰⁺ x
-sim-zero-le-positive-rational-ℝ⁰⁺ x H =
-  sim-sim-leq-ℝ
-    ( leq-zero-ℝ⁰⁺ x ,
-      saturated-leq-ℝ⁰⁺
-        ( x)
-        ( zero-ℝ⁰⁺)
-        ( λ ε → inv-tr (leq-ℝ⁰⁺ x) (left-unit-law-add-ℝ⁰⁺ _) (H ε)))
-```
-
-### The canonical embedding of nonnegative rational numbers to nonnegative real numbers preserves addition
-
-```agda
-abstract
-  add-nonnegative-real-ℚ⁰⁺ :
-    (p q : ℚ⁰⁺) →
-    nonnegative-real-ℚ⁰⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁰⁺ q ＝
-    nonnegative-real-ℚ⁰⁺ (p +ℚ⁰⁺ q)
-  add-nonnegative-real-ℚ⁰⁺ p q =
-    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁰⁺ p) (rational-ℚ⁰⁺ q))
-```
-
-### The canonical embedding of positive rational numbers to nonnegative real numbers preserves addition
-
-```agda
-abstract
-  add-nonnegative-real-ℚ⁺ :
-    (p q : ℚ⁺) →
-    nonnegative-real-ℚ⁺ p +ℝ⁰⁺ nonnegative-real-ℚ⁺ q ＝
-    nonnegative-real-ℚ⁺ (p +ℚ⁺ q)
-  add-nonnegative-real-ℚ⁺ p q =
-    eq-ℝ⁰⁺ _ _ (add-real-ℚ (rational-ℚ⁺ p) (rational-ℚ⁺ q))
+      intro-exists (q , is-positive-is-in-upper-cut-ℝ⁰⁺ x x<q) x<q
 ```
 
 ### Nonpositive rational numbers are not less than or equal to nonnegative real numbers
@@ -419,7 +223,7 @@ module _
           ( x<q)
           ( transitive-leq-ℝ (real-ℚ q) zero-ℝ (real-ℝ⁰⁺ x)
             ( is-nonnegative-real-ℝ⁰⁺ x)
-            ( preserves-leq-real-ℚ _ _ q≤0)))
+            ( preserves-leq-real-ℚ q≤0)))
 ```
 
 ### If a nonnegative real number is less than a rational number, the rational number is positive
@@ -434,107 +238,10 @@ 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
-        ( reflects-le-real-ℚ _ _
+        ( reflects-le-real-ℚ
           ( concatenate-leq-le-ℝ _ _ _ (is-nonnegative-real-ℝ⁰⁺ x) x<q))
 ```
 
-### If `x` is less than all the positive rational numbers `y` is less than, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    leq-le-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) → le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
-        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
-      leq-ℝ⁰⁺ x y
-    leq-le-positive-rational-ℝ⁰⁺ H =
-      leq-le-rational-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
-        ( λ q y<q →
-          rec-coproduct
-            ( λ 0<q →
-              let open do-syntax-trunc-Prop (le-prop-ℝ (real-ℝ⁰⁺ x) (real-ℚ q))
-              in do
-                (p , y<p , p<q) ← dense-rational-le-ℝ _ _ y<q
-                transitive-le-ℝ _ (real-ℚ p) _
-                  ( p<q)
-                  ( H
-                    ( p , is-positive-le-nonnegative-real-ℚ y p y<p)
-                    ( y<p)))
-            ( λ q≤0 → ex-falso (not-le-leq-zero-rational-ℝ⁰⁺ y q q≤0 y<q))
-            ( decide-le-leq-ℚ zero-ℚ q))
-```
-
-### If `x` is less than or equal to all the positive rational numbers `y` is less than or equal to, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    leq-leq-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) → leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q) →
-        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q)) →
-      leq-ℝ⁰⁺ x y
-    leq-leq-positive-rational-ℝ⁰⁺ H =
-      leq-le-positive-rational-ℝ⁰⁺ x y
-        ( λ q y<q →
-          let open do-syntax-trunc-Prop (le-prop-ℝ _ _)
-          in do
-            (r , y<r , r<q) ← dense-rational-le-ℝ _ _ y<q
-            concatenate-leq-le-ℝ _ _ _
-              ( H
-                ( r , is-positive-le-nonnegative-real-ℚ y r y<r)
-                ( leq-le-ℝ _ _ y<r))
-              ( r<q))
-```
-
-### If `x` is less than or equal to the same positive rational numbers `y` is less than or equal to, then `x` and `y` are similar
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    sim-leq-same-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) →
-        leq-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ leq-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
-      sim-ℝ⁰⁺ x y
-    sim-leq-same-positive-rational-ℝ⁰⁺ H =
-      sim-sim-leq-ℝ
-        ( leq-leq-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
-          leq-leq-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
-```
-
-### If `x` is less than the same positive rational numbers `y` is less than, then `x` and `y` are similar
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
-  where
-
-  abstract
-    sim-le-same-positive-rational-ℝ⁰⁺ :
-      ( (q : ℚ⁺) →
-        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
-      sim-ℝ⁰⁺ x y
-    sim-le-same-positive-rational-ℝ⁰⁺ H =
-      sim-sim-leq-ℝ
-        ( leq-le-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
-          leq-le-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
-```
-
-### The metric space of nonnegative real numbers
-
-```agda
-metric-space-ℝ⁰⁺ : (l : Level) → Metric-Space (lsuc l) l
-metric-space-ℝ⁰⁺ l = metric-space-subset-ℝ (is-nonnegative-prop-ℝ {l})
-```
-
 ### `x ≤ y` if and only if `y - x` is nonnegative
 
 ```agda
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index a334945845..850496dd46 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -43,7 +43,7 @@ On this page, we outline basic relations between
 abstract
   is-nonnegative-is-positive-ℝ :
     {l : Level} {x : ℝ l} → is-positive-ℝ x → is-nonnegative-ℝ x
-  is-nonnegative-is-positive-ℝ = leq-le-ℝ _ _
+  is-nonnegative-is-positive-ℝ = leq-le-ℝ
 
 nonnegative-ℝ⁺ : {l : Level} → ℝ⁺ l → ℝ⁰⁺ l
 nonnegative-ℝ⁺ (x , is-pos-x) = (x , is-nonnegative-is-positive-ℝ is-pos-x)
@@ -59,7 +59,7 @@ abstract
     tr
       ( le-ℝ (neg-ℝ x))
       ( neg-zero-ℝ)
-      ( neg-le-ℝ zero-ℝ x 0<x)
+      ( neg-le-ℝ 0<x)
 
 neg-ℝ⁺ : {l : Level} → ℝ⁺ l → ℝ⁻ l
 neg-ℝ⁺ (x , is-pos-x) = (neg-ℝ x , neg-is-positive-ℝ x is-pos-x)
@@ -75,7 +75,7 @@ abstract
     tr
       ( λ z → le-ℝ z (neg-ℝ x))
       ( neg-zero-ℝ)
-      ( neg-le-ℝ x zero-ℝ x<0)
+      ( neg-le-ℝ x<0)
 
 neg-ℝ⁻ : {l : Level} → ℝ⁻ l → ℝ⁺ l
 neg-ℝ⁻ (x , is-neg-x) = (neg-ℝ x , neg-is-negative-ℝ x is-neg-x)
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 1891f071d6..3e3881f93c 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -39,6 +39,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -132,8 +133,7 @@ module _
   abstract
     is-positive-iff-zero-in-lower-cut-ℝ :
       is-positive-ℝ x ↔ is-in-lower-cut-ℝ x zero-ℚ
-    is-positive-iff-zero-in-lower-cut-ℝ =
-      inv-iff (le-real-iff-lower-cut-ℚ zero-ℚ x)
+    is-positive-iff-zero-in-lower-cut-ℝ = inv-iff (le-real-iff-lower-cut-ℚ x)
 
     is-positive-zero-in-lower-cut-ℝ :
       is-in-lower-cut-ℝ x zero-ℚ → is-positive-ℝ x
@@ -254,24 +254,19 @@ abstract
 ### The canonical embedding of rational numbers preserves positivity
 
 ```agda
-preserves-is-positive-real-ℚ :
-  (q : ℚ) → is-positive-ℚ q → is-positive-ℝ (real-ℚ 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
+abstract
+  preserves-is-positive-real-ℚ :
+    {q : ℚ} → is-positive-ℚ q → is-positive-ℝ (real-ℚ q)
+  preserves-is-positive-real-ℚ pos-q =
+    preserves-le-real-ℚ (le-zero-is-positive-ℚ pos-q)
 
-  is-positive-rational-is-positive-real-ℚ :
-    (q : ℚ) → is-positive-ℝ (real-ℚ q) → is-positive-ℚ q
-  is-positive-rational-is-positive-real-ℚ q =
-    elim-exists
-      ( is-positive-prop-ℚ q)
-      ( λ r (0<r , r<q) →
-        is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ r q r<q 0<r))
+  reflects-is-positive-real-ℚ :
+    {q : ℚ} → is-positive-ℝ (real-ℚ q) → is-positive-ℚ q
+  reflects-is-positive-real-ℚ {q} 0<qℝ =
+    is-positive-le-zero-ℚ (reflects-le-real-ℚ 0<qℝ)
 
 positive-real-ℚ⁺ : ℚ⁺ → ℝ⁺ lzero
-positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ q pos-q)
+positive-real-ℚ⁺ (q , pos-q) = (real-ℚ q , preserves-is-positive-real-ℚ pos-q)
 
 one-ℝ⁺ : ℝ⁺ lzero
 one-ℝ⁺ = positive-real-ℚ⁺ one-ℚ⁺
@@ -287,13 +282,3 @@ abstract
     is-positive-le-zero-ℚ
       ( le-lower-upper-cut-ℝ x zero-ℚ q (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
 ```
-
-### Similarity of positive real numbers
-
-```agda
-sim-prop-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → Prop (l1 ⊔ l2)
-sim-prop-ℝ⁺ (x , _) (y , _) = sim-prop-ℝ x y
-
-sim-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → UU (l1 ⊔ l2)
-sim-ℝ⁺ (x , _) (y , _) = sim-ℝ x y
-```
diff --git a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
index 8af1317ffa..2782e63dab 100644
--- a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
+++ b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
@@ -154,7 +154,7 @@ module _
       map-disjunction
         ( map-raise)
         ( map-raise)
-        ( is-located-lower-upper-cut-ℝ x p q p<q)
+        ( is-located-lower-upper-cut-ℝ x p<q)
 
   raise-ℝ : ℝ (l0 ⊔ l)
   raise-ℝ =
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index 63ec308b13..c639837691 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -66,12 +66,10 @@ the [image](foundation.images.md) of this embedding
 ```agda
 is-dedekind-lower-upper-real-ℚ :
   (x : ℚ) →
-  is-dedekind-lower-upper-ℝ
-    ( lower-real-ℚ x)
-    ( upper-real-ℚ x)
+  is-dedekind-lower-upper-ℝ (lower-real-ℚ x) (upper-real-ℚ x)
 is-dedekind-lower-upper-real-ℚ x =
-  (λ q (H , K) → asymmetric-le-ℚ q x H K) ,
-  located-le-ℚ x
+  ( (λ q (H , K) → asymmetric-le-ℚ q x H K) ,
+    located-le-ℚ x)
 ```
 
 ### The canonical map from `ℚ` to `ℝ lzero`
@@ -161,7 +159,7 @@ all-eq-is-rational-ℝ x p q H H' =
         ( empty-Prop)
         ( pr1 H)
         ( pr2 H')
-        ( is-located-lower-upper-cut-ℝ x p q I))
+        ( is-located-lower-upper-cut-ℝ x I))
 
   right-case : le-ℚ q p → p ＝ q
   right-case I =
@@ -170,7 +168,7 @@ all-eq-is-rational-ℝ x p q H H' =
         ( empty-Prop)
         ( pr1 H')
         ( pr2 H)
-        ( is-located-lower-upper-cut-ℝ x q p I))
+        ( is-located-lower-upper-cut-ℝ x I))
 
 is-prop-rational-real : {l : Level} (x : ℝ l) → is-prop (Σ ℚ (is-rational-ℝ x))
 is-prop-rational-real x =
@@ -240,7 +238,7 @@ opaque
       ( lower-cut-ℝ x p)
       ( id)
       ( ex-falso ∘ q∉ux)
-      ( is-located-lower-upper-cut-ℝ x p q p<q)
+      ( is-located-lower-upper-cut-ℝ x p<q)
 
 eq-real-rational-is-rational-ℝ :
   (x : ℝ lzero) (q : ℚ) (H : is-rational-ℝ x q) → real-ℚ q ＝ x
diff --git a/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
new file mode 100644
index 0000000000..037e54c2f2
--- /dev/null
+++ b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,71 @@
+# Saturation of inequality of nonnegative real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.saturation-inequality-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-nonnegative-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.saturation-inequality-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
+```
+
+</details>
+
+## Idea
+
+If `x ≤ y + ε` for [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) `x` and `y` and every
+[positive rational](elementary-number-theory.positive-rational-numbers.md) `ε`,
+then `x ≤ y`.
+
+Despite being a property of
+[inequality of nonnegative real numbers](real-numbers.inequality-nonnegative-real-numbers.md),
+this is much easier to prove via
+[strict inequality](real-numbers.strict-inequality-nonnegative-real-numbers.md),
+so it is moved to its own file to prevent circular dependency.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    saturated-leq-ℝ⁰⁺ :
+      ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (y +ℝ⁰⁺ nonnegative-real-ℚ⁺ ε)) →
+      leq-ℝ⁰⁺ x y
+    saturated-leq-ℝ⁰⁺ = saturated-leq-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+```
+
+## Corollaries
+
+### If a nonnegative real number is less than or equal to all positive rational numbers, it is similar to zero
+
+```agda
+sim-zero-le-positive-rational-ℝ⁰⁺ :
+  {l : Level} (x : ℝ⁰⁺ l) →
+  ((ε : ℚ⁺) → leq-ℝ⁰⁺ x (nonnegative-real-ℚ⁺ ε)) →
+  sim-zero-ℝ⁰⁺ x
+sim-zero-le-positive-rational-ℝ⁰⁺ x H =
+  sim-sim-leq-ℝ
+    ( leq-zero-ℝ⁰⁺ x ,
+      saturated-leq-ℝ⁰⁺
+        ( x)
+        ( zero-ℝ⁰⁺)
+        ( λ ε → inv-tr (leq-ℝ⁰⁺ x) (left-unit-law-add-ℝ⁰⁺ _) (H ε)))
+```
diff --git a/src/real-numbers/saturation-inequality-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
index 6657f97f72..437a1a7a32 100644
--- a/src/real-numbers/saturation-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
@@ -12,12 +12,16 @@ module real-numbers.saturation-inequality-real-numbers where
 open import elementary-number-theory.positive-rational-numbers
 open import elementary-number-theory.strict-inequality-positive-rational-numbers
 
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.propositional-truncations
 open import foundation.universe-levels
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -38,6 +42,29 @@ moved to its own file to prevent circular dependency.
 
 ## Proof
 
+### If `x < y + ε` for every positive rational `ε`, then `x ≤ y`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  where
+
+  abstract
+    saturated-le-ℝ : ((ε : ℚ⁺) → le-ℝ x (y +ℝ real-ℚ⁺ ε)) → leq-ℝ x y
+    saturated-le-ℝ H =
+      leq-not-le-ℝ y x
+        ( λ y<x →
+          let open do-syntax-trunc-Prop empty-Prop
+          in do
+            (ε , y+ε<x) ←
+              exists-positive-rational-separation-le-ℝ {x = y} {y = x} y<x
+            irreflexive-le-ℝ
+              ( x)
+              ( transitive-le-ℝ x (y +ℝ real-ℚ⁺ ε) x y+ε<x (H ε)))
+```
+
+### If `x ≤ y + ε` for every positive rational `ε`, then `x ≤ y`
+
 ```agda
 module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
@@ -57,5 +84,5 @@ module _
               ( y)
               ( real-ℚ⁺ (mediant-zero-ℚ⁺ ε))
               ( real-ℚ⁺ ε)
-              ( preserves-le-real-ℚ _ _ (le-mediant-zero-ℚ⁺ ε))))
+              ( preserves-le-real-ℚ (le-mediant-zero-ℚ⁺ ε))))
 ```
diff --git a/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
new file mode 100644
index 0000000000..3686eb0a01
--- /dev/null
+++ b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
@@ -0,0 +1,70 @@
+# Similarity of nonnegative real numbers
+
+```agda
+module real-numbers.similarity-nonnegative-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.propositions
+open import foundation.universe-levels
+
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+Two [nonnegative](real-numbers.nonnegative-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are
+{{#concept "similar" Disambiguation="nonnegative real numbers" Agda=sim-ℝ⁰⁺}} if
+they are [similar](real-numbers.similarity-real-numbers.md) as real numbers.
+
+## Definition
+
+### Similarity of nonnegative real numbers
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  sim-prop-ℝ⁰⁺ : Prop (l1 ⊔ l2)
+  sim-prop-ℝ⁰⁺ = sim-prop-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+
+  sim-ℝ⁰⁺ : UU (l1 ⊔ l2)
+  sim-ℝ⁰⁺ = sim-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
+
+infix 6 _~ℝ⁰⁺_
+_~ℝ⁰⁺_ : {l1 l2 : Level} → ℝ⁰⁺ l1 → ℝ⁰⁺ l2 → UU (l1 ⊔ l2)
+_~ℝ⁰⁺_ = sim-ℝ⁰⁺
+
+sim-zero-prop-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → Prop l
+sim-zero-prop-ℝ⁰⁺ = sim-prop-ℝ⁰⁺ zero-ℝ⁰⁺
+
+sim-zero-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → UU l
+sim-zero-ℝ⁰⁺ = sim-ℝ⁰⁺ zero-ℝ⁰⁺
+
+eq-sim-ℝ⁰⁺ : {l : Level} (x y : ℝ⁰⁺ l) → sim-ℝ⁰⁺ x y → x ＝ y
+eq-sim-ℝ⁰⁺ x y x~y = eq-ℝ⁰⁺ x y (eq-sim-ℝ {x = real-ℝ⁰⁺ x} {y = real-ℝ⁰⁺ y} x~y)
+```
+
+#### Similarity is symmetric
+
+```agda
+abstract
+  symmetric-sim-ℝ⁰⁺ :
+    {l1 l2 : Level} → (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) → x ~ℝ⁰⁺ y → y ~ℝ⁰⁺ x
+  symmetric-sim-ℝ⁰⁺ _ _ = symmetric-sim-ℝ
+```
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 a9576b242d..154237168d 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -54,6 +54,7 @@ open import real-numbers.multiplication-nonnegative-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.squares-real-numbers
 ```
@@ -184,7 +185,7 @@ module _
         (p , p<r² , x<p) ←
           forward-implication (is-rounded-upper-cut-ℝ⁰⁺ x (r *ℚ r)) x<r²
         let
-          is-pos-p = is-positive-is-in-upper-cut-ℝ⁰⁺ x p x<p
+          is-pos-p = is-positive-is-in-upper-cut-ℝ⁰⁺ x x<p
         (q⁺@(q , is-pos-q) , q<r , p<q²) ←
           rounded-above-square-le-ℚ⁺ (r , is-pos-r) (p , is-pos-p) p<r²
         intro-exists
@@ -270,8 +271,6 @@ module _
             ( λ x<q² → (is-positive-le-ℚ⁰⁺ (p , is-nonneg-p) q p<q , x<q²))
             ( is-located-lower-upper-cut-ℝ
               ( real-ℝ⁰⁺ x)
-              ( p *ℚ p)
-              ( q *ℚ q)
               ( preserves-le-square-ℚ⁰⁺
                 ( p , is-nonneg-p)
                 ( q , is-nonnegative-le-ℚ⁰⁺ (p , is-nonneg-p) q p<q)
@@ -533,7 +532,7 @@ opaque
     leq-ℝ' (real-sqrt-ℝ⁰⁺ x) (real-ℝ⁰⁺ y)
   leq-unique-sqrt-ℝ⁰⁺' x⁰⁺@(x , _) y⁰⁺@(y , is-nonneg-y) (_ , x≤y²) q y<q =
     let
-      is-pos-q = is-positive-is-in-upper-cut-ℝ⁰⁺ y⁰⁺ q y<q
+      is-pos-q = is-positive-is-in-upper-cut-ℝ⁰⁺ y⁰⁺ y<q
       q⁺ = (q , is-pos-q)
       p⁻@(p , is-neg-p) = neg-ℚ⁺ q⁺
       p≤q = leq-negative-positive-ℚ p⁻ q⁺
@@ -554,7 +553,7 @@ opaque
               ( ([p,q] , p<y , y<q) , ([p,q] , p<y , y<q))
               ( leq-max-leq-both-ℚ _ _ _
                 ( leq-max-leq-both-ℚ _ _ _
-                  ( leq-eq-ℚ _ _ (negative-law-mul-ℚ q q))
+                  ( leq-eq-ℚ (negative-law-mul-ℚ q q))
                   ( leq-negative-positive-ℚ
                     ( mul-negative-positive-ℚ p⁻ q⁺)
                     ( q⁺ *ℚ⁺ q⁺)))
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 7c9c42cc2d..36ce763cb8 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -193,7 +193,7 @@ abstract
       ( square-ℝ x)
       ( x *ℝ y)
       ( square-ℝ y)
-      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ x y (leq-le-ℝ x y x<y))
+      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ x y (leq-le-ℝ x<y))
       ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x y x<y)
 ```
 
@@ -207,10 +207,9 @@ abstract
   is-in-lower-cut-square-ℝ x q⁰⁺@(q , _) q∈Lx =
     let
       qℝ = nonnegative-real-ℚ⁰⁺ q⁰⁺
-      q<x = le-real-is-in-lower-cut-ℚ q x q∈Lx
+      q<x = le-real-is-in-lower-cut-ℚ x q∈Lx
     in
       is-in-lower-cut-le-real-ℚ
-        ( square-ℚ q)
         ( square-ℝ x)
         ( tr
           ( λ y → le-ℝ y (square-ℝ x))
@@ -230,7 +229,6 @@ abstract
     is-in-upper-cut-ℝ⁰⁺ (square-ℝ⁰⁺ x) (square-ℚ q)
   is-in-upper-cut-square-ℝ x⁰⁺@(x , _) q q∈Ux =
     is-in-upper-cut-le-real-ℚ
-      ( square-ℚ q)
       ( square-ℝ x)
       ( tr
         ( le-ℝ (square-ℝ x))
@@ -238,8 +236,8 @@ abstract
         ( preserves-le-square-ℝ⁰⁺
           ( x⁰⁺)
           ( nonnegative-real-ℚ⁺
-            ( q , is-positive-is-in-upper-cut-ℝ⁰⁺ x⁰⁺ q q∈Ux))
-          ( le-real-is-in-upper-cut-ℚ q x q∈Ux)))
+            ( q , is-positive-is-in-upper-cut-ℝ⁰⁺ x⁰⁺ q∈Ux))
+          ( le-real-is-in-upper-cut-ℚ x q∈Ux)))
 ```
 
 ### If a rational `q` is in the upper cut of both `x` and `-x`, `q²` is in the upper cut of `x²`
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
new file mode 100644
index 0000000000..978356cd75
--- /dev/null
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -0,0 +1,289 @@
+# Strict inequalities about addition and subtraction on the real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.strict-inequalities-addition-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+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.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.conjunction
+open import foundation.dependent-pair-types
+open import foundation.empty-types
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.propositional-truncations
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.arithmetically-located-dedekind-cuts
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.difference-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+This file describes lemmas about
+[strict inequalities](real-numbers.strict-inequality-real-numbers.md) of
+[real numbers](real-numbers.dedekind-real-numbers.md) related to
+[addition](real-numbers.addition-real-numbers.md) and
+[subtraction](real-numbers.difference-real-numbers.md).
+
+## Lemmas
+
+### Strict inequality on the real numbers is translation invariant
+
+```agda
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  opaque
+    unfolding add-ℝ le-ℝ
+
+    preserves-le-right-add-ℝ : le-ℝ x y → le-ℝ (x +ℝ z) (y +ℝ z)
+    preserves-le-right-add-ℝ x<y =
+      let
+        open
+          do-syntax-trunc-Prop
+            ( ∃ ℚ (λ r → upper-cut-ℝ (x +ℝ z) r ∧ lower-cut-ℝ (y +ℝ z) r))
+      in do
+        ( p , x<p , p<y) ← x<y
+        ( q , p<q , q<y) ← forward-implication (is-rounded-lower-cut-ℝ y p) p<y
+        ( (r , s) , s<r+q-p , r<z , z<s) ←
+          is-arithmetically-located-ℝ
+            ( z)
+            ( positive-diff-le-ℚ p<q)
+        let
+          p-q+s<r : le-ℚ ((p -ℚ q) +ℚ s) r
+          p-q+s<r =
+            tr
+              ( le-ℚ ((p -ℚ q) +ℚ s))
+              ( equational-reasoning
+                  (p -ℚ q) +ℚ (r +ℚ (q -ℚ p))
+                  ＝ (p -ℚ q) +ℚ (r -ℚ (p -ℚ q))
+                    by
+                      ap
+                        ( λ t → (p -ℚ q) +ℚ (r +ℚ t))
+                        ( inv (distributive-neg-diff-ℚ p q))
+                  ＝ r by is-identity-right-conjugation-add-ℚ (p -ℚ q) r)
+              ( preserves-le-right-add-ℚ (p -ℚ q) s (r +ℚ (q -ℚ p)) s<r+q-p)
+        intro-exists
+          ( p +ℚ s)
+          ( intro-exists (p , s) (x<p , z<s , refl) ,
+            intro-exists
+              ( q , (p -ℚ q) +ℚ s)
+              ( q<y ,
+                le-lower-cut-ℝ z ((p -ℚ q) +ℚ s) r p-q+s<r r<z ,
+                ( equational-reasoning
+                    p +ℚ s
+                    ＝ (q +ℚ (p -ℚ q)) +ℚ s
+                      by
+                        ap
+                          ( _+ℚ s)
+                          ( inv (is-identity-right-conjugation-add-ℚ q p))
+                    ＝ q +ℚ ((p -ℚ q) +ℚ s) by associative-add-ℚ _ _ _)))
+
+    preserves-le-left-add-ℝ : le-ℝ x y → le-ℝ (z +ℝ x) (z +ℝ y)
+    preserves-le-left-add-ℝ x<y =
+      binary-tr
+        ( le-ℝ)
+        ( commutative-add-ℝ x z)
+        ( commutative-add-ℝ y z)
+        ( preserves-le-right-add-ℝ x<y)
+
+abstract
+  preserves-le-diff-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    le-ℝ x y → le-ℝ (x -ℝ z) (y -ℝ z)
+  preserves-le-diff-ℝ z = preserves-le-right-add-ℝ (neg-ℝ z)
+
+  reverses-le-diff-ℝ :
+    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
+    le-ℝ x y → le-ℝ (z -ℝ y) (z -ℝ x)
+  reverses-le-diff-ℝ z x y x<y =
+    preserves-le-left-add-ℝ z _ _ (neg-le-ℝ x<y)
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  abstract
+    reflects-le-right-add-ℝ : le-ℝ (x +ℝ z) (y +ℝ z) → le-ℝ x y
+    reflects-le-right-add-ℝ x+z<y+z =
+      preserves-le-sim-ℝ
+        ( (x +ℝ z) +ℝ neg-ℝ z)
+        ( x)
+        ( (y +ℝ z) +ℝ neg-ℝ z)
+        ( y)
+        ( cancel-right-add-diff-ℝ x z)
+        ( cancel-right-add-diff-ℝ y z)
+        ( preserves-le-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z<y+z)
+
+    reflects-le-left-add-ℝ : le-ℝ (z +ℝ x) (z +ℝ y) → le-ℝ x y
+    reflects-le-left-add-ℝ z+x<z+y =
+      reflects-le-right-add-ℝ
+        ( binary-tr
+          ( le-ℝ)
+          ( commutative-add-ℝ z x)
+          ( commutative-add-ℝ z y)
+          ( z+x<z+y))
+
+module _
+  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
+  where
+
+  iff-translate-right-le-ℝ : le-ℝ x y ↔ le-ℝ (x +ℝ z) (y +ℝ z)
+  pr1 iff-translate-right-le-ℝ = preserves-le-right-add-ℝ z x y
+  pr2 iff-translate-right-le-ℝ = reflects-le-right-add-ℝ z x y
+
+  iff-translate-left-le-ℝ : le-ℝ x y ↔ le-ℝ (z +ℝ x) (z +ℝ y)
+  pr1 iff-translate-left-le-ℝ = preserves-le-left-add-ℝ z x y
+  pr2 iff-translate-left-le-ℝ = reflects-le-left-add-ℝ z x y
+
+abstract
+  preserves-le-add-ℝ :
+    {l1 l2 l3 l4 : Level} {a : ℝ l1} {b : ℝ l2} {c : ℝ l3} {d : ℝ l4} →
+    le-ℝ a b → le-ℝ c d → le-ℝ (a +ℝ c) (b +ℝ d)
+  preserves-le-add-ℝ {a = a} {b = b} {c = c} {d = d} a≤b c≤d =
+    transitive-le-ℝ
+      ( a +ℝ c)
+      ( a +ℝ d)
+      ( b +ℝ d)
+      ( preserves-le-right-add-ℝ d a b a≤b)
+      ( preserves-le-left-add-ℝ a c d c≤d)
+```
+
+### `x + y < z` if and only if `x < z - y`
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-left-add-ℝ : le-ℝ (x +ℝ y) z → le-ℝ x (z -ℝ y)
+    le-transpose-left-add-ℝ x+y<z =
+      preserves-le-left-sim-ℝ
+        ( z -ℝ y)
+        ( (x +ℝ y) -ℝ y)
+        ( x)
+        ( cancel-right-add-diff-ℝ x y)
+        ( preserves-le-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y<z)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-left-add-ℝ' : le-ℝ (x +ℝ y) z → le-ℝ y (z -ℝ x)
+    le-transpose-left-add-ℝ' x+y<z =
+      le-transpose-left-add-ℝ y x z
+        ( tr (λ w → le-ℝ w z) (commutative-add-ℝ _ _) x+y<z)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-right-diff-ℝ : le-ℝ x (y -ℝ z) → le-ℝ (x +ℝ z) y
+    le-transpose-right-diff-ℝ x<y-z =
+      preserves-le-right-sim-ℝ
+        ( x +ℝ z)
+        ( (y -ℝ z) +ℝ z)
+        ( y)
+        ( cancel-right-diff-add-ℝ y z)
+        ( preserves-le-right-add-ℝ z x (y -ℝ z) x<y-z)
+
+    le-transpose-right-diff-ℝ' : le-ℝ x (y -ℝ z) → le-ℝ (z +ℝ x) y
+    le-transpose-right-diff-ℝ' x<y-z =
+      tr
+        ( λ w → le-ℝ w y)
+        ( commutative-add-ℝ _ _)
+        ( le-transpose-right-diff-ℝ x<y-z)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  iff-diff-right-le-ℝ : le-ℝ (x +ℝ y) z ↔ le-ℝ x (z -ℝ y)
+  iff-diff-right-le-ℝ =
+    (le-transpose-left-add-ℝ x y z , le-transpose-right-diff-ℝ x z y)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    iff-add-right-le-ℝ : le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ y)
+    iff-add-right-le-ℝ =
+      tr
+        ( λ w → le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ w))
+        ( neg-neg-ℝ y)
+        ( iff-diff-right-le-ℝ x (neg-ℝ y) z)
+
+    le-transpose-left-diff-ℝ : le-ℝ (x -ℝ y) z → le-ℝ x (z +ℝ y)
+    le-transpose-left-diff-ℝ = forward-implication iff-add-right-le-ℝ
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-right-add-ℝ : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ z) y
+    le-transpose-right-add-ℝ = backward-implication (iff-add-right-le-ℝ x z y)
+
+module _
+  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    le-transpose-right-add-ℝ' : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ y) z
+    le-transpose-right-add-ℝ' x<y+z =
+      le-transpose-right-add-ℝ x z y (tr (le-ℝ x) (commutative-add-ℝ _ _) x<y+z)
+```
+
+### If `x < y`, then there is some `ε : ℚ⁺` with `x + ε < y`
+
+```agda
+module _
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} (x<y : le-ℝ x y)
+  where
+
+  abstract
+    exists-positive-rational-separation-le-ℝ :
+      exists ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y)
+    exists-positive-rational-separation-le-ℝ =
+      let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y))
+      in do
+        (q , 0<q , q<y-x) ←
+          dense-rational-le-ℝ zero-ℝ (y -ℝ x)
+            ( preserves-le-left-sim-ℝ
+              ( y -ℝ x)
+              ( x -ℝ x)
+              ( zero-ℝ)
+              ( 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 (reflects-le-real-ℚ 0<q))
+          ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
+```
diff --git a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
index 7bf6d53efa..07f144ab5d 100644
--- a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
@@ -15,13 +15,19 @@ open import elementary-number-theory.strict-inequality-nonnegative-rational-numb
 
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
+open import foundation.function-types
+open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequality-nonnegative-real-numbers
+open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.similarity-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -59,7 +65,7 @@ abstract
   preserves-le-nonnegative-real-ℚ⁰⁺ :
     (p q : ℚ⁰⁺) →
     le-ℚ⁰⁺ p q → le-ℝ⁰⁺ (nonnegative-real-ℚ⁰⁺ p) (nonnegative-real-ℚ⁰⁺ q)
-  preserves-le-nonnegative-real-ℚ⁰⁺ p q = preserves-le-real-ℚ _ _
+  preserves-le-nonnegative-real-ℚ⁰⁺ p q = preserves-le-real-ℚ
 ```
 
 ### Similarity preserves strict inequality
@@ -75,20 +81,6 @@ module _
       preserves-le-left-sim-ℝ (real-ℝ⁰⁺ z) _ _ x~y
 ```
 
-### Addition preserves strict inequality
-
-```agda
-module _
-  {l1 l2 l3 l4 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2) (z : ℝ⁰⁺ l3) (w : ℝ⁰⁺ l4)
-  where
-
-  abstract
-    preserves-le-add-ℝ⁰⁺ :
-      le-ℝ⁰⁺ x y → le-ℝ⁰⁺ z w → le-ℝ⁰⁺ (x +ℝ⁰⁺ z) (y +ℝ⁰⁺ w)
-    preserves-le-add-ℝ⁰⁺ =
-      preserves-le-add-ℝ (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ z) (real-ℝ⁰⁺ w)
-```
-
 ### Concatenation of inequality and strict inequality
 
 ```agda
@@ -114,6 +106,24 @@ module _
       exists ℚ⁺ (λ q → le-prop-ℝ⁰⁺ x (nonnegative-real-ℚ⁺ q))
     le-some-positive-rational-ℝ⁰⁺ =
       map-tot-exists
-        ( λ (q , _) x<q → le-real-is-in-upper-cut-ℚ q (real-ℝ⁰⁺ x) x<q)
+        ( λ (q , _) x<q → le-real-is-in-upper-cut-ℚ (real-ℝ⁰⁺ x) x<q)
         ( exists-ℚ⁺-in-upper-cut-ℝ⁰⁺ x)
 ```
+
+### If `x` is less than the same positive rational numbers `y` is less than, then `x` and `y` are similar
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ⁰⁺ l1) (y : ℝ⁰⁺ l2)
+  where
+
+  abstract
+    sim-le-same-positive-rational-ℝ⁰⁺ :
+      ( (q : ℚ⁺) →
+        le-ℝ (real-ℝ⁰⁺ x) (real-ℚ⁺ q) ↔ le-ℝ (real-ℝ⁰⁺ y) (real-ℚ⁺ q)) →
+      sim-ℝ⁰⁺ x y
+    sim-le-same-positive-rational-ℝ⁰⁺ H =
+      sim-sim-leq-ℝ
+        ( leq-le-positive-rational-ℝ⁰⁺ x y (backward-implication ∘ H) ,
+          leq-le-positive-rational-ℝ⁰⁺ y x (forward-implication ∘ H))
+```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index a68d6e8cbd..dc20d798d6 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -81,20 +81,16 @@ le-prop-ℝ x y = (le-ℝ x y , is-prop-le-ℝ x y)
 ### Strict inequality on the reals implies inequality
 
 ```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  opaque
-    unfolding le-ℝ leq-ℝ
-
-    leq-le-ℝ : le-ℝ x y → leq-ℝ x y
-    leq-le-ℝ x<y p p<x =
-      elim-exists
-        ( lower-cut-ℝ y p)
-        ( λ q (x<q , q<y) →
-          le-lower-cut-ℝ y p q (le-lower-upper-cut-ℝ x p q p<x x<q) q<y)
-        ( x<y)
+opaque
+  unfolding le-ℝ leq-ℝ
+
+  leq-le-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → leq-ℝ x y
+  leq-le-ℝ {x = x} {y = y} x<y p p<x =
+    elim-exists
+      ( lower-cut-ℝ y p)
+      ( λ q (x<q , q<y) →
+        le-lower-cut-ℝ y p q (le-lower-upper-cut-ℝ x p q p<x x<q) q<y)
+      ( x<y)
 ```
 
 ### Strict inequality on the reals is irreflexive
@@ -119,9 +115,7 @@ module _
 
 ```agda
 module _
-  {l1 l2 : Level}
-  (x : ℝ l1)
-  (y : ℝ l2)
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
   opaque
@@ -176,7 +170,7 @@ module _
 
 ```agda
 module _
-  (x y : ℚ)
+  {x y : ℚ}
   where
 
   opaque
@@ -226,7 +220,7 @@ module _
 
 ```agda
 module _
-  {l : Level} (q : ℚ) (x : ℝ l)
+  {l : Level} {q : ℚ} (x : ℝ l)
   where
 
   opaque
@@ -247,7 +241,7 @@ module _
 
 ```agda
 module _
-  {l : Level} (q : ℚ) (x : ℝ l)
+  {l : Level} {q : ℚ} (x : ℝ l)
   where
 
   opaque
@@ -276,9 +270,9 @@ module _
   is-located-le-ℝ : disjunction-type (le-ℝ (real-ℚ p) x) (le-ℝ x (real-ℚ q))
   is-located-le-ℝ =
     map-disjunction
-      ( le-real-is-in-lower-cut-ℚ p x)
-      ( le-real-is-in-upper-cut-ℚ q x)
-      ( is-located-lower-upper-cut-ℝ x p q p<q)
+      ( le-real-is-in-lower-cut-ℚ x)
+      ( le-real-is-in-upper-cut-ℚ x)
+      ( is-located-lower-upper-cut-ℝ x p<q)
 ```
 
 ### Every real is less than a rational number
@@ -292,7 +286,7 @@ module _
     le-some-rational-ℝ : exists ℚ (λ q → le-prop-ℝ x (real-ℚ q))
     le-some-rational-ℝ =
       map-tot-exists
-        ( λ q → le-real-is-in-upper-cut-ℚ q x)
+        ( λ q → le-real-is-in-upper-cut-ℚ x)
         ( is-inhabited-upper-cut-ℝ x)
 ```
 
@@ -311,7 +305,7 @@ module _
         open do-syntax-trunc-Prop (∃ (ℝ lzero) (λ y → le-prop-ℝ y x))
       in do
         ( q , q<x) ← is-inhabited-lower-cut-ℝ x
-        intro-exists (real-ℚ q) (le-real-is-in-lower-cut-ℚ q x q<x)
+        intro-exists (real-ℚ q) (le-real-is-in-lower-cut-ℚ x q<x)
 
     exists-greater-ℝ : exists (ℝ lzero) (λ y → le-prop-ℝ x y)
     exists-greater-ℝ =
@@ -319,7 +313,7 @@ module _
         open do-syntax-trunc-Prop (∃ (ℝ lzero) (le-prop-ℝ x))
       in do
         ( q , x<q) ← is-inhabited-upper-cut-ℝ x
-        intro-exists (real-ℚ q) (le-real-is-in-upper-cut-ℚ q x x<q)
+        intro-exists (real-ℚ q) (le-real-is-in-upper-cut-ℚ x x<q)
 ```
 
 ### Negation reverses the strict ordering of real numbers
@@ -327,8 +321,7 @@ module _
 ```agda
 module _
   {l1 l2 : Level}
-  (x : ℝ l1)
-  (y : ℝ l2)
+  {x : ℝ l1} {y : ℝ l2}
   where
 
   opaque
@@ -385,7 +378,7 @@ module _
           ( lower-cut-ℝ x p)
           ( id)
           ( λ x<q → reductio-ad-absurdum (intro-exists q (x<q , q<y)) x≮y)
-          ( is-located-lower-upper-cut-ℝ x p q p<q)
+          ( is-located-lower-upper-cut-ℝ x p<q)
 ```
 
 ### If `x` is less than or equal to `y`, then `y` is not less than `x`
@@ -476,7 +469,7 @@ opaque
       map-disjunction
         ( λ p<z → intro-exists p (x<p , p<z))
         ( λ z<q → intro-exists q (z<q , q<y))
-        ( is-located-lower-upper-cut-ℝ z p q p<q)
+        ( is-located-lower-upper-cut-ℝ z p<q)
 ```
 
 ### Strict inequality on the real numbers is invariant under similarity
@@ -517,265 +510,6 @@ module _
       ( preserves-le-right-sim-ℝ x1 y1 y2 y1~y2 x1<y1)
 ```
 
-### Strict inequality on the real numbers is translation invariant
-
-```agda
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  opaque
-    unfolding add-ℝ le-ℝ
-
-    preserves-le-right-add-ℝ : le-ℝ x y → le-ℝ (x +ℝ z) (y +ℝ z)
-    preserves-le-right-add-ℝ x<y =
-      let
-        open
-          do-syntax-trunc-Prop
-            ( ∃ ℚ (λ r → upper-cut-ℝ (x +ℝ z) r ∧ lower-cut-ℝ (y +ℝ z) r))
-      in do
-        ( p , x<p , p<y) ← x<y
-        ( q , p<q , q<y) ← forward-implication (is-rounded-lower-cut-ℝ y p) p<y
-        ( (r , s) , s<r+q-p , r<z , z<s) ←
-          is-arithmetically-located-ℝ
-            ( z)
-            ( positive-diff-le-ℚ p q p<q)
-        let
-          p-q+s<r : le-ℚ ((p -ℚ q) +ℚ s) r
-          p-q+s<r =
-            tr
-              ( le-ℚ ((p -ℚ q) +ℚ s))
-              ( equational-reasoning
-                  (p -ℚ q) +ℚ (r +ℚ (q -ℚ p))
-                  ＝ (p -ℚ q) +ℚ (r -ℚ (p -ℚ q))
-                    by
-                      ap
-                        ( λ t → (p -ℚ q) +ℚ (r +ℚ t))
-                        ( inv (distributive-neg-diff-ℚ p q))
-                  ＝ r by is-identity-right-conjugation-add-ℚ (p -ℚ q) r)
-              ( preserves-le-right-add-ℚ (p -ℚ q) s (r +ℚ (q -ℚ p)) s<r+q-p)
-        intro-exists
-          ( p +ℚ s)
-          ( intro-exists (p , s) (x<p , z<s , refl) ,
-            intro-exists
-              ( q , (p -ℚ q) +ℚ s)
-              ( q<y ,
-                le-lower-cut-ℝ z ((p -ℚ q) +ℚ s) r p-q+s<r r<z ,
-                ( equational-reasoning
-                    p +ℚ s
-                    ＝ (q +ℚ (p -ℚ q)) +ℚ s
-                      by
-                        ap
-                          ( _+ℚ s)
-                          ( inv (is-identity-right-conjugation-add-ℚ q p))
-                    ＝ q +ℚ ((p -ℚ q) +ℚ s) by associative-add-ℚ _ _ _)))
-
-    preserves-le-left-add-ℝ : le-ℝ x y → le-ℝ (z +ℝ x) (z +ℝ y)
-    preserves-le-left-add-ℝ x<y =
-      binary-tr
-        ( le-ℝ)
-        ( commutative-add-ℝ x z)
-        ( commutative-add-ℝ y z)
-        ( preserves-le-right-add-ℝ x<y)
-
-abstract
-  preserves-le-diff-ℝ :
-    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
-    le-ℝ x y → le-ℝ (x -ℝ z) (y -ℝ z)
-  preserves-le-diff-ℝ z = preserves-le-right-add-ℝ (neg-ℝ z)
-
-  reverses-le-diff-ℝ :
-    {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
-    le-ℝ x y → le-ℝ (z -ℝ y) (z -ℝ x)
-  reverses-le-diff-ℝ z x y x<y =
-    preserves-le-left-add-ℝ z _ _ (neg-le-ℝ _ _ x<y)
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  abstract
-    reflects-le-right-add-ℝ : le-ℝ (x +ℝ z) (y +ℝ z) → le-ℝ x y
-    reflects-le-right-add-ℝ x+z<y+z =
-      preserves-le-sim-ℝ
-        ( (x +ℝ z) +ℝ neg-ℝ z)
-        ( x)
-        ( (y +ℝ z) +ℝ neg-ℝ z)
-        ( y)
-        ( cancel-right-add-diff-ℝ x z)
-        ( cancel-right-add-diff-ℝ y z)
-        ( preserves-le-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z<y+z)
-
-    reflects-le-left-add-ℝ : le-ℝ (z +ℝ x) (z +ℝ y) → le-ℝ x y
-    reflects-le-left-add-ℝ z+x<z+y =
-      reflects-le-right-add-ℝ
-        ( binary-tr
-          ( le-ℝ)
-          ( commutative-add-ℝ z x)
-          ( commutative-add-ℝ z y)
-          ( z+x<z+y))
-
-module _
-  {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
-  where
-
-  iff-translate-right-le-ℝ : le-ℝ x y ↔ le-ℝ (x +ℝ z) (y +ℝ z)
-  pr1 iff-translate-right-le-ℝ = preserves-le-right-add-ℝ z x y
-  pr2 iff-translate-right-le-ℝ = reflects-le-right-add-ℝ z x y
-
-  iff-translate-left-le-ℝ : le-ℝ x y ↔ le-ℝ (z +ℝ x) (z +ℝ y)
-  pr1 iff-translate-left-le-ℝ = preserves-le-left-add-ℝ z x y
-  pr2 iff-translate-left-le-ℝ = reflects-le-left-add-ℝ z x y
-
-abstract
-  preserves-le-add-ℝ :
-    {l1 l2 l3 l4 : Level} (a : ℝ l1) (b : ℝ l2) (c : ℝ l3) (d : ℝ l4) →
-    le-ℝ a b → le-ℝ c d → le-ℝ (a +ℝ c) (b +ℝ d)
-  preserves-le-add-ℝ a b c d a≤b c≤d =
-    transitive-le-ℝ
-      ( a +ℝ c)
-      ( a +ℝ d)
-      ( b +ℝ d)
-      ( preserves-le-right-add-ℝ d a b a≤b)
-      ( preserves-le-left-add-ℝ a c d c≤d)
-```
-
-### `x + y < z` if and only if `x < z - y`
-
-```agda
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-left-add-ℝ : le-ℝ (x +ℝ y) z → le-ℝ x (z -ℝ y)
-    le-transpose-left-add-ℝ x+y<z =
-      preserves-le-left-sim-ℝ
-        ( z -ℝ y)
-        ( (x +ℝ y) -ℝ y)
-        ( x)
-        ( cancel-right-add-diff-ℝ x y)
-        ( preserves-le-right-add-ℝ (neg-ℝ y) (x +ℝ y) z x+y<z)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-left-add-ℝ' : le-ℝ (x +ℝ y) z → le-ℝ y (z -ℝ x)
-    le-transpose-left-add-ℝ' x+y<z =
-      le-transpose-left-add-ℝ y x z
-        ( tr (λ w → le-ℝ w z) (commutative-add-ℝ _ _) x+y<z)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-right-diff-ℝ : le-ℝ x (y -ℝ z) → le-ℝ (x +ℝ z) y
-    le-transpose-right-diff-ℝ x<y-z =
-      preserves-le-right-sim-ℝ
-        ( x +ℝ z)
-        ( (y -ℝ z) +ℝ z)
-        ( y)
-        ( cancel-right-diff-add-ℝ y z)
-        ( preserves-le-right-add-ℝ z x (y -ℝ z) x<y-z)
-
-    le-transpose-right-diff-ℝ' : le-ℝ x (y -ℝ z) → le-ℝ (z +ℝ x) y
-    le-transpose-right-diff-ℝ' x<y-z =
-      tr
-        ( λ w → le-ℝ w y)
-        ( commutative-add-ℝ _ _)
-        ( le-transpose-right-diff-ℝ x<y-z)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  iff-diff-right-le-ℝ : le-ℝ (x +ℝ y) z ↔ le-ℝ x (z -ℝ y)
-  iff-diff-right-le-ℝ =
-    (le-transpose-left-add-ℝ x y z , le-transpose-right-diff-ℝ x z y)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    iff-add-right-le-ℝ : le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ y)
-    iff-add-right-le-ℝ =
-      tr
-        ( λ w → le-ℝ (x -ℝ y) z ↔ le-ℝ x (z +ℝ w))
-        ( neg-neg-ℝ y)
-        ( iff-diff-right-le-ℝ x (neg-ℝ y) z)
-
-    le-transpose-left-diff-ℝ : le-ℝ (x -ℝ y) z → le-ℝ x (z +ℝ y)
-    le-transpose-left-diff-ℝ = forward-implication iff-add-right-le-ℝ
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-right-add-ℝ : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ z) y
-    le-transpose-right-add-ℝ = backward-implication (iff-add-right-le-ℝ x z y)
-
-module _
-  {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    le-transpose-right-add-ℝ' : le-ℝ x (y +ℝ z) → le-ℝ (x -ℝ y) z
-    le-transpose-right-add-ℝ' x<y+z =
-      le-transpose-right-add-ℝ x z y (tr (le-ℝ x) (commutative-add-ℝ _ _) x<y+z)
-```
-
-### If `x < y`, then there is some `ε : ℚ⁺` with `x + ε < y`
-
-```agda
-module _
-  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} (x<y : le-ℝ x y)
-  where
-
-  abstract
-    exists-positive-rational-separation-le-ℝ :
-      exists ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y)
-    exists-positive-rational-separation-le-ℝ =
-      let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → le-prop-ℝ (x +ℝ real-ℚ⁺ q) y))
-      in do
-        (q , 0<q , q<y-x) ←
-          dense-rational-le-ℝ zero-ℝ (y -ℝ x)
-            ( preserves-le-left-sim-ℝ
-              ( y -ℝ x)
-              ( x -ℝ x)
-              ( zero-ℝ)
-              ( 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 (reflects-le-real-ℚ _ _ 0<q))
-          ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
-```
-
-### If `x < y + ε` for every positive rational `ε`, then `x ≤ y`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
-  where
-
-  abstract
-    saturated-le-ℝ : ((ε : ℚ⁺) → le-ℝ x (y +ℝ real-ℚ⁺ ε)) → leq-ℝ x y
-    saturated-le-ℝ H =
-      leq-not-le-ℝ y x
-        ( λ y<x →
-          let open do-syntax-trunc-Prop empty-Prop
-          in do
-            (ε , y+ε<x) ←
-              exists-positive-rational-separation-le-ℝ {x = y} {y = x} y<x
-            irreflexive-le-ℝ
-              ( x)
-              ( transitive-le-ℝ x (y +ℝ real-ℚ⁺ ε) x y+ε<x (H ε)))
-```
-
 ### If `x` is less than each rational number `y` is less than, then `x ≤ y`
 
 ```agda
@@ -791,8 +525,8 @@ module _
     leq-le-rational-ℝ H =
       leq-leq'-ℝ _ _
         ( λ q y<q →
-          is-in-upper-cut-le-real-ℚ q x
-            ( H q (le-real-is-in-upper-cut-ℚ q y y<q)))
+          is-in-upper-cut-le-real-ℚ x
+            ( H q (le-real-is-in-upper-cut-ℚ y y<q)))
 ```
 
 ### Two real numbers are similar if they are less than the same rational numbers
@@ -832,13 +566,12 @@ module _
     le-split-add-rational-ℝ x+y<p =
       let open do-syntax-trunc-Prop (∃ _ _)
       in do
-        ((q , r) , x<q , y<r , p=q+r) ←
-          is-in-upper-cut-le-real-ℚ p (x +ℝ y) x+y<p
+        ((q , r) , x<q , y<r , p=q+r) ← is-in-upper-cut-le-real-ℚ (x +ℝ y) x+y<p
         intro-exists
           ( q , r)
           ( p=q+r ,
-            le-real-is-in-upper-cut-ℚ q x x<q ,
-            le-real-is-in-upper-cut-ℚ r y y<r)
+            le-real-is-in-upper-cut-ℚ x x<q ,
+            le-real-is-in-upper-cut-ℚ y y<r)
 ```
 
 ## References
diff --git a/src/real-numbers/suprema-families-real-numbers.lagda.md b/src/real-numbers/suprema-families-real-numbers.lagda.md
index 4ef731d25e..f730c05498 100644
--- a/src/real-numbers/suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/suprema-families-real-numbers.lagda.md
@@ -42,6 +42,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 ```
@@ -151,7 +152,7 @@ module _
                 ( x-ε<yᵢ)
                 ( le-transpose-left-add-ℝ' _ _ _
                   ( le-transpose-right-diff-ℝ _ _ _
-                    ( le-real-is-in-lower-cut-ℚ ε (x -ℝ z) ε<x-z))))
+                    ( le-real-is-in-lower-cut-ℚ (x -ℝ z) ε<x-z))))
               ( yᵢ≤z i))
     pr2 (is-least-upper-bound-is-supremum-family-ℝ z) x≤z i =
       transitive-leq-ℝ (y i) x z x≤z
@@ -258,7 +259,7 @@ module _
             ( x-ε<yᵢ)
             ( le-transpose-left-add-ℝ' _ _ _
               ( le-transpose-right-diff-ℝ _ _ _
-                ( le-real-is-in-lower-cut-ℚ ε (x -ℝ z) ε<x-z))))
+                ( le-real-is-in-lower-cut-ℚ (x -ℝ z) ε<x-z))))
 
     le-supremum-iff-le-element-family-ℝ :
       {l4 : Level} → (z : ℝ l4) →
diff --git a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
index ac543613a6..91e5c9c0be 100644
--- a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
@@ -24,6 +24,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
@@ -56,7 +57,6 @@ module _
       is-in-lower-cut-ℝ (x -ℝ real-ℚ q) p
     transpose-add-is-in-lower-cut-ℝ p+q<x =
       is-in-lower-cut-le-real-ℚ
-        ( p)
         ( x -ℝ real-ℚ q)
         ( le-transpose-left-add-ℝ
           ( real-ℚ p)
@@ -65,13 +65,12 @@ module _
           ( inv-tr
             ( λ y → le-ℝ y x)
             ( add-real-ℚ p q)
-            ( le-real-is-in-lower-cut-ℚ (p +ℚ q) x p+q<x)))
+            ( le-real-is-in-lower-cut-ℚ x p+q<x)))
 
     transpose-is-in-lower-cut-diff-ℝ :
       is-in-lower-cut-ℝ (x -ℝ real-ℚ p) q → is-in-lower-cut-ℝ x (q +ℚ p)
     transpose-is-in-lower-cut-diff-ℝ x-p<q =
       is-in-lower-cut-le-real-ℚ
-        ( q +ℚ p)
         ( x)
         ( tr
           ( λ y → le-ℝ y x)
@@ -80,7 +79,7 @@ module _
             ( real-ℚ q)
             ( x)
             ( real-ℚ p)
-            ( le-real-is-in-lower-cut-ℚ q (x -ℝ real-ℚ p) x-p<q)))
+            ( le-real-is-in-lower-cut-ℚ (x -ℝ real-ℚ p) x-p<q)))
 
 module _
   {l : Level} (x : ℝ l) (p q : ℚ)
@@ -127,7 +126,6 @@ module _
       is-in-upper-cut-ℝ (x -ℝ real-ℚ q) p
     transpose-add-is-in-upper-cut-ℝ x<p+q =
       is-in-upper-cut-le-real-ℚ
-        ( p)
         ( x -ℝ real-ℚ q)
         ( le-transpose-right-add-ℝ
           ( x)
@@ -136,13 +134,12 @@ module _
           ( inv-tr
             ( le-ℝ x)
             ( add-real-ℚ p q)
-            ( le-real-is-in-upper-cut-ℚ (p +ℚ q) x x<p+q)))
+            ( le-real-is-in-upper-cut-ℚ x x<p+q)))
 
     transpose-is-in-upper-cut-diff-ℝ :
       is-in-upper-cut-ℝ (x -ℝ real-ℚ p) q → is-in-upper-cut-ℝ x (q +ℚ p)
     transpose-is-in-upper-cut-diff-ℝ x-p<q =
       is-in-upper-cut-le-real-ℚ
-        ( q +ℚ p)
         ( x)
         ( tr
           ( le-ℝ x)
@@ -151,7 +148,7 @@ module _
             ( x)
             ( real-ℚ p)
             ( real-ℚ q)
-            ( le-real-is-in-upper-cut-ℚ q (x -ℝ real-ℚ p) x-p<q)))
+            ( le-real-is-in-upper-cut-ℚ (x -ℝ real-ℚ p) x-p<q)))
 
 module _
   {l : Level} (x : ℝ l) (p q : ℚ)

From b5a27af88b7b41a9eb5f549baa1026529d1a31f4 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 15:01:52 -0800
Subject: [PATCH 02/21] Progress

---
 ...lue-closed-intervals-real-numbers.lagda.md |  2 --
 .../absolute-value-real-numbers.lagda.md      | 16 +++------
 .../distance-real-numbers.lagda.md            | 16 ++++-----
 ...nequalities-addition-real-numbers.lagda.md |  4 ---
 .../inequality-real-numbers.lagda.md          |  2 +-
 .../metric-space-of-real-numbers.lagda.md     |  8 -----
 ...tiplication-positive-real-numbers.lagda.md |  4 +--
 .../nonnegative-real-numbers.lagda.md         | 27 ++++++++-------
 .../positive-real-numbers.lagda.md            |  4 +--
 .../similarity-positive-real-numbers.lagda.md | 34 +++++++++++++++++++
 ...nequalities-addition-real-numbers.lagda.md |  4 ---
 .../strict-inequality-real-numbers.lagda.md   |  2 +-
 12 files changed, 67 insertions(+), 56 deletions(-)
 create mode 100644 src/real-numbers/similarity-positive-real-numbers.lagda.md

diff --git a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
index f040689b7c..634f6c9034 100644
--- a/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-closed-intervals-real-numbers.lagda.md
@@ -49,8 +49,6 @@ abstract
     leq-ℝ (abs-ℝ z) (max-abs-closed-interval-ℝ [x,y])
   leq-max-abs-is-in-closed-interval-ℝ ((x , y) , x≤y) z (x≤z , z≤y) =
     leq-abs-leq-leq-neg-ℝ
-      ( z)
-      ( max-ℝ (neg-ℝ x) y)
       ( transitive-leq-ℝ _ _ _ (leq-right-max-ℝ _ _) z≤y)
       ( transitive-leq-ℝ _ _ _ (leq-left-max-ℝ _ _) (neg-leq-ℝ x≤z))
 ```
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index d31b66d8b8..8d3c38266f 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -66,7 +66,7 @@ opaque
 ### The absolute value of zero is zero
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ
 
   abs-zero-ℝ : abs-ℝ zero-ℝ ＝ zero-ℝ
@@ -76,7 +76,7 @@ opaque
 ### The absolute value preserves similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ
 
   preserves-sim-abs-ℝ :
@@ -89,7 +89,7 @@ opaque
 ### The absolute value of a real number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ leq-ℝ max-ℝ neg-ℚ neg-ℝ real-ℚ
 
   is-nonnegative-abs-ℝ : {l : Level} → (x : ℝ l) → is-nonnegative-ℝ (abs-ℝ x)
@@ -210,10 +210,10 @@ module _
 
 ```agda
 module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
+  {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding abs-ℝ
 
     leq-abs-leq-leq-neg-ℝ : leq-ℝ x y → leq-ℝ (neg-ℝ x) y → leq-ℝ (abs-ℝ x) y
@@ -231,8 +231,6 @@ module _
     triangle-inequality-abs-ℝ : leq-ℝ (abs-ℝ (x +ℝ y)) (abs-ℝ x +ℝ abs-ℝ y)
     triangle-inequality-abs-ℝ =
       leq-abs-leq-leq-neg-ℝ
-        ( x +ℝ y)
-        ( abs-ℝ x +ℝ abs-ℝ y)
         ( preserves-leq-add-ℝ (leq-abs-ℝ x) (leq-abs-ℝ y))
         ( inv-tr
           ( λ z → leq-ℝ z (abs-ℝ x +ℝ abs-ℝ y))
@@ -259,8 +257,6 @@ module _
         ( abs-ℝ x)
         ( abs-ℝ y)
         ( leq-abs-leq-leq-neg-ℝ
-          ( x)
-          ( abs-ℝ y +ℝ real-ℚ⁺ d)
           ( transitive-leq-ℝ
             ( x)
             ( y +ℝ real-ℚ⁺ d)
@@ -286,8 +282,6 @@ module _
               ( x)
               ( right-leq-real-bound-neighborhood-ℝ d x y I))))
         ( leq-abs-leq-leq-neg-ℝ
-          ( y)
-          ( abs-ℝ x +ℝ real-ℚ⁺ d)
           ( transitive-leq-ℝ
             ( y)
             ( x +ℝ real-ℚ⁺ d)
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index df3a005ee3..1f779167f0 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -83,11 +83,13 @@ nonnegative-dist-ℝ x y = (dist-ℝ x y , is-nonnegative-dist-ℝ x y)
 
 ### Relationship to the metric space of real numbers
 
-Two real numbers `x` and `y` are in an `ε`-neighborhood of each other if and
-only if their distance is at most `ε`.
+Two real numbers `x` and `y` are in an `ε`-neighborhood of each other in the
+[metric space of real numbers](real-numbers.metric-space-of-real-numbers.md)
+[if and only if](foundation.logical-equivalences.md) their distance is
+[at most](real-numbers.inequality-real-numbers.md) `ε`.
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ neighborhood-ℝ
 
   diff-bound-neighborhood-ℝ :
@@ -125,8 +127,6 @@ abstract
     dist-ℝ x y ≤-ℝ real-ℚ⁺ d
   leq-dist-neighborhood-ℝ d⁺@(d , _) x y H =
     leq-abs-leq-leq-neg-ℝ
-      ( x -ℝ y)
-      ( real-ℚ d)
       ( diff-bound-neighborhood-ℝ d⁺ x y H)
       ( inv-tr
         ( λ z → leq-ℝ z (real-ℚ d))
@@ -236,8 +236,6 @@ abstract
     dist-ℝ x y ≤-ℝ z
   leq-dist-leq-diff-ℝ x y z x-y≤z y-x≤z =
     leq-abs-leq-leq-neg-ℝ
-      ( _)
-      ( z)
       ( x-y≤z)
       ( inv-tr (λ w → leq-ℝ w z) (distributive-neg-diff-ℝ _ _) y-x≤z)
 
@@ -259,7 +257,7 @@ abstract
   preserves-dist-left-add-ℝ :
     {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
     sim-ℝ
-      ( dist-ℝ (add-ℝ x y) (add-ℝ x z))
+      ( dist-ℝ (x +ℝ y) (x +ℝ z))
       ( dist-ℝ y z)
   preserves-dist-left-add-ℝ x y z =
     similarity-reasoning-ℝ
@@ -280,7 +278,7 @@ abstract
   preserves-dist-right-add-ℝ :
     {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) →
     sim-ℝ
-      ( dist-ℝ (add-ℝ x z) (add-ℝ y z))
+      ( dist-ℝ (x +ℝ z) (y +ℝ z))
       ( dist-ℝ x y)
   preserves-dist-right-add-ℝ z x y =
     similarity-reasoning-ℝ
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
index 654f922bc1..aa5f3ad62e 100644
--- a/src/real-numbers/inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
@@ -73,10 +73,6 @@ module _
     reflects-leq-right-add-ℝ : leq-ℝ (x +ℝ z) (y +ℝ z) → leq-ℝ x y
     reflects-leq-right-add-ℝ x+z≤y+z =
       preserves-leq-sim-ℝ
-        ( (x +ℝ z) -ℝ z)
-        ( x)
-        ( (y +ℝ z) -ℝ z)
-        ( y)
         ( cancel-right-add-diff-ℝ x z)
         ( cancel-right-add-diff-ℝ y z)
         ( preserves-leq-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z≤y+z)
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index d9b18435f1..d0a0cd049c 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -337,7 +337,7 @@ module _
 
 module _
   {l1 l2 l3 l4 : Level}
-  (x1 : ℝ l1) (x2 : ℝ l2) (y1 : ℝ l3) (y2 : ℝ l4)
+  {x1 : ℝ l1} {x2 : ℝ l2} {y1 : ℝ l3} {y2 : ℝ l4}
   (x1~x2 : sim-ℝ x1 x2) (y1~y2 : sim-ℝ y1 y2)
   where
 
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index 5afe6b7b56..7a64835572 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -355,10 +355,6 @@ module _
       ( x')
       ( y')
       ( preserves-leq-sim-ℝ
-        ( x)
-        ( x')
-        ( y +ℝ real-ℚ⁺ d)
-        ( y' +ℝ real-ℚ⁺ d)
         ( x~x')
         ( preserves-sim-right-add-ℝ
           ( real-ℚ⁺ d)
@@ -367,10 +363,6 @@ module _
           ( y~y'))
         ( left-leq-real-bound-neighborhood-ℝ d x y H))
       ( preserves-leq-sim-ℝ
-        ( y)
-        ( y')
-        ( x +ℝ real-ℚ⁺ d)
-        ( x' +ℝ real-ℚ⁺ d)
         ( y~y')
         ( preserves-sim-right-add-ℝ
           ( real-ℚ⁺ d)
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index 0c7aeb5307..ab63080b99 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -123,7 +123,7 @@ abstract
     {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
     le-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z) → le-ℝ y z
   reflects-le-left-mul-ℝ⁺ x y z xy<xz =
-    preserves-le-sim-ℝ _ _ _ _
+    preserves-le-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
       ( preserves-le-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy<xz)
@@ -169,7 +169,7 @@ abstract
     {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
     leq-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z) → leq-ℝ y z
   reflects-leq-left-mul-ℝ⁺ x y z xy≤xz =
-    preserves-leq-sim-ℝ _ _ _ _
+    preserves-leq-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
       ( preserves-leq-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy≤xz)
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 6b5a11da37..999eff4ee7 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -24,7 +24,9 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
+open import foundation.functoriality-propositional-truncation
 open import foundation.function-types
+open import foundation.inhabited-subtypes
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
@@ -43,10 +45,8 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.saturation-inequality-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
-open import real-numbers.subsets-real-numbers
 ```
 
 </details>
@@ -102,8 +102,12 @@ upper-cut-ℝ⁰⁺ (x , _) = upper-cut-ℝ x
 is-in-upper-cut-ℝ⁰⁺ : {l : Level} → ℝ⁰⁺ l → ℚ → UU l
 is-in-upper-cut-ℝ⁰⁺ (x , _) = is-in-upper-cut-ℝ x
 
+is-inhabited-upper-cut-ℝ⁰⁺ :
+  {l : Level} (x : ℝ⁰⁺ l) → is-inhabited-subtype (upper-cut-ℝ⁰⁺ x)
+is-inhabited-upper-cut-ℝ⁰⁺ (x , _) = is-inhabited-upper-cut-ℝ x
+
 is-rounded-upper-cut-ℝ⁰⁺ :
-  {l : Level} → (x : ℝ⁰⁺ l) (r : ℚ) →
+  {l : Level} (x : ℝ⁰⁺ l) (r : ℚ) →
   (is-in-upper-cut-ℝ⁰⁺ x r ↔ exists ℚ (λ q → le-ℚ-Prop q r ∧ upper-cut-ℝ⁰⁺ x q))
 is-rounded-upper-cut-ℝ⁰⁺ (x , _) = is-rounded-upper-cut-ℝ x
 ```
@@ -163,8 +167,8 @@ abstract
           ( 0≤x)
           ( le-real-is-in-upper-cut-ℚ x x<q)))
 
-opaque
-  unfolding leq-ℝ leq-ℝ' real-ℚ
+abstract opaque
+  unfolding leq-ℝ' real-ℚ
 
   is-nonnegative-is-positive-upper-cut-ℝ :
     {l : Level} → (x : ℝ l) → (upper-cut-ℝ x ⊆ is-positive-prop-ℚ) →
@@ -176,7 +180,7 @@ opaque
 ### A real number is nonnegative if and only if every negative rational number is in its lower cut
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ real-ℚ
 
   is-nonnegative-leq-negative-lower-cut-ℝ :
@@ -200,10 +204,9 @@ abstract
     {l : Level} → (x : ℝ⁰⁺ l) →
     exists ℚ⁺ (λ q → upper-cut-ℝ⁰⁺ x (rational-ℚ⁺ q))
   exists-ℚ⁺-in-upper-cut-ℝ⁰⁺ x =
-    let open do-syntax-trunc-Prop (∃ ℚ⁺ (λ q → upper-cut-ℝ⁰⁺ x (rational-ℚ⁺ q)))
-    in do
-      (q , x<q) ← is-inhabited-upper-cut-ℝ (real-ℝ⁰⁺ x)
-      intro-exists (q , is-positive-is-in-upper-cut-ℝ⁰⁺ x x<q) x<q
+    map-trunc-Prop
+      ( λ (q , x<q) → ((q , is-positive-is-in-upper-cut-ℝ⁰⁺ x x<q) , x<q))
+      ( is-inhabited-upper-cut-ℝ⁰⁺ x)
 ```
 
 ### Nonpositive rational numbers are not less than or equal to nonnegative real numbers
@@ -219,9 +222,9 @@ module _
     not-le-leq-zero-rational-ℝ⁰⁺ q≤0 x<q =
       irreflexive-le-ℝ
         ( real-ℝ⁰⁺ x)
-        ( concatenate-le-leq-ℝ _ (real-ℚ q) _
+        ( concatenate-le-leq-ℝ _ _ _
           ( x<q)
-          ( transitive-leq-ℝ (real-ℚ q) zero-ℝ (real-ℝ⁰⁺ x)
+          ( transitive-leq-ℝ _ _ _
             ( is-nonnegative-real-ℝ⁰⁺ x)
             ( preserves-leq-real-ℚ q≤0)))
 ```
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 3e3881f93c..95cc41a662 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -153,7 +153,7 @@ module _
   {l : Level} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     exists-ℚ⁺-in-lower-cut-is-positive-ℝ :
@@ -186,7 +186,7 @@ exists-ℚ⁺-in-lower-cut-ℝ⁺ = ind-Σ exists-ℚ⁺-in-lower-cut-is-positiv
 ### Addition with a positive real number is a strictly inflationary map
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ le-ℝ
 
   le-left-add-real-ℝ⁺ :
diff --git a/src/real-numbers/similarity-positive-real-numbers.lagda.md b/src/real-numbers/similarity-positive-real-numbers.lagda.md
new file mode 100644
index 0000000000..7253e34436
--- /dev/null
+++ b/src/real-numbers/similarity-positive-real-numbers.lagda.md
@@ -0,0 +1,34 @@
+# Similarity of positive real numbers
+
+```agda
+module real-numbers.similarity-positive-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+open import foundation.propositions
+open import real-numbers.positive-real-numbers
+open import real-numbers.similarity-real-numbers
+open import foundation.dependent-pair-types
+```
+
+</details>
+
+## Idea
+
+Two [positive](real-numbers.positive-real-numbers.md)
+[real numbers](real-numbers.dedekind-real-numbers.md) are
+{{#concept "similar" Disambiguation="positive real numbers" Agda=sim-ℝ⁺}} if
+they are [similar](real-numbers.similarity-real-numbers.md) as real numbers.
+
+## Definition
+
+```agda
+sim-prop-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → Prop (l1 ⊔ l2)
+sim-prop-ℝ⁺ (x , _) (y , _) = sim-prop-ℝ x y
+
+sim-ℝ⁺ : {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁺ l2 → UU (l1 ⊔ l2)
+sim-ℝ⁺ (x , _) (y , _) = sim-ℝ x y
+```
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 978356cd75..e28ec7bda1 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -131,10 +131,6 @@ module _
     reflects-le-right-add-ℝ : le-ℝ (x +ℝ z) (y +ℝ z) → le-ℝ x y
     reflects-le-right-add-ℝ x+z<y+z =
       preserves-le-sim-ℝ
-        ( (x +ℝ z) +ℝ neg-ℝ z)
-        ( x)
-        ( (y +ℝ z) +ℝ neg-ℝ z)
-        ( y)
         ( cancel-right-add-diff-ℝ x z)
         ( cancel-right-add-diff-ℝ y z)
         ( preserves-le-right-add-ℝ (neg-ℝ z) (x +ℝ z) (y +ℝ z) x+z<y+z)
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index dc20d798d6..d2a97facc6 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -496,7 +496,7 @@ module _
 
 module _
   {l1 l2 l3 l4 : Level}
-  (x1 : ℝ l1) (x2 : ℝ l2) (y1 : ℝ l3) (y2 : ℝ l4)
+  {x1 : ℝ l1} {x2 : ℝ l2} {y1 : ℝ l3} {y2 : ℝ l4}
   (x1~x2 : sim-ℝ x1 x2) (y1~y2 : sim-ℝ y1 y2)
   where
 

From 5545c34d011b95f5514e13fc0429bc6f218282f5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 15:07:38 -0800
Subject: [PATCH 03/21] Progress

---
 .../powers-positive-rational-numbers.lagda.md  |  4 ++--
 ...ic-spaces-are-uniformly-continuous.lagda.md | 18 ++++++++++--------
 src/real-numbers.lagda.md                      |  1 +
 .../nonnegative-real-numbers.lagda.md          |  4 ++--
 .../similarity-positive-real-numbers.lagda.md  |  5 +++--
 5 files changed, 18 insertions(+), 14 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 1f3932919c..08a63343eb 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -197,8 +197,8 @@ abstract
     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
+      (n , b<⟨1+q-1⟩ⁿ) =
+        bound-unbounded-power-one-plus-ℚ⁺ (positive-diff-le-ℚ 1<q) b
     in
       ( n ,
         tr
diff --git a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
index 4a4fb29d7c..c33453db28 100644
--- a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
@@ -32,7 +32,9 @@ open import order-theory.large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
+open import real-numbers.metric-space-of-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
 ```
@@ -79,15 +81,15 @@ module _
           ≤ ρ' x y' +ℝ ρ' y' y
             by is-triangular-is-metric-of-Metric-Space M ρ H x y' y
           ≤ ρ' y' y +ℝ ρ' x y'
-            by leq-eq-ℝ _ _ (commutative-add-ℝ _ _)
+            by leq-eq-ℝ (commutative-add-ℝ _ _)
           ≤ ρ' y y' +ℝ ρ' x y'
-            by leq-eq-ℝ _ _ (ap-add-ℝ (commutative-ρ' y' y) refl))
+            by leq-eq-ℝ (ap-add-ℝ (commutative-ρ' y' y) refl))
         ( chain-of-inequalities
           ρ' x y'
           ≤ ρ' x y +ℝ ρ' y y'
             by is-triangular-is-metric-of-Metric-Space M ρ H x y y'
           ≤ ρ' y y' +ℝ ρ' x y
-            by leq-eq-ℝ _ _ (commutative-add-ℝ _ _))
+            by leq-eq-ℝ (commutative-add-ℝ _ _))
 
     modulus-of-uniform-continuity-metric-of-Metric-Space : ℚ⁺ → ℚ⁺
     modulus-of-uniform-continuity-metric-of-Metric-Space =
@@ -112,7 +114,7 @@ module _
               by triangle-inequality-dist-ℝ _ _ _
             ≤ dist-ℝ (ρ' x y) (ρ' x y') +ℝ dist-ℝ (ρ' y' x) (ρ' y' x')
               by
-                leq-eq-ℝ _ _
+                leq-eq-ℝ
                   ( ap-add-ℝ
                     ( refl)
                     ( ap-binary
@@ -121,18 +123,18 @@ module _
                       ( commutative-ρ' x' y')))
             ≤ ρ' y y' +ℝ ρ' x x'
               by
-                preserves-leq-add-ℝ _ _ _ _
+                preserves-leq-add-ℝ
                   ( dist-metric-leq-metric-of-Metric-Space x y y')
                   ( dist-metric-leq-metric-of-Metric-Space y' x x')
             ≤ real-ℚ⁺ ε' +ℝ real-ℚ⁺ ε'
               by
-                preserves-leq-add-ℝ _ _ _ _
+                preserves-leq-add-ℝ
                   ( forward-implication (H ε' y y') Nε'yy')
                   ( forward-implication (H ε' x x') Nε'xx')
             ≤ real-ℚ⁺ (ε' +ℚ⁺ ε')
-              by leq-eq-ℝ _ _ (add-real-ℚ _ _)
+              by leq-eq-ℝ (add-real-ℚ _ _)
             ≤ real-ℚ⁺ ε
-              by preserves-leq-real-ℚ _ _ (leq-le-ℚ 2ε'<ε))
+              by preserves-leq-real-ℚ (leq-le-ℚ 2ε'<ε))
 
     is-uniformly-continuous-metric-of-Metric-Space :
       is-uniformly-continuous-function-Metric-Space
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 1a10077ffc..aa970733de 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -74,6 +74,7 @@ open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
+open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
 open import real-numbers.squares-real-numbers public
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index 999eff4ee7..c2f3447dd1 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -24,10 +24,10 @@ open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
-open import foundation.functoriality-propositional-truncation
 open import foundation.function-types
-open import foundation.inhabited-subtypes
+open import foundation.functoriality-propositional-truncation
 open import foundation.identity-types
+open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
diff --git a/src/real-numbers/similarity-positive-real-numbers.lagda.md b/src/real-numbers/similarity-positive-real-numbers.lagda.md
index 7253e34436..0dbfb733ac 100644
--- a/src/real-numbers/similarity-positive-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-positive-real-numbers.lagda.md
@@ -7,11 +7,12 @@ module real-numbers.similarity-positive-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.universe-levels
+open import foundation.dependent-pair-types
 open import foundation.propositions
+open import foundation.universe-levels
+
 open import real-numbers.positive-real-numbers
 open import real-numbers.similarity-real-numbers
-open import foundation.dependent-pair-types
 ```
 
 </details>

From 007d5876181591352e3536ca324d4fc69dcb13e1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 15:59:13 -0800
Subject: [PATCH 04/21] Progress

---
 ...ddition-positive-rational-numbers.lagda.md |   8 +-
 ...ive-and-negative-rational-numbers.lagda.md |   8 +-
 ...closed-intervals-rational-numbers.lagda.md |  44 ++---
 .../negative-rational-numbers.lagda.md        |  18 +-
 ...ive-and-negative-rational-numbers.lagda.md |  12 +-
 .../positive-rational-numbers.lagda.md        |   9 +-
 ...wers-nonnegative-rational-numbers.lagda.md |   2 +-
 .../squares-rational-numbers.lagda.md         |   2 +-
 ...quality-positive-rational-numbers.lagda.md |   6 +-
 ...trict-inequality-rational-numbers.lagda.md |  11 +-
 .../located-metric-spaces.lagda.md            |   5 +-
 src/real-numbers.lagda.md                     |   1 +
 .../addition-real-numbers.lagda.md            | 104 +++--------
 .../binary-maximum-real-numbers.lagda.md      | 161 ++---------------
 ...lication-nonnegative-real-numbers.lagda.md |   3 +-
 ...tiplication-positive-real-numbers.lagda.md |   6 +-
 .../multiplication-real-numbers.lagda.md      |  32 ++--
 ...ve-inverses-positive-real-numbers.lagda.md |  17 +-
 .../negation-real-numbers.lagda.md            |  13 --
 .../nonnegative-real-numbers.lagda.md         |   7 -
 ...ional-lower-dedekind-real-numbers.lagda.md |   2 +-
 ...ional-upper-dedekind-real-numbers.lagda.md |   2 +-
 ...-from-lower-dedekind-real-numbers.lagda.md |   8 +-
 ...-from-upper-dedekind-real-numbers.lagda.md |  11 +-
 ...short-binary-maximum-real-numbers.lagda.md | 166 ++++++++++++++++++
 ...re-roots-nonnegative-real-numbers.lagda.md |   8 +-
 .../squares-real-numbers.lagda.md             |   2 -
 .../strict-inequality-real-numbers.lagda.md   |   5 +-
 28 files changed, 303 insertions(+), 370 deletions(-)
 create mode 100644 src/real-numbers/short-binary-maximum-real-numbers.lagda.md

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 7e1cf13e5b..78b872ffe9 100644
--- a/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-positive-rational-numbers.lagda.md
@@ -334,16 +334,16 @@ module _
     ((d : ℚ⁺) → leq-ℚ x (y +ℚ (rational-ℚ⁺ d))) → leq-ℚ x y
   leq-leq-add-positive-ℚ H =
     rec-coproduct
-      ( λ I →
+      ( λ y<x →
         ex-falso
           ( not-leq-le-ℚ
             ( mediant-ℚ y x)
             ( x)
-            ( le-right-mediant-ℚ y x I)
+            ( le-right-mediant-ℚ y<x)
             ( tr
               ( leq-ℚ x)
-              ( right-law-positive-diff-le-ℚ (le-left-mediant-ℚ y x I))
-              ( H (positive-diff-le-ℚ (le-left-mediant-ℚ y x I))))))
+              ( right-law-positive-diff-le-ℚ (le-left-mediant-ℚ y<x))
+              ( H (positive-diff-le-ℚ (le-left-mediant-ℚ y<x))))))
       ( id)
       ( decide-le-leq-ℚ y x)
 
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 88ac145a32..8d8b52b30f 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
@@ -101,8 +101,8 @@ abstract
 ```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 =
+    (p : ℚ⁰⁺) {q : ℚ} → le-ℚ (rational-ℚ⁰⁺ p) q → is-positive-ℚ q
+  is-positive-le-ℚ⁰⁺ (p , nonneg-p) p<q =
     is-positive-le-zero-ℚ
       ( concatenate-leq-le-ℚ _ _ _ (leq-zero-is-nonnegative-ℚ nonneg-p) p<q)
 ```
@@ -112,8 +112,8 @@ abstract
 ```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 =
+    (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)
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 b05bb91cc0..ed0f43ca73 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
@@ -1,6 +1,8 @@
 # Multiplication of interior intervals of closed intervals of rational numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module elementary-number-theory.multiplication-interior-closed-intervals-rational-numbers where
 ```
 
@@ -325,12 +327,10 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=a'c')
-            is-pos-b' =
-              is-positive-le-ℚ⁰⁺ (a' , is-nonneg-a') b' a'<b'
-            is-pos-b = is-positive-le-ℚ⁺ (b' , is-pos-b') b b'<b
-            is-pos-d' =
-              is-positive-le-ℚ⁰⁺ (c' , is-nonneg-c') d' c'<d'
-            is-pos-d = is-positive-le-ℚ⁺ (d' , is-pos-d') d d'<d
+            is-pos-b' = is-positive-le-ℚ⁰⁺ (a' , is-nonneg-a') a'<b'
+            is-pos-b = is-positive-le-ℚ⁺ (b' , is-pos-b') b'<b
+            is-pos-d' = is-positive-le-ℚ⁰⁺ (c' , is-nonneg-c') c'<d'
+            is-pos-d = is-positive-le-ℚ⁺ (d' , is-pos-d') d'<d
             is-nonneg-min' =
               inv-tr
                 ( is-nonnegative-ℚ)
@@ -377,11 +377,7 @@ abstract
                         ( le-ℚ (a *ℚ c'))
                         ( min'=a'c')
                         ( preserves-le-right-mul-ℚ⁺
-                          ( c' ,
-                            is-positive-le-ℚ⁰⁺
-                              ( c , is-nonneg-c)
-                              ( c')
-                              ( c<c'))
+                          ( c' , is-positive-le-ℚ⁰⁺ (c , is-nonneg-c) c<c')
                           ( a)
                           ( a')
                           ( a<a'))))
@@ -403,10 +399,8 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=a'd')
-            is-neg-a =
-              is-negative-le-ℚ⁰⁻ (a' , is-nonpos-a') a a<a'
-            is-pos-d =
-              is-positive-le-ℚ⁰⁺ (d' , is-nonneg-d') d d'<d
+            is-neg-a = is-negative-le-ℚ⁰⁻ (a' , is-nonpos-a') a<a'
+            is-pos-d = is-positive-le-ℚ⁰⁺ (d' , is-nonneg-d') d'<d
           in
             concatenate-leq-le-ℚ
               ( min)
@@ -444,8 +438,8 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=b'c')
-            is-pos-b = is-positive-le-ℚ⁰⁺ (b' , is-nonneg-b') b b'<b
-            is-neg-c = is-negative-le-ℚ⁰⁻ (c' , is-nonpos-c') c c<c'
+            is-pos-b = is-positive-le-ℚ⁰⁺ (b' , is-nonneg-b') b'<b
+            is-neg-c = is-negative-le-ℚ⁰⁻ (c' , is-nonpos-c') c<c'
           in
             concatenate-le-leq-ℚ
               ( min)
@@ -481,12 +475,10 @@ abstract
                 ( a'<b')
                 ( c'<d')
                 ( min'=b'd')
-            is-neg-a' =
-              is-negative-le-ℚ⁰⁻ (b' , is-nonpos-b') a' a'<b'
-            is-neg-a = is-negative-le-ℚ⁻ (a' , is-neg-a') a a<a'
-            is-neg-c' =
-              is-negative-le-ℚ⁰⁻ (d' , is-nonpos-d') c' c'<d'
-            is-neg-c = is-negative-le-ℚ⁻ (c' , is-neg-c') c c<c'
+            is-neg-a' = is-negative-le-ℚ⁰⁻ (b' , is-nonpos-b') a'<b'
+            is-neg-a = is-negative-le-ℚ⁻ (a' , is-neg-a') a<a'
+            is-neg-c' = is-negative-le-ℚ⁰⁻ (d' , is-nonpos-d') c'<d'
+            is-neg-c = is-negative-le-ℚ⁻ (c' , is-neg-c') c<c'
             is-nonneg-min' =
               inv-tr
                 ( is-nonnegative-ℚ)
@@ -533,11 +525,7 @@ abstract
                           ( le-ℚ (b' *ℚ d))
                           ( min'=b'd')
                           ( reverses-le-left-mul-ℚ⁻
-                            ( b' ,
-                              is-negative-le-ℚ⁰⁻
-                                ( b , is-nonpos-b)
-                                ( b')
-                                ( b'<b))
+                            ( b' , is-negative-le-ℚ⁰⁻ (b , is-nonpos-b) b'<b)
                             ( d')
                             ( d)
                             ( d'<d)))))
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index f5754c5d86..7a41b1a752 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -201,7 +201,7 @@ module _
   {x y : ℚ} (H : le-ℚ x y)
   where
 
-  opaque
+  abstract opaque
     unfolding is-negative-ℚ
 
     is-negative-diff-le-ℚ : is-negative-ℚ (x -ℚ y)
@@ -218,7 +218,7 @@ module _
 ### Negative rational numbers are nonzero
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ
 
   is-nonzero-is-negative-ℚ : {x : ℚ} → is-negative-ℚ x → is-nonzero-ℚ x
@@ -247,30 +247,30 @@ le-ℚ⁻ p q = le-ℚ (rational-ℚ⁻ p) (rational-ℚ⁻ q)
 ```agda
 abstract
   is-negative-leq-ℚ⁻ :
-    (q : ℚ⁻) (p : ℚ) → leq-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
-  is-negative-leq-ℚ⁻ (q , neg-q) p p≤q =
+    (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
       ( 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
-  is-negative-le-ℚ⁻ q p p<q = is-negative-leq-ℚ⁻ q p (leq-le-ℚ p<q)
+    (q : ℚ⁻) {p : ℚ} → le-ℚ p (rational-ℚ⁻ q) → is-negative-ℚ p
+  is-negative-le-ℚ⁻ q p<q = is-negative-leq-ℚ⁻ q (leq-le-ℚ p<q)
 ```
 
 ### There is no greatest negative rational number
 
 ```agda
-opaque
+abstract 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 is-neg-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 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/positive-and-negative-rational-numbers.lagda.md b/src/elementary-number-theory/positive-and-negative-rational-numbers.lagda.md
index e0447a64bb..021ef28f50 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
@@ -125,7 +125,7 @@ decide-is-negative-is-positive-is-nonzero-ℚ {q} q≠0 =
 #### Any positive rational number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding is-nonnegative-ℚ is-positive-ℚ
 
   is-nonnegative-is-positive-ℚ : {q : ℚ} → is-positive-ℚ q → is-nonnegative-ℚ q
@@ -138,7 +138,7 @@ nonnegative-ℚ⁺ (q , H) = (q , is-nonnegative-is-positive-ℚ H)
 ### Any negative rational number is nonpositive
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ is-nonpositive-ℚ
 
   is-nonpositive-is-negative-ℚ : {q : ℚ} → is-negative-ℚ q → is-nonpositive-ℚ q
@@ -153,7 +153,7 @@ nonpositive-ℚ⁻ (q , H) = (q , is-nonpositive-is-negative-ℚ H)
 #### The negation of a positive rational number is negative
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ
 
   is-negative-neg-is-positive-ℚ :
@@ -168,7 +168,7 @@ neg-ℚ⁺ (q , is-pos-q) =
 #### The negation of a negative rational number is positive
 
 ```agda
-opaque
+abstract opaque
   unfolding is-negative-ℚ
 
   is-positive-neg-is-negative-ℚ :
@@ -182,7 +182,7 @@ neg-ℚ⁻ (q , is-neg-q) = (neg-ℚ q , is-positive-neg-is-negative-ℚ is-neg-
 #### The negation of a nonnegative rational number is nonpositive
 
 ```agda
-opaque
+abstract opaque
   unfolding is-nonpositive-ℚ
 
   is-nonpositive-neg-is-nonnegative-ℚ :
@@ -197,7 +197,7 @@ neg-ℚ⁰⁺ (q , is-nonneg-q) =
 #### The negation of a nonpositive rational number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding is-nonpositive-ℚ
 
   is-nonnegative-neg-is-nonpositive-ℚ :
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index dec1ae2814..f146b5f15f 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -284,13 +284,12 @@ nonzero-ℚ⁺ (x , P) = (x , is-nonzero-is-positive-ℚ P)
 ```agda
 abstract
   is-positive-leq-ℚ⁺ :
-    (p : ℚ⁺) (q : ℚ) → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
-  is-positive-leq-ℚ⁺ (p , pos-p) q p≤q =
+    (p : ℚ⁺) {q : ℚ} → leq-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
+  is-positive-leq-ℚ⁺ (p , pos-p) p≤q =
     is-positive-le-zero-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
-  is-positive-le-ℚ⁺ p q p<q =
-    is-positive-leq-ℚ⁺ p q (leq-le-ℚ p<q)
+    (p : ℚ⁺) {q : ℚ} → le-ℚ (rational-ℚ⁺ p) q → is-positive-ℚ q
+  is-positive-le-ℚ⁺ p p<q = is-positive-leq-ℚ⁺ p (leq-le-ℚ p<q)
 ```
diff --git a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
index 36c1c13e24..535de72152 100644
--- a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
@@ -122,7 +122,7 @@ abstract
       ( 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⁺ 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 p q p<q (is-nonzero-succ-ℕ _)))
diff --git a/src/elementary-number-theory/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index d9c561a017..ada7062ba5 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -236,7 +236,7 @@ abstract
       ( p *ℚ q)
       ( square-ℚ q)
       ( preserves-leq-left-mul-ℚ⁰⁺ p⁰⁺ p q (leq-le-ℚ p<q))
-      ( preserves-le-right-mul-ℚ⁺ (q , is-positive-le-ℚ⁰⁺ p⁰⁺ q p<q) p q p<q)
+      ( preserves-le-right-mul-ℚ⁺ (q , is-positive-le-ℚ⁰⁺ p⁰⁺ p<q) p q p<q)
 ```
 
 ### Squaring nonnegative rational numbers reflects inequality
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 83f68e3629..4e2a1657a3 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
@@ -82,20 +82,16 @@ strict-preorder-ℚ⁺ =
 ### There is no least positive rational number
 
 ```agda
-opaque
+abstract opaque
   mediant-zero-ℚ⁺ : ℚ⁺ → ℚ⁺
   mediant-zero-ℚ⁺ x =
     ( mediant-ℚ zero-ℚ (rational-ℚ⁺ x) ,
       is-positive-le-zero-ℚ
         ( le-left-mediant-ℚ
-          ( zero-ℚ)
-          ( 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 (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 db95edd8b7..51e9f6df63 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -519,12 +519,11 @@ abstract
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (H : le-ℚ x y)
   where
 
-  opaque
-    unfolding le-ℚ-Prop
-    unfolding mediant-ℚ
+  abstract opaque
+    unfolding le-ℚ-Prop mediant-ℚ
 
     le-left-mediant-ℚ : le-ℚ x (mediant-ℚ x y)
     le-left-mediant-ℚ =
@@ -543,7 +542,7 @@ module _
 
 ```agda
 module _
-  (x y : ℚ) (H : le-ℚ x y)
+  {x y : ℚ} (x<y : le-ℚ x y)
   where
 
   abstract
@@ -551,7 +550,7 @@ module _
     dense-le-ℚ =
       intro-exists
         ( mediant-ℚ x y)
-        ( le-left-mediant-ℚ x y H , le-right-mediant-ℚ x y H)
+        ( le-left-mediant-ℚ x<y , le-right-mediant-ℚ x<y)
 ```
 
 ### Strict inequality on the rational numbers is located
diff --git a/src/metric-spaces/located-metric-spaces.lagda.md b/src/metric-spaces/located-metric-spaces.lagda.md
index 1fef763229..a5d4416fc0 100644
--- a/src/metric-spaces/located-metric-spaces.lagda.md
+++ b/src/metric-spaces/located-metric-spaces.lagda.md
@@ -137,8 +137,7 @@ module _
           let
             r = mediant-ℚ p q
             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
+            p<r = le-left-mediant-ℚ p<q
             r⁺ =
               ( r ,
                 is-positive-le-zero-ℚ (transitive-le-ℚ zero-ℚ p r p<r 0<p))
@@ -150,7 +149,7 @@ module _
                   ((ε⁺@(ε , _) , Nεxy) , ε<p) ← p∈U
                   ¬Npxy
                     ( monotonic-neighborhood-Metric-Space M x y ε⁺ p⁺ ε<p Nεxy))
-              ( λ Nrxy → intro-exists (r⁺ , Nrxy) r<q)
+              ( λ Nrxy → intro-exists (r⁺ , Nrxy) (le-right-mediant-ℚ p<q))
               ( L x y p⁺ r⁺ p<r))
         ( λ p≤0 →
           inl-disjunction
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index aa970733de..a80d2a8bf8 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -73,6 +73,7 @@ open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
+open import real-numbers.short-binary-maximum-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index 70d1e41be9..fa74b6a4c7 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.addition-real-numbers where
 ```agda
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.addition-rational-numbers
-open import elementary-number-theory.additive-group-of-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
@@ -20,10 +19,8 @@ open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
 open import foundation.cartesian-product-types
-open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.empty-types
-open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
@@ -33,12 +30,6 @@ open import foundation.propositional-truncations
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
-open import group-theory.commutative-monoids
-open import group-theory.groups
-open import group-theory.monoids
-open import group-theory.semigroups
-
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.addition-lower-dedekind-real-numbers
@@ -46,7 +37,6 @@ open import real-numbers.addition-upper-dedekind-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
-open import real-numbers.negation-lower-upper-dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
@@ -156,48 +146,41 @@ ap-add-ℝ x=x' y=y' = ap-binary add-ℝ x=x' y=y'
 ### Addition is commutative
 
 ```agda
-module _
-  {l1 l2 : Level}
-  (x : ℝ l1) (y : ℝ l2)
-  where
+abstract opaque
+  unfolding add-ℝ
 
-  opaque
-    unfolding add-ℝ
-
-    commutative-add-ℝ : x +ℝ y ＝ y +ℝ x
-    commutative-add-ℝ =
-      eq-eq-lower-real-ℝ
-        ( x +ℝ y)
-        ( y +ℝ x)
-        ( commutative-add-lower-ℝ (lower-real-ℝ x) (lower-real-ℝ y))
+  commutative-add-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → x +ℝ y ＝ y +ℝ x
+  commutative-add-ℝ x y =
+    eq-eq-lower-real-ℝ
+      ( x +ℝ y)
+      ( y +ℝ x)
+      ( commutative-add-lower-ℝ (lower-real-ℝ x) (lower-real-ℝ y))
 ```
 
 ### Addition is associative
 
 ```agda
-module _
-  {l1 l2 l3 : Level}
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
+abstract opaque
+  unfolding add-ℝ
 
-  opaque
-    unfolding add-ℝ
-
-    associative-add-ℝ : (x +ℝ y) +ℝ z ＝ x +ℝ (y +ℝ z)
-    associative-add-ℝ =
-      eq-eq-lower-real-ℝ
-        ( (x +ℝ y) +ℝ z)
-        ( x +ℝ (y +ℝ z))
-        ( associative-add-lower-ℝ
-          ( lower-real-ℝ x)
-          ( lower-real-ℝ y)
-          ( lower-real-ℝ z))
+  associative-add-ℝ :
+    {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3) →
+    (x +ℝ y) +ℝ z ＝ x +ℝ (y +ℝ z)
+  associative-add-ℝ x y z =
+    eq-eq-lower-real-ℝ
+      ( (x +ℝ y) +ℝ z)
+      ( x +ℝ (y +ℝ z))
+      ( associative-add-lower-ℝ
+        ( lower-real-ℝ x)
+        ( lower-real-ℝ y)
+        ( lower-real-ℝ z))
 ```
 
 ### Unit laws for addition
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ real-ℚ
 
   left-unit-law-add-ℝ : {l : Level} → (x : ℝ l) → zero-ℝ +ℝ x ＝ x
@@ -218,7 +201,7 @@ opaque
 ### Inverse laws for addition
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ neg-ℝ
 
   right-inverse-law-add-ℝ :
@@ -283,7 +266,7 @@ module _
   (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ sim-ℝ
 
     preserves-sim-right-add-ℝ : sim-ℝ x y → sim-ℝ (x +ℝ z) (y +ℝ z)
@@ -406,7 +389,7 @@ module _
 ### The inclusion of rational numbers preserves addition
 
 ```agda
-opaque
+abstract opaque
   unfolding add-ℝ real-ℚ
 
   add-real-ℚ : (p q : ℚ) → real-ℚ p +ℝ real-ℚ q ＝ real-ℚ (p +ℚ q)
@@ -502,41 +485,6 @@ module _
             by sim-eq-ℝ (ap (_+ℝ neg-ℝ y) (left-unit-law-add-ℝ _)))
 ```
 
-### The Abelian group of real numbers at `lzero` under addition
-
-```agda
-semigroup-add-ℝ-lzero : Semigroup (lsuc lzero)
-semigroup-add-ℝ-lzero =
-  ( ℝ-Set lzero ,
-    add-ℝ ,
-    associative-add-ℝ)
-
-monoid-add-ℝ-lzero : Monoid (lsuc lzero)
-monoid-add-ℝ-lzero =
-  ( semigroup-add-ℝ-lzero ,
-    zero-ℝ ,
-    left-unit-law-add-ℝ ,
-    right-unit-law-add-ℝ)
-
-commutative-monoid-add-ℝ-lzero : Commutative-Monoid (lsuc lzero)
-commutative-monoid-add-ℝ-lzero =
-  ( monoid-add-ℝ-lzero ,
-    commutative-add-ℝ)
-
-group-add-ℝ-lzero : Group (lsuc lzero)
-group-add-ℝ-lzero =
-  ( ( semigroup-add-ℝ-lzero) ,
-    ( zero-ℝ , left-unit-law-add-ℝ , right-unit-law-add-ℝ) ,
-    ( neg-ℝ ,
-      eq-sim-ℝ ∘ left-inverse-law-add-ℝ ,
-      eq-sim-ℝ ∘ right-inverse-law-add-ℝ))
-
-abelian-group-add-ℝ-lzero : Ab (lsuc lzero)
-abelian-group-add-ℝ-lzero =
-  ( group-add-ℝ-lzero ,
-    commutative-add-ℝ)
-```
-
 ### If `x + y` is similar to `0`, then `y` is similar to `-x` and `x` is similar to `-y`
 
 ```agda
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 2482c2f855..34cafb59ec 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.binary-maximum-real-numbers where
 ```agda
 open import elementary-number-theory.positive-rational-numbers
 
-open import foundation.binary-transport
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
@@ -22,9 +21,6 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import metric-spaces.metric-space-of-short-functions-metric-spaces
-open import metric-spaces.short-functions-metric-spaces
-
 open import order-theory.join-semilattices
 open import order-theory.large-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
@@ -37,7 +33,6 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
 open import real-numbers.maximum-upper-dedekind-real-numbers
-open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
@@ -125,7 +120,7 @@ module _
   (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ max-ℝ
 
     is-least-binary-upper-bound-max-ℝ :
@@ -177,28 +172,24 @@ module _
 ### The binary maximum is commutative
 
 ```agda
-module _
-  {l1 l2 : Level}
-  (x : ℝ l1) (y : ℝ l2)
-  where
+abstract opaque
+  unfolding leq-ℝ sim-ℝ
 
-  opaque
-    unfolding leq-ℝ sim-ℝ
-
-    commutative-max-ℝ : max-ℝ x y ＝ max-ℝ y x
-    commutative-max-ℝ =
-      eq-sim-ℝ
-        ( sim-is-least-binary-upper-bound-Large-Poset
+  commutative-max-ℝ :
+    {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → max-ℝ x y ＝ max-ℝ y x
+  commutative-max-ℝ x y =
+    eq-sim-ℝ
+      ( sim-is-least-binary-upper-bound-Large-Poset
+        ( ℝ-Large-Poset)
+        ( x)
+        ( y)
+        ( is-least-binary-upper-bound-max-ℝ x y)
+        ( is-binary-least-upper-bound-swap-Large-Poset
           ( ℝ-Large-Poset)
-          ( x)
           ( y)
-          ( is-least-binary-upper-bound-max-ℝ x y)
-          ( is-binary-least-upper-bound-swap-Large-Poset
-            ( ℝ-Large-Poset)
-            ( y)
-            ( x)
-            ( max-ℝ y x)
-            ( is-least-binary-upper-bound-max-ℝ y x)))
+          ( x)
+          ( max-ℝ y x)
+          ( is-least-binary-upper-bound-max-ℝ y x)))
 ```
 
 ### The large poset of real numbers has joins
@@ -306,126 +297,6 @@ abstract
   preserves-sim-right-max-ℝ a = preserves-sim-max-ℝ a a (refl-sim-ℝ a)
 ```
 
-### The binary maximum preserves lower neighborhoods
-
-```agda
-module _
-  {l1 l2 l3 : Level} (d : ℚ⁺)
-  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
-  where
-
-  abstract
-    preserves-lower-neighborhood-leq-left-max-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ
-        ( max-ℝ x y)
-        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d =
-      leq-is-least-binary-upper-bound-Large-Poset
-        ( ℝ-Large-Poset)
-        ( x)
-        ( y)
-        ( is-least-binary-upper-bound-max-ℝ x y)
-        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
-        ( ( transitive-leq-ℝ
-            ( x)
-            ( max-ℝ x z)
-            ( max-ℝ x z +ℝ real-ℚ⁺ d)
-            ( leq-le-ℝ
-              ( le-left-add-real-ℝ⁺
-                ( max-ℝ x z)
-                ( positive-real-ℚ⁺ d)))
-            ( leq-left-max-ℝ x z)) ,
-          ( transitive-leq-ℝ
-            ( y)
-            ( z +ℝ real-ℚ⁺ d)
-            ( max-ℝ x z +ℝ real-ℚ⁺ d)
-            ( preserves-leq-right-add-ℝ
-              ( real-ℚ⁺ d)
-              ( z)
-              ( max-ℝ x z)
-              ( leq-right-max-ℝ x z))
-            ( z≤y+d)))
-
-    preserves-lower-neighborhood-leq-right-max-ℝ :
-      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
-      leq-ℝ
-        ( max-ℝ y x)
-        ( (max-ℝ z x) +ℝ real-ℚ⁺ d)
-    preserves-lower-neighborhood-leq-right-max-ℝ z≤y+d =
-      binary-tr
-        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
-        ( commutative-max-ℝ x y)
-        ( commutative-max-ℝ x z)
-        ( preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d)
-```
-
-### The maximum with a real number is a short function `ℝ → ℝ`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1)
-  where
-
-  abstract
-    is-short-function-left-max-ℝ :
-      is-short-function-Metric-Space
-        ( metric-space-ℝ l2)
-        ( metric-space-ℝ (l1 ⊔ l2))
-        ( max-ℝ x)
-    is-short-function-left-max-ℝ d y z Nyz =
-      neighborhood-real-bound-each-leq-ℝ
-        ( d)
-        ( max-ℝ x y)
-        ( max-ℝ x z)
-        ( preserves-lower-neighborhood-leq-left-max-ℝ d x y z
-          ( left-leq-real-bound-neighborhood-ℝ d y z Nyz))
-        ( preserves-lower-neighborhood-leq-left-max-ℝ d x z y
-          ( right-leq-real-bound-neighborhood-ℝ d y z Nyz))
-
-  short-left-max-ℝ :
-    short-function-Metric-Space
-      ( metric-space-ℝ l2)
-      ( metric-space-ℝ (l1 ⊔ l2))
-  short-left-max-ℝ =
-    (max-ℝ x , is-short-function-left-max-ℝ)
-```
-
-### The binary maximum is a short function from `ℝ` to the metric space of short functions `ℝ → ℝ`
-
-```agda
-module _
-  {l1 l2 : Level}
-  where
-
-  abstract
-    is-short-function-short-left-max-ℝ :
-      is-short-function-Metric-Space
-        ( metric-space-ℝ l1)
-        ( metric-space-of-short-functions-Metric-Space
-          ( metric-space-ℝ l2)
-          ( metric-space-ℝ (l1 ⊔ l2)))
-        ( short-left-max-ℝ)
-    is-short-function-short-left-max-ℝ d x y Nxy z =
-      neighborhood-real-bound-each-leq-ℝ
-        ( d)
-        ( max-ℝ x z)
-        ( max-ℝ y z)
-        ( preserves-lower-neighborhood-leq-right-max-ℝ d z x y
-          ( left-leq-real-bound-neighborhood-ℝ d x y Nxy))
-        ( preserves-lower-neighborhood-leq-right-max-ℝ d z y x
-          ( right-leq-real-bound-neighborhood-ℝ d x y Nxy))
-
-  short-max-ℝ :
-    short-function-Metric-Space
-      ( metric-space-ℝ l1)
-      ( metric-space-of-short-functions-Metric-Space
-        ( metric-space-ℝ l2)
-        ( metric-space-ℝ (l1 ⊔ l2)))
-  short-max-ℝ =
-    (short-left-max-ℝ , is-short-function-short-left-max-ℝ)
-```
-
 ### For any `ε : ℚ⁺`, `(max-ℝ x y - ε < x) ∨ (max-ℝ x y - ε < y)`
 
 ```agda
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index dd18d3c8e5..a20149edf8 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -47,7 +47,7 @@ nonnegative.
 ## Definition
 
 ```agda
-opaque
+abstract opaque
   unfolding mul-ℝ
 
   is-nonnegative-mul-ℝ :
@@ -67,7 +67,6 @@ opaque
             d⁺ = (d , is-positive-is-in-upper-cut-ℝ⁰⁺ (y , 0≤y) y<d)
           is-positive-le-ℚ⁺
             ( b⁺ *ℚ⁺ d⁺)
-            ( q)
             ( concatenate-leq-le-ℚ
               ( b *ℚ d)
               ( upper-bound-mul-closed-interval-ℚ [a,b] [c,d])
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index ab63080b99..63176527d9 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -53,7 +53,7 @@ module _
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding mul-ℝ
 
     is-positive-mul-ℝ :
@@ -67,9 +67,9 @@ module _
         (d , y<d) ← is-inhabited-upper-cut-ℝ y
         let
           a<b = le-lower-upper-cut-ℝ x a b a<x x<b
-          b⁺ = (b , is-positive-le-ℚ⁺ a⁺ b a<b)
+          b⁺ = (b , is-positive-le-ℚ⁺ a⁺ a<b)
           c<d = le-lower-upper-cut-ℝ y c d c<y y<d
-          d⁺ = (d , is-positive-le-ℚ⁺ c⁺ d c<d)
+          d⁺ = (d , is-positive-le-ℚ⁺ c⁺ c<d)
           [a,b] = ((a , b) , leq-le-ℚ a<b)
           [c,d] = ((c , d) , leq-le-ℚ c<d)
         is-positive-exists-ℚ⁺-in-lower-cut-ℝ
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index e26e0fb218..53040c6a1a 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import elementary-number-theory.maximum-natural-numbers
 open import elementary-number-theory.maximum-nonnegative-rational-numbers
 open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minimum-positive-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-interior-closed-intervals-rational-numbers
 open import elementary-number-theory.multiplication-nonnegative-rational-numbers
@@ -36,7 +35,6 @@ 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
 open import elementary-number-theory.strict-inequality-rational-numbers
 open import elementary-number-theory.unit-fractions-rational-numbers
 
@@ -52,19 +50,16 @@ open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.similarity-subtypes
 open import foundation.subtypes
 open import foundation.transport-along-identifications
-open import foundation.univalence
 open import foundation.universe-levels
 
 open import group-theory.groups
 
 open import logic.functoriality-existential-quantification
 
-open import order-theory.large-posets
 open import order-theory.posets
 
 open import real-numbers.addition-real-numbers
@@ -77,7 +72,6 @@ open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.upper-dedekind-real-numbers
 ```
 
@@ -255,7 +249,7 @@ module _
           do-syntax-trunc-Prop (∃ ℚ (λ r → le-ℚ-Prop q r ∧ lower-cut-mul-ℝ r))
       in do
         ((a<x<b , c<y<d) , q<[a,b][c,d]) ← q∈L
-        (r , q<r , r<[a,b][c,d]) ← dense-le-ℚ q _ q<[a,b][c,d]
+        (r , q<r , r<[a,b][c,d]) ← dense-le-ℚ q<[a,b][c,d]
         intro-exists r (q<r , intro-exists (a<x<b , c<y<d) r<[a,b][c,d])
 
     forward-implication-is-rounded-upper-cut-mul-ℝ :
@@ -267,7 +261,7 @@ module _
           do-syntax-trunc-Prop (∃ ℚ (λ q → le-ℚ-Prop q r ∧ upper-cut-mul-ℝ q))
       in do
         ((a<x<b , c<y<d) , [a,b][c,d]<r) ← r∈U
-        (q , [a,b][c,d]<q , q<r) ← dense-le-ℚ _ r [a,b][c,d]<r
+        (q , [a,b][c,d]<q , q<r) ← dense-le-ℚ [a,b][c,d]<r
         intro-exists q (q<r , intro-exists (a<x<b , c<y<d) [a,b][c,d]<q)
 
     backward-implication-is-rounded-lower-cut-mul-ℝ :
@@ -616,7 +610,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding mul-ℝ
 
     enclosing-rational-range-mul-ℝ :
@@ -717,13 +711,13 @@ module _
 ### Commutativity of multiplication
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ mul-ℝ
 
-  leq-commute-ℝ :
+  leq-commute-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) →
     leq-ℝ (x *ℝ y) (y *ℝ x)
-  leq-commute-ℝ x y q q<xy =
+  leq-commute-mul-ℝ x y q q<xy =
     let open do-syntax-trunc-Prop (lower-cut-mul-ℝ y x q)
     in do
       ((a<x<b@([a,b] , _ , _) , c<y<d@([c,d] , _ , _)) , q<[a,b][c,d]) ← q<xy
@@ -737,7 +731,7 @@ opaque
 abstract
   commutative-mul-ℝ : {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → x *ℝ y ＝ y *ℝ x
   commutative-mul-ℝ x y =
-    antisymmetric-leq-ℝ _ _ (leq-commute-ℝ x y) (leq-commute-ℝ y x)
+    antisymmetric-leq-ℝ _ _ (leq-commute-mul-ℝ x y) (leq-commute-mul-ℝ y x)
 ```
 
 ### Associativity of multiplication
@@ -747,7 +741,7 @@ module _
   {l1 l2 l3 : Level} (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ mul-ℝ
 
     leq-associative-mul-ℝ : leq-ℝ ((x *ℝ y) *ℝ z) (x *ℝ (y *ℝ z))
@@ -811,7 +805,7 @@ module _
   {l : Level} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' mul-ℝ real-ℚ
 
     leq-right-unit-law-mul-ℝ : leq-ℝ (x *ℝ one-ℝ) x
@@ -875,7 +869,7 @@ module _
 ### Distributivity of multiplication over addition
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ leq-ℝ' mul-ℝ add-ℝ
 
   leq-left-distributive-mul-add-ℝ :
@@ -1044,7 +1038,7 @@ module _
   {l : Level} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' mul-ℝ real-ℚ
 
     leq-left-zero-law-mul-ℝ : leq-ℝ (zero-ℝ *ℝ x) zero-ℝ
@@ -1109,7 +1103,7 @@ module _
 ### The inclusion of rational numbers preserves multiplication
 
 ```agda
-opaque
+abstract opaque
   unfolding mul-ℝ real-ℚ
 
   mul-real-ℚ : (p q : ℚ) → real-ℚ p *ℝ real-ℚ q ＝ real-ℚ (p *ℚ q)
@@ -1157,7 +1151,7 @@ opaque
 ### Multiplication on the real numbers preserves similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ leq-ℝ' mul-ℝ sim-ℝ
 
   leq-sim-right-mul-ℝ :
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 d17a359116..7a3823d101 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -188,7 +188,7 @@ module _
               ( is-rounded-lower-cut-ℝ⁺ x (rational-inv-ℚ⁺ q⁺))
               ( q⁻¹<x)
           intro-exists
-            ( r , is-positive-le-ℚ⁺ (inv-ℚ⁺ q⁺) r q⁻¹<r)
+            ( r , is-positive-le-ℚ⁺ (inv-ℚ⁺ q⁺) q⁻¹<r)
             ( r<x ,
               tr
                 ( λ s → le-ℚ s (q *ℚ r))
@@ -316,7 +316,7 @@ module _
       in do
         (q' , q<q' , q'∈L) ← ∃q'
         (r'⁺@(r' , _) , x<r' , q'r'<1) ←
-          q'∈L (is-positive-le-ℚ⁺ (q , is-pos-q) q' q<q')
+          q'∈L (is-positive-le-ℚ⁺ (q , is-pos-q) q<q')
         intro-exists
           ( r'⁺)
           ( x<r' ,
@@ -347,7 +347,7 @@ module _
         (r⁺@(r , _) , r<x , 1<qr) ← ∃r
         (r' , r<r' , r'<x) ←
           forward-implication (is-rounded-lower-cut-ℝ⁺ x r) r<x
-        let r'⁺ = (r' , is-positive-le-ℚ⁺ r⁺ r' r<r')
+        let r'⁺ = (r' , is-positive-le-ℚ⁺ r⁺ r<r')
         intro-exists
           ( rational-ℚ⁺ (inv-ℚ⁺ r'⁺))
           ( transitive-le-ℚ
@@ -379,7 +379,7 @@ module _
       in do
         (q' , q'<q , is-pos-q' , ∃r') ← ∃q'
         (r'⁺@(r' , _) , x<r' , 1<q'r') ← ∃r'
-        ( is-positive-le-ℚ⁺ (q' , is-pos-q') q q'<q ,
+        ( is-positive-le-ℚ⁺ (q' , is-pos-q') q'<q ,
           intro-exists
             ( r'⁺)
             ( x<r' ,
@@ -421,7 +421,7 @@ module _
         ( λ is-pos-p →
           let
             p⁺ = (p , is-pos-p)
-            is-pos-q = is-positive-le-ℚ⁺ p⁺ q p<q
+            is-pos-q = is-positive-le-ℚ⁺ p⁺ p<q
             q⁺ = (q , is-pos-q)
           in
             elim-disjunction
@@ -513,8 +513,7 @@ module _
         a' = max-ℚ a (rational-inv-ℚ⁺ d⁺)
         a'<x =
           is-in-lower-cut-max-ℚ (real-ℝ⁺ x) a (rational-inv-ℚ⁺ d⁺) a<x d⁻¹<x
-        is-pos-a' =
-          is-positive-leq-ℚ⁺ (inv-ℚ⁺ d⁺) a' (leq-right-max-ℚ _ _)
+        is-pos-a' = is-positive-leq-ℚ⁺ (inv-ℚ⁺ d⁺) (leq-right-max-ℚ _ _)
         a'⁺ = (a' , is-pos-a')
         c⁺ = (c , is-pos-c)
         x<c⁻¹ : is-in-upper-cut-ℝ⁺ x (rational-inv-ℚ⁺ c⁺)
@@ -635,10 +634,10 @@ module _
             (c'⁺@(c' , is-pos-c') , c'<x⁻¹) ← exists-ℚ⁺-in-lower-cut-inv-ℝ⁺ x
             let
               a'' = max-ℚ a a'
-              is-pos-a'' = is-positive-leq-ℚ⁺ a'⁺ a'' (leq-right-max-ℚ _ _)
+              is-pos-a'' = is-positive-leq-ℚ⁺ a'⁺ (leq-right-max-ℚ _ _)
               a''<x = is-in-lower-cut-max-ℚ (real-ℝ⁺ x) a a' a<x a'<x
               c'' = max-ℚ c c'
-              is-pos-c'' = is-positive-leq-ℚ⁺ c'⁺ c'' (leq-right-max-ℚ _ _)
+              is-pos-c'' = is-positive-leq-ℚ⁺ c'⁺ (leq-right-max-ℚ _ _)
               c''<x⁻¹ = is-in-lower-cut-max-ℚ (real-inv-ℝ⁺ x) c c' c<x⁻¹ c'<x⁻¹
               [a'',b] =
                 ( (a'' , b) , leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'' b a''<x x<b)
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 183ddbcd09..71573c09b7 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -9,37 +9,24 @@ module real-numbers.negation-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-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.cartesian-product-types
-open import foundation.conjunction
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjoint-subtypes
 open import foundation.disjunction
-open import foundation.empty-types
-open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.negation
-open import foundation.propositional-truncations
-open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.negation-lower-upper-dedekind-real-numbers
-open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.rational-upper-dedekind-real-numbers
 open import real-numbers.similarity-real-numbers
 open import real-numbers.upper-dedekind-real-numbers
 ```
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index c2f3447dd1..fcb7e126e0 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.nonnegative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-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.negative-rational-numbers
 open import elementary-number-theory.nonnegative-rational-numbers
@@ -20,17 +18,13 @@ open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.conjunction
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
-open import foundation.empty-types
 open import foundation.existential-quantification
-open import foundation.function-types
 open import foundation.functoriality-propositional-truncation
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
@@ -45,7 +39,6 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.inequalities-addition-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
index 5176279b29..3dbae48f3b 100644
--- a/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
@@ -50,7 +50,7 @@ module _
   pr1 (is-rounded-cut-lower-real-ℚ p) p<q =
     intro-exists
       ( mediant-ℚ p q)
-      ( le-left-mediant-ℚ p q p<q , le-right-mediant-ℚ p q p<q)
+      ( le-left-mediant-ℚ p<q , le-right-mediant-ℚ p<q)
   pr2 (is-rounded-cut-lower-real-ℚ p) =
     elim-exists
       ( le-ℚ-Prop p q)
diff --git a/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
index 5e897208f5..ca0acca1dc 100644
--- a/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
@@ -50,7 +50,7 @@ module _
   pr1 (is-rounded-cut-upper-real-ℚ p) q<p =
     intro-exists
       ( mediant-ℚ q p)
-      ( le-right-mediant-ℚ q p q<p , le-left-mediant-ℚ q p q<p)
+      ( le-right-mediant-ℚ q<p , le-left-mediant-ℚ q<p)
   pr2 (is-rounded-cut-upper-real-ℚ p) =
     elim-exists
       ( le-ℚ-Prop q p)
diff --git a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
index 53cc1f1c7d..227fa5c7af 100644
--- a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
@@ -94,8 +94,8 @@ module _
         (p , p<q , p∉L) ← q∈U
         intro-exists
           ( mediant-ℚ p q)
-          ( le-right-mediant-ℚ p q p<q ,
-            intro-exists p (le-left-mediant-ℚ p q p<q , p∉L))
+          ( le-right-mediant-ℚ p<q ,
+            intro-exists p (le-left-mediant-ℚ p<q , p∉L))
     pr2 (is-rounded-upper-cut-real-lower-ℝ q) ∃p =
       let open do-syntax-trunc-Prop (upper-cut-real-lower-ℝ q)
       in do
@@ -131,8 +131,8 @@ module _
       in
         map-disjunction
           ( id)
-          ( λ r∉L → intro-exists r ( le-right-mediant-ℚ p q p<q , r∉L))
-          ( L p r (le-left-mediant-ℚ p q p<q))
+          ( λ r∉L → intro-exists r ( le-right-mediant-ℚ p<q , r∉L))
+          ( L p r (le-left-mediant-ℚ p<q))
 
   real-lower-ℝ : ℝ l
   real-lower-ℝ =
diff --git a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
index 622ad1d0a4..80c6cacfa9 100644
--- a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
@@ -41,7 +41,8 @@ open import real-numbers.upper-dedekind-real-numbers
 Given an
 [upper Dedekind real number](real-numbers.upper-dedekind-real-numbers.md) `x`,
 we can convert `x` to a
-[Dedekind real number](real-numbers.dedekind-real-numbers.md) if and only if the
+[Dedekind real number](real-numbers.dedekind-real-numbers.md)
+[if and only if](foundation.logical-equivalences.md) the
 [complement](foundation.complements-subtypes.md) of the cut of `x` is
 [inhabited](foundation.inhabited-subtypes.md) and for any `p q : ℚ` with
 `p < q`, `p` is in the [complement](foundation.complements-subtypes.md) of the
@@ -89,8 +90,8 @@ module _
         (r , q<r , r∉U) ← q∈L
         intro-exists
           ( mediant-ℚ q r)
-          ( le-left-mediant-ℚ q r q<r ,
-            intro-exists r (le-right-mediant-ℚ q r q<r , r∉U))
+          ( le-left-mediant-ℚ q<r ,
+            intro-exists r (le-right-mediant-ℚ q<r , r∉U))
     pr2 (is-rounded-lower-cut-real-upper-ℝ q) ∃r =
       let open do-syntax-trunc-Prop (lower-cut-real-upper-ℝ q)
       in do
@@ -125,9 +126,9 @@ module _
         elim-disjunction
           (lower-cut-real-upper-ℝ p ∨ cut-upper-ℝ x q)
           ( λ r∉U →
-            inl-disjunction (intro-exists r (le-left-mediant-ℚ p q p<q , r∉U)))
+            inl-disjunction (intro-exists r (le-left-mediant-ℚ p<q , r∉U)))
           ( inr-disjunction)
-          ( L r q (le-right-mediant-ℚ p q p<q))
+          ( L r q (le-right-mediant-ℚ p<q))
 
   real-upper-ℝ : ℝ l
   real-upper-ℝ =
diff --git a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
new file mode 100644
index 0000000000..2ac42192c9
--- /dev/null
+++ b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
@@ -0,0 +1,166 @@
+## The binary maximum of real numbers is a short function
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.short-binary-maximum-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.positive-rational-numbers
+
+open import foundation.binary-transport
+open import foundation.dependent-pair-types
+open import foundation.universe-levels
+
+open import metric-spaces.metric-space-of-short-functions-metric-spaces
+open import metric-spaces.short-functions-metric-spaces
+
+open import order-theory.least-upper-bounds-large-posets
+
+open import real-numbers.addition-real-numbers
+open import real-numbers.binary-maximum-real-numbers
+open import real-numbers.dedekind-real-numbers
+open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequality-real-numbers
+open import real-numbers.metric-space-of-real-numbers
+open import real-numbers.positive-real-numbers
+open import real-numbers.rational-real-numbers
+open import real-numbers.strict-inequality-real-numbers
+```
+
+</details>
+
+## Idea
+
+For any `a : ℝ`, the
+[binary maximum](real-numbers.binary-maximum-real-numbers.md) with `a` is a
+[short function](metric-spaces.short-functions-metric-spaces.md) `ℝ → ℝ` for the
+[standard real metric structure](real-numbers.metric-space-of-real-numbers.md).
+Moreover, the map `x ↦ max-ℝ x` is a short function from `ℝ` into the
+[metric space of short functions](metric-spaces.metric-space-of-short-functions-metric-spaces.md)
+of `ℝ`.
+
+## Proof
+
+### The binary maximum preserves lower neighborhoods
+
+```agda
+module _
+  {l1 l2 l3 : Level} (d : ℚ⁺)
+  (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
+  where
+
+  abstract
+    preserves-lower-neighborhood-leq-left-max-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ
+        ( max-ℝ x y)
+        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d =
+      leq-is-least-binary-upper-bound-Large-Poset
+        ( ℝ-Large-Poset)
+        ( x)
+        ( y)
+        ( is-least-binary-upper-bound-max-ℝ x y)
+        ( (max-ℝ x z) +ℝ real-ℚ⁺ d)
+        ( ( transitive-leq-ℝ
+            ( x)
+            ( max-ℝ x z)
+            ( max-ℝ x z +ℝ real-ℚ⁺ d)
+            ( leq-le-ℝ
+              ( le-left-add-real-ℝ⁺
+                ( max-ℝ x z)
+                ( positive-real-ℚ⁺ d)))
+            ( leq-left-max-ℝ x z)) ,
+          ( transitive-leq-ℝ
+            ( y)
+            ( z +ℝ real-ℚ⁺ d)
+            ( max-ℝ x z +ℝ real-ℚ⁺ d)
+            ( preserves-leq-right-add-ℝ
+              ( real-ℚ⁺ d)
+              ( z)
+              ( max-ℝ x z)
+              ( leq-right-max-ℝ x z))
+            ( z≤y+d)))
+
+    preserves-lower-neighborhood-leq-right-max-ℝ :
+      leq-ℝ y (z +ℝ real-ℚ⁺ d) →
+      leq-ℝ
+        ( max-ℝ y x)
+        ( (max-ℝ z x) +ℝ real-ℚ⁺ d)
+    preserves-lower-neighborhood-leq-right-max-ℝ z≤y+d =
+      binary-tr
+        ( λ u v → leq-ℝ u (v +ℝ real-ℚ⁺ d))
+        ( commutative-max-ℝ x y)
+        ( commutative-max-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-max-ℝ z≤y+d)
+```
+
+### The maximum with a real number is a short function `ℝ → ℝ`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1)
+  where
+
+  abstract
+    is-short-function-left-max-ℝ :
+      is-short-function-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2))
+        ( max-ℝ x)
+    is-short-function-left-max-ℝ d y z Nyz =
+      neighborhood-real-bound-each-leq-ℝ
+        ( d)
+        ( max-ℝ x y)
+        ( max-ℝ x z)
+        ( preserves-lower-neighborhood-leq-left-max-ℝ d x y z
+          ( left-leq-real-bound-neighborhood-ℝ d y z Nyz))
+        ( preserves-lower-neighborhood-leq-left-max-ℝ d x z y
+          ( right-leq-real-bound-neighborhood-ℝ d y z Nyz))
+
+  short-left-max-ℝ :
+    short-function-Metric-Space
+      ( metric-space-ℝ l2)
+      ( metric-space-ℝ (l1 ⊔ l2))
+  short-left-max-ℝ =
+    (max-ℝ x , is-short-function-left-max-ℝ)
+```
+
+### The binary maximum is a short function from `ℝ` to the metric space of short functions `ℝ → ℝ`
+
+```agda
+module _
+  {l1 l2 : Level}
+  where
+
+  abstract
+    is-short-function-short-left-max-ℝ :
+      is-short-function-Metric-Space
+        ( metric-space-ℝ l1)
+        ( metric-space-of-short-functions-Metric-Space
+          ( metric-space-ℝ l2)
+          ( metric-space-ℝ (l1 ⊔ l2)))
+        ( short-left-max-ℝ)
+    is-short-function-short-left-max-ℝ d x y Nxy z =
+      neighborhood-real-bound-each-leq-ℝ
+        ( d)
+        ( max-ℝ x z)
+        ( max-ℝ y z)
+        ( preserves-lower-neighborhood-leq-right-max-ℝ d z x y
+          ( left-leq-real-bound-neighborhood-ℝ d x y Nxy))
+        ( preserves-lower-neighborhood-leq-right-max-ℝ d z y x
+          ( right-leq-real-bound-neighborhood-ℝ d x y Nxy))
+
+  short-max-ℝ :
+    short-function-Metric-Space
+      ( metric-space-ℝ l1)
+      ( metric-space-of-short-functions-Metric-Space
+        ( metric-space-ℝ l2)
+        ( metric-space-ℝ (l1 ⊔ l2)))
+  short-max-ℝ =
+    (short-left-max-ℝ , is-short-function-short-left-max-ℝ)
+```
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 154237168d..dce82321de 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -164,7 +164,6 @@ module _
                 is-pos-p =
                   is-positive-le-ℚ⁺
                     ( q *ℚ q , is-positive-mul-ℚ is-pos-q is-pos-q)
-                    ( p)
                     ( q²<p)
               (q'⁺@(q' , _) , q<q' , q'²<p) ←
                 rounded-below-square-le-ℚ⁺ (q , is-pos-q) (p , is-pos-p) q²<p
@@ -203,7 +202,7 @@ module _
         let
           is-nonneg-r =
             is-nonnegative-is-positive-ℚ
-              ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) r q<r)
+              ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) q<r)
         le-lower-cut-ℝ
           ( real-ℝ⁰⁺ x)
           ( q *ℚ q)
@@ -218,7 +217,7 @@ module _
       let open do-syntax-trunc-Prop (upper-cut-sqrt-ℝ⁰⁺ r)
       in do
         (q , q<r , is-pos-q , x<q²) ← ∃q
-        let is-pos-r = is-positive-le-ℚ⁺ (q , is-pos-q) r q<r
+        let is-pos-r = is-positive-le-ℚ⁺ (q , is-pos-q) q<r
         ( is-pos-r ,
           le-upper-cut-ℝ
             ( real-ℝ⁰⁺ x)
@@ -268,7 +267,7 @@ module _
         ( λ is-nonneg-p →
           map-disjunction
             ( λ p²<x _ → p²<x)
-            ( λ x<q² → (is-positive-le-ℚ⁰⁺ (p , is-nonneg-p) q p<q , x<q²))
+            ( λ x<q² → (is-positive-le-ℚ⁰⁺ (p , is-nonneg-p) p<q , x<q²))
             ( is-located-lower-upper-cut-ℝ
               ( real-ℝ⁰⁺ x)
               ( preserves-le-square-ℚ⁰⁺
@@ -338,7 +337,6 @@ module _
               is-pos-xy x y {r} lb1 lb2 =
                 is-positive-le-ℚ⁰⁺
                   ( q , is-nonneg-q)
-                  ( x *ℚ y)
                   ( concatenate-le-leq-ℚ
                     ( q)
                     ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 36ce763cb8..1b6faca2fd 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -119,7 +119,6 @@ opaque
               in
                 is-positive-le-ℚ⁺
                   ( a'⁻ *ℚ⁻ a'⁻)
-                  ( q)
                   ( concatenate-leq-le-ℚ
                     ( a' *ℚ a')
                     ( upper-bound-mul-closed-interval-ℚ [a',b'] [a',b'])
@@ -134,7 +133,6 @@ opaque
               in
                 is-positive-le-ℚ⁺
                   ( b'⁺ *ℚ⁺ b'⁺)
-                  ( q)
                   ( concatenate-leq-le-ℚ
                     ( b' *ℚ b')
                     ( upper-bound-mul-closed-interval-ℚ [a',b'] [a',b'])
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index d2a97facc6..b8256aa198 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -177,10 +177,7 @@ module _
     unfolding le-ℝ real-ℚ
 
     preserves-le-real-ℚ : le-ℚ x y → le-ℝ (real-ℚ x) (real-ℚ y)
-    preserves-le-real-ℚ x<y =
-      intro-exists
-        ( mediant-ℚ x y)
-        ( le-left-mediant-ℚ x y x<y , le-right-mediant-ℚ x y x<y)
+    preserves-le-real-ℚ = dense-le-ℚ
 
     reflects-le-real-ℚ : le-ℝ (real-ℚ x) (real-ℚ y) → le-ℚ x y
     reflects-le-real-ℚ =

From 6241f8cb6c5ff0c574e58e803dbf8e8ff444af4c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:01:54 -0800
Subject: [PATCH 05/21] Progress

---
 .../metrics-of-metric-spaces.lagda.md         |   1 +
 .../absolute-value-real-numbers.lagda.md      |   2 +-
 .../addition-real-numbers.lagda.md            |   4 +-
 ...thmetically-located-dedekind-cuts.lagda.md |   6 +-
 ...ompleteness-dedekind-real-numbers.lagda.md |   8 --
 .../dedekind-real-numbers.lagda.md            |   9 +-
 .../distance-real-numbers.lagda.md            |   2 +-
 ...d-rational-intervals-real-numbers.lagda.md |  13 +-
 ...nequalities-addition-real-numbers.lagda.md |   2 +-
 ...ality-lower-dedekind-real-numbers.lagda.md |   2 -
 .../inequality-real-numbers.lagda.md          |  14 +-
 ...nuity-multiplication-real-numbers.lagda.md |   2 -
 .../metric-space-of-real-numbers.lagda.md     |   5 +-
 ...tiplication-negative-real-numbers.lagda.md |  12 +-
 ...lication-nonnegative-real-numbers.lagda.md |  14 +-
 ...tiplication-positive-real-numbers.lagda.md |  32 ++---
 .../multiplication-real-numbers.lagda.md      |  24 ++--
 ...ve-inverses-positive-real-numbers.lagda.md |  31 ++--
 ...lower-upper-dedekind-real-numbers.lagda.md |  21 ++-
 .../negation-real-numbers.lagda.md            |   4 +-
 .../positive-real-numbers.lagda.md            |   8 +-
 .../rational-real-numbers.lagda.md            |   2 +-
 .../similarity-real-numbers.lagda.md          |  14 +-
 ...re-roots-nonnegative-real-numbers.lagda.md |  34 ++---
 .../squares-real-numbers.lagda.md             |  10 +-
 ...nequalities-addition-real-numbers.lagda.md |  33 ++++-
 .../strict-inequality-real-numbers.lagda.md   | 135 +++++-------------
 27 files changed, 180 insertions(+), 264 deletions(-)

diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index 7a8cea0f77..d113ff5bac 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -28,6 +28,7 @@ open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.saturation-inequality-nonnegative-real-numbers
 open import real-numbers.similarity-nonnegative-real-numbers
+open import real-numbers.strict-inequalities-addition-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 8d3c38266f..0ebbeb29d4 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -107,7 +107,7 @@ nonnegative-abs-ℝ x = (abs-ℝ x , is-nonnegative-abs-ℝ x)
 ### The absolute value of the negation of a real number is its absolute value
 
 ```agda
-opaque
+abstract opaque
   unfolding abs-ℝ
 
   abs-neg-ℝ : {l : Level} → (x : ℝ l) → abs-ℝ (neg-ℝ x) ＝ abs-ℝ x
diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index fa74b6a4c7..fef1ddfc68 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -88,8 +88,8 @@ module _
               { qx}
               { py}
               { qy}
-              ( le-lower-upper-cut-ℝ x px qx px<x x<qx)
-              ( le-lower-upper-cut-ℝ y py qy py<y y<qy)))
+              ( le-lower-upper-cut-ℝ x px<x x<qx)
+              ( le-lower-upper-cut-ℝ y py<y y<qy)))
 
     is-arithmetically-located-lower-upper-add-ℝ :
       is-arithmetically-located-lower-upper-ℝ lower-real-add-ℝ upper-real-add-ℝ
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index ffca8483f9..f70d4ca8b4 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -358,7 +358,7 @@ abstract
                 ( chain-of-inequalities
                     a
                     ≤ q
-                      by leq-lower-upper-cut-ℝ x a q a<x x<q
+                      by leq-lower-upper-cut-ℝ x a<x x<q
                     ≤ rational-abs-ℚ q
                       by leq-abs-ℚ q
                     ≤ ε +ℚ rational-abs-ℚ q
@@ -387,7 +387,7 @@ abstract
                           ( neg-ℚ b)
                           ( neg-ℚ p)
                           ( neg-leq-ℚ
-                            ( leq-lower-upper-cut-ℝ x p b p<x x<b))
+                            ( leq-lower-upper-cut-ℝ x p<x x<b))
                     ≤ rational-abs-ℚ (ε -ℚ p)
                       by leq-abs-ℚ _
                     ≤ rational-abs-ℚ ε +ℚ rational-abs-ℚ p
@@ -419,7 +419,7 @@ 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<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/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 2a3e529451..57f47ac4dd 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -207,8 +207,6 @@ module _
             ( ε⁺ , θ⁺)
             ( le-lower-cut-ℝ
               ( xε)
-              ( q +ℚ (ε +ℚ θ))
-              ( r +ℚ (ε +ℚ θ))
               ( preserves-le-left-add-ℚ (ε +ℚ θ) q r q<r)
               ( r+ε+θ<xε))
 
@@ -274,8 +272,6 @@ module _
             ( ε⁺ , θ⁺)
             ( le-upper-cut-ℝ
               ( xε)
-              ( p -ℚ (ε +ℚ θ))
-              ( q -ℚ (ε +ℚ θ))
               ( preserves-le-left-add-ℚ (neg-ℚ (ε +ℚ θ)) p q p<q)
               ( xε<p-ε-θ))
 
@@ -344,8 +340,6 @@ module _
           ( q -ℚ εu)
           ( le-lower-cut-ℝ
               ( xεu)
-              ( q -ℚ εu)
-              ( (q -ℚ εu) +ℚ θl)
               ( le-right-add-rational-ℚ⁺ (q -ℚ εu) θl⁺)
               ( q-εu+θl<xεu) ,
             tr
@@ -359,8 +353,6 @@ module _
                 ＝ q -ℚ εu by is-section-diff-ℚ θu _)
               ( le-upper-cut-ℝ
                 ( xεu)
-                ( q -ℚ (εu +ℚ θu))
-                ( (q -ℚ (εu +ℚ θu)) +ℚ θu)
                 ( le-right-add-rational-ℚ⁺ (q -ℚ (εu +ℚ θu)) θu⁺)
                 ( xεu<q-εu-θu)))
 
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index a6183b2bc4..0c1531e91f 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -222,7 +222,7 @@ abstract
 
 ```agda
 module _
-  {l : Level} (x : ℝ l) (p q : ℚ)
+  {l : Level} (x : ℝ l) {p q : ℚ}
   where
 
   le-lower-cut-ℝ :
@@ -254,7 +254,7 @@ module _
 
 ```agda
 module _
-  {l : Level} (x : ℝ l) (p q : ℚ)
+  {l : Level} (x : ℝ l) {p q : ℚ}
   where
 
   abstract
@@ -265,10 +265,7 @@ module _
     le-lower-upper-cut-ℝ H H' =
       rec-coproduct
         ( id)
-        ( λ I →
-          ex-falso
-            ( is-disjoint-cut-ℝ x p
-                ( H , leq-upper-cut-ℝ x q p I H')))
+        ( λ I → ex-falso (is-disjoint-cut-ℝ x p (H , leq-upper-cut-ℝ x I H')))
         ( decide-le-leq-ℚ p q)
 
     leq-lower-upper-cut-ℝ :
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 1f779167f0..eff38740f5 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -149,7 +149,7 @@ abstract
           ( transpose-add-is-in-lower-cut-ℝ x q d q+d<x))
         ( swap-right-diff-leq-ℝ x y (real-ℚ d) x-y≤d))
 
-opaque
+abstract opaque
   unfolding neighborhood-ℝ
 
   neighborhood-dist-ℝ :
diff --git a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
index 1f9272b8db..77d3ab8c6b 100644
--- a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
@@ -8,15 +8,11 @@ module real-numbers.enclosing-closed-rational-intervals-real-numbers where
 
 ```agda
 open import elementary-number-theory.closed-intervals-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.intersections-closed-intervals-rational-numbers
-open import elementary-number-theory.maximum-rational-numbers
-open import elementary-number-theory.minimum-rational-numbers
 open import elementary-number-theory.rational-numbers
 
 open import foundation.conjunction
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.inhabited-types
@@ -24,7 +20,6 @@ open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
@@ -81,7 +76,7 @@ module _
         (p , p<x) ← is-inhabited-lower-cut-ℝ x
         (q , x<q) ← is-inhabited-upper-cut-ℝ x
         intro-exists
-          ( (p , q) , leq-lower-upper-cut-ℝ x p q p<x x<q)
+          ( (p , q) , leq-lower-upper-cut-ℝ x p<x x<q)
           ( p<x , x<q)
 ```
 
@@ -117,7 +112,7 @@ module _
         (b' , b'<b , x<b') ←
           forward-implication (is-rounded-upper-cut-ℝ x b) x<b
         intro-exists
-          ( (a' , b') , leq-lower-upper-cut-ℝ x a' b' a'<x x<b')
+          ( (a' , b') , leq-lower-upper-cut-ℝ x a'<x x<b')
           ( ( a'<x , x<b') , (a<a' , b'<b))
 ```
 
@@ -136,8 +131,8 @@ module _
       intersect-closed-interval-ℚ [a,b] [c,d]
     intersect-enclosing-closed-rational-interval-ℝ
       [a,b]@((a , b) , _) [c,d]@((c , d) , _) (a<x , x<b) (c<x , x<d) =
-      ( leq-lower-upper-cut-ℝ x a d a<x x<d ,
-        leq-lower-upper-cut-ℝ x c b c<x x<b)
+      ( leq-lower-upper-cut-ℝ x a<x x<d ,
+        leq-lower-upper-cut-ℝ x c<x x<b)
 
     is-enclosing-intersection-enclosing-closed-rational-interval-ℝ :
       ([a,b] [c,d] : closed-interval-ℚ) →
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
index aa5f3ad62e..b5af4c0787 100644
--- a/src/real-numbers/inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/inequalities-addition-real-numbers.lagda.md
@@ -48,7 +48,7 @@ module _
   {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ leq-ℝ
 
     preserves-leq-right-add-ℝ : leq-ℝ x y → leq-ℝ (x +ℝ z) (y +ℝ z)
diff --git a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
index 5f31c60805..7e9c14d77b 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -18,8 +18,6 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.logical-equivalences
-open import foundation.negation
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.unit-type
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index d0a0cd049c..89c0fbbb57 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -10,20 +10,15 @@ module real-numbers.inequality-real-numbers where
 
 ```agda
 open import elementary-number-theory.inequality-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.cartesian-product-types
 open import foundation.complements-subtypes
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.function-types
-open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.negation
@@ -33,25 +28,18 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import order-theory.large-posets
 open import order-theory.large-preorders
 open import order-theory.posets
 open import order-theory.preorders
 open import order-theory.similarity-of-elements-large-posets
 
-open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-lower-dedekind-real-numbers
 open import real-numbers.inequality-upper-dedekind-real-numbers
-open import real-numbers.lower-dedekind-real-numbers
-open import real-numbers.negation-lower-upper-dedekind-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
@@ -62,7 +50,7 @@ The {{#concept "standard ordering" Disambiguation="real numbers" Agda=leq-ℝ}}
 the [real numbers](real-numbers.dedekind-real-numbers.md) is defined as the
 [lower cut](real-numbers.lower-dedekind-real-numbers.md) of one being a
 [subset](foundation-core.subtypes.md) of the lower cut of the other. I.e.,
-`x ≤ y` if `lower-cut x ⊆ lower-cut y `. This is the definition used in
+`x ≤ y` if `lower-cut-ℝ x ⊆ lower-cut-ℝ y `. This is the definition used in
 {{#cite UF13}}, section 11.2.1.
 
 Inequality of the real numbers is equivalently described by the _upper_ cut of
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index 7b3d8f8c66..ed960956fe 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -258,8 +258,6 @@ module _
                   by
                     preserves-leq-right-mul-ℝ⁰⁺
                       ( nonnegative-real-ℚ⁺ ε)
-                      ( my +ℝ mx)
-                      ( real-ℚ q)
                       ( leq-le-ℝ (le-real-is-in-upper-cut-ℚ (my +ℝ mx) my+mx<q))
                 ≤ real-ℚ⁺ (q⁺ *ℚ⁺ ε)
                   by leq-eq-ℝ (mul-real-ℚ q (rational-ℚ⁺ ε))))
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index 7a64835572..d199e3ef54 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -132,10 +132,7 @@ module _
       diagonal-product
         ( (r : ℚ) →
           is-in-lower-cut-ℝ x (r +ℚ (rational-ℚ⁺ d)) → is-in-lower-cut-ℝ x r)
-        ( λ r →
-          le-lower-cut-ℝ x r
-            ( r +ℚ rational-ℚ⁺ d)
-            ( le-right-add-rational-ℚ⁺ r d))
+        ( λ r → le-lower-cut-ℝ x (le-right-add-rational-ℚ⁺ r d))
 
     is-symmetric-neighborhood-ℝ :
       is-symmetric-Rational-Neighborhood-Relation (neighborhood-prop-ℝ l)
diff --git a/src/real-numbers/multiplication-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
index e13cdd5cc2..7d2e4c8289 100644
--- a/src/real-numbers/multiplication-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-negative-real-numbers.lagda.md
@@ -69,14 +69,14 @@ _*ℝ⁻_ = mul-ℝ⁻
 ```agda
 abstract
   reverses-le-left-mul-ℝ⁻ :
-    {l1 l2 l3 : Level} (x : ℝ⁻ l1) (y : ℝ l2) (z : ℝ l3) → le-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁻ l1) {y : ℝ l2} {z : ℝ l3} → le-ℝ y z →
     le-ℝ (real-ℝ⁻ x *ℝ z) (real-ℝ⁻ x *ℝ y)
-  reverses-le-left-mul-ℝ⁻ x y z y<z =
+  reverses-le-left-mul-ℝ⁻ x y<z =
     binary-tr
       ( le-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-le-ℝ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-le-ℝ (preserves-le-left-mul-ℝ⁺ (neg-ℝ⁻ x) y<z))
 ```
 
 ### Multiplication by a negative real number reverses inequality
@@ -84,12 +84,12 @@ abstract
 ```agda
 abstract
   reverses-leq-left-mul-ℝ⁻ :
-    {l1 l2 l3 : Level} (x : ℝ⁻ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁻ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (real-ℝ⁻ x *ℝ z) (real-ℝ⁻ x *ℝ y)
-  reverses-leq-left-mul-ℝ⁻ x y z y<z =
+  reverses-leq-left-mul-ℝ⁻ x y<z =
     binary-tr
       ( leq-ℝ)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
       ( ap neg-ℝ (left-negative-law-mul-ℝ _ _) ∙ neg-neg-ℝ _)
-      ( neg-leq-ℝ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y z y<z))
+      ( neg-leq-ℝ (preserves-leq-left-mul-ℝ⁺ (neg-ℝ⁻ x) y<z))
 ```
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index a20149edf8..a1bb60ef8e 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -96,9 +96,9 @@ ap-mul-ℝ⁰⁺ = ap-binary mul-ℝ⁰⁺
 ```agda
 abstract
   preserves-leq-left-mul-ℝ⁰⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (real-ℝ⁰⁺ x *ℝ y) (real-ℝ⁰⁺ x *ℝ z)
-  preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺@(x , 0≤x) y z y≤z =
+  preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺@(x , 0≤x) {y} {z} y≤z =
     leq-is-nonnegative-diff-ℝ
       ( x *ℝ y)
       ( x *ℝ z)
@@ -110,14 +110,14 @@ abstract
           ( is-nonnegative-diff-leq-ℝ y≤z)))
 
   preserves-leq-right-mul-ℝ⁰⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁰⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (y *ℝ real-ℝ⁰⁺ x) (z *ℝ real-ℝ⁰⁺ x)
-  preserves-leq-right-mul-ℝ⁰⁺ x y z y≤z =
+  preserves-leq-right-mul-ℝ⁰⁺ x y≤z =
     binary-tr
       ( leq-ℝ)
       ( commutative-mul-ℝ _ _)
       ( commutative-mul-ℝ _ _)
-      ( preserves-leq-left-mul-ℝ⁰⁺ x y z y≤z)
+      ( preserves-leq-left-mul-ℝ⁰⁺ x y≤z)
 
   preserves-leq-mul-ℝ⁰⁺ :
     {l1 l2 l3 l4 : Level} →
@@ -125,6 +125,6 @@ abstract
     leq-ℝ⁰⁺ x x' → leq-ℝ⁰⁺ y y' → leq-ℝ⁰⁺ (x *ℝ⁰⁺ y) (x' *ℝ⁰⁺ y')
   preserves-leq-mul-ℝ⁰⁺ x x' y y' x≤x' y≤y' =
     transitive-leq-ℝ _ _ _
-      ( preserves-leq-right-mul-ℝ⁰⁺ y' (real-ℝ⁰⁺ x) (real-ℝ⁰⁺ x') x≤x')
-      ( preserves-leq-left-mul-ℝ⁰⁺ x (real-ℝ⁰⁺ y) (real-ℝ⁰⁺ y') y≤y')
+      ( preserves-leq-right-mul-ℝ⁰⁺ y' x≤x')
+      ( preserves-leq-left-mul-ℝ⁰⁺ x y≤y')
 ```
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index 63176527d9..7972741f76 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -66,9 +66,9 @@ module _
         (c⁺@(c , _) , c<y) ← exists-ℚ⁺-in-lower-cut-is-positive-ℝ y 0<y
         (d , y<d) ← is-inhabited-upper-cut-ℝ y
         let
-          a<b = le-lower-upper-cut-ℝ x a b a<x x<b
+          a<b = le-lower-upper-cut-ℝ x a<x x<b
           b⁺ = (b , is-positive-le-ℚ⁺ a⁺ a<b)
-          c<d = le-lower-upper-cut-ℝ y c d c<y y<d
+          c<d = le-lower-upper-cut-ℝ y c<y y<d
           d⁺ = (d , is-positive-le-ℚ⁺ c⁺ c<d)
           [a,b] = ((a , b) , leq-le-ℚ a<b)
           [c,d] = ((c , d) , leq-le-ℚ c<d)
@@ -110,9 +110,9 @@ abstract
 ```agda
 abstract
   preserves-le-left-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → le-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → le-ℝ y z →
     le-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z)
-  preserves-le-left-mul-ℝ⁺ x⁺@(x , 0<x) y z y<z =
+  preserves-le-left-mul-ℝ⁺ x⁺@(x , 0<x) {y} {z} y<z =
     le-is-positive-diff-ℝ
       ( tr
         ( is-positive-ℝ)
@@ -120,29 +120,29 @@ abstract
         ( is-positive-mul-ℝ 0<x (is-positive-diff-le-ℝ y<z)))
 
   reflects-le-left-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} →
     le-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z) → le-ℝ y z
-  reflects-le-left-mul-ℝ⁺ x y z xy<xz =
+  reflects-le-left-mul-ℝ⁺ x {y} {z} xy<xz =
     preserves-le-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
-      ( preserves-le-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy<xz)
+      ( preserves-le-left-mul-ℝ⁺ (inv-ℝ⁺ x) xy<xz)
 
   preserves-le-right-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → le-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → le-ℝ y z →
     le-ℝ (y *ℝ real-ℝ⁺ x) (z *ℝ real-ℝ⁺ x)
-  preserves-le-right-mul-ℝ⁺ x y z y<z =
+  preserves-le-right-mul-ℝ⁺ x y<z =
     binary-tr
       ( le-ℝ)
       ( commutative-mul-ℝ _ _)
       ( commutative-mul-ℝ _ _)
-      ( preserves-le-left-mul-ℝ⁺ x y z y<z)
+      ( preserves-le-left-mul-ℝ⁺ x y<z)
 
   reflects-le-right-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} →
     le-ℝ (y *ℝ real-ℝ⁺ x) (z *ℝ real-ℝ⁺ x) → le-ℝ y z
-  reflects-le-right-mul-ℝ⁺ x y z yx<zx =
-    reflects-le-left-mul-ℝ⁺ x y z
+  reflects-le-right-mul-ℝ⁺ x yx<zx =
+    reflects-le-left-mul-ℝ⁺ x
       ( binary-tr
         ( le-ℝ)
         ( commutative-mul-ℝ _ _)
@@ -155,12 +155,12 @@ abstract
 ```agda
 abstract
   preserves-leq-left-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (real-ℝ⁺ x *ℝ y) (real-ℝ⁺ x *ℝ z)
   preserves-leq-left-mul-ℝ⁺ x⁺ = preserves-leq-left-mul-ℝ⁰⁺ (nonnegative-ℝ⁺ x⁺)
 
   preserves-leq-right-mul-ℝ⁺ :
-    {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) → leq-ℝ y z →
+    {l1 l2 l3 : Level} (x : ℝ⁺ l1) {y : ℝ l2} {z : ℝ l3} → leq-ℝ y z →
     leq-ℝ (y *ℝ real-ℝ⁺ x) (z *ℝ real-ℝ⁺ x)
   preserves-leq-right-mul-ℝ⁺ x⁺ =
     preserves-leq-right-mul-ℝ⁰⁺ (nonnegative-ℝ⁺ x⁺)
@@ -172,7 +172,7 @@ abstract
     preserves-leq-sim-ℝ
       ( cancel-left-div-mul-ℝ⁺ x y)
       ( cancel-left-div-mul-ℝ⁺ x z)
-      ( preserves-leq-left-mul-ℝ⁺ (inv-ℝ⁺ x) _ _ xy≤xz)
+      ( preserves-leq-left-mul-ℝ⁺ (inv-ℝ⁺ x) xy≤xz)
 
   reflects-leq-right-mul-ℝ⁺ :
     {l1 l2 l3 : Level} (x : ℝ⁺ l1) (y : ℝ l2) (z : ℝ l3) →
diff --git a/src/real-numbers/multiplication-real-numbers.lagda.md b/src/real-numbers/multiplication-real-numbers.lagda.md
index 53040c6a1a..b389211c2d 100644
--- a/src/real-numbers/multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-real-numbers.lagda.md
@@ -160,8 +160,8 @@ module _
               ( [c',d'])
               ( a<a' , b'<b)
               ( c<c' , d'<d)
-              ( le-lower-upper-cut-ℝ x a' b' a'<x x<b')
-              ( le-lower-upper-cut-ℝ y c' d' c'<y y<d')))
+              ( le-lower-upper-cut-ℝ x a'<x x<b')
+              ( le-lower-upper-cut-ℝ y c'<y y<d')))
 
     leq-upper-cut-mul-ℝ'-upper-cut-mul-ℝ : upper-cut-mul-ℝ' ⊆ upper-cut-mul-ℝ
     leq-upper-cut-mul-ℝ'-upper-cut-mul-ℝ q q∈U' =
@@ -190,8 +190,8 @@ module _
               ( [c',d'])
               ( a<a' , b'<b)
               ( c<c' , d'<d)
-              ( le-lower-upper-cut-ℝ x a' b' a'<x x<b')
-              ( le-lower-upper-cut-ℝ y c' d' c'<y y<d'))
+              ( le-lower-upper-cut-ℝ x a'<x x<b')
+              ( le-lower-upper-cut-ℝ y c'<y y<d'))
             ( [a,b][c,d]≤q))
 
     eq-lower-cut-mul-ℝ' : lower-cut-mul-ℝ ＝ lower-cut-mul-ℝ'
@@ -336,11 +336,11 @@ module _
                     ( c'')
                     ( leq-right-max-ℚ a a' ,
                       leq-max-leq-both-ℚ b' a a'
-                        ( leq-lower-upper-cut-ℝ x a b' a<x x<b')
+                        ( leq-lower-upper-cut-ℝ x a<x x<b')
                         ( a'≤b'))
                     ( leq-right-max-ℚ c c' ,
                       leq-max-leq-both-ℚ d' c c'
-                        ( leq-lower-upper-cut-ℝ y c d' c<y y<d')
+                        ( leq-lower-upper-cut-ℝ y c<y y<d')
                         ( c'≤d'))))
                 ( pr1
                   ( is-in-mul-interval-mul-is-in-closed-interval-ℚ
@@ -351,11 +351,11 @@ module _
                     ( leq-left-max-ℚ a a' ,
                       leq-max-leq-both-ℚ b a a'
                         ( a≤b)
-                        ( leq-lower-upper-cut-ℝ x a' b a'<x x<b))
+                        ( leq-lower-upper-cut-ℝ x a'<x x<b))
                     ( leq-left-max-ℚ c c' ,
                       leq-max-leq-both-ℚ d c c'
                         ( c≤d)
-                        ( leq-lower-upper-cut-ℝ y c' d c'<y y<d)))))
+                        ( leq-lower-upper-cut-ℝ y c'<y y<d)))))
               ( [a',b'][c',d']<q))
             ( q<[a,b][c,d]))
 
@@ -427,8 +427,8 @@ module _
         ((p , q) , q<p+εx , p<x , x<q) ← is-arithmetically-located-ℝ x εx
         ((r , s) , s<r+εy , r<y , y<s) ← is-arithmetically-located-ℝ y εy
         let
-          p≤q = leq-lower-upper-cut-ℝ x p q p<x x<q
-          r≤s = leq-lower-upper-cut-ℝ y r s r<y y<s
+          p≤q = leq-lower-upper-cut-ℝ x p<x x<q
+          r≤s = leq-lower-upper-cut-ℝ y r<y y<s
           q-p<εx : le-ℚ (q -ℚ p) (rational-ℚ⁺ εx)
           q-p<εx =
             le-transpose-right-add-ℚ _ _ _
@@ -815,7 +815,7 @@ module _
         ( ( ax<x<bx@([ax,bx]@((ax , bx) , _) , ax<x , x<bx) ,
             a₁<1<b₁@([a₁,b₁]@((a₁ , b₁) , _) , a₁<1 , 1<b₁)) ,
           q<[ax,bx][a₁,b₁]) ← q<x1
-        le-lower-cut-ℝ x q ax
+        le-lower-cut-ℝ x
           ( concatenate-le-leq-ℚ _ _ _
             ( q<[ax,bx][a₁,b₁])
             ( tr
@@ -840,7 +840,7 @@ module _
             ( ( ax<x<bx@([ax,bx]@((ax , bx) , _) , ax<x , x<bx) ,
                 a₁<1<b₁@([a₁,b₁]@((a₁ , b₁) , _) , a₁<1 , 1<b₁)) ,
               [ax,bx][a₁,b₁]<q) ← x1<q
-            le-upper-cut-ℝ x bx q
+            le-upper-cut-ℝ x
               ( concatenate-leq-le-ℚ _ _ _
                 ( tr
                   ( λ p →
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 7a3823d101..490dd19044 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -117,8 +117,6 @@ module _
         (r⁺@(r , _) , x<r , qr<1) ← q∈L is-pos-q
         le-upper-cut-ℝ
           ( real-ℝ⁺ x)
-          ( r)
-          ( rational-inv-ℚ⁺ q⁺)
           ( reflects-le-left-mul-ℚ⁺
             ( q⁺)
             ( r)
@@ -140,8 +138,6 @@ module _
           (r⁺@(r , _) , r<x , 1<qr) ← ∃r
           le-lower-cut-ℝ
             ( real-ℝ⁺ x)
-            ( rational-inv-ℚ⁺ q⁺)
-            ( r)
             ( reflects-le-left-mul-ℚ⁺
               ( q⁺)
               ( rational-inv-ℚ⁺ q⁺)
@@ -166,7 +162,7 @@ module _
             ( is-rounded-upper-cut-ℝ⁺ x (rational-inv-ℚ⁺ q⁺))
             ( q∈L' is-pos-q)
         intro-exists
-          ( r , is-positive-is-in-upper-cut-ℝ⁺ x r x<r)
+          ( r , is-positive-is-in-upper-cut-ℝ⁺ x x<r)
           ( x<r ,
             tr
               ( le-ℚ (q *ℚ r))
@@ -216,7 +212,7 @@ module _
       let open do-syntax-trunc-Prop (∃ ℚ⁺ (lower-cut-inv-ℝ⁺ ∘ rational-ℚ⁺))
       in do
         (q , x<q) ← is-inhabited-upper-cut-ℝ⁺ x
-        let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x q x<q)
+        let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x x<q)
         intro-exists
           ( inv-ℚ⁺ q⁺)
           ( leq-lower-cut-inv-ℝ⁺'-lower-cut-inv-ℝ⁺
@@ -274,7 +270,7 @@ module _
                 forward-implication (is-rounded-upper-cut-ℝ⁺ x r) x<r
               let
                 r⁻¹ = inv-ℚ⁺ r⁺
-                r'⁺ = (r' , is-positive-is-in-upper-cut-ℝ⁺ x r' x<r')
+                r'⁺ = (r' , is-positive-is-in-upper-cut-ℝ⁺ x x<r')
                 r'⁻¹ = inv-ℚ⁺ r'⁺
               intro-exists
                 ( rational-ℚ⁺ r'⁻¹)
@@ -411,7 +407,7 @@ module _
             ( _)
             ( _)
             ( transitive-le-ℚ (q *ℚ rₗ) one-ℚ (q *ℚ rᵤ) 1<qrᵤ qrₗ<1))
-          ( le-lower-upper-cut-ℝ⁺ x rᵤ rₗ rᵤ<x x<rₗ)
+          ( le-lower-upper-cut-ℝ⁺ x rᵤ<x x<rₗ)
 
     is-located-lower-upper-cut-inv-ℝ⁺ :
       (p q : ℚ) → le-ℚ p q →
@@ -506,7 +502,7 @@ module _
       [a,b]@((a , b) , a≤b) [c,d]@((c , d) , c≤d) (a<x , x<b) (c<x⁻¹ , x⁻¹<d)
       is-pos-a is-pos-c =
       let
-        is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x b x<b
+        is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x x<b
         (is-pos-d , d⁻¹<x) =
           leq-upper-cut-inv-ℝ⁺-upper-cut-inv-ℝ⁺' x d x⁻¹<d
         d⁺ = (d , is-pos-d)
@@ -531,12 +527,7 @@ module _
             ( inv-leq-ℚ⁺
               ( a'⁺)
               ( inv-ℚ⁺ c⁺)
-              ( leq-lower-upper-cut-ℝ
-                ( real-ℝ⁺ x)
-                ( a')
-                ( rational-inv-ℚ⁺ c⁺)
-                ( a'<x)
-                ( x<c⁻¹)))
+              ( leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'<x x<c⁻¹))
       in
         tr
           ( is-in-closed-interval-ℚ
@@ -548,7 +539,7 @@ module _
             ( a')
             ( rational-inv-ℚ⁺ a'⁺)
             ( leq-left-max-ℚ _ _ ,
-              leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a' b a'<x x<b)
+              leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'<x x<b)
             ( c≤a'⁻¹ , a'⁻¹≤d))
 
     leq-real-right-inverse-law-mul-ℝ⁺ :
@@ -560,7 +551,7 @@ module _
             ([c,d]@((c , d) , c≤d) , c<x⁻¹ , x⁻¹<d)) ,
           q<[a,b][c,d]) ← q<xx⁻¹
         let
-          is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x b x<b
+          is-pos-b = is-positive-is-in-upper-cut-ℝ⁺ x x<b
           (is-pos-d , d⁻¹<x) =
             leq-upper-cut-inv-ℝ⁺-upper-cut-inv-ℝ⁺' x d x⁻¹<d
           case-is-nonpos-a is-nonpos-a =
@@ -640,10 +631,10 @@ module _
               is-pos-c'' = is-positive-leq-ℚ⁺ c'⁺ (leq-right-max-ℚ _ _)
               c''<x⁻¹ = is-in-lower-cut-max-ℚ (real-inv-ℝ⁺ x) c c' c<x⁻¹ c'<x⁻¹
               [a'',b] =
-                ( (a'' , b) , leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a'' b a''<x x<b)
+                ( (a'' , b) , leq-lower-upper-cut-ℝ (real-ℝ⁺ x) a''<x x<b)
               [c'',d] =
                 ( (c'' , d) ,
-                  leq-lower-upper-cut-ℝ (real-inv-ℝ⁺ x) c'' d c''<x⁻¹ x⁻¹<d)
+                leq-lower-upper-cut-ℝ (real-inv-ℝ⁺ x) c''<x⁻¹ x⁻¹<d)
             concatenate-leq-le-ℚ
               ( one-ℚ)
               ( upper-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -718,7 +709,7 @@ opaque
     let open do-syntax-trunc-Prop (le-prop-ℝ⁺ (inv-ℝ⁺ y) (inv-ℝ⁺ x))
     in do
       (q , x<q , q<y) ← x<y
-      let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x q x<q)
+      let q⁺ = (q , is-positive-is-in-upper-cut-ℝ⁺ x x<q)
       intro-exists
         ( rational-inv-ℚ⁺ q⁺)
         ( leq-upper-cut-inv-ℝ⁺'-upper-cut-inv-ℝ⁺
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 0bf3ac519c..ab010a21b3 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
@@ -19,7 +19,6 @@ open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.retractions
 open import foundation.sections
 open import foundation.similarity-subtypes
@@ -29,8 +28,6 @@ open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
-open import real-numbers.inequality-lower-dedekind-real-numbers
-open import real-numbers.inequality-upper-dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
@@ -177,16 +174,18 @@ abstract
           (λ p → tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)))))
 
   is-equiv-neg-lower-ℝ : (l : Level) → is-equiv (neg-lower-ℝ {l})
-  pr1 (pr1 (is-equiv-neg-lower-ℝ l)) = neg-upper-ℝ
-  pr1 (pr2 (is-equiv-neg-lower-ℝ l)) = neg-upper-ℝ
-  pr2 (pr1 (is-equiv-neg-lower-ℝ l)) = is-section-neg-upper-ℝ l
-  pr2 (pr2 (is-equiv-neg-lower-ℝ l)) = is-retraction-neg-upper-ℝ l
+  is-equiv-neg-lower-ℝ l =
+    is-equiv-is-invertible
+      ( neg-upper-ℝ)
+      ( is-section-neg-upper-ℝ l)
+      ( is-retraction-neg-upper-ℝ l)
 
   is-equiv-neg-upper-ℝ : (l : Level) → is-equiv (neg-upper-ℝ {l})
-  pr1 (pr1 (is-equiv-neg-upper-ℝ l)) = neg-lower-ℝ
-  pr1 (pr2 (is-equiv-neg-upper-ℝ l)) = neg-lower-ℝ
-  pr2 (pr1 (is-equiv-neg-upper-ℝ l)) = is-retraction-neg-upper-ℝ l
-  pr2 (pr2 (is-equiv-neg-upper-ℝ l)) = is-section-neg-upper-ℝ l
+  is-equiv-neg-upper-ℝ l =
+    is-equiv-is-invertible
+      ( neg-lower-ℝ)
+      ( is-retraction-neg-upper-ℝ l)
+      ( is-section-neg-upper-ℝ l)
 ```
 
 ### The negation of a rational projected to a lower real is the projection of its negation as an upper real
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 71573c09b7..514703a9e3 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -87,7 +87,7 @@ module _
 ### The negation function on real numbers is an involution
 
 ```agda
-opaque
+abstract opaque
   unfolding neg-ℝ
 
   neg-neg-ℝ : {l : Level} → (x : ℝ l) → neg-ℝ (neg-ℝ x) ＝ x
@@ -130,7 +130,7 @@ abstract
 ### Negation preserves similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding neg-ℝ
 
   preserves-sim-neg-ℝ :
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 95cc41a662..dcad10a676 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -111,7 +111,7 @@ is-rounded-upper-cut-ℝ⁺ :
 is-rounded-upper-cut-ℝ⁺ (x , _) = is-rounded-upper-cut-ℝ x
 
 le-lower-upper-cut-ℝ⁺ :
-  {l : Level} (x : ℝ⁺ l) (p q : ℚ) →
+  {l : Level} (x : ℝ⁺ l) {p q : ℚ} →
   is-in-lower-cut-ℝ⁺ x p → is-in-upper-cut-ℝ⁺ x q → le-ℚ p q
 le-lower-upper-cut-ℝ⁺ (x , _) = le-lower-upper-cut-ℝ x
 ```
@@ -277,8 +277,8 @@ one-ℝ⁺ = positive-real-ℚ⁺ one-ℚ⁺
 ```agda
 abstract
   is-positive-is-in-upper-cut-ℝ⁺ :
-    {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 =
+    {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 , _) x<q =
     is-positive-le-zero-ℚ
-      ( le-lower-upper-cut-ℝ x zero-ℚ q (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
+      ( le-lower-upper-cut-ℝ x (zero-in-lower-cut-ℝ⁺ x⁺) x<q)
 ```
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index c639837691..df01919a62 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -232,7 +232,7 @@ opaque
       ( q)
       ( id)
       ( λ p=q → ex-falso (q∉lx (tr (is-in-lower-cut-ℝ x) p=q p∈lx)))
-      ( λ q<p → ex-falso (q∉lx (le-lower-cut-ℝ x q p q<p p∈lx)))
+      ( λ q<p → ex-falso (q∉lx (le-lower-cut-ℝ x q<p p∈lx)))
   pr2 (sim-rational-ℝ (x , q , q∉lx , q∉ux)) p p<q =
     elim-disjunction
       ( lower-cut-ℝ x p)
diff --git a/src/real-numbers/similarity-real-numbers.lagda.md b/src/real-numbers/similarity-real-numbers.lagda.md
index 3f8a1ef83c..5664b3a49b 100644
--- a/src/real-numbers/similarity-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-real-numbers.lagda.md
@@ -70,7 +70,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding sim-ℝ
 
     sim-lower-cut-iff-sim-ℝ :
@@ -85,7 +85,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding sim-ℝ
 
     sim-sim-upper-cut-ℝ : sim-subtype (upper-cut-ℝ x) (upper-cut-ℝ y) → (x ~ℝ y)
@@ -102,7 +102,7 @@ module _
 ### Reflexivity
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   refl-sim-ℝ : {l : Level} → (x : ℝ l) → x ~ℝ x
@@ -115,7 +115,7 @@ opaque
 ### Symmetry
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   symmetric-sim-ℝ :
@@ -127,7 +127,7 @@ opaque
 ### Transitivity
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   transitive-sim-ℝ :
@@ -141,7 +141,7 @@ opaque
 ### Similar real numbers in the same universe are equal
 
 ```agda
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   eq-sim-ℝ : {l : Level} → {x y : ℝ l} → x ~ℝ y → x ＝ y
@@ -185,7 +185,7 @@ similarity-reasoning-ℝ
 infixl 1 similarity-reasoning-ℝ_
 infixl 0 step-similarity-reasoning-ℝ
 
-opaque
+abstract opaque
   unfolding sim-ℝ
 
   similarity-reasoning-ℝ_ :
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 dce82321de..d73dc98a0f 100644
--- a/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/square-roots-nonnegative-real-numbers.lagda.md
@@ -105,7 +105,7 @@ module _
         (p⁺@(p , is-pos-p) , q<p²) ← square-le-ℚ⁺' q⁺
         intro-exists
           ( p)
-          ( is-pos-p , le-upper-cut-ℝ (real-ℝ⁰⁺ x) q (p *ℚ p) q<p² x<q)
+          ( is-pos-p , le-upper-cut-ℝ (real-ℝ⁰⁺ x) q<p² x<q)
 
     forward-implication-is-rounded-lower-cut-sqrt-ℝ⁰⁺ :
       (q : ℚ) → is-in-subtype lower-cut-sqrt-ℝ⁰⁺ q →
@@ -152,8 +152,7 @@ module _
                     ( λ r → le-ℚ r p)
                     ( q=0)
                     ( 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 _ → le-lower-cut-ℝ (real-ℝ⁰⁺ x) p²<q' q'<x))
           ( λ is-pos-q →
             do
               (p , q²<p , p<x) ←
@@ -169,8 +168,7 @@ module _
                 rounded-below-square-le-ℚ⁺ (q , is-pos-q) (p , is-pos-p) q²<p
               intro-exists
                 ( q')
-                ( q<q' ,
-                  λ _ → le-lower-cut-ℝ (real-ℝ⁰⁺ x) (q' *ℚ q') p q'²<p p<x))
+                ( q<q' , λ _ → le-lower-cut-ℝ (real-ℝ⁰⁺ x) q'²<p p<x))
 
     forward-implication-is-rounded-upper-cut-sqrt-ℝ⁰⁺ :
       (r : ℚ) → is-in-subtype upper-cut-sqrt-ℝ⁰⁺ r →
@@ -189,7 +187,7 @@ module _
           rounded-above-square-le-ℚ⁺ (r , is-pos-r) (p , is-pos-p) p<r²
         intro-exists
           ( q)
-          ( q<r , is-pos-q , le-upper-cut-ℝ (real-ℝ⁰⁺ x) p (q *ℚ q) p<q² x<p)
+          ( q<r , is-pos-q , le-upper-cut-ℝ (real-ℝ⁰⁺ x) p<q² x<p)
 
     backward-implication-is-rounded-lower-cut-sqrt-ℝ⁰⁺ :
       (q : ℚ) → exists ℚ (λ r → le-ℚ-Prop q r ∧ lower-cut-sqrt-ℝ⁰⁺ r) →
@@ -205,8 +203,6 @@ module _
               ( is-positive-le-ℚ⁰⁺ (q , is-nonneg-q) q<r)
         le-lower-cut-ℝ
           ( real-ℝ⁰⁺ x)
-          ( q *ℚ q)
-          ( r *ℚ r)
           ( preserves-le-square-ℚ⁰⁺ (q , is-nonneg-q) (r , is-nonneg-r) q<r)
           ( r<√x is-nonneg-r)
 
@@ -221,8 +217,6 @@ module _
         ( is-pos-r ,
           le-upper-cut-ℝ
             ( real-ℝ⁰⁺ x)
-            ( q *ℚ q)
-            ( r *ℚ r)
             ( preserves-le-square-ℚ⁰⁺
               ( q , is-nonnegative-is-positive-ℚ is-pos-q)
               ( r , is-nonnegative-is-positive-ℚ is-pos-r)
@@ -298,7 +292,7 @@ module _
   {l : Level} (x : ℝ⁰⁺ l)
   where
 
-  opaque
+  abstract opaque
     unfolding real-ℚ real-sqrt-ℝ⁰⁺
 
     is-nonnegative-real-sqrt-ℝ⁰⁺ : is-nonnegative-ℝ (real-sqrt-ℝ⁰⁺ x)
@@ -316,7 +310,7 @@ module _
   {l : Level} (x : ℝ⁰⁺ l)
   where
 
-  opaque
+  abstract opaque
     unfolding mul-ℝ real-sqrt-ℝ⁰⁺ leq-ℝ leq-ℝ'
 
     leq-square-sqrt-ℝ⁰⁺ :
@@ -390,8 +384,6 @@ module _
                   ( linear-leq-ℚ a c)
             le-lower-cut-ℝ
               ( real-ℝ⁰⁺ x)
-              ( q)
-              ( a' *ℚ a')
               ( concatenate-le-leq-ℚ
                 ( q)
                 ( lower-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -406,15 +398,11 @@ module _
                     ( leq-left-max-ℚ a c ,
                       leq-lower-upper-cut-ℝ
                         ( real-sqrt-ℝ⁰⁺ x)
-                        ( a')
-                        ( b)
                         ( λ _ → a'²<x)
                         ( √x<b))
                     ( leq-right-max-ℚ a c ,
                       leq-lower-upper-cut-ℝ
                         ( real-sqrt-ℝ⁰⁺ x)
-                        ( a')
-                        ( d)
                         ( λ _ → a'²<x)
                         ( √x<d)))))
               ( a'²<x))
@@ -459,8 +447,6 @@ module _
           √x<b' = (is-pos-b' , x<b'²)
         le-upper-cut-ℝ
           ( real-ℝ⁰⁺ x)
-          ( b' *ℚ b')
-          ( q)
           ( concatenate-leq-le-ℚ
             ( b' *ℚ b')
             ( upper-bound-mul-closed-interval-ℚ [a,b] [c,d])
@@ -471,9 +457,9 @@ module _
                 ( [c,d])
                 ( b')
                 ( b')
-                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) a b' a<√x √x<b' ,
+                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) a<√x √x<b' ,
                   leq-left-min-ℚ b d)
-                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) c b' c<√x √x<b' ,
+                ( leq-lower-upper-cut-ℝ (real-sqrt-ℝ⁰⁺ x) c<√x √x<b' ,
                   leq-right-min-ℚ b d)))
             ( [a,b][c,d]<q))
           ( x<b'²)
@@ -491,7 +477,7 @@ module _
 ### Square roots of nonnegative real numbers are unique
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ leq-ℝ' mul-ℝ sim-ℝ real-sqrt-ℝ⁰⁺
 
   leq-unique-sqrt-ℝ⁰⁺ :
@@ -504,7 +490,7 @@ opaque
     in do
       (r , y<r) ← is-inhabited-upper-cut-ℝ y
       let
-        q≤r = leq-lower-upper-cut-ℝ y q r q<y y<r
+        q≤r = leq-lower-upper-cut-ℝ y q<y y<r
         [q,r] = ((q , r) , q≤r)
         q⁰⁺ = (q , is-nonneg-q)
         r⁰⁺ = (r , is-nonnegative-leq-ℚ⁰⁺ q⁰⁺ r q≤r)
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index 1b6faca2fd..773a8b9066 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -70,7 +70,7 @@ square-ℝ x = x *ℝ x
 ### The square of a real number is nonnegative
 
 ```agda
-opaque
+abstract opaque
   unfolding mul-ℝ
 
   is-nonnegative-square-ℝ :
@@ -142,7 +142,7 @@ opaque
                       ( leq-right-max-ℚ _ _))
                     ( [a',b'][a',b']<q)))
             ( located-le-ℚ zero-ℚ a' b'
-              ( le-lower-upper-cut-ℝ x a' b' a'<x x<b')))
+              ( le-lower-upper-cut-ℝ x a'<x x<b')))
 
 nonnegative-square-ℝ : {l : Level} → ℝ l → ℝ⁰⁺ l
 nonnegative-square-ℝ x = (square-ℝ x , is-nonnegative-square-ℝ x)
@@ -191,8 +191,8 @@ abstract
       ( square-ℝ x)
       ( x *ℝ y)
       ( square-ℝ y)
-      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ x y (leq-le-ℝ x<y))
-      ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x y x<y)
+      ( preserves-leq-left-mul-ℝ⁰⁺ x⁰⁺ (leq-le-ℝ x<y))
+      ( preserves-le-right-mul-ℝ⁺ (y , is-positive-le-ℝ⁰⁺ x⁰⁺ y x<y) x<y)
 ```
 
 ### If a nonnegative rational `q` is in the lower cut of `x`, `q²` is in the lower cut of `x²`
@@ -250,7 +250,7 @@ abstract opaque
     is-in-upper-cut-ℝ (square-ℝ x) (square-ℚ q)
   is-in-upper-cut-square-pos-neg-ℝ x q x<q -q<x =
     let
-      [-q,q] = ((neg-ℚ q , q) , leq-lower-upper-cut-ℝ x (neg-ℚ q) q -q<x x<q)
+      [-q,q] = ((neg-ℚ q , q) , leq-lower-upper-cut-ℝ x -q<x x<q)
     in
       leq-upper-cut-mul-ℝ'-upper-cut-mul-ℝ x x (square-ℚ q)
         ( intro-exists
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index e28ec7bda1..6db92caf63 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -18,6 +18,7 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
+open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.empty-types
@@ -25,6 +26,7 @@ open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
+open import foundation.sets
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
@@ -93,7 +95,7 @@ module _
             intro-exists
               ( q , (p -ℚ q) +ℚ s)
               ( q<y ,
-                le-lower-cut-ℝ z ((p -ℚ q) +ℚ s) r p-q+s<r r<z ,
+                le-lower-cut-ℝ z p-q+s<r r<z ,
                 ( equational-reasoning
                     p +ℚ s
                     ＝ (q +ℚ (p -ℚ q)) +ℚ s
@@ -283,3 +285,32 @@ module _
           ( q , is-positive-le-zero-ℚ (reflects-le-real-ℚ 0<q))
           ( le-transpose-right-diff-ℝ' _ _ _ q<y-x)
 ```
+
+### If `x + y < p` for some rational `p`, then there exist `q r : ℚ` such that `p = q + r`, `x < q`, `y < r`
+
+```agda
+module _
+  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) (p : ℚ)
+  where
+
+  opaque
+    unfolding add-ℝ
+
+    le-split-add-rational-ℝ :
+      le-ℝ (x +ℝ y) (real-ℚ p) →
+      exists
+        ( ℚ × ℚ)
+        ( λ (q , r) →
+          Id-Prop ℚ-Set p (q +ℚ r) ∧
+          le-prop-ℝ x (real-ℚ q) ∧
+          le-prop-ℝ y (real-ℚ r))
+    le-split-add-rational-ℝ x+y<p =
+      let open do-syntax-trunc-Prop (∃ _ _)
+      in do
+        ((q , r) , x<q , y<r , p=q+r) ← is-in-upper-cut-le-real-ℚ (x +ℝ y) x+y<p
+        intro-exists
+          ( q , r)
+          ( p=q+r ,
+            le-real-is-in-upper-cut-ℚ x x<q ,
+            le-real-is-in-upper-cut-ℚ y y<r)
+```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index b8256aa198..96296b0ca7 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -9,16 +9,9 @@ module real-numbers.strict-inequality-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-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.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.cartesian-product-types
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
@@ -28,26 +21,19 @@ open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-disjunction
-open import foundation.identity-types
 open import foundation.large-binary-relations
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.sets
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
-
 open import logic.functoriality-existential-quantification
 
-open import real-numbers.addition-real-numbers
-open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
@@ -81,34 +67,28 @@ le-prop-ℝ x y = (le-ℝ x y , is-prop-le-ℝ x y)
 ### Strict inequality on the reals implies inequality
 
 ```agda
-opaque
+abstract opaque
   unfolding le-ℝ leq-ℝ
 
   leq-le-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → le-ℝ x y → leq-ℝ x y
   leq-le-ℝ {x = x} {y = y} x<y p p<x =
     elim-exists
       ( lower-cut-ℝ y p)
-      ( λ q (x<q , q<y) →
-        le-lower-cut-ℝ y p q (le-lower-upper-cut-ℝ x p q p<x x<q) q<y)
+      ( λ q (x<q , q<y) → le-lower-cut-ℝ y (le-lower-upper-cut-ℝ x p<x x<q) q<y)
       ( x<y)
 ```
 
 ### Strict inequality on the reals is irreflexive
 
 ```agda
-module _
-  {l : Level}
-  (x : ℝ l)
-  where
-
-  opaque
-    unfolding le-ℝ
+abstract opaque
+  unfolding le-ℝ
 
-    irreflexive-le-ℝ : ¬ (le-ℝ x x)
-    irreflexive-le-ℝ =
-      elim-exists
-        ( empty-Prop)
-        ( λ q (x<q , q<x) → is-disjoint-cut-ℝ x q (q<x , x<q))
+  irreflexive-le-ℝ : {l : Level} (x : ℝ l) → ¬ (le-ℝ x x)
+  irreflexive-le-ℝ x =
+    elim-exists
+      ( empty-Prop)
+      ( λ q (x<q , q<x) → is-disjoint-cut-ℝ x q (q<x , x<q))
 ```
 
 ### Strict inequality on the reals is asymmetric
@@ -118,7 +98,7 @@ module _
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ
 
     asymmetric-le-ℝ : le-ℝ x y → ¬ (le-ℝ y x)
@@ -132,11 +112,11 @@ module _
           ( asymmetric-le-ℚ
             ( q)
             ( p)
-            ( le-lower-upper-cut-ℝ x q p q<x x<p))
+            ( le-lower-upper-cut-ℝ x q<x x<p))
           ( not-leq-le-ℚ
             ( p)
             ( q)
-            ( le-lower-upper-cut-ℝ y p q p<y y<q))
+            ( le-lower-upper-cut-ℝ y p<y y<q))
           ( decide-le-leq-ℚ p q)
 ```
 
@@ -150,7 +130,7 @@ module _
   (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ
 
     transitive-le-ℝ : le-ℝ y z → le-ℝ x y → le-ℝ x z
@@ -162,8 +142,7 @@ module _
         ( q , y<q , q<z) ← y<z
         intro-exists
           ( p)
-          ( x<p ,
-            le-lower-cut-ℝ z p q (le-lower-upper-cut-ℝ y p q p<y y<q) q<z)
+          ( x<p , le-lower-cut-ℝ z (le-lower-upper-cut-ℝ y p<y y<q) q<z)
 ```
 
 ### The canonical map from rationals to reals preserves and reflects strict inequality
@@ -173,7 +152,7 @@ module _
   {x y : ℚ}
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     preserves-le-real-ℚ : le-ℚ x y → le-ℝ (real-ℚ x) (real-ℚ y)
@@ -200,7 +179,7 @@ module _
   (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ leq-ℝ leq-ℝ'
 
     concatenate-le-leq-ℝ : le-ℝ x y → leq-ℝ y z → le-ℝ x z
@@ -220,13 +199,12 @@ module _
   {l : Level} {q : ℚ} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     le-real-iff-lower-cut-ℚ : is-in-lower-cut-ℝ x q ↔ le-ℝ (real-ℚ q) x
     le-real-iff-lower-cut-ℚ = is-rounded-lower-cut-ℝ x q
 
-  abstract
     le-real-is-in-lower-cut-ℚ : is-in-lower-cut-ℝ x q → le-ℝ (real-ℚ q) x
     le-real-is-in-lower-cut-ℚ = forward-implication le-real-iff-lower-cut-ℚ
 
@@ -241,7 +219,7 @@ module _
   {l : Level} {q : ℚ} (x : ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ real-ℚ
 
     le-iff-upper-cut-real-ℚ : is-in-upper-cut-ℝ x q ↔ le-ℝ x (real-ℚ q)
@@ -249,7 +227,6 @@ module _
       iff-tot-exists (λ _ → iff-equiv commutative-product) ∘iff
       is-rounded-upper-cut-ℝ x q
 
-  abstract
     le-real-is-in-upper-cut-ℚ : is-in-upper-cut-ℝ x q → le-ℝ x (real-ℚ q)
     le-real-is-in-upper-cut-ℚ = forward-implication le-iff-upper-cut-real-ℚ
 
@@ -264,12 +241,13 @@ module _
   {l : Level} (x : ℝ l) (p q : ℚ) (p<q : le-ℚ p q)
   where
 
-  is-located-le-ℝ : disjunction-type (le-ℝ (real-ℚ p) x) (le-ℝ x (real-ℚ q))
-  is-located-le-ℝ =
-    map-disjunction
-      ( le-real-is-in-lower-cut-ℚ x)
-      ( le-real-is-in-upper-cut-ℚ x)
-      ( is-located-lower-upper-cut-ℝ x p<q)
+  abstract
+    is-located-le-ℝ : disjunction-type (le-ℝ (real-ℚ p) x) (le-ℝ x (real-ℚ q))
+    is-located-le-ℝ =
+      map-disjunction
+        ( le-real-is-in-lower-cut-ℚ x)
+        ( le-real-is-in-upper-cut-ℚ x)
+        ( is-located-lower-upper-cut-ℝ x p<q)
 ```
 
 ### Every real is less than a rational number
@@ -321,7 +299,7 @@ module _
   {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ neg-ℝ
 
     neg-le-ℝ : le-ℝ x y → le-ℝ (neg-ℝ y) (neg-ℝ x)
@@ -343,7 +321,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ leq-ℝ
 
     not-leq-le-ℝ : le-ℝ x y → ¬ (leq-ℝ y x)
@@ -361,7 +339,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ leq-ℝ
 
     leq-not-le-ℝ : ¬ (le-ℝ x y) → leq-ℝ y x
@@ -410,24 +388,18 @@ module _
   (y : ℝ l2)
   where
 
-  opaque
-    unfolding le-ℝ real-ℚ
+  abstract opaque
+    unfolding le-ℝ
 
     dense-rational-le-ℝ :
       le-ℝ x y →
       exists ℚ (λ z → le-prop-ℝ x (real-ℚ z) ∧ le-prop-ℝ (real-ℚ z) y)
-    dense-rational-le-ℝ x<y =
-      let
-        open
-          do-syntax-trunc-Prop
-            ( ∃ ℚ (λ z → le-prop-ℝ x (real-ℚ z) ∧ le-prop-ℝ (real-ℚ z) y))
-      in do
-        ( q , x<q , q<y) ← x<y
-        ( p , p<q , x<p) ← forward-implication (is-rounded-upper-cut-ℝ x q) x<q
-        ( r , q<r , r<y) ← forward-implication (is-rounded-lower-cut-ℝ y q) q<y
-        intro-exists
-          ( q)
-          ( intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
+    dense-rational-le-ℝ =
+      map-tot-exists
+        ( λ q →
+          map-product
+            ( le-real-is-in-upper-cut-ℚ x)
+            ( le-real-is-in-lower-cut-ℚ y))
 ```
 
 ### Strict inequality on the real numbers is dense
@@ -453,7 +425,7 @@ module _
 ### Strict inequality on the real numbers is cotransitive
 
 ```agda
-opaque
+abstract opaque
   unfolding le-ℝ
 
   cotransitive-le-ℝ : is-cotransitive-Large-Relation-Prop ℝ le-prop-ℝ
@@ -476,7 +448,7 @@ module _
   {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3) (x~y : sim-ℝ x y)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ sim-ℝ
 
     preserves-le-left-sim-ℝ : le-ℝ x z → le-ℝ y z
@@ -514,7 +486,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ'
 
     leq-le-rational-ℝ :
@@ -542,35 +514,6 @@ module _
           leq-le-rational-ℝ y x (forward-implication ∘ H))
 ```
 
-### If `x + y < p` for some rational `p`, then there exist `q r : ℚ` such that `p = q + r`, `x < q`, `y < r`
-
-```agda
-module _
-  {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) (p : ℚ)
-  where
-
-  opaque
-    unfolding add-ℝ
-
-    le-split-add-rational-ℝ :
-      le-ℝ (x +ℝ y) (real-ℚ p) →
-      exists
-        ( ℚ × ℚ)
-        ( λ (q , r) →
-          Id-Prop ℚ-Set p (q +ℚ r) ∧
-          le-prop-ℝ x (real-ℚ q) ∧
-          le-prop-ℝ y (real-ℚ r))
-    le-split-add-rational-ℝ x+y<p =
-      let open do-syntax-trunc-Prop (∃ _ _)
-      in do
-        ((q , r) , x<q , y<r , p=q+r) ← is-in-upper-cut-le-real-ℚ (x +ℝ y) x+y<p
-        intro-exists
-          ( q , r)
-          ( p=q+r ,
-            le-real-is-in-upper-cut-ℚ x x<q ,
-            le-real-is-in-upper-cut-ℚ y y<r)
-```
-
 ## References
 
 {{#bibliography}}

From 13ddd47231bc1218b7fb161e662e9f26482f0520 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:07:50 -0800
Subject: [PATCH 06/21] Mark more things abstract opaque

---
 .../binary-minimum-real-numbers.lagda.md      |  4 ++--
 ...ompleteness-dedekind-real-numbers.lagda.md |  4 ++--
 .../closed-intervals-real-numbers.lagda.md    |  2 +-
 .../inequality-real-numbers.lagda.md          | 22 +++++++++----------
 4 files changed, 16 insertions(+), 16 deletions(-)

diff --git a/src/real-numbers/binary-minimum-real-numbers.lagda.md b/src/real-numbers/binary-minimum-real-numbers.lagda.md
index 121bf5d42b..b0ae4c0640 100644
--- a/src/real-numbers/binary-minimum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-minimum-real-numbers.lagda.md
@@ -66,7 +66,7 @@ module _
   (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ min-ℝ
 
     is-greatest-binary-lower-bound-min-ℝ :
@@ -155,7 +155,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding min-ℝ
 
     approximate-above-min-ℝ :
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 57f47ac4dd..ab073ba3c2 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -299,7 +299,7 @@ module _
     is-inhabited-upper-cut-lim-cauchy-approximation-ℝ ,
     is-rounded-upper-cut-lim-cauchy-approximation-ℝ
 
-  opaque
+  abstract opaque
     unfolding neighborhood-ℝ
 
     is-disjoint-cut-lim-cauchy-approximation-ℝ :
@@ -411,7 +411,7 @@ module _
   {l : Level} (x : cauchy-approximation-ℝ l)
   where
 
-  opaque
+  abstract opaque
     unfolding le-ℝ lim-cauchy-approximation-ℝ
 
     is-limit-lim-cauchy-approximation-ℝ :
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index 846087aa48..d8460f5bdc 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -90,7 +90,7 @@ unit-closed-interval-ℝ =
 ### Closed intervals in the real numbers are closed in the metric space of real numbers
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ neighborhood-ℝ
 
   is-closed-subset-closed-interval-ℝ :
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index 89c0fbbb57..4ece5ae59a 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -127,7 +127,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ'
 
     leq'-leq-ℝ : leq-ℝ x y → leq-ℝ' x y
@@ -175,7 +175,7 @@ module _
 ### Inequality on the real numbers is reflexive
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ
 
   refl-leq-ℝ : {l : Level} (x : ℝ l) → leq-ℝ x x
@@ -184,7 +184,7 @@ opaque
   leq-eq-ℝ : {l : Level} {x y : ℝ l} → x ＝ y → leq-ℝ x y
   leq-eq-ℝ {x = x} refl = refl-leq-ℝ x
 
-opaque
+abstract opaque
   unfolding leq-ℝ sim-ℝ
 
   leq-sim-ℝ : {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2} → sim-ℝ x y → leq-ℝ x y
@@ -194,7 +194,7 @@ opaque
 ### Inequality on the real numbers is antisymmetric
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ sim-ℝ
 
   sim-antisymmetric-leq-ℝ :
@@ -215,7 +215,7 @@ module _
   (x : ℝ l1) (y : ℝ l2) (z : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ
 
     transitive-leq-ℝ : leq-ℝ y z → leq-ℝ x y → leq-ℝ x z
@@ -244,7 +244,7 @@ antisymmetric-leq-Large-Poset ℝ-Large-Poset = antisymmetric-leq-ℝ
 ### Similarity in the large poset of real numbers is equivalent to similarity
 
 ```agda
-opaque
+abstract opaque
   unfolding leq-ℝ sim-ℝ
 
   sim-sim-leq-ℝ :
@@ -280,7 +280,7 @@ module _
   {x y : ℚ}
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ real-ℚ
 
     preserves-leq-real-ℚ : leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
@@ -300,7 +300,7 @@ module _
   {l1 l2 : Level} {x : ℝ l1} {y : ℝ l2}
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' neg-ℝ
 
     neg-leq-ℝ : leq-ℝ x y → leq-ℝ (neg-ℝ y) (neg-ℝ x)
@@ -314,7 +314,7 @@ module _
   {l1 l2 l3 : Level} {z : ℝ l1} {x : ℝ l2} {y : ℝ l3} (x~y : sim-ℝ x y)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ sim-ℝ
 
     preserves-leq-left-sim-ℝ : leq-ℝ x z → leq-ℝ y z
@@ -343,7 +343,7 @@ module _
   {l : Level} (x : ℝ l) (q : ℚ)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ leq-ℝ' real-ℚ
 
     not-in-lower-cut-leq-ℝ : leq-ℝ x (real-ℚ q) → ¬ (is-in-lower-cut-ℝ x q)
@@ -376,7 +376,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2)
   where
 
-  opaque
+  abstract opaque
     unfolding leq-ℝ'
 
     leq-leq-rational-ℝ :

From 04ccb85e348ae65c7afe3dd091e99585f0a20d4a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:09:00 -0800
Subject: [PATCH 07/21] Fix indent

---
 src/real-numbers/addition-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index fef1ddfc68..356d51654f 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -97,7 +97,7 @@ module _
       let
         open
           do-syntax-trunc-Prop
-            (∃
+            ( ∃
               ( ℚ × ℚ)
               ( close-bounds-lower-upper-ℝ
                 ( lower-real-add-ℝ)

From 1d65b0a47540f785cc9142871450b6018706fbcb Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:13:56 -0800
Subject: [PATCH 08/21] Fix title

---
 src/real-numbers/short-binary-maximum-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
index 2ac42192c9..5dd32621e1 100644
--- a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
@@ -1,4 +1,4 @@
-## The binary maximum of real numbers is a short function
+# The binary maximum of real numbers is a short function
 
 ```agda
 {-# OPTIONS --lossy-unification #-}

From 84dec2a5fc910c14f47edcb4457754f5cbe8d797 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 7 Nov 2025 17:14:54 -0800
Subject: [PATCH 09/21] Fix

---
 src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md | 2 --
 1 file changed, 2 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index f70d4ca8b4..1592876b21 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -298,8 +298,6 @@ module _
             ( p<x)
             ( le-upper-cut-ℝ
               ( x)
-              ( q)
-              ( p +ℚ (nℚ *ℚ ε'))
               ( tr
                 ( le-ℚ q)
                 ( commutative-add-ℚ (nℚ *ℚ ε') p)

From 139c5ace7b8ea8352b162266b8d22917e96cc85d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:53:39 -0800
Subject: [PATCH 10/21] Update src/real-numbers/dedekind-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 src/real-numbers/dedekind-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 0c1531e91f..d846a3556d 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -312,7 +312,7 @@ module _
           ( lower-cut-ℝ x p)
           ( upper-cut-ℝ x q)
           ( pr2 I)
-          ( is-located-lower-upper-cut-ℝ x ( pr1 I)))
+          ( is-located-lower-upper-cut-ℝ x (pr1 I)))
 
   subset-lower-complement-upper-cut-lower-cut-ℝ :
     lower-cut-ℝ x ⊆ lower-complement-upper-cut-ℝ x

From d9aeb7fda4ea62f8502e4183945b386f770f1137 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:55:14 -0800
Subject: [PATCH 11/21] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../strict-inequalities-addition-real-numbers.lagda.md          | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 6db92caf63..1c8789dd67 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -293,7 +293,7 @@ module _
   {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) (p : ℚ)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ
 
     le-split-add-rational-ℝ :

From defa2134d99b91f3aaf396f1200f6e6c47458ecd Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:55:32 -0800
Subject: [PATCH 12/21] Update
 src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../strict-inequalities-addition-real-numbers.lagda.md          | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 1c8789dd67..9403114e3a 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -59,7 +59,7 @@ module _
   {l1 l2 l3 : Level} (z : ℝ l1) (x : ℝ l2) (y : ℝ l3)
   where
 
-  opaque
+  abstract opaque
     unfolding add-ℝ le-ℝ
 
     preserves-le-right-add-ℝ : le-ℝ x y → le-ℝ (x +ℝ z) (y +ℝ z)

From fa31c521b7092f0e46eea6d3cf8904418d952271 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:55:52 -0800
Subject: [PATCH 13/21] Update
 src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../strict-inequalities-addition-real-numbers.lagda.md          | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
index 9403114e3a..24417dab87 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
@@ -44,7 +44,7 @@ open import real-numbers.strict-inequality-real-numbers
 
 ## Idea
 
-This file describes lemmas about
+On this page we describe lemmas about
 [strict inequalities](real-numbers.strict-inequality-real-numbers.md) of
 [real numbers](real-numbers.dedekind-real-numbers.md) related to
 [addition](real-numbers.addition-real-numbers.md) and

From 45757e8f51bf8fccacfd913a0fed7853b7eb9726 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:57:09 -0800
Subject: [PATCH 14/21] Update
 src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../saturation-inequality-nonnegative-real-numbers.lagda.md     | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
index 037e54c2f2..69f3a5fb11 100644
--- a/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/saturation-inequality-nonnegative-real-numbers.lagda.md
@@ -36,7 +36,7 @@ Despite being a property of
 [inequality of nonnegative real numbers](real-numbers.inequality-nonnegative-real-numbers.md),
 this is much easier to prove via
 [strict inequality](real-numbers.strict-inequality-nonnegative-real-numbers.md),
-so it is moved to its own file to prevent circular dependency.
+so this page is dedicated to just this property to prevent circular dependency.
 
 ## Definition
 

From 70cef9b710f6564e4559bff99dae0a07f25883ca Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 11:59:33 -0800
Subject: [PATCH 15/21] Rename files

---
 ...trics-of-metric-spaces-are-uniformly-continuous.lagda.md | 2 +-
 src/metric-spaces/metrics-of-metric-spaces.lagda.md         | 2 +-
 src/real-numbers.lagda.md                                   | 6 +++---
 src/real-numbers/absolute-value-real-numbers.lagda.md       | 2 +-
 src/real-numbers/addition-nonnegative-real-numbers.lagda.md | 4 ++--
 src/real-numbers/binary-maximum-real-numbers.lagda.md       | 2 +-
 .../cauchy-completeness-dedekind-real-numbers.lagda.md      | 2 +-
 src/real-numbers/distance-real-numbers.lagda.md             | 2 +-
 ...ualities-addition-and-subtraction-real-numbers.lagda.md} | 2 +-
 src/real-numbers/infima-families-real-numbers.lagda.md      | 2 +-
 .../inhabited-totally-bounded-subsets-real-numbers.lagda.md | 4 ++--
 src/real-numbers/isometry-addition-real-numbers.lagda.md    | 2 +-
 src/real-numbers/isometry-negation-real-numbers.lagda.md    | 2 +-
 ...ipschitz-continuity-multiplication-real-numbers.lagda.md | 2 +-
 .../maximum-finite-families-real-numbers.lagda.md           | 2 +-
 src/real-numbers/metric-space-of-real-numbers.lagda.md      | 4 ++--
 src/real-numbers/nonnegative-real-numbers.lagda.md          | 2 +-
 src/real-numbers/positive-real-numbers.lagda.md             | 2 +-
 .../saturation-inequality-real-numbers.lagda.md             | 2 +-
 ... => short-function-binary-maximum-real-numbers.lagda.md} | 4 ++--
 ...ualities-addition-and-subtraction-real-numbers.lagda.md} | 2 +-
 src/real-numbers/suprema-families-real-numbers.lagda.md     | 2 +-
 ...addition-subtraction-cuts-dedekind-real-numbers.lagda.md | 2 +-
 23 files changed, 29 insertions(+), 29 deletions(-)
 rename src/real-numbers/{inequalities-addition-real-numbers.lagda.md => inequalities-addition-and-subtraction-real-numbers.lagda.md} (98%)
 rename src/real-numbers/{short-binary-maximum-real-numbers.lagda.md => short-function-binary-maximum-real-numbers.lagda.md} (97%)
 rename src/real-numbers/{strict-inequalities-addition-real-numbers.lagda.md => strict-inequalities-addition-and-subtraction-real-numbers.lagda.md} (99%)

diff --git a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
index c33453db28..0eed19b179 100644
--- a/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces-are-uniformly-continuous.lagda.md
@@ -32,7 +32,7 @@ open import order-theory.large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
diff --git a/src/metric-spaces/metrics-of-metric-spaces.lagda.md b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
index d113ff5bac..271f974d53 100644
--- a/src/metric-spaces/metrics-of-metric-spaces.lagda.md
+++ b/src/metric-spaces/metrics-of-metric-spaces.lagda.md
@@ -28,7 +28,7 @@ open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.saturation-inequality-nonnegative-real-numbers
 open import real-numbers.similarity-nonnegative-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 876511f445..3b84f6c0b0 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -23,7 +23,7 @@ open import real-numbers.difference-real-numbers public
 open import real-numbers.distance-real-numbers public
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers public
 open import real-numbers.finitely-enumerable-subsets-real-numbers public
-open import real-numbers.inequalities-addition-real-numbers public
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
 open import real-numbers.inequality-nonnegative-real-numbers public
 open import real-numbers.inequality-positive-real-numbers public
@@ -75,13 +75,13 @@ open import real-numbers.real-numbers-from-lower-dedekind-real-numbers public
 open import real-numbers.real-numbers-from-upper-dedekind-real-numbers public
 open import real-numbers.saturation-inequality-nonnegative-real-numbers public
 open import real-numbers.saturation-inequality-real-numbers public
-open import real-numbers.short-binary-maximum-real-numbers public
+open import real-numbers.short-function-binary-maximum-real-numbers public
 open import real-numbers.similarity-nonnegative-real-numbers public
 open import real-numbers.similarity-positive-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.square-roots-nonnegative-real-numbers public
 open import real-numbers.squares-real-numbers public
-open import real-numbers.strict-inequalities-addition-real-numbers public
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers public
 open import real-numbers.strict-inequality-nonnegative-real-numbers public
 open import real-numbers.strict-inequality-positive-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index ba2031c002..4f77b3aad2 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -28,7 +28,7 @@ open import metric-spaces.short-functions-metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.isometry-negation-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
index 0735a72392..5298362a87 100644
--- a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -21,12 +21,12 @@ open import foundation.universe-levels
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index 69b9224d86..a321c82356 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -28,7 +28,7 @@ open import order-theory.least-upper-bounds-large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index ab073ba3c2..9d4f3567a1 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -45,7 +45,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.distance-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index eff38740f5..8e430fb8b0 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -28,7 +28,7 @@ open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.multiplication-real-numbers
diff --git a/src/real-numbers/inequalities-addition-real-numbers.lagda.md b/src/real-numbers/inequalities-addition-and-subtraction-real-numbers.lagda.md
similarity index 98%
rename from src/real-numbers/inequalities-addition-real-numbers.lagda.md
rename to src/real-numbers/inequalities-addition-and-subtraction-real-numbers.lagda.md
index b5af4c0787..61de6c5cfb 100644
--- a/src/real-numbers/inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/inequalities-addition-and-subtraction-real-numbers.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module real-numbers.inequalities-addition-real-numbers where
+module real-numbers.inequalities-addition-and-subtraction-real-numbers where
 ```
 
 <details><summary>Imports</summary>
diff --git a/src/real-numbers/infima-families-real-numbers.lagda.md b/src/real-numbers/infima-families-real-numbers.lagda.md
index add144950d..0b338cbe37 100644
--- a/src/real-numbers/infima-families-real-numbers.lagda.md
+++ b/src/real-numbers/infima-families-real-numbers.lagda.md
@@ -41,7 +41,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
diff --git a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
index 21124540b7..8a3677def8 100644
--- a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
@@ -44,7 +44,7 @@ open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.closed-intervals-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.infima-and-suprema-families-real-numbers
@@ -55,7 +55,7 @@ open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index a7d3686d03..c42c49edba 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index 7e5f7562a2..e36974427a 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import metric-spaces.metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index ed960956fe..1b56e2c244 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -36,7 +36,7 @@ open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.enclosing-closed-rational-intervals-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 5e757a72e7..34e2ed1f44 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -43,7 +43,7 @@ open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.suprema-families-real-numbers
 
diff --git a/src/real-numbers/metric-space-of-real-numbers.lagda.md b/src/real-numbers/metric-space-of-real-numbers.lagda.md
index d199e3ef54..7d78c3c437 100644
--- a/src/real-numbers/metric-space-of-real-numbers.lagda.md
+++ b/src/real-numbers/metric-space-of-real-numbers.lagda.md
@@ -44,12 +44,12 @@ open import metric-spaces.triangular-rational-neighborhood-relations
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.transposition-addition-subtraction-cuts-dedekind-real-numbers
 ```
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index fcb7e126e0..d8219df9e7 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -36,7 +36,7 @@ open import metric-spaces.metric-spaces
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index 668e1644d9..e32729d0f7 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -40,7 +40,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/saturation-inequality-real-numbers.lagda.md b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
index 437a1a7a32..fad1d8e16c 100644
--- a/src/real-numbers/saturation-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/saturation-inequality-real-numbers.lagda.md
@@ -21,7 +21,7 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 
diff --git a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
similarity index 97%
rename from src/real-numbers/short-binary-maximum-real-numbers.lagda.md
rename to src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
index 5dd32621e1..9cca40873e 100644
--- a/src/real-numbers/short-binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/short-function-binary-maximum-real-numbers.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module real-numbers.short-binary-maximum-real-numbers where
+module real-numbers.short-function-binary-maximum-real-numbers where
 ```
 
 <details><summary>Imports</summary>
@@ -23,7 +23,7 @@ open import order-theory.least-upper-bounds-large-posets
 open import real-numbers.addition-real-numbers
 open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequalities-addition-real-numbers
+open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.positive-real-numbers
diff --git a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
similarity index 99%
rename from src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
rename to src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
index 24417dab87..26b7ac1b7d 100644
--- a/src/real-numbers/strict-inequalities-addition-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
@@ -3,7 +3,7 @@
 ```agda
 {-# OPTIONS --lossy-unification #-}
 
-module real-numbers.strict-inequalities-addition-real-numbers where
+module real-numbers.strict-inequalities-addition-and-subtraction-real-numbers where
 ```
 
 <details><summary>Imports</summary>
diff --git a/src/real-numbers/suprema-families-real-numbers.lagda.md b/src/real-numbers/suprema-families-real-numbers.lagda.md
index f730c05498..d51a2e2a29 100644
--- a/src/real-numbers/suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/suprema-families-real-numbers.lagda.md
@@ -42,7 +42,7 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 open import real-numbers.subsets-real-numbers
 ```
diff --git a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
index 91e5c9c0be..82a733d139 100644
--- a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
@@ -24,7 +24,7 @@ open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequalities-addition-real-numbers
+open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
 

From a85cdcbaa33e6a6f4c989c05584e592e304701de Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 12:04:12 -0800
Subject: [PATCH 16/21] Fix

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

diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index 5b51936f6a..1398ef233b 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -97,5 +97,5 @@ antisymmetric-leq-ℚ⁺ = antisymmetric-leq-Poset poset-ℚ⁺
 
 ```agda
 leq-eq-ℚ⁺ : {x y : ℚ⁺} → x ＝ y → leq-ℚ⁺ x y
-leq-eq-ℚ⁺ x=y = leq-eq-ℚ _ _ (ap rational-ℚ⁺ x=y)
+leq-eq-ℚ⁺ x=y = leq-eq-ℚ (ap rational-ℚ⁺ x=y)
 ```

From 1d5f0a217083dd72cdb145026a765ae94b94b38a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 12:25:44 -0800
Subject: [PATCH 17/21] Fix

---
 src/real-numbers/absolute-value-real-numbers.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index 4f77b3aad2..fc75c7ed59 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -132,7 +132,7 @@ abstract opaque
           ( zero-ℝ)
           ( x)
           ( 0≤x)
-          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ _ _ 0≤x))))
+          ( tr (leq-ℝ (neg-ℝ x)) neg-zero-ℝ (neg-leq-ℝ 0≤x))))
 
   abs-real-ℝ⁺ : {l : Level} (x : ℝ⁺ l) → abs-ℝ (real-ℝ⁺ x) ＝ real-ℝ⁺ x
   abs-real-ℝ⁺ x = abs-real-ℝ⁰⁺ (nonnegative-ℝ⁺ x)

From 399e3c37bb884b897b35304ec9133218680f5b6f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Nov 2025 12:35:18 -0800
Subject: [PATCH 18/21] Fix build

---
 .../powers-positive-rational-numbers.lagda.md          |  8 ++++----
 ...ication-positive-and-negative-real-numbers.lagda.md |  2 +-
 src/real-numbers/powers-real-numbers.lagda.md          | 10 +++++-----
 3 files changed, 10 insertions(+), 10 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 847d6dbbba..05641432b4 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -294,9 +294,9 @@ abstract
             ( _)
             ( preserves-leq-power-ℚ⁺ k ε one-ℚ⁺ ε≤1)
         ≤ one-ℚ *ℚ rational-power-ℚ⁺ m ε
-          by leq-eq-ℚ _ _ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
+          by leq-eq-ℚ (ap-mul-ℚ (ap rational-ℚ⁺ (power-one-ℚ⁺ k)) refl)
         ≤ rational-power-ℚ⁺ m ε
-          by leq-eq-ℚ _ _ (left-unit-law-mul-ℚ _)
+          by leq-eq-ℚ (left-unit-law-mul-ℚ _)
 ```
 
 ### If `ε` is a positive rational number less than 1, `εⁿ` approaches 0
@@ -327,10 +327,10 @@ abstract
                   rational-dist-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε)) zero-ℚ
                   ≤ rational-abs-ℚ (rational-ℚ⁺ (power-ℚ⁺ n ε))
                     by
-                    leq-eq-ℚ _ _ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
+                    leq-eq-ℚ (ap rational-ℚ⁰⁺ (right-zero-law-dist-ℚ _))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ n ε)
                     by
-                    leq-eq-ℚ _ _
+                    leq-eq-ℚ
                       ( ap rational-ℚ⁰⁺
                         ( abs-rational-ℚ⁰⁺ (nonnegative-ℚ⁺ (power-ℚ⁺ n ε))))
                   ≤ rational-ℚ⁺ (power-ℚ⁺ m ε)
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index 7ecc0768e3..c0fea89695 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -44,7 +44,7 @@ abstract
       ( x *ℝ zero-ℝ)
       ( zero-ℝ)
       ( right-zero-law-mul-ℝ x)
-      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) y zero-ℝ is-neg-y)
+      ( preserves-le-left-mul-ℝ⁺ (x , is-pos-x) is-neg-y)
 
 mul-positive-negative-ℝ :
   {l1 l2 : Level} → ℝ⁺ l1 → ℝ⁻ l2 → ℝ⁻ (l1 ⊔ l2)
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 497f3b7c04..8038140ac0 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -293,7 +293,7 @@ abstract
     {l1 l2 : Level} (n : ℕ) (x : ℝ l1) (y : ℝ l2) → leq-ℝ (abs-ℝ x) (abs-ℝ y) →
     leq-ℝ (abs-ℝ (power-ℝ n x)) (abs-ℝ (power-ℝ n y))
   preserves-leq-abs-power-ℝ 0 _ _ _ =
-    preserves-leq-sim-ℝ _ _ _ _
+    preserves-leq-sim-ℝ
       ( inv-tr
         ( sim-ℝ one-ℝ)
         ( abs-real-ℝ⁺ (raise-ℝ⁺ _ one-ℝ⁺))
@@ -310,9 +310,9 @@ abstract
     chain-of-inequalities
       abs-ℝ (power-ℝ (succ-ℕ n) x)
       ≤ abs-ℝ (power-ℝ n x *ℝ x)
-        by leq-eq-ℝ _ _ (ap abs-ℝ (power-succ-ℝ n x))
+        by leq-eq-ℝ (ap abs-ℝ (power-succ-ℝ n x))
       ≤ abs-ℝ (power-ℝ n x) *ℝ abs-ℝ x
-        by leq-eq-ℝ _ _ (abs-mul-ℝ _ _)
+        by leq-eq-ℝ (abs-mul-ℝ _ _)
       ≤ abs-ℝ (power-ℝ n y) *ℝ abs-ℝ y
         by
           preserves-leq-mul-ℝ⁰⁺
@@ -323,7 +323,7 @@ abstract
             ( preserves-leq-abs-power-ℝ n x y |x|≤|y|)
             ( |x|≤|y|)
       ≤ abs-ℝ (power-ℝ n y *ℝ y)
-        by leq-eq-ℝ _ _ (inv (abs-mul-ℝ _ _))
+        by leq-eq-ℝ (inv (abs-mul-ℝ _ _))
       ≤ abs-ℝ (power-ℝ (succ-ℕ n) y)
-        by leq-eq-ℝ _ _ (ap abs-ℝ (inv (power-succ-ℝ n y)))
+        by leq-eq-ℝ (ap abs-ℝ (inv (power-succ-ℝ n y)))
 ```

From 8569df8582c3d35496e39a140b7ed34d00d965cd Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 9 Nov 2025 13:06:21 +0100
Subject: [PATCH 19/21] chore: optimize imports `real-numbers`

---
 src/real-numbers/absolute-value-real-numbers.lagda.md |  1 -
 .../addition-lower-dedekind-real-numbers.lagda.md     |  5 -----
 .../addition-nonnegative-real-numbers.lagda.md        |  1 -
 src/real-numbers/addition-real-numbers.lagda.md       |  1 -
 .../addition-upper-dedekind-real-numbers.lagda.md     |  2 --
 src/real-numbers/apartness-real-numbers.lagda.md      |  3 ---
 .../arithmetically-located-dedekind-cuts.lagda.md     |  2 --
 src/real-numbers/binary-maximum-real-numbers.lagda.md |  2 --
 ...cauchy-completeness-dedekind-real-numbers.lagda.md |  4 ----
 .../cauchy-sequences-real-numbers.lagda.md            |  4 ----
 .../closed-intervals-real-numbers.lagda.md            |  4 ----
 src/real-numbers/dedekind-real-numbers.lagda.md       |  9 ---------
 src/real-numbers/difference-real-numbers.lagda.md     |  1 -
 src/real-numbers/distance-real-numbers.lagda.md       |  2 --
 ...ng-closed-rational-intervals-real-numbers.lagda.md |  1 -
 .../inequality-nonnegative-real-numbers.lagda.md      |  1 -
 .../inequality-positive-real-numbers.lagda.md         |  1 -
 .../inequality-upper-dedekind-real-numbers.lagda.md   |  1 -
 .../infima-and-suprema-families-real-numbers.lagda.md |  2 --
 ...-finitely-enumerable-subsets-real-numbers.lagda.md |  1 -
 ...ited-totally-bounded-subsets-real-numbers.lagda.md |  1 -
 .../isometry-addition-real-numbers.lagda.md           |  5 -----
 .../isometry-negation-real-numbers.lagda.md           |  1 -
 .../large-additive-group-of-real-numbers.lagda.md     |  2 --
 ...rge-multiplicative-monoid-of-real-numbers.lagda.md |  2 --
 src/real-numbers/large-ring-of-real-numbers.lagda.md  |  4 ----
 .../limits-sequences-real-numbers.lagda.md            |  2 --
 ...tz-continuity-multiplication-real-numbers.lagda.md |  7 -------
 src/real-numbers/lower-dedekind-real-numbers.lagda.md |  1 -
 .../maximum-finite-families-real-numbers.lagda.md     |  3 ---
 .../maximum-upper-dedekind-real-numbers.lagda.md      |  6 ------
 ...-finitely-enumerable-subsets-real-numbers.lagda.md |  3 ---
 .../minimum-lower-dedekind-real-numbers.lagda.md      |  1 -
 .../minimum-upper-dedekind-real-numbers.lagda.md      |  2 --
 .../multiplication-nonnegative-real-numbers.lagda.md  |  1 -
 ...cation-positive-and-negative-real-numbers.lagda.md |  1 -
 .../multiplication-positive-real-numbers.lagda.md     |  1 -
 ...iplicative-inverses-negative-real-numbers.lagda.md |  1 -
 ...tiplicative-inverses-nonzero-real-numbers.lagda.md |  4 ----
 ...iplicative-inverses-positive-real-numbers.lagda.md |  3 ---
 src/real-numbers/negative-real-numbers.lagda.md       |  1 -
 src/real-numbers/nonnegative-real-numbers.lagda.md    |  2 --
 .../positive-and-negative-real-numbers.lagda.md       |  3 ---
 src/real-numbers/positive-real-numbers.lagda.md       |  9 ---------
 src/real-numbers/powers-real-numbers.lagda.md         |  1 -
 .../raising-universe-levels-real-numbers.lagda.md     |  3 ---
 src/real-numbers/rational-real-numbers.lagda.md       | 11 -----------
 ...-numbers-from-lower-dedekind-real-numbers.lagda.md |  3 ---
 ...-numbers-from-upper-dedekind-real-numbers.lagda.md |  3 ---
 .../similarity-nonnegative-real-numbers.lagda.md      |  7 -------
 src/real-numbers/similarity-real-numbers.lagda.md     |  9 ---------
 src/real-numbers/squares-real-numbers.lagda.md        |  7 -------
 ...ies-addition-and-subtraction-real-numbers.lagda.md |  2 --
 ...trict-inequality-nonnegative-real-numbers.lagda.md |  1 -
 .../strict-inequality-positive-real-numbers.lagda.md  |  1 -
 .../strict-inequality-real-numbers.lagda.md           |  1 -
 src/real-numbers/subsets-real-numbers.lagda.md        |  3 ---
 .../totally-bounded-subsets-real-numbers.lagda.md     |  3 ---
 ...on-subtraction-cuts-dedekind-real-numbers.lagda.md |  1 -
 src/real-numbers/upper-dedekind-real-numbers.lagda.md |  1 -
 60 files changed, 171 deletions(-)

diff --git a/src/real-numbers/absolute-value-real-numbers.lagda.md b/src/real-numbers/absolute-value-real-numbers.lagda.md
index fc75c7ed59..1956f14116 100644
--- a/src/real-numbers/absolute-value-real-numbers.lagda.md
+++ b/src/real-numbers/absolute-value-real-numbers.lagda.md
@@ -19,7 +19,6 @@ open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
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 bd1e58abfe..6b2493249e 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -21,27 +21,22 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositional-truncations
-open import foundation.sets
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
 open import group-theory.commutative-monoids
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 open import group-theory.monoids
 open import group-theory.semigroups
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 ```
diff --git a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
index 5298362a87..6c29699df8 100644
--- a/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/addition-nonnegative-real-numbers.lagda.md
@@ -28,7 +28,6 @@ open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-nonnegative-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/addition-real-numbers.lagda.md b/src/real-numbers/addition-real-numbers.lagda.md
index 356d51654f..8f67041ac9 100644
--- a/src/real-numbers/addition-real-numbers.lagda.md
+++ b/src/real-numbers/addition-real-numbers.lagda.md
@@ -11,7 +11,6 @@ module real-numbers.addition-real-numbers where
 ```agda
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.addition-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/real-numbers/addition-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
index 616124b951..7ada6a30a2 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -19,7 +19,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
@@ -30,7 +29,6 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
 open import group-theory.commutative-monoids
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
diff --git a/src/real-numbers/apartness-real-numbers.lagda.md b/src/real-numbers/apartness-real-numbers.lagda.md
index 023123e63e..242133133b 100644
--- a/src/real-numbers/apartness-real-numbers.lagda.md
+++ b/src/real-numbers/apartness-real-numbers.lagda.md
@@ -7,18 +7,15 @@ module real-numbers.apartness-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import foundation.apartness-relations
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.function-types
 open import foundation.functoriality-disjunction
-open import foundation.identity-types
 open import foundation.large-apartness-relations
 open import foundation.large-binary-relations
 open import foundation.negated-equality
 open import foundation.negation
 open import foundation.propositions
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 1592876b21..d2abf4e12a 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -27,7 +27,6 @@ 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.cartesian-product-types
 open import foundation.conjunction
 open import foundation.coproduct-types
@@ -37,7 +36,6 @@ open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
diff --git a/src/real-numbers/binary-maximum-real-numbers.lagda.md b/src/real-numbers/binary-maximum-real-numbers.lagda.md
index a321c82356..d4da7d69c0 100644
--- a/src/real-numbers/binary-maximum-real-numbers.lagda.md
+++ b/src/real-numbers/binary-maximum-real-numbers.lagda.md
@@ -25,10 +25,8 @@ open import order-theory.join-semilattices
 open import order-theory.large-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
 
-open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.maximum-lower-dedekind-real-numbers
diff --git a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
index 9d4f3567a1..6e102ad464 100644
--- a/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-completeness-dedekind-real-numbers.lagda.md
@@ -21,7 +21,6 @@ open import foundation.action-on-identifications-functions
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
-open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.functoriality-disjunction
@@ -34,19 +33,16 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cauchy-approximations-metric-spaces
-open import metric-spaces.cauchy-sequences-complete-metric-spaces
 open import metric-spaces.complete-metric-spaces
 open import metric-spaces.convergent-cauchy-approximations-metric-spaces
 open import metric-spaces.limits-of-cauchy-approximations-metric-spaces
 open import metric-spaces.metric-spaces
 
-open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.distance-real-numbers
 open import real-numbers.inequalities-addition-and-subtraction-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.negation-real-numbers
diff --git a/src/real-numbers/cauchy-sequences-real-numbers.lagda.md b/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
index cac5f34c62..6876201386 100644
--- a/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/cauchy-sequences-real-numbers.lagda.md
@@ -11,14 +11,10 @@ module real-numbers.cauchy-sequences-real-numbers where
 ```agda
 open import foundation.universe-levels
 
-open import lists.sequences
-
 open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.cauchy-sequences-complete-metric-spaces
 open import metric-spaces.cauchy-sequences-metric-spaces
-open import metric-spaces.convergent-sequences-metric-spaces
 
-open import real-numbers.addition-real-numbers
 open import real-numbers.cauchy-completeness-dedekind-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-addition-real-numbers
diff --git a/src/real-numbers/closed-intervals-real-numbers.lagda.md b/src/real-numbers/closed-intervals-real-numbers.lagda.md
index d8460f5bdc..09d4fada9e 100644
--- a/src/real-numbers/closed-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/closed-intervals-real-numbers.lagda.md
@@ -11,9 +11,6 @@ open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.inequality-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
-open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.logical-equivalences
@@ -26,7 +23,6 @@ open import foundation.universe-levels
 open import metric-spaces.closed-subsets-metric-spaces
 open import metric-spaces.complete-metric-spaces
 open import metric-spaces.metric-spaces
-open import metric-spaces.subspaces-metric-spaces
 
 open import order-theory.closed-intervals-large-posets
 
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index d846a3556d..22ece70191 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -15,8 +15,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
-open import foundation.complements-subtypes
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
@@ -24,28 +22,21 @@ open import foundation.disjoint-subtypes
 open import foundation.disjunction
 open import foundation.embeddings
 open import foundation.empty-types
-open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-dependent-pair-types
 open import foundation.identity-types
-open import foundation.inhabited-types
 open import foundation.logical-equivalences
 open import foundation.negation
-open import foundation.powersets
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.sets
 open import foundation.similarity-subtypes
 open import foundation.subtypes
 open import foundation.transport-along-identifications
-open import foundation.truncated-types
 open import foundation.universal-quantification
 open import foundation.universe-levels
 
-open import foundation-core.truncation-levels
-
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.lower-dedekind-real-numbers
diff --git a/src/real-numbers/difference-real-numbers.lagda.md b/src/real-numbers/difference-real-numbers.lagda.md
index e3a1cd036e..0d13f8b33f 100644
--- a/src/real-numbers/difference-real-numbers.lagda.md
+++ b/src/real-numbers/difference-real-numbers.lagda.md
@@ -14,7 +14,6 @@ open import elementary-number-theory.rational-numbers
 
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
-open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/distance-real-numbers.lagda.md b/src/real-numbers/distance-real-numbers.lagda.md
index 8e430fb8b0..4bc37c1536 100644
--- a/src/real-numbers/distance-real-numbers.lagda.md
+++ b/src/real-numbers/distance-real-numbers.lagda.md
@@ -9,11 +9,9 @@ module real-numbers.distance-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-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.positive-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
diff --git a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
index 77d3ab8c6b..977e33b224 100644
--- a/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
+++ b/src/real-numbers/enclosing-closed-rational-intervals-real-numbers.lagda.md
@@ -10,7 +10,6 @@ module real-numbers.enclosing-closed-rational-intervals-real-numbers where
 open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.interior-closed-intervals-rational-numbers
 open import elementary-number-theory.intersections-closed-intervals-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.conjunction
 open import foundation.dependent-pair-types
diff --git a/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
index fa71aea44a..dc71c190c0 100644
--- a/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-nonnegative-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
 open import real-numbers.rational-real-numbers
diff --git a/src/real-numbers/inequality-positive-real-numbers.lagda.md b/src/real-numbers/inequality-positive-real-numbers.lagda.md
index 2ef1d3a686..15343cc094 100644
--- a/src/real-numbers/inequality-positive-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-positive-real-numbers.lagda.md
@@ -12,7 +12,6 @@ module real-numbers.inequality-positive-real-numbers where
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.positive-real-numbers
 ```
diff --git a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
index 2b8a3de889..48dc2d04d4 100644
--- a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
@@ -18,7 +18,6 @@ open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.unit-type
diff --git a/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md b/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md
index f889ec1a31..6a0b26fdf4 100644
--- a/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md
+++ b/src/real-numbers/infima-and-suprema-families-real-numbers.lagda.md
@@ -8,7 +8,6 @@ module real-numbers.infima-and-suprema-families-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.existential-quantification
 open import foundation.propositional-truncations
 open import foundation.subtypes
 open import foundation.universe-levels
@@ -17,7 +16,6 @@ open import real-numbers.closed-intervals-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.infima-families-real-numbers
-open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.subsets-real-numbers
 open import real-numbers.suprema-families-real-numbers
 ```
diff --git a/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
index 945abdb69c..a5eb36703a 100644
--- a/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
@@ -20,7 +20,6 @@ open import real-numbers.negation-real-numbers
 open import real-numbers.subsets-real-numbers
 
 open import univalent-combinatorics.finitely-enumerable-subtypes
-open import univalent-combinatorics.finitely-enumerable-types
 open import univalent-combinatorics.inhabited-finitely-enumerable-subtypes
 ```
 
diff --git a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
index 8a3677def8..3b65a9823d 100644
--- a/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/inhabited-totally-bounded-subsets-real-numbers.lagda.md
@@ -32,7 +32,6 @@ open import metric-spaces.approximations-metric-spaces
 open import metric-spaces.inhabited-totally-bounded-subspaces-metric-spaces
 open import metric-spaces.metric-spaces
 open import metric-spaces.nets-metric-spaces
-open import metric-spaces.subspaces-metric-spaces
 open import metric-spaces.totally-bounded-metric-spaces
 
 open import order-theory.upper-bounds-large-posets
diff --git a/src/real-numbers/isometry-addition-real-numbers.lagda.md b/src/real-numbers/isometry-addition-real-numbers.lagda.md
index c42c49edba..3b1ca98943 100644
--- a/src/real-numbers/isometry-addition-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-addition-real-numbers.lagda.md
@@ -11,25 +11,20 @@ module real-numbers.isometry-addition-real-numbers where
 ```agda
 open import foundation.dependent-pair-types
 open import foundation.function-extensionality
-open import foundation.function-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.isometries-metric-spaces
 open import metric-spaces.metric-space-of-isometries-metric-spaces
-open import metric-spaces.metric-spaces
 open import metric-spaces.modulated-uniformly-continuous-functions-metric-spaces
-open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequalities-addition-and-subtraction-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/isometry-negation-real-numbers.lagda.md b/src/real-numbers/isometry-negation-real-numbers.lagda.md
index e36974427a..3c2b449432 100644
--- a/src/real-numbers/isometry-negation-real-numbers.lagda.md
+++ b/src/real-numbers/isometry-negation-real-numbers.lagda.md
@@ -20,7 +20,6 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-spaces
 
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/large-additive-group-of-real-numbers.lagda.md b/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
index 77808a473a..8e876af34b 100644
--- a/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-additive-group-of-real-numbers.lagda.md
@@ -9,10 +9,8 @@ module real-numbers.large-additive-group-of-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.universe-levels
diff --git a/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md b/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
index 87f9524dc3..b5d2a73c59 100644
--- a/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-multiplicative-monoid-of-real-numbers.lagda.md
@@ -11,9 +11,7 @@ module real-numbers.large-multiplicative-monoid-of-real-numbers where
 ```agda
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
 open import group-theory.commutative-monoids
-open import group-theory.large-abelian-groups
 open import group-theory.large-commutative-monoids
 open import group-theory.large-monoids
 open import group-theory.large-semigroups
diff --git a/src/real-numbers/large-ring-of-real-numbers.lagda.md b/src/real-numbers/large-ring-of-real-numbers.lagda.md
index 5786d7c56c..98fd1e8782 100644
--- a/src/real-numbers/large-ring-of-real-numbers.lagda.md
+++ b/src/real-numbers/large-ring-of-real-numbers.lagda.md
@@ -13,16 +13,12 @@ open import commutative-algebra.commutative-rings
 open import commutative-algebra.homomorphisms-commutative-rings
 open import commutative-algebra.large-commutative-rings
 
-open import elementary-number-theory.multiplication-rational-numbers
 open import elementary-number-theory.ring-of-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.universe-levels
 
-open import group-theory.large-monoids
-
 open import real-numbers.large-additive-group-of-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
diff --git a/src/real-numbers/limits-sequences-real-numbers.lagda.md b/src/real-numbers/limits-sequences-real-numbers.lagda.md
index f40aab460f..f7ab605183 100644
--- a/src/real-numbers/limits-sequences-real-numbers.lagda.md
+++ b/src/real-numbers/limits-sequences-real-numbers.lagda.md
@@ -10,7 +10,6 @@ module real-numbers.limits-sequences-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.propositional-truncations
 open import foundation.universe-levels
 
 open import lists.sequences
@@ -19,7 +18,6 @@ open import metric-spaces.cartesian-products-metric-spaces
 open import metric-spaces.limits-of-sequences-metric-spaces
 
 open import real-numbers.addition-real-numbers
-open import real-numbers.cauchy-sequences-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-addition-real-numbers
 open import real-numbers.metric-space-of-real-numbers
diff --git a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
index 1b56e2c244..b44914841b 100644
--- a/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
+++ b/src/real-numbers/lipschitz-continuity-multiplication-real-numbers.lagda.md
@@ -12,14 +12,10 @@ module real-numbers.lipschitz-continuity-multiplication-real-numbers where
 open import elementary-number-theory.multiplication-positive-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
 
-open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
-open import foundation.function-extensionality
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositional-truncations
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import metric-spaces.cartesian-products-metric-spaces
@@ -28,14 +24,11 @@ open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import order-theory.large-posets
 
-open import real-numbers.absolute-value-closed-intervals-real-numbers
 open import real-numbers.absolute-value-real-numbers
 open import real-numbers.addition-nonnegative-real-numbers
 open import real-numbers.addition-real-numbers
-open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.distance-real-numbers
-open import real-numbers.enclosing-closed-rational-intervals-real-numbers
 open import real-numbers.inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.inhabited-totally-bounded-subsets-real-numbers
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 6cfa5efab4..0372e6da37 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -24,7 +24,6 @@ open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.sets
 open import foundation.similarity-subtypes
diff --git a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
index 34e2ed1f44..543950300e 100644
--- a/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-finite-families-real-numbers.lagda.md
@@ -9,7 +9,6 @@ module real-numbers.maximum-finite-families-real-numbers where
 ```agda
 open import elementary-number-theory.addition-positive-rational-numbers
 open import elementary-number-theory.natural-numbers
-open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
@@ -30,7 +29,6 @@ open import lists.finite-sequences
 
 open import logic.functoriality-existential-quantification
 
-open import order-theory.join-semilattices
 open import order-theory.joins-finite-families-join-semilattices
 open import order-theory.least-upper-bounds-large-posets
 open import order-theory.upper-bounds-large-posets
@@ -40,7 +38,6 @@ open import real-numbers.binary-maximum-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
diff --git a/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md
index b4eb768f8b..8819476f7c 100644
--- a/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/maximum-upper-dedekind-real-numbers.lagda.md
@@ -16,15 +16,9 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
-open import foundation.disjunction
 open import foundation.existential-quantification
-open import foundation.function-types
-open import foundation.functoriality-cartesian-product-types
-open import foundation.inhabited-types
 open import foundation.intersections-subtypes
 open import foundation.logical-equivalences
-open import foundation.powersets
-open import foundation.propositional-truncations
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md b/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
index 5909da62ca..57f9d1bf80 100644
--- a/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/minimum-inhabited-finitely-enumerable-subsets-real-numbers.lagda.md
@@ -21,9 +21,6 @@ open import real-numbers.infima-families-real-numbers
 open import real-numbers.inhabited-finitely-enumerable-subsets-real-numbers
 open import real-numbers.maximum-inhabited-finitely-enumerable-subsets-real-numbers
 open import real-numbers.negation-real-numbers
-open import real-numbers.subsets-real-numbers
-
-open import univalent-combinatorics.finitely-enumerable-subtypes
 ```
 
 </details>
diff --git a/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md
index 860e3cee2a..e0c6900799 100644
--- a/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/minimum-lower-dedekind-real-numbers.lagda.md
@@ -26,7 +26,6 @@ open import logic.functoriality-existential-quantification
 
 open import order-theory.greatest-lower-bounds-large-posets
 open import order-theory.large-meet-semilattices
-open import order-theory.lower-bounds-large-posets
 
 open import real-numbers.inequality-lower-dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
diff --git a/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md
index 590cb5a2e9..6ea8a8c76e 100644
--- a/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/minimum-upper-dedekind-real-numbers.lagda.md
@@ -21,7 +21,6 @@ open import foundation.functoriality-cartesian-product-types
 open import foundation.functoriality-disjunction
 open import foundation.inhabited-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositional-truncations
 open import foundation.subtypes
 open import foundation.unions-subtypes
@@ -30,7 +29,6 @@ open import foundation.universe-levels
 open import logic.functoriality-existential-quantification
 
 open import order-theory.greatest-lower-bounds-large-posets
-open import order-theory.large-inflattices
 open import order-theory.lower-bounds-large-posets
 
 open import real-numbers.inequality-upper-dedekind-real-numbers
diff --git a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
index a1bb60ef8e..b4af60d1c5 100644
--- a/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-nonnegative-real-numbers.lagda.md
@@ -26,7 +26,6 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.difference-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
diff --git a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
index c0fea89695..9e9e66cc37 100644
--- a/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-and-negative-real-numbers.lagda.md
@@ -8,7 +8,6 @@ module real-numbers.multiplication-positive-and-negative-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
diff --git a/src/real-numbers/multiplication-positive-real-numbers.lagda.md b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
index 7972741f76..c151f0ecd2 100644
--- a/src/real-numbers/multiplication-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplication-positive-real-numbers.lagda.md
@@ -9,7 +9,6 @@ module real-numbers.multiplication-positive-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.closed-intervals-rational-numbers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.minimum-positive-rational-numbers
 open import elementary-number-theory.multiplication-closed-intervals-rational-numbers
diff --git a/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md b/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md
index 8acc9d09e7..564230fd9b 100644
--- a/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-negative-real-numbers.lagda.md
@@ -17,7 +17,6 @@ open import foundation.universe-levels
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.positive-and-negative-real-numbers
 open import real-numbers.rational-real-numbers
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 f91c1d2bfc..ad287640f8 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -19,23 +19,19 @@ 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.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
 open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
-open import foundation.unit-type
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.multiplication-real-numbers
 open import real-numbers.multiplicative-inverses-negative-real-numbers
 open import real-numbers.multiplicative-inverses-positive-real-numbers
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 490dd19044..47b08b6c2a 100644
--- a/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-positive-real-numbers.lagda.md
@@ -12,9 +12,7 @@ module real-numbers.multiplicative-inverses-positive-real-numbers where
 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
-open import elementary-number-theory.minimum-positive-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-positive-and-negative-rational-numbers
@@ -39,7 +37,6 @@ open import foundation.empty-types
 open import foundation.equivalences
 open import foundation.existential-quantification
 open import foundation.function-types
-open import foundation.functoriality-cartesian-product-types
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
diff --git a/src/real-numbers/negative-real-numbers.lagda.md b/src/real-numbers/negative-real-numbers.lagda.md
index fd87ef7b8c..75b3944778 100644
--- a/src/real-numbers/negative-real-numbers.lagda.md
+++ b/src/real-numbers/negative-real-numbers.lagda.md
@@ -10,7 +10,6 @@ module real-numbers.negative-real-numbers where
 
 ```agda
 open import elementary-number-theory.negative-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
diff --git a/src/real-numbers/nonnegative-real-numbers.lagda.md b/src/real-numbers/nonnegative-real-numbers.lagda.md
index d8219df9e7..c7b56ff0af 100644
--- a/src/real-numbers/nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/nonnegative-real-numbers.lagda.md
@@ -31,8 +31,6 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import metric-spaces.metric-spaces
-
 open import real-numbers.addition-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
diff --git a/src/real-numbers/positive-and-negative-real-numbers.lagda.md b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
index 850496dd46..d1d0920e34 100644
--- a/src/real-numbers/positive-and-negative-real-numbers.lagda.md
+++ b/src/real-numbers/positive-and-negative-real-numbers.lagda.md
@@ -10,13 +10,10 @@ module real-numbers.positive-and-negative-real-numbers where
 
 ```agda
 open import foundation.dependent-pair-types
-open import foundation.identity-types
-open import foundation.propositions
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.negative-real-numbers
 open import real-numbers.nonnegative-real-numbers
diff --git a/src/real-numbers/positive-real-numbers.lagda.md b/src/real-numbers/positive-real-numbers.lagda.md
index e32729d0f7..9b8c6101ca 100644
--- a/src/real-numbers/positive-real-numbers.lagda.md
+++ b/src/real-numbers/positive-real-numbers.lagda.md
@@ -9,31 +9,22 @@ module real-numbers.positive-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-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.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.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import group-theory.abelian-groups
-open import group-theory.groups
-
 open import real-numbers.addition-real-numbers
-open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
diff --git a/src/real-numbers/powers-real-numbers.lagda.md b/src/real-numbers/powers-real-numbers.lagda.md
index 8038140ac0..55ff07d63d 100644
--- a/src/real-numbers/powers-real-numbers.lagda.md
+++ b/src/real-numbers/powers-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import elementary-number-theory.ring-of-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
-open import foundation.disjunction
 open import foundation.identity-types
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
diff --git a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
index 2782e63dab..c3f1b1d417 100644
--- a/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
+++ b/src/real-numbers/raising-universe-levels-real-numbers.lagda.md
@@ -28,9 +28,6 @@ open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
-open import metric-spaces.isometries-metric-spaces
-open import metric-spaces.metric-space-of-rational-numbers
-
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.similarity-real-numbers
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index df01919a62..285729c4a4 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -9,7 +9,6 @@ module real-numbers.rational-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integers
 open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -17,20 +16,14 @@ 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.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.disjunction
-open import foundation.embeddings
 open import foundation.empty-types
 open import foundation.equivalences
-open import foundation.existential-quantification
 open import foundation.function-types
-open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.negation
-open import foundation.propositional-truncations
 open import foundation.propositions
 open import foundation.retractions
 open import foundation.sections
@@ -38,15 +31,11 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.raising-universe-levels-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
index 227fa5c7af..1f1c549e9a 100644
--- a/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-lower-dedekind-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.real-numbers-from-lower-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.addition-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
 
@@ -27,7 +25,6 @@ open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
-open import foundation.propositions
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
index 80c6cacfa9..d1d478c1a3 100644
--- a/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/real-numbers-from-upper-dedekind-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.real-numbers-from-upper-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.difference-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
 
@@ -25,7 +23,6 @@ open import foundation.inhabited-subtypes
 open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
-open import foundation.propositions
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
index 3686eb0a01..6b03b09e6f 100644
--- a/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-nonnegative-real-numbers.lagda.md
@@ -7,19 +7,12 @@ module real-numbers.similarity-nonnegative-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.positive-rational-numbers
-
 open import foundation.identity-types
-open import foundation.logical-equivalences
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
-open import real-numbers.rational-real-numbers
 open import real-numbers.similarity-real-numbers
-open import real-numbers.strict-inequality-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/similarity-real-numbers.lagda.md b/src/real-numbers/similarity-real-numbers.lagda.md
index 5664b3a49b..2c959f7d46 100644
--- a/src/real-numbers/similarity-real-numbers.lagda.md
+++ b/src/real-numbers/similarity-real-numbers.lagda.md
@@ -7,26 +7,17 @@ module real-numbers.similarity-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.strict-inequality-rational-numbers
-
-open import foundation.complements-subtypes
 open import foundation.dependent-pair-types
-open import foundation.disjunction
-open import foundation.empty-types
-open import foundation.function-types
 open import foundation.identity-types
 open import foundation.large-equivalence-relations
 open import foundation.large-similarity-relations
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.similarity-subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import order-theory.large-posets
 open import order-theory.large-preorders
-open import order-theory.similarity-of-elements-large-posets
 
 open import real-numbers.dedekind-real-numbers
 ```
diff --git a/src/real-numbers/squares-real-numbers.lagda.md b/src/real-numbers/squares-real-numbers.lagda.md
index e9995706b5..86d106d8c6 100644
--- a/src/real-numbers/squares-real-numbers.lagda.md
+++ b/src/real-numbers/squares-real-numbers.lagda.md
@@ -9,30 +9,23 @@ module real-numbers.squares-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.intersections-closed-intervals-rational-numbers
 open import elementary-number-theory.maximum-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-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.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.squares-rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 open import foundation.disjunction
-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
diff --git a/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
index 26b7ac1b7d..20fe2fbe23 100644
--- a/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequalities-addition-and-subtraction-real-numbers.lagda.md
@@ -21,7 +21,6 @@ open import foundation.binary-transport
 open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-types
-open import foundation.empty-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
@@ -34,7 +33,6 @@ open import real-numbers.addition-real-numbers
 open import real-numbers.arithmetically-located-dedekind-cuts
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
-open import real-numbers.inequality-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
 open import real-numbers.strict-inequality-real-numbers
diff --git a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
index 07f144ab5d..af7f51e493 100644
--- a/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-nonnegative-real-numbers.lagda.md
@@ -22,7 +22,6 @@ open import foundation.universe-levels
 
 open import logic.functoriality-existential-quantification
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.inequality-nonnegative-real-numbers
 open import real-numbers.inequality-real-numbers
 open import real-numbers.nonnegative-real-numbers
diff --git a/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md b/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md
index a0b7199f32..e24b886aea 100644
--- a/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-positive-real-numbers.lagda.md
@@ -12,7 +12,6 @@ module real-numbers.strict-inequality-positive-real-numbers where
 open import foundation.propositions
 open import foundation.universe-levels
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.positive-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index 96296b0ca7..d632b09a54 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -26,7 +26,6 @@ open import foundation.logical-equivalences
 open import foundation.negation
 open import foundation.propositional-truncations
 open import foundation.propositions
-open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.type-arithmetic-cartesian-product-types
 open import foundation.universe-levels
diff --git a/src/real-numbers/subsets-real-numbers.lagda.md b/src/real-numbers/subsets-real-numbers.lagda.md
index b83bc66f44..c42f82f45e 100644
--- a/src/real-numbers/subsets-real-numbers.lagda.md
+++ b/src/real-numbers/subsets-real-numbers.lagda.md
@@ -20,11 +20,8 @@ open import foundation.involutions
 open import foundation.logical-equivalences
 open import foundation.sets
 open import foundation.subtypes
-open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import metric-spaces.equality-of-metric-spaces
 open import metric-spaces.images-isometries-metric-spaces
 open import metric-spaces.isometries-metric-spaces
diff --git a/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md b/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md
index 3f060650bf..02cfcd06db 100644
--- a/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md
+++ b/src/real-numbers/totally-bounded-subsets-real-numbers.lagda.md
@@ -9,8 +9,6 @@ module real-numbers.totally-bounded-subsets-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
-open import elementary-number-theory.positive-rational-numbers
-
 open import foundation.dependent-pair-types
 open import foundation.propositions
 open import foundation.subtypes
@@ -21,7 +19,6 @@ open import metric-spaces.metric-spaces
 open import metric-spaces.subspaces-metric-spaces
 open import metric-spaces.totally-bounded-metric-spaces
 
-open import real-numbers.dedekind-real-numbers
 open import real-numbers.isometry-negation-real-numbers
 open import real-numbers.metric-space-of-real-numbers
 open import real-numbers.subsets-real-numbers
diff --git a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
index 82a733d139..7c96ab1c56 100644
--- a/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/transposition-addition-subtraction-cuts-dedekind-real-numbers.lagda.md
@@ -23,7 +23,6 @@ open import real-numbers.dedekind-real-numbers
 open import real-numbers.difference-real-numbers
 open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
-open import real-numbers.similarity-real-numbers
 open import real-numbers.strict-inequalities-addition-and-subtraction-real-numbers
 open import real-numbers.strict-inequality-real-numbers
 ```
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index 133ba99ea0..b9ed0136c8 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -24,7 +24,6 @@ open import foundation.existential-quantification
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.powersets
 open import foundation.propositions
 open import foundation.sets
 open import foundation.similarity-subtypes

From f11b6e2ecc5abdd4eae20346a460807ea293f811 Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 9 Nov 2025 13:09:19 +0100
Subject: [PATCH 20/21] fix

---
 .../multiplicative-inverses-nonzero-real-numbers.lagda.md     | 4 ++--
 1 file changed, 2 insertions(+), 2 deletions(-)

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 ad287640f8..51ab87da43 100644
--- a/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
+++ b/src/real-numbers/multiplicative-inverses-nonzero-real-numbers.lagda.md
@@ -132,8 +132,8 @@ module _
 ### If a real number has a multiplicative inverse, it is nonzero
 
 ```agda
-opaque
-  unfolding leq-ℝ mul-ℝ real-ℚ sim-ℝ
+abstract opaque
+  unfolding mul-ℝ real-ℚ sim-ℝ
 
   is-nonzero-has-right-inverse-mul-ℝ :
     {l1 l2 : Level} (x : ℝ l1) (y : ℝ l2) → sim-ℝ (x *ℝ y) one-ℝ →

From 2c0bdbcfc81c2e2f20941257b3fbfdec7d3dbdac Mon Sep 17 00:00:00 2001
From: Fredrik Bakke <fredrbak@gmail.com>
Date: Sun, 9 Nov 2025 13:30:32 +0100
Subject: [PATCH 21/21] chore: optimize imports rational numbers

---
 .../addition-rational-numbers.lagda.md                       | 1 -
 .../difference-rational-numbers.lagda.md                     | 1 -
 .../distance-rational-numbers.lagda.md                       | 2 --
 .../geometric-sequences-rational-numbers.lagda.md            | 4 ----
 .../inequality-positive-rational-numbers.lagda.md            | 1 -
 .../metric-additive-group-of-rational-numbers.lagda.md       | 2 --
 .../multiplicative-group-of-rational-numbers.lagda.md        | 2 --
 .../negative-rational-numbers.lagda.md                       | 2 --
 .../positive-and-negative-rational-numbers.lagda.md          | 2 --
 .../positive-rational-numbers.lagda.md                       | 4 ----
 .../powers-nonnegative-rational-numbers.lagda.md             | 2 --
 .../powers-positive-rational-numbers.lagda.md                | 1 -
 .../powers-rational-numbers.lagda.md                         | 2 --
 .../square-roots-positive-rational-numbers.lagda.md          | 1 -
 .../squares-rational-numbers.lagda.md                        | 2 --
 .../strict-inequality-rational-numbers.lagda.md              | 5 -----
 src/elementary-number-theory/triangular-numbers.lagda.md     | 4 ----
 17 files changed, 38 deletions(-)

diff --git a/src/elementary-number-theory/addition-rational-numbers.lagda.md b/src/elementary-number-theory/addition-rational-numbers.lagda.md
index 4e3854dddf..72024841c0 100644
--- a/src/elementary-number-theory/addition-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/addition-rational-numbers.lagda.md
@@ -23,7 +23,6 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.equivalences
 open import foundation.identity-types
-open import foundation.injective-maps
 open import foundation.interchange-law
 open import foundation.retractions
 open import foundation.sections
diff --git a/src/elementary-number-theory/difference-rational-numbers.lagda.md b/src/elementary-number-theory/difference-rational-numbers.lagda.md
index e4830f7cff..cc782a4c7c 100644
--- a/src/elementary-number-theory/difference-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/difference-rational-numbers.lagda.md
@@ -17,7 +17,6 @@ open import elementary-number-theory.rational-numbers
 open import foundation.action-on-identifications-binary-functions
 open import foundation.action-on-identifications-functions
 open import foundation.identity-types
-open import foundation.interchange-law
 ```
 
 </details>
diff --git a/src/elementary-number-theory/distance-rational-numbers.lagda.md b/src/elementary-number-theory/distance-rational-numbers.lagda.md
index 3926d248dc..ccc7b5f8b3 100644
--- a/src/elementary-number-theory/distance-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/distance-rational-numbers.lagda.md
@@ -21,13 +21,11 @@ open import elementary-number-theory.minimum-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.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-binary-functions
 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.function-types
diff --git a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
index 21695e9d15..07004973a8 100644
--- a/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/geometric-sequences-rational-numbers.lagda.md
@@ -29,9 +29,7 @@ open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.ring-of-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.function-extensionality
 open import foundation.identity-types
 open import foundation.negated-equality
@@ -45,8 +43,6 @@ open import lists.sequences
 open import metric-spaces.limits-of-sequences-metric-spaces
 open import metric-spaces.metric-space-of-rational-numbers
 open import metric-spaces.uniformly-continuous-functions-metric-spaces
-
-open import order-theory.strictly-increasing-sequences-strictly-preordered-sets
 ```
 
 </details>
diff --git a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
index 1398ef233b..bd42b31bd9 100644
--- a/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-positive-rational-numbers.lagda.md
@@ -13,7 +13,6 @@ open import elementary-number-theory.positive-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
-open import foundation.dependent-pair-types
 open import foundation.identity-types
 open import foundation.propositions
 open import foundation.universe-levels
diff --git a/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
index ba29b98878..c692847c8c 100644
--- a/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/metric-additive-group-of-rational-numbers.lagda.md
@@ -9,9 +9,7 @@ module elementary-number-theory.metric-additive-group-of-rational-numbers where
 ```agda
 open import analysis.metric-abelian-groups
 
-open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
-open import elementary-number-theory.rational-numbers
 
 open import foundation.dependent-pair-types
 open import foundation.universe-levels
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 30f54cea5a..9b527c5f53 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
@@ -16,7 +16,6 @@ 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.negative-rational-numbers
-open import elementary-number-theory.nonnegative-rational-numbers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-rational-numbers
 open import elementary-number-theory.positive-rational-numbers
@@ -27,7 +26,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
-open import foundation.empty-types
 open import foundation.function-types
 open import foundation.identity-types
 open import foundation.negated-equality
diff --git a/src/elementary-number-theory/negative-rational-numbers.lagda.md b/src/elementary-number-theory/negative-rational-numbers.lagda.md
index 7a41b1a752..15c00ecac4 100644
--- a/src/elementary-number-theory/negative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/negative-rational-numbers.lagda.md
@@ -17,13 +17,11 @@ open import elementary-number-theory.negative-integer-fractions
 open import elementary-number-theory.negative-integers
 open import elementary-number-theory.nonzero-rational-numbers
 open import elementary-number-theory.positive-and-negative-integers
-open import elementary-number-theory.positive-integer-fractions
 open import elementary-number-theory.positive-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.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.binary-transport
 open import foundation.dependent-pair-types
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 021ef28f50..09a711a99e 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
@@ -11,13 +11,11 @@ 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
diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index f146b5f15f..eda656f821 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -15,17 +15,13 @@ 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.negative-integers
 open import elementary-number-theory.nonzero-natural-numbers
 open import elementary-number-theory.nonzero-rational-numbers
-open import elementary-number-theory.positive-and-negative-integers
 open import elementary-number-theory.positive-integer-fractions
 open import elementary-number-theory.positive-integers
 open import elementary-number-theory.rational-numbers
 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
diff --git a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
index 535de72152..b07d9baf6d 100644
--- a/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-nonnegative-rational-numbers.lagda.md
@@ -17,7 +17,6 @@ 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
@@ -27,7 +26,6 @@ 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>
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 05641432b4..1e1dbbcd56 100644
--- a/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-positive-rational-numbers.lagda.md
@@ -25,7 +25,6 @@ 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.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
diff --git a/src/elementary-number-theory/powers-rational-numbers.lagda.md b/src/elementary-number-theory/powers-rational-numbers.lagda.md
index ea1efba98d..51e308477d 100644
--- a/src/elementary-number-theory/powers-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/powers-rational-numbers.lagda.md
@@ -14,9 +14,7 @@ open import elementary-number-theory.inequality-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 elementary-number-theory.natural-numbers
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 7687c6d3d4..c6621bc79b 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
@@ -14,7 +14,6 @@ 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.inequality-rational-numbers
-open import elementary-number-theory.maximum-rational-numbers
 open import elementary-number-theory.minimum-positive-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/squares-rational-numbers.lagda.md b/src/elementary-number-theory/squares-rational-numbers.lagda.md
index ada7062ba5..41b169790a 100644
--- a/src/elementary-number-theory/squares-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/squares-rational-numbers.lagda.md
@@ -18,7 +18,6 @@ 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
 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
@@ -31,7 +30,6 @@ open import elementary-number-theory.strict-inequality-rational-numbers
 
 open import foundation.action-on-identifications-functions
 open import foundation.coproduct-types
-open import foundation.decidable-types
 open import foundation.dependent-pair-types
 open import foundation.empty-types
 open import foundation.function-types
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 ddaa7ba616..e237a00ca0 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -13,10 +13,7 @@ open import elementary-number-theory.addition-integer-fractions
 open import elementary-number-theory.addition-rational-numbers
 open import elementary-number-theory.additive-group-of-rational-numbers
 open import elementary-number-theory.cross-multiplication-difference-integer-fractions
-open import elementary-number-theory.difference-integers
 open import elementary-number-theory.difference-rational-numbers
-open import elementary-number-theory.inequality-integer-fractions
-open import elementary-number-theory.inequality-integers
 open import elementary-number-theory.inequality-rational-numbers
 open import elementary-number-theory.integer-fractions
 open import elementary-number-theory.integers
@@ -26,7 +23,6 @@ open import elementary-number-theory.natural-numbers
 open import elementary-number-theory.nonnegative-integers
 open import elementary-number-theory.nonpositive-integers
 open import elementary-number-theory.positive-and-negative-integers
-open import elementary-number-theory.positive-integers
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.reduced-integer-fractions
 open import elementary-number-theory.strict-inequality-integer-fractions
@@ -36,7 +32,6 @@ open import elementary-number-theory.strict-inequality-natural-numbers
 open import foundation.action-on-identifications-functions
 open import foundation.binary-relations
 open import foundation.binary-transport
-open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.decidable-propositions
diff --git a/src/elementary-number-theory/triangular-numbers.lagda.md b/src/elementary-number-theory/triangular-numbers.lagda.md
index 451206ca03..23ae32125c 100644
--- a/src/elementary-number-theory/triangular-numbers.lagda.md
+++ b/src/elementary-number-theory/triangular-numbers.lagda.md
@@ -35,20 +35,16 @@ open import foundation.dependent-pair-types
 open import foundation.function-extensionality
 open import foundation.homotopies
 open import foundation.identity-types
-open import foundation.transport-along-identifications
 
 open import group-theory.abelian-groups
 open import group-theory.groups
 
-open import lists.sequences
-
 open import metric-spaces.limits-of-sequences-metric-spaces
 open import metric-spaces.metric-space-of-rational-numbers
 open import metric-spaces.rational-sequences-approximating-zero
 open import metric-spaces.uniformly-continuous-functions-metric-spaces
 
 open import ring-theory.partial-sums-sequences-semirings
-open import ring-theory.sums-of-finite-sequences-of-elements-semirings
 ```
 
 </details>