From 03bf44ad7a0a2f1cedef740c7b79932569cb3dfd Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Feb 2025 19:38:46 -0800
Subject: [PATCH 01/73] Do syntax for elim-exists chains, including application
 to strict real inequality

---
 .../existential-quantification.lagda.md       |  18 +++
 .../strict-inequality-real-numbers.lagda.md   | 152 +++++++-----------
 2 files changed, 79 insertions(+), 91 deletions(-)

diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md
index a15d2258a9..1cd6eb79f9 100644
--- a/src/foundation/existential-quantification.lagda.md
+++ b/src/foundation/existential-quantification.lagda.md
@@ -331,6 +331,24 @@ module _
     eq-distributive-product-exists-structure P
 ```
 
+## Existential quantification `do` syntax
+
+When eliminating a chain of existential quantifications, which may be
+interdependent, Agda's `do` syntax can eliminate many levels of nesting.
+
+```agda
+module elim-exists-do
+  {l : Level}
+  (P : Prop l)
+  where
+
+  _>>=_ :
+    {l1 l2 : Level}  {A : UU l1} {B : A -> UU l2} →
+    exists-structure A B →
+    (Σ A B -> type-Prop P) -> type-Prop P
+  x >>= f = elim-exists P (ev-pair f) x
+```
+
 ## See also
 
 - Existential quantification is the indexed counterpart to
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index c37393a0ca..da7259e97c 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -20,10 +20,12 @@ open import foundation.disjunction
 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
 open import foundation.propositions
+open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
@@ -90,28 +92,24 @@ module _
 
   asymmetric-le-ℝ : le-ℝ x y → ¬ (le-ℝ y x)
   asymmetric-le-ℝ x<y y<x =
-    elim-exists
-      ( empty-Prop)
-      ( λ p (p-in-ux , p-in-ly) →
-        elim-exists
-          ( empty-Prop)
-          ( λ q (q-in-uy , q-in-lx) →
-            rec-coproduct
-              ( λ p<q →
-                asymmetric-le-ℚ
-                  ( p)
-                  ( q)
-                  ( p<q)
-                  ( le-lower-upper-cut-ℝ x q p q-in-lx p-in-ux))
-              ( λ q≤p →
-                not-leq-le-ℚ
-                  ( p)
-                  ( q)
-                  ( le-lower-upper-cut-ℝ y p q p-in-ly q-in-uy)
-                  ( q≤p))
-              ( decide-le-leq-ℚ p q))
-          ( y<x))
-      ( x<y)
+    do
+      p , x<p , p<y ← x<y
+      q , y<q , q<x ← y<x
+      rec-coproduct
+        ( λ p<q →
+          asymmetric-le-ℚ
+            ( p)
+            ( q)
+            ( p<q)
+            ( le-lower-upper-cut-ℝ x q p q<x x<p))
+        ( λ q≤p →
+          not-leq-le-ℚ
+            ( p)
+            ( q)
+            ( le-lower-upper-cut-ℝ y p q p<y y<q)
+            ( q≤p))
+        ( decide-le-leq-ℚ p q)
+    where open elim-exists-do empty-Prop
 ```
 
 ### Strict inequality on the reals is transitive
@@ -125,23 +123,15 @@ module _
   where
 
   transitive-le-ℝ : le-ℝ y z → le-ℝ x y → le-ℝ x z
-  transitive-le-ℝ y<z =
-    elim-exists
-      ( le-ℝ-Prop x z)
-      ( λ p (p-in-ux , p-in-ly) →
-        elim-exists
-          (le-ℝ-Prop x z)
-          (λ q (q-in-uy , q-in-lz) →
-            intro-exists
-              p
-              ( p-in-ux ,
-                le-lower-cut-ℝ
-                  ( z)
-                  ( p)
-                  ( q)
-                  ( le-lower-upper-cut-ℝ y p q p-in-ly q-in-uy)
-                  ( q-in-lz)))
-          ( y<z))
+  transitive-le-ℝ y<z x<y =
+    do
+      p , x<p , p<y ← x<y
+      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)
+    where open elim-exists-do (le-ℝ-Prop x z)
 ```
 
 ### The canonical map from rationals to reals preserves and reflects strict inequality
@@ -180,26 +170,12 @@ module _
 
   concatenate-le-leq-ℝ : le-ℝ x y → leq-ℝ y z → le-ℝ x z
   concatenate-le-leq-ℝ x<y y≤z =
-    elim-exists
-      ( le-ℝ-Prop x z)
-      ( λ p (p-in-upper-x , p-in-lower-y) →
-        intro-exists p (p-in-upper-x , y≤z p p-in-lower-y))
-      ( x<y)
+    map-tot-exists (λ p → map-product id (y≤z p)) x<y
 
   concatenate-leq-le-ℝ : leq-ℝ x y → le-ℝ y z → le-ℝ x z
-  concatenate-leq-le-ℝ x≤y y<z =
-    elim-exists
-      ( le-ℝ-Prop x z)
-      ( λ p (p-in-upper-y , p-in-lower-z) →
-        intro-exists
-          ( p)
-          ( forward-implication
-            ( leq-iff-ℝ' x y)
-            ( x≤y)
-            ( p)
-            ( p-in-upper-y) ,
-        p-in-lower-z))
-      ( y<z)
+  concatenate-leq-le-ℝ x≤y =
+    map-tot-exists
+      ( λ p → map-product (forward-implication (leq-iff-ℝ' x y) x≤y p) id)
 ```
 
 ### The reals have no lower or upper bound
@@ -212,26 +188,19 @@ module _
 
   exists-lesser-ℝ : exists (ℝ lzero) (λ y → le-ℝ-Prop y x)
   exists-lesser-ℝ =
-    elim-exists
-      ( ∃ (ℝ lzero) (λ y → le-ℝ-Prop y x))
-      ( λ q q-in-lx →
-        intro-exists
-          ( real-ℚ q)
-          ( forward-implication (is-rounded-lower-cut-ℝ x q) q-in-lx))
+    map-exists
+      ( λ y → le-ℝ y x)
+      ( real-ℚ)
+      ( λ q → forward-implication (is-rounded-lower-cut-ℝ x q))
       ( is-inhabited-lower-cut-ℝ x)
 
   exists-greater-ℝ : exists (ℝ lzero) (λ y → le-ℝ-Prop x y)
   exists-greater-ℝ =
-    elim-exists
-      ( ∃ (ℝ lzero) (λ y → le-ℝ-Prop x y))
-      ( λ q q-in-ux →
-        intro-exists
-          ( real-ℚ q)
-          ( elim-exists
-              ( le-ℝ-Prop x (real-ℚ q))
-              ( λ r (r<q , r-in-ux) → intro-exists r (r-in-ux , r<q))
-              ( forward-implication (is-rounded-upper-cut-ℝ x q) q-in-ux)))
-      ( is-inhabited-upper-cut-ℝ x)
+    do
+      q , x<q ← is-inhabited-upper-cut-ℝ x
+      r , r<q , x<r ← forward-implication (is-rounded-upper-cut-ℝ x q) x<q
+      intro-exists (real-ℚ q) (intro-exists r (x<r , r<q))
+    where open elim-exists-do (∃ (ℝ lzero) (le-ℝ-Prop x))
 ```
 
 ### Negation reverses the strict ordering of real numbers
@@ -245,13 +214,15 @@ module _
 
   reverses-order-neg-ℝ : le-ℝ x y → le-ℝ (neg-ℝ y) (neg-ℝ x)
   reverses-order-neg-ℝ =
-    elim-exists
-      ( le-ℝ-Prop (neg-ℝ y) (neg-ℝ x))
-      ( λ p (p-in-ux , p-in-ly) →
-        intro-exists
-          ( neg-ℚ p)
-          ( tr (is-in-lower-cut-ℝ y) (inv (neg-neg-ℚ p)) p-in-ly ,
-            tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ p)) p-in-ux))
+    map-exists
+      ( λ p →
+        is-in-subtype (upper-cut-neg-ℝ y) p ×
+        is-in-subtype (lower-cut-neg-ℝ x) p)
+      ( neg-ℚ)
+      ( λ p (x<p , p<y) →
+        tr (is-in-lower-cut-ℝ y) (inv (neg-neg-ℚ p)) p<y ,
+        tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ p)) x<p)
+    where open elim-exists-do (le-ℝ-Prop (neg-ℝ y) (neg-ℝ x))
 ```
 
 ### If `x` is less than `y`, then `y` is not less than or equal to `x`
@@ -364,17 +335,16 @@ module _
   where
 
   dense-le-ℝ : le-ℝ x y → exists (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y)
-  dense-le-ℝ =
-    elim-exists
-      ( ∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y))
-      ( λ q (q∈ux , q∈ly) →
-        map-binary-exists
-          ( λ z → le-ℝ x z × le-ℝ z y)
-          ( λ _ _ → real-ℚ q)
-          ( λ p r (p<q , p∈ux) (q<r , r∈ly) →
-            intro-exists p (p∈ux , p<q) , intro-exists r (q<r , r∈ly))
-          ( forward-implication (is-rounded-upper-cut-ℝ x q) q∈ux)
-          ( forward-implication (is-rounded-lower-cut-ℝ y q) q∈ly))
+  dense-le-ℝ x<y =
+    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
+        (real-ℚ q)
+        (intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
+    where
+      open elim-exists-do (∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y))
 ```
 
 ## References

From a49369f7a1f8e9ce9def64d41298598884a82c77 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Feb 2025 19:42:58 -0800
Subject: [PATCH 02/73] Simplifications and indentation

---
 .../strict-inequality-real-numbers.lagda.md      | 16 ++++++----------
 1 file changed, 6 insertions(+), 10 deletions(-)

diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index da7259e97c..d4ff970765 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -96,18 +96,14 @@ module _
       p , x<p , p<y ← x<y
       q , y<q , q<x ← y<x
       rec-coproduct
-        ( λ p<q →
-          asymmetric-le-ℚ
-            ( p)
+        ( asymmetric-le-ℚ
             ( q)
-            ( p<q)
+            ( p)
             ( le-lower-upper-cut-ℝ x q p q<x x<p))
-        ( λ q≤p →
-          not-leq-le-ℚ
+        ( not-leq-le-ℚ
             ( p)
             ( q)
-            ( le-lower-upper-cut-ℝ y p q p<y y<q)
-            ( q≤p))
+            ( le-lower-upper-cut-ℝ y p q p<y y<q))
         ( decide-le-leq-ℚ p q)
     where open elim-exists-do empty-Prop
 ```
@@ -341,8 +337,8 @@ module _
       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
-        (real-ℚ q)
-        (intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
+        ( real-ℚ q)
+        ( intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
     where
       open elim-exists-do (∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y))
 ```

From f053354b06de0327a645580b9ed5884b2656994c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sat, 8 Feb 2025 19:44:11 -0800
Subject: [PATCH 03/73] make pre-commit

---
 src/foundation/existential-quantification.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md
index 1cd6eb79f9..eaf1a65d1c 100644
--- a/src/foundation/existential-quantification.lagda.md
+++ b/src/foundation/existential-quantification.lagda.md
@@ -343,7 +343,7 @@ module elim-exists-do
   where
 
   _>>=_ :
-    {l1 l2 : Level}  {A : UU l1} {B : A -> UU l2} →
+    {l1 l2 : Level} {A : UU l1} {B : A -> UU l2} →
     exists-structure A B →
     (Σ A B -> type-Prop P) -> type-Prop P
   x >>= f = elim-exists P (ev-pair f) x

From 5481940316c92867e55b94c467eb6c4187b9009b Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 09:25:08 -0800
Subject: [PATCH 04/73] Updates

---
 .../existential-quantification.lagda.md       | 26 +++++++++++++++--
 .../strict-inequality-real-numbers.lagda.md   | 28 +++++++++----------
 2 files changed, 38 insertions(+), 16 deletions(-)

diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md
index eaf1a65d1c..23d53d9417 100644
--- a/src/foundation/existential-quantification.lagda.md
+++ b/src/foundation/existential-quantification.lagda.md
@@ -334,10 +334,32 @@ module _
 ## Existential quantification `do` syntax
 
 When eliminating a chain of existential quantifications, which may be
-interdependent, Agda's `do` syntax can eliminate many levels of nesting.
+interdependent, [Agda's `do` syntax](https://agda.readthedocs.io/en/stable/language/syntactic-sugar.html#do-notation) can eliminate many levels of nesting.
+
+For example, suppose we have
+
+```text
+R : Prop l
+
+exists-a-p : exists A P
+a-p-implies-b-q-exists : (a : A) → P a → exists B Q
+a-b-p-q-implies-r : (a : A) (b : B) → P a → Q b → R
+```
+
+To deduce `R`, we could write
+
+```text
+do
+  a , pa ← exists-a-p
+  b , qb ← a-p-implies-b-q-exists a pa
+  a-b-p-q-implies-r a b pa qb
+where open do-syntax-elim-exists R
+```
+
+instead of a nested chain of `elim-exists`.
 
 ```agda
-module elim-exists-do
+module do-syntax-elim-exists
   {l : Level}
   (P : Prop l)
   where
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index d4ff970765..e6c6041f4b 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -105,7 +105,7 @@ module _
             ( q)
             ( le-lower-upper-cut-ℝ y p q p<y y<q))
         ( decide-le-leq-ℚ p q)
-    where open elim-exists-do empty-Prop
+    where open do-syntax-elim-exists empty-Prop
 ```
 
 ### Strict inequality on the reals is transitive
@@ -127,7 +127,7 @@ module _
         ( p)
         ( x<p ,
           le-lower-cut-ℝ z p q (le-lower-upper-cut-ℝ y p q p<y y<q) q<z)
-    where open elim-exists-do (le-ℝ-Prop x z)
+    where open do-syntax-elim-exists (le-ℝ-Prop x z)
 ```
 
 ### The canonical map from rationals to reals preserves and reflects strict inequality
@@ -196,7 +196,7 @@ module _
       q , x<q ← is-inhabited-upper-cut-ℝ x
       r , r<q , x<r ← forward-implication (is-rounded-upper-cut-ℝ x q) x<q
       intro-exists (real-ℚ q) (intro-exists r (x<r , r<q))
-    where open elim-exists-do (∃ (ℝ lzero) (le-ℝ-Prop x))
+    where open do-syntax-elim-exists (∃ (ℝ lzero) (le-ℝ-Prop x))
 ```
 
 ### Negation reverses the strict ordering of real numbers
@@ -209,16 +209,14 @@ module _
   where
 
   reverses-order-neg-ℝ : le-ℝ x y → le-ℝ (neg-ℝ y) (neg-ℝ x)
-  reverses-order-neg-ℝ =
-    map-exists
-      ( λ p →
-        is-in-subtype (upper-cut-neg-ℝ y) p ×
-        is-in-subtype (lower-cut-neg-ℝ x) p)
-      ( neg-ℚ)
-      ( λ p (x<p , p<y) →
-        tr (is-in-lower-cut-ℝ y) (inv (neg-neg-ℚ p)) p<y ,
-        tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ p)) x<p)
-    where open elim-exists-do (le-ℝ-Prop (neg-ℝ y) (neg-ℝ x))
+  reverses-order-neg-ℝ x<y =
+    do
+      p , x<p , p<y ← x<y
+      intro-exists
+        (neg-ℚ p)
+        ( tr (is-in-lower-cut-ℝ y) (inv (neg-neg-ℚ p)) p<y ,
+          tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ p)) x<p)
+    where open do-syntax-elim-exists (le-ℝ-Prop (neg-ℝ y) (neg-ℝ x))
 ```
 
 ### If `x` is less than `y`, then `y` is not less than or equal to `x`
@@ -340,7 +338,9 @@ module _
         ( real-ℚ q)
         ( intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
     where
-      open elim-exists-do (∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y))
+      open
+        do-syntax-elim-exists
+          ( ∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y))
 ```
 
 ## References

From 96b05c1cf27de02e809c99e3362143e8319ee1a7 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 09:26:52 -0800
Subject: [PATCH 05/73] make pre-commit

---
 src/foundation/existential-quantification.lagda.md | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md
index 23d53d9417..04483c575f 100644
--- a/src/foundation/existential-quantification.lagda.md
+++ b/src/foundation/existential-quantification.lagda.md
@@ -334,7 +334,9 @@ module _
 ## Existential quantification `do` syntax
 
 When eliminating a chain of existential quantifications, which may be
-interdependent, [Agda's `do` syntax](https://agda.readthedocs.io/en/stable/language/syntactic-sugar.html#do-notation) can eliminate many levels of nesting.
+interdependent,
+[Agda's `do` syntax](https://agda.readthedocs.io/en/stable/language/syntactic-sugar.html#do-notation)
+can eliminate many levels of nesting.
 
 For example, suppose we have
 

From e84f3ce826860e6f38873509b2a003be10f799a2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 10:22:00 -0800
Subject: [PATCH 06/73] Define Minkowski multiplication for semigroups

---
 ...nkowski-multiplication-semigroups.lagda.md | 144 ++++++++++++++++++
 1 file changed, 144 insertions(+)
 create mode 100644 src/group-theory/minkowski-multiplication-semigroups.lagda.md

diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
new file mode 100644
index 0000000000..1978d3b786
--- /dev/null
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -0,0 +1,144 @@
+# Minkowski multiplication of semigroup subtypes
+
+```agda
+module group-theory.minkowski-multiplication-semigroups where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+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.existential-quantification
+open import foundation.identity-types
+open import foundation.propositions
+open import foundation.sets
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import group-theory.semigroups
+```
+
+</details>
+
+## Idea
+
+For two subtypes `A`, `B` of a semigroup `S`, the Minkowski multiplication of `A` and `B` is the set
+of elements that can be formed by multiplying an element of `A` and an element
+of `B`.  (This is more usually referred to as a Minkowski sum, but as the
+operation on semigroups is referred to as `mul`, we use multiplicative
+terminology.)
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (G : Semigroup l1)
+  (A : subtype l2 (type-Semigroup G))
+  (B : subtype l3 (type-Semigroup G))
+  where
+
+  minkowski-mul-Semigroup : subtype (l1 ⊔ l2 ⊔ l3) (type-Semigroup G)
+  minkowski-mul-Semigroup c =
+    ∃
+      ( type-Semigroup G × type-Semigroup G)
+      ( λ (a , b) →
+        A a ∧ B b ∧ Id-Prop (set-Semigroup G) c (mul-Semigroup G a b))
+```
+
+## Properties
+
+### Minkowski multiplication of semigroup subtypes is associative
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (G : Semigroup l1)
+  (A : subtype l2 (type-Semigroup G))
+  (B : subtype l3 (type-Semigroup G))
+  (C : subtype l4 (type-Semigroup G))
+  where
+
+  associative-minkowski-mul-has-same-elements-subtype-Semigroup :
+    has-same-elements-subtype
+      ( minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C)
+      ( minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C))
+  pr1 (associative-minkowski-mul-has-same-elements-subtype-Semigroup x) =
+    elim-exists
+      ( claim)
+      ( λ (ab , c) (ab∈AB , c∈C , x=ab*c) →
+        elim-exists
+          ( claim)
+          ( λ (a , b) (a∈A , b∈B , ab=a*b) →
+            intro-exists
+              ( a , mul-Semigroup G b c)
+              ( a∈A ,
+                intro-exists (b , c) (b∈B , c∈C , refl) ,
+                ( equational-reasoning
+                  x
+                  ＝ mul-Semigroup G ab c by x=ab*c
+                  ＝ mul-Semigroup G (mul-Semigroup G a b) c
+                    by ap (mul-Semigroup' G c) ab=a*b
+                  ＝ mul-Semigroup G a (mul-Semigroup G b c)
+                    by associative-mul-Semigroup G a b c)))
+          ( ab∈AB))
+    where
+      claim =
+        minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C) x
+  pr2 (associative-minkowski-mul-has-same-elements-subtype-Semigroup x) =
+    elim-exists
+      ( claim)
+      ( λ (a , bc) (a∈A , bc∈BC , x=a*bc) →
+        elim-exists
+          ( claim)
+          ( λ (b , c) (b∈B , c∈C , bc=b*c) →
+            intro-exists
+              ( mul-Semigroup G a b , c)
+              ( intro-exists (a , b) (a∈A , b∈B , refl) ,
+                c∈C ,
+                ( equational-reasoning
+                  x
+                  ＝ mul-Semigroup G a bc by x=a*bc
+                  ＝ mul-Semigroup G a (mul-Semigroup G b c)
+                    by ap (mul-Semigroup G a) bc=b*c
+                  ＝ mul-Semigroup G (mul-Semigroup G a b) c
+                    by inv (associative-mul-Semigroup G a b c))))
+          ( bc∈BC))
+    where
+      claim =
+        minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C x
+
+  associative-minkowski-mul-Semigroup :
+    minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C ＝
+    minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C)
+  associative-minkowski-mul-Semigroup =
+    eq-has-same-elements-subtype
+      ( minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C)
+      ( minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C))
+      ( associative-minkowski-mul-has-same-elements-subtype-Semigroup)
+```
+
+### Minkowski multiplication of subtypes of a semigroup forms a semigroup
+
+```agda
+module _
+  {l1 : Level}
+  (G : Semigroup l1)
+  where
+
+  semigroup-minkowski-mul-Semigroup :
+    (l : Level) → Semigroup (lsuc l1 ⊔ lsuc l)
+  pr1 (semigroup-minkowski-mul-Semigroup l) =
+    subtype-Set (l1 ⊔ l) (type-Semigroup G)
+  pr1 (pr2 (semigroup-minkowski-mul-Semigroup l)) = minkowski-mul-Semigroup G
+  pr2 (pr2 (semigroup-minkowski-mul-Semigroup l)) =
+    associative-minkowski-mul-Semigroup G
+```
+
+## External links
+
+- [Minkowski addition](https://en.wikipedia.org/wiki/Minkowski_addition) at
+  Wikipedia

From 4f547c83691fc800d61a68d06f0518ac0ad1b514 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 10:22:39 -0800
Subject: [PATCH 07/73] make pre-commit

---
 src/group-theory.lagda.md                              |  1 +
 .../minkowski-multiplication-semigroups.lagda.md       | 10 +++++-----
 2 files changed, 6 insertions(+), 5 deletions(-)

diff --git a/src/group-theory.lagda.md b/src/group-theory.lagda.md
index 962a96a816..1233b0af98 100644
--- a/src/group-theory.lagda.md
+++ b/src/group-theory.lagda.md
@@ -116,6 +116,7 @@ open import group-theory.large-semigroups public
 open import group-theory.loop-groups-sets public
 open import group-theory.mere-equivalences-concrete-group-actions public
 open import group-theory.mere-equivalences-group-actions public
+open import group-theory.minkowski-multiplication-semigroups public
 open import group-theory.monoid-actions public
 open import group-theory.monoids public
 open import group-theory.monomorphisms-concrete-groups public
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index 1978d3b786..e4a81a386c 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -25,11 +25,11 @@ open import group-theory.semigroups
 
 ## Idea
 
-For two subtypes `A`, `B` of a semigroup `S`, the Minkowski multiplication of `A` and `B` is the set
-of elements that can be formed by multiplying an element of `A` and an element
-of `B`.  (This is more usually referred to as a Minkowski sum, but as the
-operation on semigroups is referred to as `mul`, we use multiplicative
-terminology.)
+For two subtypes `A`, `B` of a semigroup `S`, the Minkowski multiplication of
+`A` and `B` is the set of elements that can be formed by multiplying an element
+of `A` and an element of `B`. (This is more usually referred to as a Minkowski
+sum, but as the operation on semigroups is referred to as `mul`, we use
+multiplicative terminology.)
 
 ## Definition
 

From ab87dac060d95050de28e5aa81fd4830ceb99c50 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 11:15:39 -0800
Subject: [PATCH 08/73] Add monoids and commutative monoids

---
 ...ultiplication-commutative-monoids.lagda.md | 116 +++++++++++++++
 .../minkowski-multiplication-monoids.lagda.md | 138 ++++++++++++++++++
 ...nkowski-multiplication-semigroups.lagda.md |  17 +--
 src/group-theory/monoids.lagda.md             |   6 +
 4 files changed, 268 insertions(+), 9 deletions(-)
 create mode 100644 src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
 create mode 100644 src/group-theory/minkowski-multiplication-monoids.lagda.md

diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
new file mode 100644
index 0000000000..f85d234c4a
--- /dev/null
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -0,0 +1,116 @@
+# Minkowski multiplication of commutative monoid subtypes
+
+```agda
+module group-theory.minkowski-multiplication-commutative-monoids where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.powersets
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import group-theory.minkowski-multiplication-monoids
+open import group-theory.commutative-monoids
+open import group-theory.monoids
+```
+
+</details>
+
+## Idea
+
+For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
+[commutative monoid](group-theory.commutative-monoids.md) `M`, the Minkowski multiplication of
+`A` and `B` is the set of elements that can be formed by multiplying an element
+of `A` and an element of `B`. (This is more usually referred to as a Minkowski
+sum, but as the operation on commutative monoids is referred to as `mul`, we use
+multiplicative terminology.)
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (M : Commutative-Monoid l1)
+  (A : subtype l2 (type-Commutative-Monoid M))
+  (B : subtype l3 (type-Commutative-Monoid M))
+  where
+
+  minkowski-mul-Commutative-Monoid :
+    subtype (l1 ⊔ l2 ⊔ l3) (type-Commutative-Monoid M)
+  minkowski-mul-Commutative-Monoid =
+    minkowski-mul-Monoid (monoid-Commutative-Monoid M) A B
+```
+
+## Properties
+
+### Minkowski multiplication on a commutative monoid is commutative
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (M : Commutative-Monoid l1)
+  (A : subtype l2 (type-Commutative-Monoid M))
+  (B : subtype l3 (type-Commutative-Monoid M))
+  where
+
+  commutative-minkowski-mul-leq-Commutative-Monoid :
+    minkowski-mul-Commutative-Monoid M A B ⊆
+    minkowski-mul-Commutative-Monoid M B A
+  commutative-minkowski-mul-leq-Commutative-Monoid x =
+    elim-exists
+      (minkowski-mul-Commutative-Monoid M B A x)
+      ( λ (a , b) (a∈A , b∈B , x=ab) →
+        intro-exists
+          ( b , a)
+          ( b∈B , a∈A , x=ab ∙ commutative-mul-Commutative-Monoid M a b))
+
+module _
+  {l1 l2 l3 : Level}
+  (M : Commutative-Monoid l1)
+  (A : subtype l2 (type-Commutative-Monoid M))
+  (B : subtype l3 (type-Commutative-Monoid M))
+  where
+
+  commutative-minkowski-mul-Commutative-Monoid :
+    minkowski-mul-Commutative-Monoid M A B ＝
+    minkowski-mul-Commutative-Monoid M B A
+  commutative-minkowski-mul-Commutative-Monoid =
+    antisymmetric-sim-subtype
+      ( minkowski-mul-Commutative-Monoid M A B)
+      ( minkowski-mul-Commutative-Monoid M B A)
+      ( commutative-minkowski-mul-leq-Commutative-Monoid M A B ,
+        commutative-minkowski-mul-leq-Commutative-Monoid M B A)
+```
+
+### Minkowski multiplication on a commutative monoid is a commutative monoid
+
+```agda
+module _
+  {l : Level}
+  (M : Commutative-Monoid l)
+  where
+
+  commutative-monoid-minkowski-mul-Commutative-Monoid :
+    Commutative-Monoid (lsuc l)
+  pr1 commutative-monoid-minkowski-mul-Commutative-Monoid =
+    monoid-minkowski-mul-Monoid (monoid-Commutative-Monoid M)
+  pr2 commutative-monoid-minkowski-mul-Commutative-Monoid =
+    commutative-minkowski-mul-Commutative-Monoid M
+```
+
+## See also
+
+- Minkowski multiplication on semigroups is defined in
+  [`group-theory.minkowski-multiplication-semigroups`](group-theory.minkowski-multiplication-semigroups.md).
+- Minkowski multiplication on monoids is defined in
+  [`group-theory.minkowski-multiplication-monoids`](group-theory.minkowski-multiplication-monoids.md).
+
+## External links
+
+- [Minkowski addition](https://en.wikipedia.org/wiki/Minkowski_addition) at
+  Wikipedia
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
new file mode 100644
index 0000000000..88de396de6
--- /dev/null
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -0,0 +1,138 @@
+# Minkowski multiplication of monoid subtypes
+
+```agda
+module group-theory.minkowski-multiplication-monoids where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.action-on-identifications-functions
+open import foundation.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.powersets
+open import foundation.unital-binary-operations
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import group-theory.minkowski-multiplication-semigroups
+open import group-theory.monoids
+open import group-theory.semigroups
+```
+
+</details>
+
+## Idea
+
+For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
+[monoid](group-theory.monoids.md) `M`, the Minkowski multiplication of
+`A` and `B` is the set of elements that can be formed by multiplying an element
+of `A` and an element of `B`. (This is more usually referred to as a Minkowski
+sum, but as the operation on monoids is referred to as `mul`, we use
+multiplicative terminology.)
+
+## Definition
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (M : Monoid l1)
+  (A : subtype l2 (type-Monoid M))
+  (B : subtype l3 (type-Monoid M))
+  where
+
+  minkowski-mul-Monoid : subtype (l1 ⊔ l2 ⊔ l3) (type-Monoid M)
+  minkowski-mul-Monoid = minkowski-mul-Semigroup (semigroup-Monoid M) A B
+```
+
+## Properties
+
+### Unit laws for Minkowski multiplication of semigroup subtypes
+
+```agda
+module _
+  {l1 l2 : Level}
+  (M : Monoid l1)
+  (A : subtype l2 (type-Monoid M))
+  where
+
+  left-unit-law-minkowski-mul-Monoid :
+    sim-subtype (minkowski-mul-Monoid M (is-unit-Monoid-Prop M) A) A
+  pr1 left-unit-law-minkowski-mul-Monoid a =
+    elim-exists
+      ( A a)
+      ( λ (z , a') (z=1 , a'∈A , a=za') →
+        tr
+          ( is-in-subtype A)
+          ( inv
+            ( equational-reasoning
+              a
+              ＝ mul-Monoid M z a' by a=za'
+              ＝ mul-Monoid M (unit-Monoid M) a' by ap (mul-Monoid' M a') z=1
+              ＝ a' by left-unit-law-mul-Monoid M a'))
+          ( a'∈A))
+  pr2 left-unit-law-minkowski-mul-Monoid a a∈A =
+    intro-exists
+      ( unit-Monoid M , a)
+      ( refl , a∈A , inv (left-unit-law-mul-Monoid M a))
+
+  right-unit-law-minkowski-mul-Monoid :
+    sim-subtype (minkowski-mul-Monoid M A (is-unit-Monoid-Prop M)) A
+  pr1 right-unit-law-minkowski-mul-Monoid a =
+    elim-exists
+      ( A a)
+      ( λ (a' , z) (a'∈A , z=1 , a=a'z) →
+        tr
+          ( is-in-subtype A)
+          ( inv
+            ( equational-reasoning
+              a
+              ＝ mul-Monoid M a' z by a=a'z
+              ＝ mul-Monoid M a' (unit-Monoid M) by ap (mul-Monoid M a') z=1
+              ＝ a' by right-unit-law-mul-Monoid M a'))
+          ( a'∈A))
+  pr2 right-unit-law-minkowski-mul-Monoid a a∈A =
+    intro-exists
+      ( a , unit-Monoid M)
+      ( a∈A , refl , inv (right-unit-law-mul-Monoid M a))
+```
+
+### Minkowski multiplication of subtypes of a monoid forms a monoid
+
+```agda
+module _
+  {l : Level}
+  (M : Monoid l)
+  where
+
+  is-unital-minkowski-mul-Monoid :
+    is-unital (minkowski-mul-Monoid {l} {l} {l} M)
+  pr1 is-unital-minkowski-mul-Monoid = is-unit-Monoid-Prop M
+  pr1 (pr2 is-unital-minkowski-mul-Monoid) A =
+    antisymmetric-sim-subtype
+      ( minkowski-mul-Monoid M (is-unit-Monoid-Prop M) A)
+      ( A)
+      ( left-unit-law-minkowski-mul-Monoid M A)
+  pr2 (pr2 is-unital-minkowski-mul-Monoid) A =
+    antisymmetric-sim-subtype
+      ( minkowski-mul-Monoid M A (is-unit-Monoid-Prop M))
+      ( A)
+      ( right-unit-law-minkowski-mul-Monoid M A)
+
+  monoid-minkowski-mul-Monoid : Monoid (lsuc l)
+  pr1 monoid-minkowski-mul-Monoid =
+    semigroup-minkowski-mul-Semigroup (semigroup-Monoid M)
+  pr2 monoid-minkowski-mul-Monoid = is-unital-minkowski-mul-Monoid
+```
+
+## See also
+
+- Minkowski multiplication on semigroups is defined in
+  [`group-theory.minkowski-multiplication-semigroups`](group-theory.minkowski-multiplication-semigroups.md).
+
+## External links
+
+- [Minkowski addition](https://en.wikipedia.org/wiki/Minkowski_addition) at
+  Wikipedia
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index e4a81a386c..4fd589d0d2 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -25,7 +25,8 @@ open import group-theory.semigroups
 
 ## Idea
 
-For two subtypes `A`, `B` of a semigroup `S`, the Minkowski multiplication of
+For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
+[semigroup](group-theory.semigroups.md) `S`, the Minkowski multiplication of
 `A` and `B` is the set of elements that can be formed by multiplying an element
 of `A` and an element of `B`. (This is more usually referred to as a Minkowski
 sum, but as the operation on semigroups is referred to as `mul`, we use
@@ -125,16 +126,14 @@ module _
 
 ```agda
 module _
-  {l1 : Level}
-  (G : Semigroup l1)
+  {l : Level}
+  (G : Semigroup l)
   where
 
-  semigroup-minkowski-mul-Semigroup :
-    (l : Level) → Semigroup (lsuc l1 ⊔ lsuc l)
-  pr1 (semigroup-minkowski-mul-Semigroup l) =
-    subtype-Set (l1 ⊔ l) (type-Semigroup G)
-  pr1 (pr2 (semigroup-minkowski-mul-Semigroup l)) = minkowski-mul-Semigroup G
-  pr2 (pr2 (semigroup-minkowski-mul-Semigroup l)) =
+  semigroup-minkowski-mul-Semigroup : Semigroup (lsuc l)
+  pr1 semigroup-minkowski-mul-Semigroup = subtype-Set l (type-Semigroup G)
+  pr1 (pr2 semigroup-minkowski-mul-Semigroup) = minkowski-mul-Semigroup G
+  pr2 (pr2 semigroup-minkowski-mul-Semigroup) =
     associative-minkowski-mul-Semigroup G
 ```
 
diff --git a/src/group-theory/monoids.lagda.md b/src/group-theory/monoids.lagda.md
index 1e76dbc225..6a0c542cfc 100644
--- a/src/group-theory/monoids.lagda.md
+++ b/src/group-theory/monoids.lagda.md
@@ -82,6 +82,12 @@ module _
   unit-Monoid : type-Monoid
   unit-Monoid = pr1 has-unit-Monoid
 
+  is-unit-Monoid-Prop : type-Monoid → Prop l
+  is-unit-Monoid-Prop x = Id-Prop set-Monoid x unit-Monoid
+
+  is-unit-Monoid : type-Monoid → UU l
+  is-unit-Monoid x = x ＝ unit-Monoid
+
   left-unit-law-mul-Monoid : (x : type-Monoid) → mul-Monoid unit-Monoid x ＝ x
   left-unit-law-mul-Monoid = pr1 (pr2 has-unit-Monoid)
 

From 1f8bb30ec86e99692a98eecfcc9f829029aa45ac Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 11:16:52 -0800
Subject: [PATCH 09/73] make pre-commit

---
 src/group-theory.lagda.md                            |  2 ++
 ...owski-multiplication-commutative-monoids.lagda.md | 12 ++++++------
 .../minkowski-multiplication-monoids.lagda.md        | 12 ++++++------
 .../minkowski-multiplication-semigroups.lagda.md     |  8 ++++----
 4 files changed, 18 insertions(+), 16 deletions(-)

diff --git a/src/group-theory.lagda.md b/src/group-theory.lagda.md
index 1233b0af98..3c4fb0e591 100644
--- a/src/group-theory.lagda.md
+++ b/src/group-theory.lagda.md
@@ -116,6 +116,8 @@ open import group-theory.large-semigroups public
 open import group-theory.loop-groups-sets public
 open import group-theory.mere-equivalences-concrete-group-actions public
 open import group-theory.mere-equivalences-group-actions public
+open import group-theory.minkowski-multiplication-commutative-monoids public
+open import group-theory.minkowski-multiplication-monoids public
 open import group-theory.minkowski-multiplication-semigroups public
 open import group-theory.monoid-actions public
 open import group-theory.monoids public
diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
index f85d234c4a..90d8c9c2a6 100644
--- a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -14,8 +14,8 @@ open import foundation.powersets
 open import foundation.subtypes
 open import foundation.universe-levels
 
-open import group-theory.minkowski-multiplication-monoids
 open import group-theory.commutative-monoids
+open import group-theory.minkowski-multiplication-monoids
 open import group-theory.monoids
 ```
 
@@ -24,11 +24,11 @@ open import group-theory.monoids
 ## Idea
 
 For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
-[commutative monoid](group-theory.commutative-monoids.md) `M`, the Minkowski multiplication of
-`A` and `B` is the set of elements that can be formed by multiplying an element
-of `A` and an element of `B`. (This is more usually referred to as a Minkowski
-sum, but as the operation on commutative monoids is referred to as `mul`, we use
-multiplicative terminology.)
+[commutative monoid](group-theory.commutative-monoids.md) `M`, the Minkowski
+multiplication of `A` and `B` is the set of elements that can be formed by
+multiplying an element of `A` and an element of `B`. (This is more usually
+referred to as a Minkowski sum, but as the operation on commutative monoids is
+referred to as `mul`, we use multiplicative terminology.)
 
 ## Definition
 
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
index 88de396de6..34ceb6b961 100644
--- a/src/group-theory/minkowski-multiplication-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -12,9 +12,9 @@ open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.powersets
-open import foundation.unital-binary-operations
 open import foundation.subtypes
 open import foundation.transport-along-identifications
+open import foundation.unital-binary-operations
 open import foundation.universe-levels
 
 open import group-theory.minkowski-multiplication-semigroups
@@ -27,11 +27,11 @@ open import group-theory.semigroups
 ## Idea
 
 For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
-[monoid](group-theory.monoids.md) `M`, the Minkowski multiplication of
-`A` and `B` is the set of elements that can be formed by multiplying an element
-of `A` and an element of `B`. (This is more usually referred to as a Minkowski
-sum, but as the operation on monoids is referred to as `mul`, we use
-multiplicative terminology.)
+[monoid](group-theory.monoids.md) `M`, the Minkowski multiplication of `A` and
+`B` is the set of elements that can be formed by multiplying an element of `A`
+and an element of `B`. (This is more usually referred to as a Minkowski sum, but
+as the operation on monoids is referred to as `mul`, we use multiplicative
+terminology.)
 
 ## Definition
 
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index 4fd589d0d2..3c6e307454 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -26,10 +26,10 @@ open import group-theory.semigroups
 ## Idea
 
 For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
-[semigroup](group-theory.semigroups.md) `S`, the Minkowski multiplication of
-`A` and `B` is the set of elements that can be formed by multiplying an element
-of `A` and an element of `B`. (This is more usually referred to as a Minkowski
-sum, but as the operation on semigroups is referred to as `mul`, we use
+[semigroup](group-theory.semigroups.md) `S`, the Minkowski multiplication of `A`
+and `B` is the set of elements that can be formed by multiplying an element of
+`A` and an element of `B`. (This is more usually referred to as a Minkowski sum,
+but as the operation on semigroups is referred to as `mul`, we use
 multiplicative terminology.)
 
 ## Definition

From 4581333c10e4b668df65a2f0a33378d7c1e28e78 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 11:27:12 -0800
Subject: [PATCH 10/73] Use subset syntax

---
 ...ultiplication-commutative-monoids.lagda.md | 17 +++++-----
 .../minkowski-multiplication-monoids.lagda.md | 13 ++++----
 ...nkowski-multiplication-semigroups.lagda.md | 32 ++++++++++---------
 3 files changed, 33 insertions(+), 29 deletions(-)

diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
index 90d8c9c2a6..1d5453d267 100644
--- a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -17,13 +17,14 @@ open import foundation.universe-levels
 open import group-theory.commutative-monoids
 open import group-theory.minkowski-multiplication-monoids
 open import group-theory.monoids
+open import group-theory.subsets-commutative-monoids
 ```
 
 </details>
 
 ## Idea
 
-For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
+For two [subsets](group-theory.subsets-commutative-monoids.md) `A`, `B` of a
 [commutative monoid](group-theory.commutative-monoids.md) `M`, the Minkowski
 multiplication of `A` and `B` is the set of elements that can be formed by
 multiplying an element of `A` and an element of `B`. (This is more usually
@@ -36,12 +37,12 @@ referred to as `mul`, we use multiplicative terminology.)
 module _
   {l1 l2 l3 : Level}
   (M : Commutative-Monoid l1)
-  (A : subtype l2 (type-Commutative-Monoid M))
-  (B : subtype l3 (type-Commutative-Monoid M))
+  (A : subset-Commutative-Monoid l2 M)
+  (B : subset-Commutative-Monoid l3 M)
   where
 
   minkowski-mul-Commutative-Monoid :
-    subtype (l1 ⊔ l2 ⊔ l3) (type-Commutative-Monoid M)
+    subset-Commutative-Monoid (l1 ⊔ l2 ⊔ l3) M
   minkowski-mul-Commutative-Monoid =
     minkowski-mul-Monoid (monoid-Commutative-Monoid M) A B
 ```
@@ -54,8 +55,8 @@ module _
 module _
   {l1 l2 l3 : Level}
   (M : Commutative-Monoid l1)
-  (A : subtype l2 (type-Commutative-Monoid M))
-  (B : subtype l3 (type-Commutative-Monoid M))
+  (A : subset-Commutative-Monoid l2 M)
+  (B : subset-Commutative-Monoid l3 M)
   where
 
   commutative-minkowski-mul-leq-Commutative-Monoid :
@@ -72,8 +73,8 @@ module _
 module _
   {l1 l2 l3 : Level}
   (M : Commutative-Monoid l1)
-  (A : subtype l2 (type-Commutative-Monoid M))
-  (B : subtype l3 (type-Commutative-Monoid M))
+  (A : subset-Commutative-Monoid l2 M)
+  (B : subset-Commutative-Monoid l3 M)
   where
 
   commutative-minkowski-mul-Commutative-Monoid :
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
index 34ceb6b961..6d51224108 100644
--- a/src/group-theory/minkowski-multiplication-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -20,13 +20,14 @@ open import foundation.universe-levels
 open import group-theory.minkowski-multiplication-semigroups
 open import group-theory.monoids
 open import group-theory.semigroups
+open import group-theory.subsets-monoids
 ```
 
 </details>
 
 ## Idea
 
-For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
+For two [subsets](group-theory.subsets-monoids.md) `A`, `B` of a
 [monoid](group-theory.monoids.md) `M`, the Minkowski multiplication of `A` and
 `B` is the set of elements that can be formed by multiplying an element of `A`
 and an element of `B`. (This is more usually referred to as a Minkowski sum, but
@@ -39,23 +40,23 @@ terminology.)
 module _
   {l1 l2 l3 : Level}
   (M : Monoid l1)
-  (A : subtype l2 (type-Monoid M))
-  (B : subtype l3 (type-Monoid M))
+  (A : subset-Monoid l2 M)
+  (B : subset-Monoid l3 M)
   where
 
-  minkowski-mul-Monoid : subtype (l1 ⊔ l2 ⊔ l3) (type-Monoid M)
+  minkowski-mul-Monoid : subset-Monoid (l1 ⊔ l2 ⊔ l3) M
   minkowski-mul-Monoid = minkowski-mul-Semigroup (semigroup-Monoid M) A B
 ```
 
 ## Properties
 
-### Unit laws for Minkowski multiplication of semigroup subtypes
+### Unit laws for Minkowski multiplication of monoid subsets
 
 ```agda
 module _
   {l1 l2 : Level}
   (M : Monoid l1)
-  (A : subtype l2 (type-Monoid M))
+  (A : subset-Monoid l2 M)
   where
 
   left-unit-law-minkowski-mul-Monoid :
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index 3c6e307454..0587640fe3 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -13,19 +13,21 @@ open import foundation.conjunction
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
+open import foundation.powersets
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.universe-levels
 
 open import group-theory.semigroups
+open import group-theory.subsets-semigroups
 ```
 
 </details>
 
 ## Idea
 
-For two [subtypes](foundation-core.subtypes.md) `A`, `B` of a
+For two [subsets](group-theory.subsets-semigroups.md) `A`, `B` of a
 [semigroup](group-theory.semigroups.md) `S`, the Minkowski multiplication of `A`
 and `B` is the set of elements that can be formed by multiplying an element of
 `A` and an element of `B`. (This is more usually referred to as a Minkowski sum,
@@ -38,11 +40,11 @@ multiplicative terminology.)
 module _
   {l1 l2 l3 : Level}
   (G : Semigroup l1)
-  (A : subtype l2 (type-Semigroup G))
-  (B : subtype l3 (type-Semigroup G))
+  (A : subset-Semigroup l2 G)
+  (B : subset-Semigroup l3 G)
   where
 
-  minkowski-mul-Semigroup : subtype (l1 ⊔ l2 ⊔ l3) (type-Semigroup G)
+  minkowski-mul-Semigroup : subset-Semigroup (l1 ⊔ l2 ⊔ l3) G
   minkowski-mul-Semigroup c =
     ∃
       ( type-Semigroup G × type-Semigroup G)
@@ -52,22 +54,22 @@ module _
 
 ## Properties
 
-### Minkowski multiplication of semigroup subtypes is associative
+### Minkowski multiplication of semigroup subsets is associative
 
 ```agda
 module _
   {l1 l2 l3 l4 : Level}
   (G : Semigroup l1)
-  (A : subtype l2 (type-Semigroup G))
-  (B : subtype l3 (type-Semigroup G))
-  (C : subtype l4 (type-Semigroup G))
+  (A : subset-Semigroup l2 G)
+  (B : subset-Semigroup l3 G)
+  (C : subset-Semigroup l4 G)
   where
 
-  associative-minkowski-mul-has-same-elements-subtype-Semigroup :
-    has-same-elements-subtype
+  associative-minkowski-mul-sim-Semigroup :
+    sim-subtype
       ( minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C)
       ( minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C))
-  pr1 (associative-minkowski-mul-has-same-elements-subtype-Semigroup x) =
+  pr1 associative-minkowski-mul-sim-Semigroup x =
     elim-exists
       ( claim)
       ( λ (ab , c) (ab∈AB , c∈C , x=ab*c) →
@@ -89,7 +91,7 @@ module _
     where
       claim =
         minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C) x
-  pr2 (associative-minkowski-mul-has-same-elements-subtype-Semigroup x) =
+  pr2 associative-minkowski-mul-sim-Semigroup x =
     elim-exists
       ( claim)
       ( λ (a , bc) (a∈A , bc∈BC , x=a*bc) →
@@ -116,13 +118,13 @@ module _
     minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C ＝
     minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C)
   associative-minkowski-mul-Semigroup =
-    eq-has-same-elements-subtype
+    antisymmetric-sim-subtype
       ( minkowski-mul-Semigroup G (minkowski-mul-Semigroup G A B) C)
       ( minkowski-mul-Semigroup G A (minkowski-mul-Semigroup G B C))
-      ( associative-minkowski-mul-has-same-elements-subtype-Semigroup)
+      ( associative-minkowski-mul-sim-Semigroup)
 ```
 
-### Minkowski multiplication of subtypes of a semigroup forms a semigroup
+### Minkowski multiplication of subsets of a semigroup forms a semigroup
 
 ```agda
 module _

From 6a9f8370e6b7eeb750fce2777014954461265d16 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 11:54:20 -0800
Subject: [PATCH 11/73] Define lower Dedekind cuts/reals

---
 .../lower-dedekind-real-numbers.lagda.md      | 115 ++++++++++++++++++
 1 file changed, 115 insertions(+)
 create mode 100644 src/real-numbers/lower-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..58b8384159
--- /dev/null
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -0,0 +1,115 @@
+# Lower Dedekind real numbers
+
+```agda
+module real-numbers.lower-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+```agda
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.dependent-pair-types
+open import foundation.subtypes
+open import foundation.conjunction
+open import foundation.identity-types
+open import foundation.existential-quantification
+open import foundation.propositions
+open import foundation.logical-equivalences
+open import foundation.universe-levels
+open import foundation.universal-quantification
+```
+
+</details>
+## Idea
+
+A lower
+{{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
+consists of a
+[subtype](foundation-core.subtypes.md) `L` of
+[the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
+satisfying the following two conditions:
+
+1. _Inhabitedness_. `L` is [inhabited](foundation.inhabited-subtypes.md).
+2. _Roundedness_. A rational number `q` is in `L`
+   [if and only if](foundation.logical-equivalences.md) there
+   [exists](foundation.existential-quantification.md) `q < r` such that `r ∈ L`.
+
+The type of all lower Dedekind real numbers is the type of all lower Dedekind cuts.
+A lower Dedekind cut is part of a [Dedekind real number](real-numbers.dedekind-real-numbers.md)
+
+## Definition
+
+### Lower Dedekind cuts
+
+```agda
+module _
+  {l : Level}
+  (L : subtype l ℚ)
+  where
+
+  is-lower-dedekind-cut-Prop : Prop l
+  is-lower-dedekind-cut-Prop =
+    (∃ ℚ L) ∧
+    (∀' ℚ (λ q → L q ⇔ (∃ ℚ (λ r → le-ℚ-Prop q r ∧ L r))))
+
+  is-lower-dedekind-cut : UU l
+  is-lower-dedekind-cut = type-Prop is-lower-dedekind-cut-Prop
+```
+
+## The lower Dedekind real numbers
+
+```agda
+lower-ℝ : (l : Level) → UU (lsuc l)
+lower-ℝ l = Σ (subtype l ℚ) is-lower-dedekind-cut
+
+module _
+  {l : Level}
+  (x : lower-ℝ l)
+  where
+
+  cut-lower-ℝ : subtype l ℚ
+  cut-lower-ℝ = pr1 x
+
+  is-in-cut-lower-ℝ : ℚ → UU l
+  is-in-cut-lower-ℝ = is-in-subtype cut-lower-ℝ
+
+  is-inhabited-lower-ℝ : exists ℚ cut-lower-ℝ
+  is-inhabited-lower-ℝ = pr1 (pr2 x)
+
+  is-rounded-lower-ℝ :
+    (q : ℚ) →
+    is-in-cut-lower-ℝ q ↔ exists ℚ (λ r → le-ℚ-Prop q r ∧ cut-lower-ℝ r)
+  is-rounded-lower-ℝ = pr2 (pr2 x)
+```
+
+## Properties
+
+### Lower Dedekind cuts are closed under the standard ordering on the rationals
+
+```agda
+module _
+  {l : Level} (x : lower-ℝ l) (p q : ℚ)
+  where
+
+  le-cut-lower-ℝ : le-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
+  le-cut-lower-ℝ p<q q<x =
+    backward-implication (is-rounded-lower-ℝ x p) (intro-exists q (p<q , q<x))
+```
+
+### Two lower real numbers with the same cut are equal
+
+```agda
+module _
+  {l : Level} (x y : lower-ℝ l)
+  where
+
+  eq-eq-cut-lower-ℝ : cut-lower-ℝ x ＝ cut-lower-ℝ y → x ＝ y
+  eq-eq-cut-lower-ℝ = eq-type-subtype is-lower-dedekind-cut-Prop
+```
+
+## References
+
+This page follows the terminology used in the exercises of Section 11 in {{#cite UF13}}.
+
+{{#bibliography}}

From ef39dd336152ff6d348dc6f6ee0d55e892c60184 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 11:54:56 -0800
Subject: [PATCH 12/73] make pre-commit

---
 src/real-numbers.lagda.md                     |  1 +
 .../lower-dedekind-real-numbers.lagda.md      | 22 ++++++++++---------
 2 files changed, 13 insertions(+), 10 deletions(-)

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index b2aaa1a1f4..6fc691eef0 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.arithmetically-located-dedekind-cuts public
 open import real-numbers.dedekind-real-numbers public
 open import real-numbers.inequality-real-numbers public
+open import real-numbers.lower-dedekind-real-numbers public
 open import real-numbers.metric-space-of-real-numbers public
 open import real-numbers.negation-real-numbers public
 open import real-numbers.rational-real-numbers public
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 58b8384159..5d99526167 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -5,19 +5,20 @@ module real-numbers.lower-dedekind-real-numbers where
 ```
 
 <details><summary>Imports</summary>
+
 ```agda
 open import elementary-number-theory.rational-numbers
 open import elementary-number-theory.strict-inequality-rational-numbers
 
-open import foundation.dependent-pair-types
-open import foundation.subtypes
 open import foundation.conjunction
-open import foundation.identity-types
+open import foundation.dependent-pair-types
 open import foundation.existential-quantification
-open import foundation.propositions
+open import foundation.identity-types
 open import foundation.logical-equivalences
-open import foundation.universe-levels
+open import foundation.propositions
+open import foundation.subtypes
 open import foundation.universal-quantification
+open import foundation.universe-levels
 ```
 
 </details>
@@ -25,8 +26,7 @@ open import foundation.universal-quantification
 
 A lower
 {{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
-consists of a
-[subtype](foundation-core.subtypes.md) `L` of
+consists of a [subtype](foundation-core.subtypes.md) `L` of
 [the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
 satisfying the following two conditions:
 
@@ -35,8 +35,9 @@ satisfying the following two conditions:
    [if and only if](foundation.logical-equivalences.md) there
    [exists](foundation.existential-quantification.md) `q < r` such that `r ∈ L`.
 
-The type of all lower Dedekind real numbers is the type of all lower Dedekind cuts.
-A lower Dedekind cut is part of a [Dedekind real number](real-numbers.dedekind-real-numbers.md)
+The type of all lower Dedekind real numbers is the type of all lower Dedekind
+cuts. A lower Dedekind cut is part of a
+[Dedekind real number](real-numbers.dedekind-real-numbers.md)
 
 ## Definition
 
@@ -110,6 +111,7 @@ module _
 
 ## References
 
-This page follows the terminology used in the exercises of Section 11 in {{#cite UF13}}.
+This page follows the terminology used in the exercises of Section 11 in
+{{#cite UF13}}.
 
 {{#bibliography}}

From 59195c9edc8dfc2117df76628621a2767981ea31 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 11:59:55 -0800
Subject: [PATCH 13/73] Define upper Dedekind reals

---
 .../lower-dedekind-real-numbers.lagda.md      |  10 +-
 .../upper-dedekind-real-numbers.lagda.md      | 124 ++++++++++++++++++
 2 files changed, 132 insertions(+), 2 deletions(-)
 create mode 100644 src/real-numbers/upper-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 5d99526167..f603657ab7 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -36,8 +36,7 @@ satisfying the following two conditions:
    [exists](foundation.existential-quantification.md) `q < r` such that `r ∈ L`.
 
 The type of all lower Dedekind real numbers is the type of all lower Dedekind
-cuts. A lower Dedekind cut is part of a
-[Dedekind real number](real-numbers.dedekind-real-numbers.md)
+cuts.
 
 ## Definition
 
@@ -109,6 +108,13 @@ module _
   eq-eq-cut-lower-ℝ = eq-type-subtype is-lower-dedekind-cut-Prop
 ```
 
+## See also
+
+- Upper Dedekind cuts, the dual structure, are defined in
+  [`real-numbers.upper-dedekind-real-numbers`](real-numbers.upper-dedekind-real-numbers.md).
+- Dedekind cuts, which form the usual real numbers, are defined in
+  [`real-numbers.dedekind-real-numbers`](real-numbers.dedekind-real-numbers.md)
+
 ## References
 
 This page follows the terminology used in the exercises of Section 11 in
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..bdf0a5525f
--- /dev/null
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -0,0 +1,124 @@
+# Upper Dedekind real numbers
+
+```agda
+module real-numbers.upper-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+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.propositions
+open import foundation.subtypes
+open import foundation.universal-quantification
+open import foundation.universe-levels
+```
+
+</details>
+
+## Idea
+
+A upper
+{{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
+consists of a [subtype](foundation-core.subtypes.md) `U` of
+[the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
+satisfying the following two conditions:
+
+1. _Inhabitedness_. `U` is [inhabited](foundation.inhabited-subtypes.md).
+2. _Roundedness_. A rational number `q` is in `U`
+   [if and only if](foundation.logical-equivalences.md) there
+   [exists](foundation.existential-quantification.md) `p < q` such that `p ∈ U`.
+
+The type of all upper Dedekind real numbers is the type of all upper Dedekind
+cuts.
+
+## Definition
+
+### Upper Dedekind cuts
+
+```agda
+module _
+  {l : Level}
+  (U : subtype l ℚ)
+  where
+
+  is-upper-dedekind-cut-Prop : Prop l
+  is-upper-dedekind-cut-Prop =
+    (∃ ℚ U) ∧
+    (∀' ℚ (λ q → U q ⇔ (∃ ℚ (λ p → le-ℚ-Prop p q ∧ U p))))
+
+  is-upper-dedekind-cut : UU l
+  is-upper-dedekind-cut = type-Prop is-upper-dedekind-cut-Prop
+```
+
+## The upper Dedekind real numbers
+
+```agda
+upper-ℝ : (l : Level) → UU (lsuc l)
+upper-ℝ l = Σ (subtype l ℚ) is-upper-dedekind-cut
+
+module _
+  {l : Level}
+  (x : upper-ℝ l)
+  where
+
+  cut-upper-ℝ : subtype l ℚ
+  cut-upper-ℝ = pr1 x
+
+  is-in-cut-upper-ℝ : ℚ → UU l
+  is-in-cut-upper-ℝ = is-in-subtype cut-upper-ℝ
+
+  is-inhabited-upper-ℝ : exists ℚ cut-upper-ℝ
+  is-inhabited-upper-ℝ = pr1 (pr2 x)
+
+  is-rounded-upper-ℝ :
+    (q : ℚ) →
+    is-in-cut-upper-ℝ q ↔ exists ℚ (λ p → le-ℚ-Prop p q ∧ cut-upper-ℝ p)
+  is-rounded-upper-ℝ = pr2 (pr2 x)
+```
+
+## Properties
+
+### Upper Dedekind cuts are closed under the standard ordering on the rationals
+
+```agda
+module _
+  {l : Level} (x : upper-ℝ l) (p q : ℚ)
+  where
+
+  le-cut-upper-ℝ : le-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
+  le-cut-upper-ℝ p<q p<x =
+    backward-implication (is-rounded-upper-ℝ x q) (intro-exists p (p<q , p<x))
+```
+
+### Two upper real numbers with the same cut are equal
+
+```agda
+module _
+  {l : Level} (x y : upper-ℝ l)
+  where
+
+  eq-eq-cut-upper-ℝ : cut-upper-ℝ x ＝ cut-upper-ℝ y → x ＝ y
+  eq-eq-cut-upper-ℝ = eq-type-subtype is-upper-dedekind-cut-Prop
+```
+
+## See also
+
+- Lower Dedekind cuts, the dual structure, are defined in
+  [`real-numbers.lower-dedekind-real-numbers`](real-numbers.lower-dedekind-real-numbers.md).
+- Dedekind cuts, which form the usual real numbers, are defined in
+  [`real-numbers.dedekind-real-numbers`](real-numbers.dedekind-real-numbers.md)
+
+## References
+
+This page follows the terminology used in the exercises of Section 11 in
+{{#cite UF13}}.
+
+{{#bibliography}}

From 4233e2871669bc59e9bbf0a8edaadf9f7bacdae0 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 12:00:27 -0800
Subject: [PATCH 14/73] make pre-commit

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

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 6fc691eef0..b110b77299 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -14,4 +14,5 @@ open import real-numbers.negation-real-numbers public
 open import real-numbers.rational-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
+open import real-numbers.upper-dedekind-real-numbers public
 ```

From 0bd35b87219cd5d8601ed56d78fe283e9015b258 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 12:08:53 -0800
Subject: [PATCH 15/73] Start defining lower addition

---
 ...dditive-group-of-rational-numbers.lagda.md |  7 ++-
 ...ition-lower-dedekind-real-numbers.lagda.md | 58 +++++++++++++++++++
 .../lower-dedekind-real-numbers.lagda.md      |  1 +
 3 files changed, 65 insertions(+), 1 deletion(-)
 create mode 100644 src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md

diff --git a/src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md b/src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md
index 60aeaf1fb9..e0e8513c49 100644
--- a/src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/additive-group-of-rational-numbers.lagda.md
@@ -17,6 +17,7 @@ open import foundation.unital-binary-operations
 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
@@ -60,9 +61,13 @@ pr2 (pr2 (pr2 (pr2 group-add-ℚ))) = right-inverse-law-add-ℚ
 
 ## Properties
 
-### Tha additive group of rational numbers is commutative
+### The additive group of rational numbers is commutative
 
 ```agda
+commutative-monoid-add-ℚ : Commutative-Monoid lzero
+pr1 commutative-monoid-add-ℚ = monoid-add-ℚ
+pr2 commutative-monoid-add-ℚ = commutative-add-ℚ
+
 abelian-group-add-ℚ : Ab lzero
 pr1 abelian-group-add-ℚ = group-add-ℚ
 pr2 abelian-group-add-ℚ = commutative-add-ℚ
diff --git a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..03096f38d0
--- /dev/null
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -0,0 +1,58 @@
+# Addition of lower Dedekind real numbers
+
+```agda
+module real-numbers.addition-lower-dedekind-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.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.existential-quantification
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import group-theory.minkowski-multiplication-commutative-monoids
+
+open import real-numbers.lower-dedekind-real-numbers
+```
+
+</details>
+
+## Idea
+
+The sum of two [lower Dedekind real numbers](real-numbers.lower-dedekind-real-numbers.md)
+is the Minkowski sum of their cuts.
+
+```agda
+module _
+  {l1 l2 : Level}
+  (x : lower-ℝ l1)
+  (y : lower-ℝ l2)
+  where
+
+  cut-add-lower-ℝ : subtype (l1 ⊔ l2) ℚ
+  cut-add-lower-ℝ =
+    minkowski-mul-Commutative-Monoid
+      ( commutative-monoid-add-ℚ)
+      ( cut-lower-ℝ x)
+      ( cut-lower-ℝ y)
+
+  is-in-add-lower-ℝ-cut : ℚ → UU (l1 ⊔ l2)
+  is-in-add-lower-ℝ-cut = is-in-subtype cut-add-lower-ℝ
+
+  -- is-inhabited-add-lower-ℝ : exists ℚ cut-add-lower-ℝ
+  -- is-inhabited-add-lower-ℝ
+```
+
+## Properties
+
+### Addition of lower Dedekind real numbers is commutative
+
+```agda
+-- commutative-add-lower-ℝ :
+```
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index f603657ab7..58c0fa6a7b 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -22,6 +22,7 @@ open import foundation.universe-levels
 ```
 
 </details>
+
 ## Idea
 
 A lower

From 258ec997f9af6eddb40b6ff3afbe6f53dfb53537 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 12:19:23 -0800
Subject: [PATCH 16/73] Inhabitedness

---
 ...ultiplication-commutative-monoids.lagda.md | 20 ++++++++++++++++
 .../minkowski-multiplication-monoids.lagda.md | 21 ++++++++++++++++
 ...nkowski-multiplication-semigroups.lagda.md | 24 +++++++++++++++++++
 3 files changed, 65 insertions(+)

diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
index 1d5453d267..1500a3439a 100644
--- a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -11,6 +11,7 @@ open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.powersets
+open import foundation.inhabited-subtypes
 open import foundation.subtypes
 open import foundation.universe-levels
 
@@ -104,6 +105,25 @@ module _
     commutative-minkowski-mul-Commutative-Monoid M
 ```
 
+### The Minkowski multiplication of two inhabited subsets of a commutative monoid is inhabited
+
+```agda
+module _
+  {l1 : Level}
+  (M : Commutative-Monoid l1)
+  where
+
+  minkowski-mul-inhabited-is-inhabited-Commutative-Monoid :
+    {l2 l3 : Level} →
+    (A : subset-Commutative-Monoid l2 M) →
+    (B : subset-Commutative-Monoid l3 M) →
+    is-inhabited-subtype A →
+    is-inhabited-subtype B →
+    is-inhabited-subtype (minkowski-mul-Commutative-Monoid M A B)
+  minkowski-mul-inhabited-is-inhabited-Commutative-Monoid =
+    minkowski-mul-inhabited-is-inhabited-Monoid (monoid-Commutative-Monoid M)
+```
+
 ## See also
 
 - Minkowski multiplication on semigroups is defined in
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
index 6d51224108..5152319062 100644
--- a/src/group-theory/minkowski-multiplication-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -13,6 +13,7 @@ open import foundation.existential-quantification
 open import foundation.identity-types
 open import foundation.powersets
 open import foundation.subtypes
+open import foundation.inhabited-subtypes
 open import foundation.transport-along-identifications
 open import foundation.unital-binary-operations
 open import foundation.universe-levels
@@ -128,6 +129,26 @@ module _
   pr2 monoid-minkowski-mul-Monoid = is-unital-minkowski-mul-Monoid
 ```
 
+### The Minkowski multiplication of two inhabited subsets of a monoid is inhabited
+
+```agda
+module _
+  {l1 : Level}
+  (M : Monoid l1)
+  where
+
+  minkowski-mul-inhabited-is-inhabited-Monoid :
+    {l2 l3 : Level} →
+    (A : subset-Monoid l2 M) →
+    (B : subset-Monoid l3 M) →
+    is-inhabited-subtype A →
+    is-inhabited-subtype B →
+    is-inhabited-subtype (minkowski-mul-Monoid M A B)
+  minkowski-mul-inhabited-is-inhabited-Monoid =
+    minkowski-mul-inhabited-is-inhabited-Semigroup (semigroup-Monoid M)
+```
+
+
 ## See also
 
 - Minkowski multiplication on semigroups is defined in
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index 0587640fe3..ea819dde6b 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -13,12 +13,15 @@ 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.powersets
 open import foundation.propositions
 open import foundation.sets
 open import foundation.subtypes
 open import foundation.universe-levels
 
+open import logic.functoriality-existential-quantification
+
 open import group-theory.semigroups
 open import group-theory.subsets-semigroups
 ```
@@ -139,6 +142,27 @@ module _
     associative-minkowski-mul-Semigroup G
 ```
 
+### The Minkowski multiplication of two inhabited subsets of a semigroup is inhabited
+
+```agda
+module _
+  {l1 l2 l3 : Level}
+  (G : Semigroup l1)
+  (A : subset-Semigroup l2 G)
+  (B : subset-Semigroup l3 G)
+  where
+
+  minkowski-mul-inhabited-is-inhabited-Semigroup :
+    is-inhabited-subtype A →
+    is-inhabited-subtype B →
+    is-inhabited-subtype (minkowski-mul-Semigroup G A B)
+  minkowski-mul-inhabited-is-inhabited-Semigroup =
+    map-binary-exists
+      ( is-in-subtype (minkowski-mul-Semigroup G A B))
+      ( mul-Semigroup G)
+      λ a b a∈A b∈B → intro-exists (a , b) (a∈A , b∈B , refl)
+```
+
 ## External links
 
 - [Minkowski addition](https://en.wikipedia.org/wiki/Minkowski_addition) at

From 40a9e33195feaa76211760edcf9138d3b4d5f192 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 12:20:00 -0800
Subject: [PATCH 17/73] make pre-commit

---
 .../minkowski-multiplication-commutative-monoids.lagda.md     | 2 +-
 src/group-theory/minkowski-multiplication-monoids.lagda.md    | 3 +--
 src/group-theory/minkowski-multiplication-semigroups.lagda.md | 4 ++--
 3 files changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
index 1500a3439a..1e0ad307f4 100644
--- a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -10,8 +10,8 @@ module group-theory.minkowski-multiplication-commutative-monoids where
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
-open import foundation.powersets
 open import foundation.inhabited-subtypes
+open import foundation.powersets
 open import foundation.subtypes
 open import foundation.universe-levels
 
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
index 5152319062..b2e67b3464 100644
--- a/src/group-theory/minkowski-multiplication-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -11,9 +11,9 @@ open import foundation.action-on-identifications-functions
 open import foundation.dependent-pair-types
 open import foundation.existential-quantification
 open import foundation.identity-types
+open import foundation.inhabited-subtypes
 open import foundation.powersets
 open import foundation.subtypes
-open import foundation.inhabited-subtypes
 open import foundation.transport-along-identifications
 open import foundation.unital-binary-operations
 open import foundation.universe-levels
@@ -148,7 +148,6 @@ module _
     minkowski-mul-inhabited-is-inhabited-Semigroup (semigroup-Monoid M)
 ```
 
-
 ## See also
 
 - Minkowski multiplication on semigroups is defined in
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index ea819dde6b..dae5ef51d5 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -20,10 +20,10 @@ open import foundation.sets
 open import foundation.subtypes
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import group-theory.semigroups
 open import group-theory.subsets-semigroups
+
+open import logic.functoriality-existential-quantification
 ```
 
 </details>

From f9006be947c95c2221923825e11ebf1c55351921 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 12:22:27 -0800
Subject: [PATCH 18/73] Progress

---
 .../addition-lower-dedekind-real-numbers.lagda.md      | 10 ++++++++--
 1 file changed, 8 insertions(+), 2 deletions(-)

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 03096f38d0..c0aa126a41 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -45,8 +45,14 @@ module _
   is-in-add-lower-ℝ-cut : ℚ → UU (l1 ⊔ l2)
   is-in-add-lower-ℝ-cut = is-in-subtype cut-add-lower-ℝ
 
-  -- is-inhabited-add-lower-ℝ : exists ℚ cut-add-lower-ℝ
-  -- is-inhabited-add-lower-ℝ
+  is-inhabited-add-lower-ℝ : exists ℚ cut-add-lower-ℝ
+  is-inhabited-add-lower-ℝ =
+    minkowski-mul-inhabited-is-inhabited-Commutative-Monoid
+      ( commutative-monoid-add-ℚ)
+      (cut-lower-ℝ x)
+      (cut-lower-ℝ y)
+      {!  is-inhabited-cut-lower-ℝ x !}
+      {!   !}
 ```
 
 ## Properties

From 6ad62d519db2b3740365e3f0695413940e5850f8 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 12:23:46 -0800
Subject: [PATCH 19/73] Rename things

---
 .../lower-dedekind-real-numbers.lagda.md             | 10 +++++-----
 .../upper-dedekind-real-numbers.lagda.md             | 12 +++++++-----
 2 files changed, 12 insertions(+), 10 deletions(-)

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index f603657ab7..6608478238 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -74,13 +74,13 @@ module _
   is-in-cut-lower-ℝ : ℚ → UU l
   is-in-cut-lower-ℝ = is-in-subtype cut-lower-ℝ
 
-  is-inhabited-lower-ℝ : exists ℚ cut-lower-ℝ
-  is-inhabited-lower-ℝ = pr1 (pr2 x)
+  is-inhabited-cut-lower-ℝ : exists ℚ cut-lower-ℝ
+  is-inhabited-cut-lower-ℝ = pr1 (pr2 x)
 
-  is-rounded-lower-ℝ :
+  is-rounded-cut-lower-ℝ :
     (q : ℚ) →
     is-in-cut-lower-ℝ q ↔ exists ℚ (λ r → le-ℚ-Prop q r ∧ cut-lower-ℝ r)
-  is-rounded-lower-ℝ = pr2 (pr2 x)
+  is-rounded-cut-lower-ℝ = pr2 (pr2 x)
 ```
 
 ## Properties
@@ -94,7 +94,7 @@ module _
 
   le-cut-lower-ℝ : le-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
   le-cut-lower-ℝ p<q q<x =
-    backward-implication (is-rounded-lower-ℝ x p) (intro-exists q (p<q , q<x))
+    backward-implication (is-rounded-cut-lower-ℝ x p) (intro-exists q (p<q , q<x))
 ```
 
 ### Two lower real numbers with the same cut are equal
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index bdf0a5525f..c9dbf0f8f4 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -75,13 +75,13 @@ module _
   is-in-cut-upper-ℝ : ℚ → UU l
   is-in-cut-upper-ℝ = is-in-subtype cut-upper-ℝ
 
-  is-inhabited-upper-ℝ : exists ℚ cut-upper-ℝ
-  is-inhabited-upper-ℝ = pr1 (pr2 x)
+  is-inhabited-cut-upper-ℝ : exists ℚ cut-upper-ℝ
+  is-inhabited-cut-upper-ℝ = pr1 (pr2 x)
 
-  is-rounded-upper-ℝ :
+  is-rounded-cut-upper-ℝ :
     (q : ℚ) →
     is-in-cut-upper-ℝ q ↔ exists ℚ (λ p → le-ℚ-Prop p q ∧ cut-upper-ℝ p)
-  is-rounded-upper-ℝ = pr2 (pr2 x)
+  is-rounded-cut-upper-ℝ = pr2 (pr2 x)
 ```
 
 ## Properties
@@ -95,7 +95,9 @@ module _
 
   le-cut-upper-ℝ : le-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
   le-cut-upper-ℝ p<q p<x =
-    backward-implication (is-rounded-upper-ℝ x q) (intro-exists p (p<q , p<x))
+    backward-implication
+      ( is-rounded-cut-upper-ℝ x q)
+      ( intro-exists p (p<q , p<x))
 ```
 
 ### Two upper real numbers with the same cut are equal

From 426df7174b025fe3ec93bc40e677643720019f24 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 13:43:23 -0800
Subject: [PATCH 20/73] Finish defining addition on lower reals

---
 .../positive-rational-numbers.lagda.md        |  28 ++--
 ...trict-inequality-rational-numbers.lagda.md |  32 +++--
 src/group-theory/abelian-groups.lagda.md      |  23 ++--
 ...ition-lower-dedekind-real-numbers.lagda.md | 128 +++++++++++++++++-
 4 files changed, 165 insertions(+), 46 deletions(-)

diff --git a/src/elementary-number-theory/positive-rational-numbers.lagda.md b/src/elementary-number-theory/positive-rational-numbers.lagda.md
index 63ae446cfa..88b15da00e 100644
--- a/src/elementary-number-theory/positive-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/positive-rational-numbers.lagda.md
@@ -190,13 +190,14 @@ module _
   (x y : ℚ) (H : le-ℚ x y)
   where
 
-  is-positive-diff-le-ℚ : is-positive-ℚ (y -ℚ x)
-  is-positive-diff-le-ℚ =
-    is-positive-le-zero-ℚ
-      ( y -ℚ x)
-      ( backward-implication
-        ( iff-translate-diff-le-zero-ℚ x y)
-        ( H))
+  abstract
+    is-positive-diff-le-ℚ : is-positive-ℚ (y -ℚ x)
+    is-positive-diff-le-ℚ =
+      is-positive-le-zero-ℚ
+        ( y -ℚ x)
+        ( backward-implication
+          ( iff-translate-diff-le-zero-ℚ x y)
+          ( H))
 
   positive-diff-le-ℚ : ℚ⁺
   positive-diff-le-ℚ = y -ℚ x , is-positive-diff-le-ℚ
@@ -539,12 +540,13 @@ mediant-zero-ℚ⁺ x =
         ( rational-ℚ⁺ x)
         ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))))
 
-le-mediant-zero-ℚ⁺ : (x : ℚ⁺) → le-ℚ⁺ (mediant-zero-ℚ⁺ x) x
-le-mediant-zero-ℚ⁺ x =
-  le-right-mediant-ℚ
-    ( zero-ℚ)
-    ( rational-ℚ⁺ x)
-    ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))
+abstract
+  le-mediant-zero-ℚ⁺ : (x : ℚ⁺) → le-ℚ⁺ (mediant-zero-ℚ⁺ x) x
+  le-mediant-zero-ℚ⁺ x =
+    le-right-mediant-ℚ
+      ( zero-ℚ)
+      ( rational-ℚ⁺ x)
+      ( le-zero-is-positive-ℚ (rational-ℚ⁺ x) (is-positive-rational-ℚ⁺ x))
 ```
 
 ### Any positive rational number is the sum of two positive rational numbers
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 99d4619055..53e20051b7 100644
--- a/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/strict-inequality-rational-numbers.lagda.md
@@ -122,12 +122,13 @@ module _
   (x y z : ℚ)
   where
 
-  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)
+  abstract
+    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)
 ```
 
 ### Concatenation rules for inequality and strict inequality on the rational numbers
@@ -315,15 +316,16 @@ module _
 ### Addition on the rational numbers preserves strict inequality
 
 ```agda
-preserves-le-add-ℚ :
-  {a b c d : ℚ} → le-ℚ a b → le-ℚ c d → le-ℚ (a +ℚ c) (b +ℚ d)
-preserves-le-add-ℚ {a} {b} {c} {d} H K =
-  transitive-le-ℚ
-    ( a +ℚ c)
-    ( b +ℚ c)
-    ( b +ℚ d)
-    ( preserves-le-right-add-ℚ b c d K)
-    ( preserves-le-left-add-ℚ c a b H)
+abstract
+  preserves-le-add-ℚ :
+    {a b c d : ℚ} → le-ℚ a b → le-ℚ c d → le-ℚ (a +ℚ c) (b +ℚ d)
+  preserves-le-add-ℚ {a} {b} {c} {d} H K =
+    transitive-le-ℚ
+      ( a +ℚ c)
+      ( b +ℚ c)
+      ( b +ℚ d)
+      ( preserves-le-right-add-ℚ b c d K)
+      ( preserves-le-left-add-ℚ c a b H)
 ```
 
 ### The rational numbers have no lower or upper bound
diff --git a/src/group-theory/abelian-groups.lagda.md b/src/group-theory/abelian-groups.lagda.md
index 6023e6276d..7033d0b8db 100644
--- a/src/group-theory/abelian-groups.lagda.md
+++ b/src/group-theory/abelian-groups.lagda.md
@@ -612,17 +612,18 @@ module _
   {l : Level} (A : Ab l)
   where
 
-  is-identity-left-conjugation-Ab :
-    (x : type-Ab A) → left-conjugation-Ab A x ~ id
-  is-identity-left-conjugation-Ab x y =
-    ( ap (add-Ab' A (neg-Ab A x)) (commutative-add-Ab A x y)) ∙
-    ( is-retraction-right-subtraction-Ab A x y)
-
-  is-identity-right-conjugation-Ab :
-    (x : type-Ab A) → right-conjugation-Ab A x ~ id
-  is-identity-right-conjugation-Ab x =
-    inv-htpy (left-right-conjugation-Ab A x) ∙h
-    is-identity-left-conjugation-Ab x
+  abstract
+    is-identity-left-conjugation-Ab :
+      (x : type-Ab A) → left-conjugation-Ab A x ~ id
+    is-identity-left-conjugation-Ab x y =
+      ( ap (add-Ab' A (neg-Ab A x)) (commutative-add-Ab A x y)) ∙
+      ( is-retraction-right-subtraction-Ab A x y)
+
+    is-identity-right-conjugation-Ab :
+      (x : type-Ab A) → right-conjugation-Ab A x ~ id
+    is-identity-right-conjugation-Ab x =
+      inv-htpy (left-right-conjugation-Ab A x) ∙h
+      is-identity-left-conjugation-Ab x
 
   is-identity-conjugation-Ab :
     (x : type-Ab A) → conjugation-Ab A x ~ id
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 c0aa126a41..573b1e9ec3 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -1,6 +1,8 @@
 # Addition of lower Dedekind real numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module real-numbers.addition-lower-dedekind-real-numbers where
 ```
 
@@ -9,13 +11,27 @@ module real-numbers.addition-lower-dedekind-real-numbers where
 ```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.dependent-pair-types
 open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.logical-equivalences
 open import foundation.subtypes
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import logic.functoriality-existential-quantification
+
+open import group-theory.abelian-groups
+open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 
 open import real-numbers.lower-dedekind-real-numbers
@@ -42,17 +58,111 @@ module _
       ( cut-lower-ℝ x)
       ( cut-lower-ℝ y)
 
-  is-in-add-lower-ℝ-cut : ℚ → UU (l1 ⊔ l2)
-  is-in-add-lower-ℝ-cut = is-in-subtype cut-add-lower-ℝ
+  is-in-cut-add-lower-ℝ : ℚ → UU (l1 ⊔ l2)
+  is-in-cut-add-lower-ℝ = is-in-subtype cut-add-lower-ℝ
 
   is-inhabited-add-lower-ℝ : exists ℚ cut-add-lower-ℝ
   is-inhabited-add-lower-ℝ =
     minkowski-mul-inhabited-is-inhabited-Commutative-Monoid
       ( commutative-monoid-add-ℚ)
-      (cut-lower-ℝ x)
-      (cut-lower-ℝ y)
-      {!  is-inhabited-cut-lower-ℝ x !}
-      {!   !}
+      ( cut-lower-ℝ x)
+      ( cut-lower-ℝ y)
+      ( is-inhabited-cut-lower-ℝ x)
+      ( is-inhabited-cut-lower-ℝ y)
+
+  abstract
+    is-rounded-cut-add-lower-ℝ :
+      (q : ℚ) →
+      is-in-cut-add-lower-ℝ q ↔
+      exists ℚ (λ r → le-ℚ-Prop q r ∧ cut-add-lower-ℝ r)
+    pr1 (is-rounded-cut-add-lower-ℝ q) =
+      elim-exists
+        ( ∃ ℚ (λ r → le-ℚ-Prop q r ∧ cut-add-lower-ℝ r))
+        ( λ (lx , ly) (lx<x , ly<y , q=lx+ly) →
+          map-binary-exists
+            ( λ r → le-ℚ q r × is-in-cut-add-lower-ℝ r)
+            ( add-ℚ)
+            ( λ lx' ly' (lx<lx' , lx'<x) (ly<ly' , ly'<y) →
+              tr
+                ( λ p → le-ℚ p (lx' +ℚ ly'))
+                ( inv q=lx+ly)
+                ( preserves-le-add-ℚ {lx} {lx'} {ly} {ly'} lx<lx' ly<ly') ,
+              intro-exists (lx' , ly') (lx'<x , ly'<y , refl))
+            ( forward-implication (is-rounded-cut-lower-ℝ x lx) lx<x)
+            ( forward-implication (is-rounded-cut-lower-ℝ y ly) ly<y))
+    pr2 (is-rounded-cut-add-lower-ℝ q) =
+      elim-exists
+        ( cut-add-lower-ℝ q)
+        ( λ r (q<r , r<x+y) →
+          elim-exists
+            ( cut-add-lower-ℝ q)
+            ( λ (rx , ry) (rx<x , ry<y , r=rx+ry) →
+              let
+                r-q⁺ : ℚ⁺
+                r-q⁺ = positive-diff-le-ℚ q r q<r
+                ε⁺ : ℚ⁺
+                ε⁺ = mediant-zero-ℚ⁺ r-q⁺
+                ε : ℚ
+                ε = rational-ℚ⁺ ε⁺
+                ε<r-q : le-ℚ ε (r -ℚ q)
+                ε<r-q = le-mediant-zero-ℚ⁺ r-q⁺
+              in
+              intro-exists
+                ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
+                ( le-cut-lower-ℝ
+                    ( x)
+                    ( rx -ℚ ε)
+                    ( rx)
+                    ( le-diff-rational-ℚ⁺ rx ε⁺)
+                    ( rx<x) ,
+                  le-cut-lower-ℝ
+                    ( y)
+                    ( q -ℚ (rx -ℚ ε))
+                    ( ry)
+                    ( binary-tr
+                        ( le-ℚ)
+                        ( equational-reasoning
+                          (q -ℚ rx) +ℚ ε
+                          ＝ q +ℚ (neg-ℚ rx +ℚ ε)
+                            by associative-add-ℚ q (neg-ℚ rx) ε
+                          ＝ q +ℚ (ε -ℚ rx)
+                            by ap (q +ℚ_) (commutative-add-ℚ (neg-ℚ rx) ε)
+                          ＝ q -ℚ (rx -ℚ ε)
+                            by ap (q +ℚ_) (inv (distributive-neg-diff-ℚ rx ε)))
+                        ( equational-reasoning
+                          (q -ℚ rx) +ℚ (r -ℚ q)
+                          ＝ (q -ℚ rx) +ℚ (neg-ℚ q +ℚ r)
+                            by
+                              ap ((q -ℚ rx) +ℚ_) (commutative-add-ℚ r (neg-ℚ q))
+                          ＝ (q -ℚ q) +ℚ (neg-ℚ rx +ℚ r)
+                            by
+                              interchange-law-add-add-ℚ q (neg-ℚ rx) (neg-ℚ q) r
+                          ＝ zero-ℚ +ℚ (neg-ℚ rx +ℚ r)
+                            by
+                              ap
+                                ( _+ℚ (neg-ℚ rx +ℚ r))
+                                ( right-inverse-law-add-ℚ q)
+                          ＝ neg-ℚ rx +ℚ r by left-unit-law-add-ℚ (neg-ℚ rx +ℚ r)
+                          ＝ neg-ℚ rx +ℚ (rx +ℚ ry) by ap (neg-ℚ rx +ℚ_) r=rx+ry
+                          ＝ ry
+                            by is-retraction-left-div-Group group-add-ℚ rx ry)
+                        ( preserves-le-right-add-ℚ
+                          ( q -ℚ rx)
+                          ( ε)
+                          ( r -ℚ q)
+                          ( ε<r-q)))
+                    ( ry<y) ,
+                  inv
+                    ( is-identity-right-conjugation-Ab
+                      ( abelian-group-add-ℚ)
+                      ( rx -ℚ ε)
+                      ( q))))
+            ( r<x+y))
+
+    add-lower-ℝ : lower-ℝ (l1 ⊔ l2)
+    pr1 add-lower-ℝ = cut-add-lower-ℝ
+    pr1 (pr2 add-lower-ℝ) = is-inhabited-add-lower-ℝ
+    pr2 (pr2 add-lower-ℝ) = is-rounded-cut-add-lower-ℝ
 ```
 
 ## Properties
@@ -60,5 +170,9 @@ module _
 ### Addition of lower Dedekind real numbers is commutative
 
 ```agda
--- commutative-add-lower-ℝ :
+module _
+  {l1 l2 : Level} (x : lower-ℝ l1) (y : lower-ℝ l2)
+  where
+
+--   commutative-add-lower-ℝ :
 ```

From 1875b506421ed75253c707629504bf8d347243c6 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 13:44:27 -0800
Subject: [PATCH 21/73] Fix naming

---
 .../addition-lower-dedekind-real-numbers.lagda.md           | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

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 573b1e9ec3..6ec4f92d57 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -61,8 +61,8 @@ module _
   is-in-cut-add-lower-ℝ : ℚ → UU (l1 ⊔ l2)
   is-in-cut-add-lower-ℝ = is-in-subtype cut-add-lower-ℝ
 
-  is-inhabited-add-lower-ℝ : exists ℚ cut-add-lower-ℝ
-  is-inhabited-add-lower-ℝ =
+  is-inhabited-cut-add-lower-ℝ : exists ℚ cut-add-lower-ℝ
+  is-inhabited-cut-add-lower-ℝ =
     minkowski-mul-inhabited-is-inhabited-Commutative-Monoid
       ( commutative-monoid-add-ℚ)
       ( cut-lower-ℝ x)
@@ -161,7 +161,7 @@ module _
 
     add-lower-ℝ : lower-ℝ (l1 ⊔ l2)
     pr1 add-lower-ℝ = cut-add-lower-ℝ
-    pr1 (pr2 add-lower-ℝ) = is-inhabited-add-lower-ℝ
+    pr1 (pr2 add-lower-ℝ) = is-inhabited-cut-add-lower-ℝ
     pr2 (pr2 add-lower-ℝ) = is-rounded-cut-add-lower-ℝ
 ```
 

From d7f6e2d518fec56f063419e9c8cefb022ce51c8a Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:09:53 -0800
Subject: [PATCH 22/73] Similarity and containment

---
 ...ultiplication-commutative-monoids.lagda.md | 80 +++++++++++++++++++
 .../minkowski-multiplication-monoids.lagda.md | 66 +++++++++++++++
 ...nkowski-multiplication-semigroups.lagda.md | 75 +++++++++++++++++
 3 files changed, 221 insertions(+)

diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
index 1e0ad307f4..8956fa2608 100644
--- a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -124,6 +124,86 @@ module _
     minkowski-mul-inhabited-is-inhabited-Monoid (monoid-Commutative-Monoid M)
 ```
 
+### Containment is preserved by Minkowski multiplication of monoid subsets
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (M : Commutative-Monoid l1)
+  (B : subset-Commutative-Monoid l2 M)
+  (A : subset-Commutative-Monoid l3 M)
+  (A' : subset-Commutative-Monoid l4 M)
+  where
+
+  preserves-leq-left-minkowski-mul-Commutative-Monoid :
+    A ⊆ A' →
+    minkowski-mul-Commutative-Monoid M A B ⊆
+    minkowski-mul-Commutative-Monoid M A' B
+  preserves-leq-left-minkowski-mul-Commutative-Monoid =
+    preserves-leq-left-minkowski-mul-Monoid (monoid-Commutative-Monoid M) B A A'
+
+  preserves-leq-right-minkowski-mul-Commutative-Monoid :
+    A ⊆ A' →
+    minkowski-mul-Commutative-Monoid M B A ⊆
+    minkowski-mul-Commutative-Monoid M B A'
+  preserves-leq-right-minkowski-mul-Commutative-Monoid =
+    preserves-leq-right-minkowski-mul-Monoid
+      ( monoid-Commutative-Monoid M)
+      ( B)
+      ( A)
+      ( A')
+```
+
+### Similarity is preserved by Minkowski multiplication of monoid subsets
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (M : Commutative-Monoid l1)
+  (B : subset-Commutative-Monoid l2 M)
+  (A : subset-Commutative-Monoid l3 M)
+  (A' : subset-Commutative-Monoid l4 M)
+  where
+
+  preserves-sim-left-minkowski-mul-Commutative-Monoid :
+    sim-subtype A A' →
+    sim-subtype
+      ( minkowski-mul-Commutative-Monoid M A B)
+      ( minkowski-mul-Commutative-Monoid M A' B)
+  preserves-sim-left-minkowski-mul-Commutative-Monoid =
+    preserves-sim-left-minkowski-mul-Monoid (monoid-Commutative-Monoid M) B A A'
+
+  preserves-sim-right-minkowski-mul-Commutative-Monoid :
+    sim-subtype A A' →
+    sim-subtype
+      ( minkowski-mul-Commutative-Monoid M B A)
+      ( minkowski-mul-Commutative-Monoid M B A')
+  preserves-sim-right-minkowski-mul-Commutative-Monoid =
+    preserves-sim-right-minkowski-mul-Monoid
+      ( monoid-Commutative-Monoid M)
+      ( B)
+      ( A)
+      ( A')
+
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (M : Commutative-Monoid l1)
+  (A : subset-Commutative-Monoid l2 M)
+  (A' : subset-Commutative-Monoid l3 M)
+  (B : subset-Commutative-Monoid l4 M)
+  (B' : subset-Commutative-Monoid l5 M)
+  where
+
+  preserves-sim-minkowski-mul-Commutative-Monoid :
+    sim-subtype A A' →
+    sim-subtype B B' →
+    sim-subtype
+      ( minkowski-mul-Commutative-Monoid M A B)
+      ( minkowski-mul-Commutative-Monoid M A' B')
+  preserves-sim-minkowski-mul-Commutative-Monoid =
+    preserves-sim-minkowski-mul-Monoid (monoid-Commutative-Monoid M) A A' B B'
+```
+
 ## See also
 
 - Minkowski multiplication on semigroups is defined in
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
index b2e67b3464..a9302b06e3 100644
--- a/src/group-theory/minkowski-multiplication-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -148,10 +148,76 @@ module _
     minkowski-mul-inhabited-is-inhabited-Semigroup (semigroup-Monoid M)
 ```
 
+### Containment is preserved by Minkowski multiplication of monoid subsets
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (M : Monoid l1)
+  (B : subset-Monoid l2 M)
+  (A : subset-Monoid l3 M)
+  (A' : subset-Monoid l4 M)
+  where
+
+  preserves-leq-left-minkowski-mul-Monoid :
+    A ⊆ A' → minkowski-mul-Monoid M A B ⊆ minkowski-mul-Monoid M A' B
+  preserves-leq-left-minkowski-mul-Monoid =
+    preserves-leq-left-minkowski-mul-Semigroup (semigroup-Monoid M) B A A'
+
+  preserves-leq-right-minkowski-mul-Monoid :
+    A ⊆ A' → minkowski-mul-Monoid M B A ⊆ minkowski-mul-Monoid M B A'
+  preserves-leq-right-minkowski-mul-Monoid =
+    preserves-leq-right-minkowski-mul-Semigroup (semigroup-Monoid M) B A A'
+```
+
+### Similarity is preserved by Minkowski multiplication of monoid subsets
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (M : Monoid l1)
+  (B : subset-Monoid l2 M)
+  (A : subset-Monoid l3 M)
+  (A' : subset-Monoid l4 M)
+  where
+
+  preserves-sim-left-minkowski-mul-Monoid :
+    sim-subtype A A' →
+    sim-subtype (minkowski-mul-Monoid M A B) (minkowski-mul-Monoid M A' B)
+  preserves-sim-left-minkowski-mul-Monoid =
+    preserves-sim-left-minkowski-mul-Semigroup (semigroup-Monoid M) B A A'
+
+  preserves-sim-right-minkowski-mul-Monoid :
+    sim-subtype A A' →
+    sim-subtype (minkowski-mul-Monoid M B A) (minkowski-mul-Monoid M B A')
+  preserves-sim-right-minkowski-mul-Monoid =
+    preserves-sim-right-minkowski-mul-Semigroup (semigroup-Monoid M) B A A'
+
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (M : Monoid l1)
+  (A : subset-Monoid l2 M)
+  (A' : subset-Monoid l3 M)
+  (B : subset-Monoid l4 M)
+  (B' : subset-Monoid l5 M)
+  where
+
+  preserves-sim-minkowski-mul-Monoid :
+    sim-subtype A A' →
+    sim-subtype B B' →
+    sim-subtype
+      ( minkowski-mul-Monoid M A B)
+      ( minkowski-mul-Monoid M A' B')
+  preserves-sim-minkowski-mul-Monoid =
+    preserves-sim-minkowski-mul-Semigroup (semigroup-Monoid M) A A' B B'
+```
+
 ## See also
 
 - Minkowski multiplication on semigroups is defined in
   [`group-theory.minkowski-multiplication-semigroups`](group-theory.minkowski-multiplication-semigroups.md).
+- Minkowski multiplication on commutative monoids is defined in
+  [`group-theory.minkowski-multiplication-commutative-monoids`](group-theory.minkowski-multiplication-commutative-monoids.md).
 
 ## External links
 
diff --git a/src/group-theory/minkowski-multiplication-semigroups.lagda.md b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
index dae5ef51d5..8b233f4d0e 100644
--- a/src/group-theory/minkowski-multiplication-semigroups.lagda.md
+++ b/src/group-theory/minkowski-multiplication-semigroups.lagda.md
@@ -12,6 +12,8 @@ open import foundation.cartesian-product-types
 open import foundation.conjunction
 open import foundation.dependent-pair-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.inhabited-subtypes
 open import foundation.powersets
@@ -163,6 +165,79 @@ module _
       λ a b a∈A b∈B → intro-exists (a , b) (a∈A , b∈B , refl)
 ```
 
+### Containment is preserved by Minkowski multiplication of semigroup subtypes
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (G : Semigroup l1)
+  (B : subset-Semigroup l2 G)
+  (A : subset-Semigroup l3 G)
+  (A' : subset-Semigroup l4 G)
+  where
+
+  preserves-leq-left-minkowski-mul-Semigroup :
+    A ⊆ A' → minkowski-mul-Semigroup G A B ⊆ minkowski-mul-Semigroup G A' B
+  preserves-leq-left-minkowski-mul-Semigroup A⊆A' x =
+    map-tot-exists (λ (a , b) → map-product (A⊆A' a) id)
+
+  preserves-leq-right-minkowski-mul-Semigroup :
+    A ⊆ A' → minkowski-mul-Semigroup G B A ⊆ minkowski-mul-Semigroup G B A'
+  preserves-leq-right-minkowski-mul-Semigroup A⊆A' x =
+    map-tot-exists (λ (b , a) → map-product id (map-product (A⊆A' a) id))
+```
+
+### Similarity is preserved by Minkowski multiplication of semigroup subtypes
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (G : Semigroup l1)
+  (B : subset-Semigroup l2 G)
+  (A : subset-Semigroup l3 G)
+  (A' : subset-Semigroup l4 G)
+  where
+
+  preserves-sim-left-minkowski-mul-Semigroup :
+    sim-subtype A A' →
+    sim-subtype (minkowski-mul-Semigroup G A B) (minkowski-mul-Semigroup G A' B)
+  pr1 (preserves-sim-left-minkowski-mul-Semigroup (A⊆A' , _)) =
+    preserves-leq-left-minkowski-mul-Semigroup G B A A' A⊆A'
+  pr2 (preserves-sim-left-minkowski-mul-Semigroup (_ , A'⊆A)) =
+    preserves-leq-left-minkowski-mul-Semigroup G B A' A A'⊆A
+
+  preserves-sim-right-minkowski-mul-Semigroup :
+    sim-subtype A A' →
+    sim-subtype (minkowski-mul-Semigroup G B A) (minkowski-mul-Semigroup G B A')
+  pr1 (preserves-sim-right-minkowski-mul-Semigroup (A⊆A' , _)) =
+    preserves-leq-right-minkowski-mul-Semigroup G B A A' A⊆A'
+  pr2 (preserves-sim-right-minkowski-mul-Semigroup (_ , A'⊆A)) =
+    preserves-leq-right-minkowski-mul-Semigroup G B A' A A'⊆A
+
+module _
+  {l1 l2 l3 l4 l5 : Level}
+  (G : Semigroup l1)
+  (A : subset-Semigroup l2 G)
+  (A' : subset-Semigroup l3 G)
+  (B : subset-Semigroup l4 G)
+  (B' : subset-Semigroup l5 G)
+  where
+
+  preserves-sim-minkowski-mul-Semigroup :
+    sim-subtype A A' →
+    sim-subtype B B' →
+    sim-subtype
+      ( minkowski-mul-Semigroup G A B)
+      ( minkowski-mul-Semigroup G A' B')
+  preserves-sim-minkowski-mul-Semigroup A≈A' B≈B' =
+    transitive-sim-subtype
+      ( minkowski-mul-Semigroup G A B)
+      ( minkowski-mul-Semigroup G A B')
+      ( minkowski-mul-Semigroup G A' B')
+      ( preserves-sim-left-minkowski-mul-Semigroup G B' A A' A≈A')
+      ( preserves-sim-right-minkowski-mul-Semigroup G A B B' B≈B')
+```
+
 ## External links
 
 - [Minkowski addition](https://en.wikipedia.org/wiki/Minkowski_addition) at

From fd4d1929dfe44d5f950237950629e2c22f37affe Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:22:00 -0800
Subject: [PATCH 23/73] Progress

---
 ...ition-lower-dedekind-real-numbers.lagda.md | 32 ++++++++++++++++---
 1 file changed, 27 insertions(+), 5 deletions(-)

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 6ec4f92d57..40fcfc78da 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -159,10 +159,10 @@ module _
                       ( q))))
             ( r<x+y))
 
-    add-lower-ℝ : lower-ℝ (l1 ⊔ l2)
-    pr1 add-lower-ℝ = cut-add-lower-ℝ
-    pr1 (pr2 add-lower-ℝ) = is-inhabited-cut-add-lower-ℝ
-    pr2 (pr2 add-lower-ℝ) = is-rounded-cut-add-lower-ℝ
+  add-lower-ℝ : lower-ℝ (l1 ⊔ l2)
+  pr1 add-lower-ℝ = cut-add-lower-ℝ
+  pr1 (pr2 add-lower-ℝ) = is-inhabited-cut-add-lower-ℝ
+  pr2 (pr2 add-lower-ℝ) = is-rounded-cut-add-lower-ℝ
 ```
 
 ## Properties
@@ -174,5 +174,27 @@ module _
   {l1 l2 : Level} (x : lower-ℝ l1) (y : lower-ℝ l2)
   where
 
---   commutative-add-lower-ℝ :
+  commutative-add-lower-ℝ : add-lower-ℝ x y ＝ add-lower-ℝ y x
+  commutative-add-lower-ℝ =
+    eq-eq-cut-lower-ℝ
+      ( add-lower-ℝ x y)
+      ( add-lower-ℝ y x)
+      ( commutative-minkowski-mul-Commutative-Monoid
+        ( commutative-monoid-add-ℚ)
+        ( cut-lower-ℝ x)
+        ( cut-lower-ℝ y))
+```
+
+### Addition of lower Dedekind real numbers is associative
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : lower-ℝ l1) (y : lower-ℝ l2) (z : lower-ℝ l3)
+  where
+{-
+  associative-add-lower-ℝ :
+    add-lower-ℝ (add-lower-ℝ x y) z ＝ add-lower-ℝ x (add-lower-ℝ y z)
+  associative-add-lower-ℝ =
+    eq-eq-cut-lower-ℝ
+      --(add-lower-ℝ (add-lower-ℝ  -}
 ```

From ca7ab8b6abb4e8c2f0273511bf159c92de70ec1e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:30:13 -0800
Subject: [PATCH 24/73] Propagate properties

---
 ...ultiplication-commutative-monoids.lagda.md | 69 +++++++++++++++++++
 .../minkowski-multiplication-monoids.lagda.md | 18 +++++
 2 files changed, 87 insertions(+)

diff --git a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
index 8956fa2608..ad3624e6ed 100644
--- a/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-commutative-monoids.lagda.md
@@ -13,6 +13,7 @@ open import foundation.identity-types
 open import foundation.inhabited-subtypes
 open import foundation.powersets
 open import foundation.subtypes
+open import foundation.unital-binary-operations
 open import foundation.universe-levels
 
 open import group-theory.commutative-monoids
@@ -50,6 +51,74 @@ module _
 
 ## Properties
 
+### Minkowski multiplication of commutative monoid subsets is associative
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (M : Commutative-Monoid l1)
+  (A : subset-Commutative-Monoid l2 M)
+  (B : subset-Commutative-Monoid l3 M)
+  (C : subset-Commutative-Monoid l4 M)
+  where
+
+  associative-minkowski-mul-Commutative-Monoid :
+    minkowski-mul-Commutative-Monoid
+      ( M)
+      ( minkowski-mul-Commutative-Monoid M A B)
+      ( C) ＝
+    minkowski-mul-Commutative-Monoid
+      ( M)
+      ( A)
+      ( minkowski-mul-Commutative-Monoid M B C)
+  associative-minkowski-mul-Commutative-Monoid =
+    associative-minkowski-mul-Monoid (monoid-Commutative-Monoid M) A B C
+```
+
+### Minkowski multiplication of commutative monoid subsets is unital
+
+```agda
+module _
+  {l1 l2 : Level}
+  (M : Commutative-Monoid l1)
+  (A : subset-Commutative-Monoid l2 M)
+  where
+
+  left-unit-law-minkowski-mul-Commutative-Monoid :
+    sim-subtype
+      ( minkowski-mul-Commutative-Monoid
+        ( M)
+        ( is-unit-prop-Commutative-Monoid M)
+        ( A))
+      ( A)
+  left-unit-law-minkowski-mul-Commutative-Monoid =
+    left-unit-law-minkowski-mul-Monoid (monoid-Commutative-Monoid M) A
+
+  right-unit-law-minkowski-mul-Commutative-Monoid :
+    sim-subtype
+      ( minkowski-mul-Commutative-Monoid
+        ( M)
+        ( A)
+        ( is-unit-prop-Commutative-Monoid M))
+      ( A)
+  right-unit-law-minkowski-mul-Commutative-Monoid =
+    right-unit-law-minkowski-mul-Monoid (monoid-Commutative-Monoid M) A
+```
+
+### Minkowski multiplication on a commutative monoid is unital
+
+```agda
+module _
+  {l : Level}
+  (M : Commutative-Monoid l)
+  where
+
+  is-unital-minkowski-mul-Commutative-Monoid :
+    is-unital (minkowski-mul-Commutative-Monoid {l} {l} {l} M)
+  is-unital-minkowski-mul-Commutative-Monoid =
+    is-unital-minkowski-mul-Monoid (monoid-Commutative-Monoid M)
+```
+
 ### Minkowski multiplication on a commutative monoid is commutative
 
 ```agda
diff --git a/src/group-theory/minkowski-multiplication-monoids.lagda.md b/src/group-theory/minkowski-multiplication-monoids.lagda.md
index a9302b06e3..b9f33264cf 100644
--- a/src/group-theory/minkowski-multiplication-monoids.lagda.md
+++ b/src/group-theory/minkowski-multiplication-monoids.lagda.md
@@ -51,6 +51,24 @@ module _
 
 ## Properties
 
+### Minkowski multiplication of monoid subsets is associative
+
+```agda
+module _
+  {l1 l2 l3 l4 : Level}
+  (M : Monoid l1)
+  (A : subset-Monoid l2 M)
+  (B : subset-Monoid l3 M)
+  (C : subset-Monoid l4 M)
+  where
+
+  associative-minkowski-mul-Monoid :
+    minkowski-mul-Monoid M (minkowski-mul-Monoid M A B) C ＝
+    minkowski-mul-Monoid M A (minkowski-mul-Monoid M B C)
+  associative-minkowski-mul-Monoid =
+    associative-minkowski-mul-Semigroup (semigroup-Monoid M) A B C
+```
+
 ### Unit laws for Minkowski multiplication of monoid subsets
 
 ```agda

From 1a5f82f4610bac227d74f89f2b4b638ba4b598fb Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:32:24 -0800
Subject: [PATCH 25/73] Addition on lower reals is associative and commutative

---
 .../addition-lower-dedekind-real-numbers.lagda.md      | 10 ++++++++--
 1 file changed, 8 insertions(+), 2 deletions(-)

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 40fcfc78da..c45c75d99b 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -191,10 +191,16 @@ module _
 module _
   {l1 l2 l3 : Level} (x : lower-ℝ l1) (y : lower-ℝ l2) (z : lower-ℝ l3)
   where
-{-
+
   associative-add-lower-ℝ :
     add-lower-ℝ (add-lower-ℝ x y) z ＝ add-lower-ℝ x (add-lower-ℝ y z)
   associative-add-lower-ℝ =
     eq-eq-cut-lower-ℝ
-      --(add-lower-ℝ (add-lower-ℝ  -}
+      ( add-lower-ℝ (add-lower-ℝ x y) z)
+      ( add-lower-ℝ x (add-lower-ℝ y z))
+      ( associative-minkowski-mul-Commutative-Monoid
+        ( commutative-monoid-add-ℚ)
+        ( cut-lower-ℝ x)
+        ( cut-lower-ℝ y)
+        ( cut-lower-ℝ z))
 ```

From 6bfb61920a9444ea881df44ddd4b2824e9b038dc Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:33:30 -0800
Subject: [PATCH 26/73] Formatting

---
 src/real-numbers/lower-dedekind-real-numbers.lagda.md | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 6608478238..167a6805a5 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -22,6 +22,7 @@ open import foundation.universe-levels
 ```
 
 </details>
+
 ## Idea
 
 A lower

From fd8f3846dafbef038616b7b992081d60ec276300 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:44:51 -0800
Subject: [PATCH 27/73] Add new file

---
 ...ional-lower-dedekind-real-numbers.lagda.md | 59 +++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100644 src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..4c9fca0bf0
--- /dev/null
+++ b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
@@ -0,0 +1,59 @@
+# Rational lower Dedekind real numbers
+
+```agda
+module real-numbers.rational-lower-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.conjunction
+open import foundation.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.logical-equivalences
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import real-numbers.lower-dedekind-real-numbers
+```
+
+</details>
+
+## Idea
+
+There is a canonical mapping from the [rational numbers](elementary-number-theory.rational-numbers.md)
+to the [lower Dedekind real numbers](real-numbers.lower-dedekind-real-numbers.md).
+
+## Definition
+
+```agda
+module _ (q : ℚ) where
+  cut-lower-real-ℚ : subtype lzero ℚ
+  cut-lower-real-ℚ p = le-ℚ-Prop p q
+
+  is-in-cut-lower-real-ℚ : ℚ → UU lzero
+  is-in-cut-lower-real-ℚ p = le-ℚ p q
+
+  is-inhabited-cut-lower-real-ℚ : exists ℚ cut-lower-real-ℚ
+  is-inhabited-cut-lower-real-ℚ = exists-lesser-ℚ q
+
+  is-rounded-cut-lower-real-ℚ :
+    (p : ℚ) →
+    le-ℚ p q ↔ exists ℚ (λ r → le-ℚ-Prop p r ∧ le-ℚ-Prop r q)
+  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)
+  pr2 (is-rounded-cut-lower-real-ℚ p) =
+    elim-exists
+      ( le-ℚ-Prop p q)
+      ( λ r (p<r , r<q) → transitive-le-ℚ p r q r<q p<r)
+
+  lower-real-ℚ : lower-ℝ lzero
+  pr1 lower-real-ℚ = cut-lower-real-ℚ
+  pr1 (pr2 lower-real-ℚ) = is-inhabited-cut-lower-real-ℚ
+  pr2 (pr2 lower-real-ℚ) = is-rounded-cut-lower-real-ℚ
+```

From fb0d5d35c9a83ac94af5695e66f14e63e9c016c6 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:45:17 -0800
Subject: [PATCH 28/73] Fix line length

---
 src/real-numbers/lower-dedekind-real-numbers.lagda.md | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 167a6805a5..8d9097a7ec 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -95,7 +95,9 @@ module _
 
   le-cut-lower-ℝ : le-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
   le-cut-lower-ℝ p<q q<x =
-    backward-implication (is-rounded-cut-lower-ℝ x p) (intro-exists q (p<q , q<x))
+    backward-implication
+      ( is-rounded-cut-lower-ℝ x p)
+      ( intro-exists q (p<q , q<x))
 ```
 
 ### Two lower real numbers with the same cut are equal

From f21b948451b0b13686a0212506d373525eb73404 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 14:49:24 -0800
Subject: [PATCH 29/73] Add rational upper reals

---
 ...ional-upper-dedekind-real-numbers.lagda.md | 59 +++++++++++++++++++
 1 file changed, 59 insertions(+)
 create mode 100644 src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..c6e1c589de
--- /dev/null
+++ b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
@@ -0,0 +1,59 @@
+# Rational upper Dedekind real numbers
+
+```agda
+module real-numbers.rational-upper-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.rational-numbers
+open import elementary-number-theory.strict-inequality-rational-numbers
+
+open import foundation.conjunction
+open import foundation.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.logical-equivalences
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import real-numbers.upper-dedekind-real-numbers
+```
+
+</details>
+
+## Idea
+
+There is a canonical mapping from the [rational numbers](elementary-number-theory.rational-numbers.md)
+to the [upper Dedekind real numbers](real-numbers.upper-dedekind-real-numbers.md).
+
+## Definition
+
+```agda
+module _ (q : ℚ) where
+  cut-upper-real-ℚ : subtype lzero ℚ
+  cut-upper-real-ℚ = le-ℚ-Prop q
+
+  is-in-cut-upper-real-ℚ : ℚ → UU lzero
+  is-in-cut-upper-real-ℚ = le-ℚ q
+
+  is-inhabited-cut-upper-real-ℚ : exists ℚ cut-upper-real-ℚ
+  is-inhabited-cut-upper-real-ℚ = exists-greater-ℚ q
+
+  is-rounded-cut-upper-real-ℚ :
+    (p : ℚ) →
+    le-ℚ q p ↔ exists ℚ (λ r → le-ℚ-Prop r p ∧ le-ℚ-Prop q r)
+  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)
+  pr2 (is-rounded-cut-upper-real-ℚ p) =
+    elim-exists
+      ( le-ℚ-Prop q p)
+      ( λ r (r<p , q<r) → transitive-le-ℚ q r p r<p q<r)
+
+  upper-real-ℚ : upper-ℝ lzero
+  pr1 upper-real-ℚ = cut-upper-real-ℚ
+  pr1 (pr2 upper-real-ℚ) = is-inhabited-cut-upper-real-ℚ
+  pr2 (pr2 upper-real-ℚ) = is-rounded-cut-upper-real-ℚ
+```

From a3d3af983f0b5974cffa53ccb31b3b933bb59a77 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 15:00:35 -0800
Subject: [PATCH 30/73] Progress

---
 ...ality-lower-dedekind-real-numbers.lagda.md | 56 +++++++++++++++++++
 1 file changed, 56 insertions(+)
 create mode 100644 src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..4980bd1fdb
--- /dev/null
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -0,0 +1,56 @@
+# Inequality on the lower Dedekind real numbers
+
+```agda
+module real-numbers.inequality-lower-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.rational-numbers
+
+open import foundation.powersets
+open import foundation.propositions
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import order-theory.large-posets
+open import order-theory.large-preorders
+
+open import real-numbers.lower-dedekind-real-numbers
+```
+
+</details>
+
+## Idea
+
+The {{#concept "standard ordering" Disambiguation="lower Dedekind real numbers" Agda=leq-lower-ℝ}} on
+the [lower real numbers](real-numbers.lower-dedekind-real-numbers.md) is defined as the
+cut of one being a subset of the cut of the other.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (x : lower-ℝ l1)
+  (y : lower-ℝ l2)
+  where
+
+  leq-lower-ℝ-Prop : Prop (l1 ⊔ l2)
+  leq-lower-ℝ-Prop = leq-prop-subtype (cut-lower-ℝ x) (cut-lower-ℝ y)
+```
+
+### Inequality on lower Dedekind reals is a large poset
+
+```agda
+lower-ℝ-Large-Preorder : Large-Preorder lsuc _⊔_
+lower-ℝ-Large-Preorder = powerset-Large-Preorder ℚ
+
+lower-ℝ-Large-Poset : Large-Poset lsuc _⊔_
+lower-ℝ-Large-Poset = powerset-Large-Poset ℚ
+```
+
+## References
+
+{{#bibliography}}

From fb18a45e9d71a0dc482e958b7bee3aba7dfd9750 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 15:09:04 -0800
Subject: [PATCH 31/73] Preservation of inequality

---
 ...ality-lower-dedekind-real-numbers.lagda.md | 31 +++++++++++++++++++
 1 file changed, 31 insertions(+)

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 4980bd1fdb..52dd94fe52 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -7,8 +7,14 @@ module real-numbers.inequality-lower-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequality-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.logical-equivalences
 open import foundation.powersets
 open import foundation.propositions
 open import foundation.subtypes
@@ -18,6 +24,7 @@ open import order-theory.large-posets
 open import order-theory.large-preorders
 
 open import real-numbers.lower-dedekind-real-numbers
+open import real-numbers.rational-lower-dedekind-real-numbers
 ```
 
 </details>
@@ -39,8 +46,13 @@ module _
 
   leq-lower-ℝ-Prop : Prop (l1 ⊔ l2)
   leq-lower-ℝ-Prop = leq-prop-subtype (cut-lower-ℝ x) (cut-lower-ℝ y)
+
+  leq-lower-ℝ : UU (l1 ⊔ l2)
+  leq-lower-ℝ = type-Prop leq-lower-ℝ-Prop
 ```
 
+## Properties
+
 ### Inequality on lower Dedekind reals is a large poset
 
 ```agda
@@ -51,6 +63,25 @@ lower-ℝ-Large-Poset : Large-Poset lsuc _⊔_
 lower-ℝ-Large-Poset = powerset-Large-Poset ℚ
 ```
 
+### The canonical map from the rational numbers to the lower reals preserves inequality
+
+```agda
+preserves-leq-lower-real-ℚ :
+  (p q : ℚ) → leq-ℚ p q → leq-lower-ℝ (lower-real-ℚ p) (lower-real-ℚ q)
+preserves-leq-lower-real-ℚ p q p≤q r r<p = concatenate-le-leq-ℚ r p q r<p p≤q
+
+reflects-leq-lower-real-ℚ :
+  (p q : ℚ) → leq-lower-ℝ (lower-real-ℚ p) (lower-real-ℚ q) → leq-ℚ p q
+reflects-leq-lower-real-ℚ p q r<p→r<q with decide-le-leq-ℚ q p
+... | inr p≤q = p≤q
+... | inl q<p = ex-falso (irreflexive-le-ℚ q (r<p→r<q q q<p))
+
+iff-leq-lower-real-ℚ :
+  (p q : ℚ) → leq-ℚ p q ↔ leq-lower-ℝ (lower-real-ℚ p) (lower-real-ℚ q)
+pr1 (iff-leq-lower-real-ℚ p q) = preserves-leq-lower-real-ℚ p q
+pr2 (iff-leq-lower-real-ℚ p q) = reflects-leq-lower-real-ℚ p q
+```
+
 ## References
 
 {{#bibliography}}

From 0aa663c11e0f2ca91b8e37a45fee466d9d72c6aa Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 15:19:08 -0800
Subject: [PATCH 32/73] Define normal Dedekind reals in terms of lower and
 upper cuts

---
 .../dedekind-real-numbers.lagda.md            | 57 ++++++++-----------
 1 file changed, 25 insertions(+), 32 deletions(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 929862f718..0824ab205d 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -39,6 +39,9 @@ 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
+open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
@@ -48,19 +51,12 @@ open import logic.functoriality-existential-quantification
 A
 {{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
 consists of a [pair](foundation.dependent-pair-types.md) `(L , U)` of
-[subtypes](foundation-core.subtypes.md) of
-[the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
-satisfying the following four conditions
-
-1. _Inhabitedness_. Both `L` and `U` are
-   [inhabited](foundation.inhabited-subtypes.md) subtypes of `ℚ`.
-2. _Roundedness_. A rational number `q` is in `L`
-   [if and only if](foundation.logical-equivalences.md) there
-   [exists](foundation.existential-quantification.md) `q < r` such that `r ∈ L`,
-   and a rational number `r` is in `U` if and only if there exists `q < r` such
-   that `q ∈ U`.
-3. _Disjointness_. `L` and `U` are disjoint subsets of `ℚ`.
-4. _Locatedness_. If `q < r` then `q ∈ L` or `r ∈ U`.
+a [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers)
+and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers)
+that also satisfy the following conditions:
+
+1. _Disjointness_. `L` and `U` are disjoint subsets of `ℚ`.
+2. _Locatedness_. If `q < r` then `q ∈ L` or `r ∈ U`.
 
 The type of {{#concept "Dedekind real numbers" Agda=ℝ}} is the type of all
 Dedekind cuts. The Dedekind real numbers will be taken as the standard
@@ -77,15 +73,10 @@ module _
 
   is-dedekind-cut-Prop : Prop (l1 ⊔ l2)
   is-dedekind-cut-Prop =
-    conjunction-Prop
-      ( (∃ ℚ L) ∧ (∃ ℚ U))
-      ( conjunction-Prop
-        ( conjunction-Prop
-          ( ∀' ℚ ( λ q → L q ⇔ ∃ ℚ (λ r → le-ℚ-Prop q r ∧ L r)))
-          ( ∀' ℚ ( λ r → U r ⇔ ∃ ℚ (λ q → le-ℚ-Prop q r ∧ U q))))
-        ( conjunction-Prop
-          ( ∀' ℚ (λ q → ¬' (L q ∧ U q)))
-          ( ∀' ℚ (λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (L q ∨ U r))))))
+    is-lower-dedekind-cut-Prop L ∧
+    is-upper-dedekind-cut-Prop U ∧
+    ( ∀' ℚ (λ q → ¬' (L q ∧ U q))) ∧
+    ( ∀' ℚ (λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (L q ∨ U r))))
 
   is-dedekind-cut : UU (l1 ⊔ l2)
   is-dedekind-cut = type-Prop is-dedekind-cut-Prop
@@ -113,6 +104,12 @@ module _
   upper-cut-ℝ : subtype l ℚ
   upper-cut-ℝ = pr1 (pr2 x)
 
+  lower-real-ℝ : lower-ℝ l
+  lower-real-ℝ = lower-cut-ℝ , pr1 (pr2 (pr2 x))
+
+  upper-real-ℝ : upper-ℝ l
+  upper-real-ℝ = upper-cut-ℝ , pr1 (pr2 (pr2 (pr2 x)))
+
   is-in-lower-cut-ℝ : ℚ → UU l
   is-in-lower-cut-ℝ = is-in-subtype lower-cut-ℝ
 
@@ -123,34 +120,30 @@ module _
   is-dedekind-cut-cut-ℝ = pr2 (pr2 x)
 
   is-inhabited-lower-cut-ℝ : exists ℚ lower-cut-ℝ
-  is-inhabited-lower-cut-ℝ = pr1 (pr1 is-dedekind-cut-cut-ℝ)
+  is-inhabited-lower-cut-ℝ = is-inhabited-cut-lower-ℝ lower-real-ℝ
 
   is-inhabited-upper-cut-ℝ : exists ℚ upper-cut-ℝ
-  is-inhabited-upper-cut-ℝ = pr2 (pr1 is-dedekind-cut-cut-ℝ)
+  is-inhabited-upper-cut-ℝ = is-inhabited-cut-upper-ℝ upper-real-ℝ
 
   is-rounded-lower-cut-ℝ :
     (q : ℚ) →
     is-in-lower-cut-ℝ q ↔
     exists ℚ (λ r → (le-ℚ-Prop q r) ∧ (lower-cut-ℝ r))
-  is-rounded-lower-cut-ℝ =
-    pr1 (pr1 (pr2 is-dedekind-cut-cut-ℝ))
+  is-rounded-lower-cut-ℝ = is-rounded-cut-lower-ℝ lower-real-ℝ
 
   is-rounded-upper-cut-ℝ :
     (r : ℚ) →
     is-in-upper-cut-ℝ r ↔
     exists ℚ (λ q → (le-ℚ-Prop q r) ∧ (upper-cut-ℝ q))
-  is-rounded-upper-cut-ℝ =
-    pr2 (pr1 (pr2 is-dedekind-cut-cut-ℝ))
+  is-rounded-upper-cut-ℝ = is-rounded-cut-upper-ℝ upper-real-ℝ
 
   is-disjoint-cut-ℝ : (q : ℚ) → ¬ (is-in-lower-cut-ℝ q × is-in-upper-cut-ℝ q)
-  is-disjoint-cut-ℝ =
-    pr1 (pr2 (pr2 is-dedekind-cut-cut-ℝ))
+  is-disjoint-cut-ℝ = pr1 (pr2 (pr2 (pr2 (pr2 x))))
 
   is-located-lower-upper-cut-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 is-dedekind-cut-cut-ℝ))
+  is-located-lower-upper-cut-ℝ = pr2 (pr2 (pr2 (pr2 (pr2 x))))
 
   cut-ℝ : subtype l ℚ
   cut-ℝ q =

From 7499c7c9baad51f54cb31f7f4c52733a038fdc22 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 15:21:33 -0800
Subject: [PATCH 33/73] Start negation

---
 ...lower-upper-dedekind-real-numbers.lagda.md | 20 +++++++++++++++++++
 1 file changed, 20 insertions(+)
 create mode 100644 src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md

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
new file mode 100644
index 0000000000..485c61bb9c
--- /dev/null
+++ b/src/real-numbers/negation-lower-upper-dedekind-real-numbers.lagda.md
@@ -0,0 +1,20 @@
+# Negation of lower and upper Dedekind real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.negation-lower-upper-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import foundation.universe-levels
+
+open import real-numbers.lower-dedekind-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
+```
+
+</details>
+
+## Idea

From 2185e189ba4af56d44a7c3d5da0861682d7271de Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Sun, 9 Feb 2025 15:26:36 -0800
Subject: [PATCH 34/73] Inequality on upper reals

---
 ...ality-upper-dedekind-real-numbers.lagda.md | 83 +++++++++++++++++++
 1 file changed, 83 insertions(+)
 create mode 100644 src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..4e6802b471
--- /dev/null
+++ b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
@@ -0,0 +1,83 @@
+# Inequality on the upper Dedekind real numbers
+
+```agda
+module real-numbers.inequality-upper-dedekind-real-numbers where
+```
+
+<details><summary>Imports</summary>
+
+```agda
+open import elementary-number-theory.inequality-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.logical-equivalences
+open import foundation.powersets
+open import foundation.propositions
+open import foundation.subtypes
+open import foundation.universe-levels
+
+open import order-theory.large-posets
+open import order-theory.large-preorders
+
+open import real-numbers.upper-dedekind-real-numbers
+open import real-numbers.rational-upper-dedekind-real-numbers
+```
+
+</details>
+
+## Idea
+
+The {{#concept "standard ordering" Disambiguation="upper Dedekind real numbers" Agda=leq-upper-ℝ}} on
+the [upper real numbers](real-numbers.upper-dedekind-real-numbers.md) is defined as the
+cut of the second being a subset of the cut of the first.
+
+## Definition
+
+```agda
+module _
+  {l1 l2 : Level}
+  (x : upper-ℝ l1)
+  (y : upper-ℝ l2)
+  where
+
+  leq-upper-ℝ-Prop : Prop (l1 ⊔ l2)
+  leq-upper-ℝ-Prop = leq-prop-subtype (cut-upper-ℝ y) (cut-upper-ℝ x)
+
+  leq-upper-ℝ : UU (l1 ⊔ l2)
+  leq-upper-ℝ = type-Prop leq-upper-ℝ-Prop
+```
+
+## Properties
+
+### Inequality on upper Dedekind reals is a large poset
+
+```agda
+upper-ℝ-Large-Preorder : Large-Preorder lsuc _⊔_
+upper-ℝ-Large-Preorder = powerset-Large-Preorder ℚ
+
+upper-ℝ-Large-Poset : Large-Poset lsuc _⊔_
+upper-ℝ-Large-Poset = powerset-Large-Poset ℚ
+```
+
+### The canonical map from the rational numbers to the upper reals preserves inequality
+
+```agda
+preserves-leq-upper-real-ℚ :
+  (p q : ℚ) → leq-ℚ p q → leq-upper-ℝ (upper-real-ℚ p) (upper-real-ℚ q)
+preserves-leq-upper-real-ℚ p q p≤q r = concatenate-leq-le-ℚ p q r p≤q
+
+reflects-leq-upper-real-ℚ :
+  (p q : ℚ) → leq-upper-ℝ (upper-real-ℚ p) (upper-real-ℚ q) → leq-ℚ p q
+reflects-leq-upper-real-ℚ p q q<r→p<r with decide-le-leq-ℚ q p
+... | inr p≤q = p≤q
+... | inl q<p = ex-falso (irreflexive-le-ℚ p (q<r→p<r p q<p))
+
+iff-leq-upper-real-ℚ :
+  (p q : ℚ) → leq-ℚ p q ↔ leq-upper-ℝ (upper-real-ℚ p) (upper-real-ℚ q)
+pr1 (iff-leq-upper-real-ℚ p q) = preserves-leq-upper-real-ℚ p q
+pr2 (iff-leq-upper-real-ℚ p q) = reflects-leq-upper-real-ℚ p q
+```

From f2da896f447efde34e0c9347c51637446385cb16 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 08:32:11 -0800
Subject: [PATCH 35/73] overhaul

---
 src/real-numbers.lagda.md                     |   5 +
 ...thmetically-located-dedekind-cuts.lagda.md |  76 +++----
 .../dedekind-real-numbers.lagda.md            |  33 ++-
 ...ality-lower-dedekind-real-numbers.lagda.md |  21 +-
 .../inequality-real-numbers.lagda.md          |  31 +--
 ...ality-upper-dedekind-real-numbers.lagda.md |  23 +-
 .../lower-dedekind-real-numbers.lagda.md      |   3 +
 ...lower-upper-dedekind-real-numbers.lagda.md | 205 ++++++++++++++++++
 .../negation-real-numbers.lagda.md            |  88 ++------
 ...ional-lower-dedekind-real-numbers.lagda.md |   5 +-
 .../rational-real-numbers.lagda.md            |  31 +--
 ...ional-upper-dedekind-real-numbers.lagda.md |   5 +-
 .../upper-dedekind-real-numbers.lagda.md      |   3 +
 13 files changed, 362 insertions(+), 167 deletions(-)

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index b110b77299..1cd86ceaa3 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -7,11 +7,16 @@ module real-numbers where
 
 open import real-numbers.arithmetically-located-dedekind-cuts public
 open import real-numbers.dedekind-real-numbers public
+open import real-numbers.inequality-lower-dedekind-real-numbers public
 open import real-numbers.inequality-real-numbers public
+open import real-numbers.inequality-upper-dedekind-real-numbers public
 open import real-numbers.lower-dedekind-real-numbers public
 open import real-numbers.metric-space-of-real-numbers public
+open import real-numbers.negation-lower-upper-dedekind-real-numbers public
 open import real-numbers.negation-real-numbers public
+open import real-numbers.rational-lower-dedekind-real-numbers public
 open import real-numbers.rational-real-numbers public
+open import real-numbers.rational-upper-dedekind-real-numbers public
 open import real-numbers.similarity-real-numbers public
 open import real-numbers.strict-inequality-real-numbers public
 open import real-numbers.upper-dedekind-real-numbers public
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index b33f8a86fc..31d707215b 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -17,6 +17,7 @@ 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.conjunction
 open import foundation.coproduct-types
@@ -31,18 +32,23 @@ open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
 open import group-theory.abelian-groups
+
+open import real-numbers.lower-dedekind-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
+open import real-numbers.dedekind-real-numbers
 ```
 
 </details>
 
 ## Definition
 
-A [Dedekind cut](real-numbers.dedekind-real-numbers.md) `(L, U)` is
-{{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if for any
-positive [rational number](elementary-number-theory.rational-numbers.md)
+A pair of a [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md)
+`L` and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`
+is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if for
+any positive [rational number](elementary-number-theory.rational-numbers.md)
 `ε : ℚ`, there exist `p, q : ℚ` such that `0 < q - p < ε`, `p ∈ L`, and `q ∈ U`.
-Intuitively, when `L , U` represent the Dedekind cuts of a real number `x`, `p`
-and `q` are rational approximations of `x` to within `ε`. This follows parts of
+Intuitively, when `L , U` represent the cuts of a real number `x`, `p` and `q`
+are rational approximations of `x` to within `ε`. This follows parts of
 Section 11 in {{#cite BauerTaylor2009}}.
 
 ## Definitions
@@ -51,16 +57,17 @@ Section 11 in {{#cite BauerTaylor2009}}.
 
 ```agda
 module _
-  {l : Level} (L : subtype l ℚ) (U : subtype l ℚ)
+  {l : Level} (L : lower-ℝ l) (U : upper-ℝ l)
   where
 
-  is-arithmetically-located-pair-of-subtypes-ℚ : UU l
-  is-arithmetically-located-pair-of-subtypes-ℚ =
-    (ε : ℚ) →
-    is-positive-ℚ ε →
+  is-arithmetically-located-pair-of-cuts-ℚ : UU l
+  is-arithmetically-located-pair-of-cuts-ℚ =
+    (ε⁺ : ℚ⁺) →
     exists
       ( ℚ × ℚ)
-      ( λ (p , q) → le-ℚ-Prop p q ∧ le-ℚ-Prop q (p +ℚ ε) ∧ L p ∧ U q)
+      ( λ (p , q) → le-ℚ-Prop q (p +ℚ rational-ℚ⁺ ε⁺) ∧
+        cut-lower-ℝ L p ∧
+        cut-upper-ℝ U q)
 ```
 
 ## Properties
@@ -72,50 +79,37 @@ rational numbers, it is also located.
 
 ```agda
 module _
-  {l : Level} (L : subtype l ℚ) (U : subtype l ℚ)
+  {l : Level} (L : lower-ℝ l) (U : upper-ℝ l)
   where
 
   abstract
-    is-located-is-arithmetically-located-pair-of-subtypes-ℚ :
-      is-arithmetically-located-pair-of-subtypes-ℚ L U →
-      ((p q : ℚ) → le-ℚ p q → is-in-subtype L q → is-in-subtype L p) →
-      ((p q : ℚ) → le-ℚ p q → is-in-subtype U p → is-in-subtype U q) →
-      (p q : ℚ) → le-ℚ p q → type-disjunction-Prop (L p) (U q)
-    is-located-is-arithmetically-located-pair-of-subtypes-ℚ
-      arithmetically-located lower-closed upper-closed p q p<q =
+    is-located-is-arithmetically-located-pair-of-cuts-ℚ :
+      is-arithmetically-located-pair-of-cuts-ℚ L U →
+      is-located-cut (cut-lower-ℝ L) (cut-upper-ℝ U)
+    is-located-is-arithmetically-located-pair-of-cuts-ℚ
+      arithmetically-located p q p<q =
       elim-exists
-        ( L p ∨ U q)
-        ( λ (p' , q') (p'<q' , q'<p'+ε , p'-in-l , q'-in-u) →
+        ( cut-lower-ℝ L p ∨ cut-upper-ℝ U q)
+        ( λ (p' , q') (q'<p'+q-p , p'∈L , q'∈U) →
           rec-coproduct
-            ( λ p<p' → inl-disjunction (lower-closed p p' p<p' p'-in-l))
+            ( λ p<p' →
+              inl-disjunction
+                ( le-cut-lower-ℝ L p p' p<p' p'∈L))
             ( λ p'≤p →
               inr-disjunction
-                ( upper-closed
+                (le-cut-upper-ℝ
+                  ( U)
                   ( q')
                   ( q)
                   ( concatenate-le-leq-ℚ
                     ( q')
                     ( p' +ℚ (q -ℚ p))
                     ( q)
-                    ( q'<p'+ε)
-                    ( tr
-                      ( leq-ℚ (p' +ℚ q -ℚ p))
-                      ( equational-reasoning
-                        p +ℚ (q -ℚ p)
-                        ＝ (p +ℚ q) -ℚ p
-                          by inv (associative-add-ℚ p q (neg-ℚ p))
-                        ＝ q
-                          by is-identity-conjugation-Ab abelian-group-add-ℚ p q)
-                      ( backward-implication
-                        ( iff-translate-right-leq-ℚ (q -ℚ p) p' p)
-                        ( p'≤p))))
-                  ( q'-in-u)))
+                    ( q'<p'+q-p)
+                    {! binary-tr leq-ℚ ? ? (preserves-le-left-add-ℚ q (p' -ℚ p) zero-ℚ ?)  !})
+                  ( q'∈U)))
             ( decide-le-leq-ℚ p p'))
-        ( arithmetically-located
-          ( q -ℚ p)
-          ( is-positive-le-zero-ℚ
-            ( q -ℚ p)
-            ( backward-implication (iff-translate-diff-le-zero-ℚ p q) p<q)))
+        ( arithmetically-located (positive-diff-le-ℚ p q p<q))
 ```
 
 ## References
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 0824ab205d..f8515d6123 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -50,10 +50,10 @@ open import real-numbers.upper-dedekind-real-numbers
 
 A
 {{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
-consists of a [pair](foundation.dependent-pair-types.md) `(L , U)` of
-a [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers)
-and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers)
-that also satisfy the following conditions:
+consists of a [pair](foundation.dependent-pair-types.md) `(L , U)` of a
+[lower Dedekind cut](real-numbers.lower-dedekind-real-numbers) and an
+[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers) that also satisfy
+the following conditions:
 
 1. _Disjointness_. `L` and `U` are disjoint subsets of `ℚ`.
 2. _Locatedness_. If `q < r` then `q ∈ L` or `r ∈ U`.
@@ -71,12 +71,24 @@ module _
   {l1 l2 : Level} (L : subtype l1 ℚ) (U : subtype l2 ℚ)
   where
 
+  is-disjoint-cut-Prop : Prop (l1 ⊔ l2)
+  is-disjoint-cut-Prop = ∀' ℚ (λ q → ¬' (L q ∧ U q))
+
+  is-disjoint-cut : UU (l1 ⊔ l2)
+  is-disjoint-cut = type-Prop is-disjoint-cut-Prop
+
+  is-located-cut-Prop : Prop (l1 ⊔ l2)
+  is-located-cut-Prop = ∀' ℚ (λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (L q ∨ U r)))
+
+  is-located-cut : UU (l1 ⊔ l2)
+  is-located-cut = type-Prop is-located-cut-Prop
+
   is-dedekind-cut-Prop : Prop (l1 ⊔ l2)
   is-dedekind-cut-Prop =
     is-lower-dedekind-cut-Prop L ∧
     is-upper-dedekind-cut-Prop U ∧
-    ( ∀' ℚ (λ q → ¬' (L q ∧ U q))) ∧
-    ( ∀' ℚ (λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (L q ∨ U r))))
+    is-disjoint-cut-Prop ∧
+    is-located-cut-Prop
 
   is-dedekind-cut : UU (l1 ⊔ l2)
   is-dedekind-cut = type-Prop is-dedekind-cut-Prop
@@ -94,6 +106,15 @@ module _
 real-dedekind-cut : {l : Level} (L U : subtype l ℚ) → is-dedekind-cut L U → ℝ l
 real-dedekind-cut L U H = L , U , H
 
+real-lower-upper-ℝ :
+  {l : Level} →
+  (L : lower-ℝ l) (U : upper-ℝ l) →
+  is-disjoint-cut (cut-lower-ℝ L) (cut-upper-ℝ U) →
+  is-located-cut (cut-lower-ℝ L) (cut-upper-ℝ U) →
+  ℝ l
+real-lower-upper-ℝ (L , lower-dedekind-L) (U , lower-dedekind-U) H K =
+  L , U , lower-dedekind-L , lower-dedekind-U , H , K
+
 module _
   {l : Level} (x : ℝ l)
   where
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 52dd94fe52..9cd1a90e4c 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -31,9 +31,10 @@ open import real-numbers.rational-lower-dedekind-real-numbers
 
 ## Idea
 
-The {{#concept "standard ordering" Disambiguation="lower Dedekind real numbers" Agda=leq-lower-ℝ}} on
-the [lower real numbers](real-numbers.lower-dedekind-real-numbers.md) is defined as the
-cut of one being a subset of the cut of the other.
+The
+{{#concept "standard ordering" Disambiguation="lower Dedekind real numbers" Agda=leq-lower-ℝ}}
+on the [lower real numbers](real-numbers.lower-dedekind-real-numbers.md) is
+defined as the cut of one being a subset of the cut of the other.
 
 ## Definition
 
@@ -57,10 +58,20 @@ module _
 
 ```agda
 lower-ℝ-Large-Preorder : Large-Preorder lsuc _⊔_
-lower-ℝ-Large-Preorder = powerset-Large-Preorder ℚ
+type-Large-Preorder lower-ℝ-Large-Preorder = lower-ℝ
+leq-prop-Large-Preorder lower-ℝ-Large-Preorder = leq-lower-ℝ-Prop
+refl-leq-Large-Preorder lower-ℝ-Large-Preorder x =
+  refl-leq-subtype (cut-lower-ℝ x)
+transitive-leq-Large-Preorder lower-ℝ-Large-Preorder x y z =
+  transitive-leq-subtype (cut-lower-ℝ x) (cut-lower-ℝ y) (cut-lower-ℝ z)
 
 lower-ℝ-Large-Poset : Large-Poset lsuc _⊔_
-lower-ℝ-Large-Poset = powerset-Large-Poset ℚ
+large-preorder-Large-Poset lower-ℝ-Large-Poset = lower-ℝ-Large-Preorder
+antisymmetric-leq-Large-Poset lower-ℝ-Large-Poset x y x≤y y≤x =
+  eq-eq-cut-lower-ℝ
+    ( x)
+    ( y)
+    ( antisymmetric-leq-subtype (cut-lower-ℝ x) (cut-lower-ℝ y) x≤y y≤x)
 ```
 
 ### The canonical map from the rational numbers to the lower reals preserves inequality
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index 92b1fe356a..83978ab7da 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -27,6 +27,12 @@ open import order-theory.large-preorders
 open import order-theory.posets
 open import order-theory.preorders
 
+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.upper-dedekind-real-numbers
+open import real-numbers.negation-real-numbers
+open import real-numbers.negation-lower-upper-dedekind-real-numbers
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.rational-real-numbers
 ```
@@ -48,7 +54,7 @@ module _
   where
 
   leq-ℝ-Prop : Prop (l1 ⊔ l2)
-  leq-ℝ-Prop = leq-prop-subtype (lower-cut-ℝ x) (lower-cut-ℝ y)
+  leq-ℝ-Prop = leq-lower-ℝ-Prop (lower-real-ℝ x) (lower-real-ℝ y)
 
   leq-ℝ : UU (l1 ⊔ l2)
   leq-ℝ = type-Prop leq-ℝ-Prop
@@ -64,7 +70,7 @@ module _
   where
 
   leq-ℝ-Prop' : Prop (l1 ⊔ l2)
-  leq-ℝ-Prop' = leq-prop-subtype (upper-cut-ℝ y) (upper-cut-ℝ x)
+  leq-ℝ-Prop' = leq-upper-ℝ-Prop (upper-real-ℝ x) (upper-real-ℝ y)
 
   leq-ℝ' : UU (l1 ⊔ l2)
   leq-ℝ' = type-Prop leq-ℝ-Prop'
@@ -73,7 +79,7 @@ module _
   pr1 (leq-iff-ℝ') lx⊆ly q q-in-uy =
     elim-exists
       ( upper-cut-ℝ x q)
-      ( λ p (p<q , p∉ly) →
+      ( λ p (p<q , p≮y) →
         subset-upper-cut-upper-complement-lower-cut-ℝ
           ( x)
           ( q)
@@ -85,12 +91,12 @@ module _
                 ( lower-cut-ℝ y)
                 ( lx⊆ly)
                 ( p)
-                ( p∉ly))))
+                ( p≮y))))
       ( subset-upper-complement-lower-cut-upper-cut-ℝ y q q-in-uy)
-  pr2 (leq-iff-ℝ') uy⊆ux p p-in-lx =
+  pr2 leq-iff-ℝ' uy⊆ux p p-in-lx =
     elim-exists
       ( lower-cut-ℝ y p)
-      ( λ q (p<q , q∉ux) →
+      ( λ q (p<q , x≮q) →
         subset-lower-cut-lower-complement-upper-cut-ℝ
           ( y)
           ( p)
@@ -102,7 +108,7 @@ module _
                 ( upper-cut-ℝ x)
                 ( uy⊆ux)
                 ( q)
-                ( q∉ux))))
+                ( x≮q))))
       ( subset-lower-complement-upper-cut-lower-cut-ℝ x p p-in-lx)
 ```
 
@@ -110,7 +116,7 @@ module _
 
 ```agda
 refl-leq-ℝ : {l : Level} → (x : ℝ l) → leq-ℝ x x
-refl-leq-ℝ x = refl-leq-subtype (lower-cut-ℝ x)
+refl-leq-ℝ x = refl-leq-Large-Preorder lower-ℝ-Large-Preorder (lower-real-ℝ x)
 ```
 
 ### Inequality on the real numbers is antisymmetric
@@ -171,16 +177,13 @@ antisymmetric-leq-Large-Poset ℝ-Large-Poset = antisymmetric-leq-ℝ
 
 ```agda
 preserves-leq-real-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℝ (real-ℚ x) (real-ℚ y)
-preserves-leq-real-ℚ x y x≤y q q<x = concatenate-le-leq-ℚ q x y q<x x≤y
+preserves-leq-real-ℚ = preserves-leq-lower-real-ℚ
 
 reflects-leq-real-ℚ : (x y : ℚ) → leq-ℝ (real-ℚ x) (real-ℚ y) → leq-ℚ x y
-reflects-leq-real-ℚ x y rx≤ry with decide-le-leq-ℚ y x
-... | inl y<x = ex-falso (irreflexive-le-ℚ y (rx≤ry y y<x))
-... | inr x≤y = x≤y
+reflects-leq-real-ℚ = reflects-leq-lower-real-ℚ
 
 iff-leq-real-ℚ : (x y : ℚ) → leq-ℚ x y ↔ leq-ℝ (real-ℚ x) (real-ℚ y)
-pr1 (iff-leq-real-ℚ x y) = preserves-leq-real-ℚ x y
-pr2 (iff-leq-real-ℚ x y) = reflects-leq-real-ℚ x y
+iff-leq-real-ℚ = iff-leq-lower-real-ℚ
 ```
 
 ## References
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 4e6802b471..dc92a54484 100644
--- a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
@@ -23,17 +23,18 @@ open import foundation.universe-levels
 open import order-theory.large-posets
 open import order-theory.large-preorders
 
-open import real-numbers.upper-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
 
 ## Idea
 
-The {{#concept "standard ordering" Disambiguation="upper Dedekind real numbers" Agda=leq-upper-ℝ}} on
-the [upper real numbers](real-numbers.upper-dedekind-real-numbers.md) is defined as the
-cut of the second being a subset of the cut of the first.
+The
+{{#concept "standard ordering" Disambiguation="upper Dedekind real numbers" Agda=leq-upper-ℝ}}
+on the [upper real numbers](real-numbers.upper-dedekind-real-numbers.md) is
+defined as the cut of the second being a subset of the cut of the first.
 
 ## Definition
 
@@ -57,10 +58,20 @@ module _
 
 ```agda
 upper-ℝ-Large-Preorder : Large-Preorder lsuc _⊔_
-upper-ℝ-Large-Preorder = powerset-Large-Preorder ℚ
+type-Large-Preorder upper-ℝ-Large-Preorder = upper-ℝ
+leq-prop-Large-Preorder upper-ℝ-Large-Preorder = leq-upper-ℝ-Prop
+refl-leq-Large-Preorder upper-ℝ-Large-Preorder x =
+  refl-leq-subtype (cut-upper-ℝ x)
+transitive-leq-Large-Preorder upper-ℝ-Large-Preorder x y z y≤z x≤y =
+  transitive-leq-subtype (cut-upper-ℝ z) (cut-upper-ℝ y) (cut-upper-ℝ x) x≤y y≤z
 
 upper-ℝ-Large-Poset : Large-Poset lsuc _⊔_
-upper-ℝ-Large-Poset = powerset-Large-Poset ℚ
+large-preorder-Large-Poset upper-ℝ-Large-Poset = upper-ℝ-Large-Preorder
+antisymmetric-leq-Large-Poset upper-ℝ-Large-Poset x y x≤y y≤x =
+  eq-eq-cut-upper-ℝ
+    ( x)
+    ( y)
+    ( antisymmetric-leq-subtype (cut-upper-ℝ x) (cut-upper-ℝ y) y≤x x≤y)
 ```
 
 ### The canonical map from the rational numbers to the upper reals preserves inequality
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 8d9097a7ec..6bc94cd66d 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -75,6 +75,9 @@ module _
   is-in-cut-lower-ℝ : ℚ → UU l
   is-in-cut-lower-ℝ = is-in-subtype cut-lower-ℝ
 
+  is-lower-dedekind-cut-lower-ℝ : is-lower-dedekind-cut cut-lower-ℝ
+  is-lower-dedekind-cut-lower-ℝ = pr2 x
+
   is-inhabited-cut-lower-ℝ : exists ℚ cut-lower-ℝ
   is-inhabited-cut-lower-ℝ = pr1 (pr2 x)
 
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 485c61bb9c..3f6366d7ac 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
@@ -9,12 +9,217 @@ module real-numbers.negation-lower-upper-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+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.conjunction
+open import foundation.dependent-pair-types
+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.subtypes
+open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
+open import logic.functoriality-existential-quantification
+
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.upper-dedekind-real-numbers
+open import real-numbers.rational-lower-dedekind-real-numbers
+open import real-numbers.rational-upper-dedekind-real-numbers
 ```
 
 </details>
 
 ## Idea
+
+The negation of a lower Dedekind real is an upper Dedekind real containing
+the negations of elements in the lower cut, and vice versa.
+
+## Definition
+
+### The negation of a lower Dedekind real, as an upper Dedekind real
+
+```agda
+module _
+  {l : Level} (x : lower-ℝ l)
+  where
+
+  cut-neg-lower-ℝ : subtype l ℚ
+  cut-neg-lower-ℝ p = cut-lower-ℝ x (neg-ℚ p)
+
+  is-in-cut-neg-lower-ℝ : ℚ → UU l
+  is-in-cut-neg-lower-ℝ = is-in-subtype cut-neg-lower-ℝ
+
+  abstract
+    is-inhabited-cut-neg-lower-ℝ : exists ℚ cut-neg-lower-ℝ
+    is-inhabited-cut-neg-lower-ℝ =
+      map-exists
+        ( is-in-cut-neg-lower-ℝ)
+        ( neg-ℚ)
+        ( λ p → tr (is-in-cut-lower-ℝ x) (inv (neg-neg-ℚ p)))
+        ( is-inhabited-cut-lower-ℝ x)
+
+    is-rounded-cut-neg-lower-ℝ :
+      (q : ℚ) →
+      is-in-cut-neg-lower-ℝ q ↔
+      exists ℚ (λ p → le-ℚ-Prop p q ∧ cut-neg-lower-ℝ p)
+    pr1 (is-rounded-cut-neg-lower-ℝ q) -q<x =
+      map-exists
+        (λ p → le-ℚ p q × is-in-cut-neg-lower-ℝ p)
+        ( neg-ℚ)
+        ( λ p (-q<p , p<x) →
+          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ (neg-ℚ q) p -q<p) ,
+          tr (is-in-cut-lower-ℝ x) (inv (neg-neg-ℚ p)) p<x)
+        ( forward-implication (is-rounded-cut-lower-ℝ x (neg-ℚ q)) -q<x)
+    pr2 (is-rounded-cut-neg-lower-ℝ q) =
+      elim-exists
+        ( cut-neg-lower-ℝ q)
+        ( λ p (p<q , -q<x) →
+          backward-implication
+            ( is-rounded-cut-lower-ℝ x (neg-ℚ q))
+            ( intro-exists (neg-ℚ p) (neg-le-ℚ p q p<q , -q<x)))
+
+  neg-lower-ℝ : upper-ℝ l
+  pr1 neg-lower-ℝ = cut-neg-lower-ℝ
+  pr1 (pr2 neg-lower-ℝ) = is-inhabited-cut-neg-lower-ℝ
+  pr2 (pr2 neg-lower-ℝ) = is-rounded-cut-neg-lower-ℝ
+```
+
+### The negation of an upper Dedekind real, as a lower Dedekind real
+
+```agda
+module _
+  {l : Level} (x : upper-ℝ l)
+  where
+
+  cut-neg-upper-ℝ : subtype l ℚ
+  cut-neg-upper-ℝ q = cut-upper-ℝ x (neg-ℚ q)
+
+  is-in-cut-neg-upper-ℝ : ℚ → UU l
+  is-in-cut-neg-upper-ℝ = is-in-subtype cut-neg-upper-ℝ
+
+  abstract
+    is-inhabited-cut-neg-upper-ℝ : exists ℚ cut-neg-upper-ℝ
+    is-inhabited-cut-neg-upper-ℝ =
+      map-exists
+        ( is-in-cut-neg-upper-ℝ)
+        ( neg-ℚ)
+        ( λ p → tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)))
+        ( is-inhabited-cut-upper-ℝ x)
+
+    is-rounded-cut-neg-upper-ℝ :
+      (q : ℚ) →
+      is-in-cut-neg-upper-ℝ q ↔
+      exists ℚ (λ r → le-ℚ-Prop q r ∧ cut-neg-upper-ℝ r)
+    pr1 (is-rounded-cut-neg-upper-ℝ q) x<-q =
+      map-exists
+        ( λ r → le-ℚ q r × is-in-cut-neg-upper-ℝ r)
+        ( neg-ℚ)
+        ( λ p (p<-q , x<p) →
+          tr
+            ( λ x → le-ℚ x (neg-ℚ p))
+            ( neg-neg-ℚ q)
+            ( neg-le-ℚ p (neg-ℚ q) p<-q) ,
+          tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)) x<p )
+        ( forward-implication (is-rounded-cut-upper-ℝ x (neg-ℚ q)) x<-q)
+    pr2 (is-rounded-cut-neg-upper-ℝ q) =
+      elim-exists
+        ( cut-neg-upper-ℝ q)
+        ( λ r (q<r , x<-r) →
+          backward-implication
+            ( is-rounded-cut-upper-ℝ x (neg-ℚ q))
+            ( intro-exists (neg-ℚ r) (neg-le-ℚ q r q<r , x<-r)))
+
+  neg-upper-ℝ : lower-ℝ l
+  pr1 neg-upper-ℝ = cut-neg-upper-ℝ
+  pr1 (pr2 neg-upper-ℝ) = is-inhabited-cut-neg-upper-ℝ
+  pr2 (pr2 neg-upper-ℝ) = is-rounded-cut-neg-upper-ℝ
+```
+
+### The negation of lower and upper Dedekind reals are mutual inverses
+
+```agda
+abstract
+  is-retraction-neg-upper-ℝ :
+    (l : Level) → is-retraction (neg-lower-ℝ {l}) (neg-upper-ℝ {l})
+  is-retraction-neg-upper-ℝ l x =
+    eq-eq-cut-lower-ℝ
+      ( neg-upper-ℝ (neg-lower-ℝ x))
+      ( x)
+      ( antisymmetric-sim-subtype
+        ( cut-neg-upper-ℝ (neg-lower-ℝ x))
+        ( cut-lower-ℝ x)
+        ( (λ p → tr (is-in-cut-lower-ℝ x) (neg-neg-ℚ p)) ,
+          (λ p → tr (is-in-cut-lower-ℝ x) (inv (neg-neg-ℚ p)))))
+
+  is-section-neg-upper-ℝ :
+    (l : Level) → is-section (neg-lower-ℝ {l}) (neg-upper-ℝ {l})
+  is-section-neg-upper-ℝ l x =
+    eq-eq-cut-upper-ℝ
+      ( neg-lower-ℝ (neg-upper-ℝ x))
+      ( x)
+      ( antisymmetric-sim-subtype
+        ( cut-neg-lower-ℝ (neg-upper-ℝ x))
+        ( cut-upper-ℝ x)
+        ( (λ p → tr (is-in-cut-upper-ℝ x) (neg-neg-ℚ p)) ,
+          (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-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
+```
+
+### The negation of a rational projected to a lower real is the projection of its negation as an upper real
+
+```agda
+neg-lower-real-ℚ : (q : ℚ) → neg-lower-ℝ (lower-real-ℚ q) ＝ upper-real-ℚ (neg-ℚ q)
+neg-lower-real-ℚ q =
+  eq-eq-cut-upper-ℝ
+    ( neg-lower-ℝ (lower-real-ℚ q))
+    ( upper-real-ℚ (neg-ℚ q))
+    ( antisymmetric-sim-subtype
+      ( cut-neg-lower-ℝ (lower-real-ℚ q))
+      ( cut-upper-real-ℚ (neg-ℚ q))
+      ( (λ p -p<q →
+          tr (le-ℚ (neg-ℚ q)) (neg-neg-ℚ p) (neg-le-ℚ (neg-ℚ p) q -p<q)) ,
+        (λ p -q<p →
+          tr (le-ℚ (neg-ℚ p)) (neg-neg-ℚ q) (neg-le-ℚ (neg-ℚ q) p -q<p))))
+```
+
+### The negation of a rational projected to an upper real is the projection of its negation as a lower real
+
+```agda
+neg-upper-real-ℚ :
+  (q : ℚ) → neg-upper-ℝ (upper-real-ℚ q) ＝ lower-real-ℚ (neg-ℚ q)
+neg-upper-real-ℚ q =
+  eq-eq-cut-lower-ℝ
+    ( neg-upper-ℝ (upper-real-ℚ q))
+    ( lower-real-ℚ (neg-ℚ q))
+    ( antisymmetric-sim-subtype
+      ( cut-neg-upper-ℝ (upper-real-ℚ q))
+      ( cut-lower-real-ℚ (neg-ℚ q))
+      ( (λ p q<-p →
+          tr
+            ( λ r → le-ℚ r (neg-ℚ q))
+            ( neg-neg-ℚ p)
+            ( neg-le-ℚ q (neg-ℚ p) q<-p)) ,
+        (λ p p<-q →
+          tr
+            ( λ r → le-ℚ r (neg-ℚ p))
+            ( neg-neg-ℚ q)
+            ( neg-le-ℚ p (neg-ℚ q) p<-q))))
+```
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index 90dcedd529..b4995b61fe 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -34,6 +34,11 @@ open import logic.functoriality-existential-quantification
 
 open import real-numbers.dedekind-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.lower-dedekind-real-numbers
+open import real-numbers.upper-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-upper-dedekind-real-numbers
 ```
 
 </details>
@@ -50,72 +55,17 @@ module _
   {l : Level} (x : ℝ l)
   where
 
+  lower-real-neg-ℝ : lower-ℝ l
+  lower-real-neg-ℝ = neg-upper-ℝ (upper-real-ℝ x)
+
+  upper-real-neg-ℝ : upper-ℝ l
+  upper-real-neg-ℝ = neg-lower-ℝ (lower-real-ℝ x)
+
   lower-cut-neg-ℝ : subtype l ℚ
-  lower-cut-neg-ℝ q = upper-cut-ℝ x (neg-ℚ q)
+  lower-cut-neg-ℝ = cut-lower-ℝ lower-real-neg-ℝ
 
   upper-cut-neg-ℝ : subtype l ℚ
-  upper-cut-neg-ℝ q = lower-cut-ℝ x (neg-ℚ q)
-
-  is-inhabited-lower-cut-neg-ℝ : exists ℚ lower-cut-neg-ℝ
-  is-inhabited-lower-cut-neg-ℝ =
-    map-exists
-      ( is-in-subtype lower-cut-neg-ℝ)
-      ( neg-ℚ)
-      ( λ q → tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ q)))
-      ( is-inhabited-upper-cut-ℝ x)
-
-  is-inhabited-upper-cut-neg-ℝ : exists ℚ upper-cut-neg-ℝ
-  is-inhabited-upper-cut-neg-ℝ =
-    map-exists
-      ( is-in-subtype upper-cut-neg-ℝ)
-      ( neg-ℚ)
-      ( λ q → tr (is-in-lower-cut-ℝ x) (inv (neg-neg-ℚ q)))
-      ( is-inhabited-lower-cut-ℝ x)
-
-  is-rounded-lower-cut-neg-ℝ :
-    (q : ℚ) →
-    is-in-subtype lower-cut-neg-ℝ q ↔
-    exists ℚ (λ r → (le-ℚ-Prop q r) ∧ (lower-cut-neg-ℝ r))
-  pr1 (is-rounded-lower-cut-neg-ℝ q) in-neg-lower =
-    map-exists
-      ( λ r → le-ℚ q r × is-in-subtype lower-cut-neg-ℝ r)
-      ( neg-ℚ)
-      ( λ r (r<-q , in-upper-r) →
-        ( ( tr
-            ( λ x → le-ℚ x (neg-ℚ r))
-            ( neg-neg-ℚ q)
-            ( neg-le-ℚ r (neg-ℚ q) r<-q)) ,
-          ( tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ r)) in-upper-r)))
-      ( forward-implication (is-rounded-upper-cut-ℝ x (neg-ℚ q)) in-neg-lower)
-  pr2 (is-rounded-lower-cut-neg-ℝ q) exists-r =
-    backward-implication
-      ( is-rounded-upper-cut-ℝ x (neg-ℚ q))
-      ( map-exists
-        ( λ r → le-ℚ r (neg-ℚ q) × is-in-upper-cut-ℝ x r)
-        ( neg-ℚ)
-        ( λ r (q<r , in-neg-lower-r) → (neg-le-ℚ q r q<r , in-neg-lower-r))
-        ( exists-r))
-
-  is-rounded-upper-cut-neg-ℝ :
-    (r : ℚ) →
-    is-in-subtype upper-cut-neg-ℝ r ↔
-    exists ℚ (λ q → (le-ℚ-Prop q r) ∧ (upper-cut-neg-ℝ q))
-  pr1 (is-rounded-upper-cut-neg-ℝ r) in-neg-upper =
-    map-exists
-      ( λ q → le-ℚ q r × is-in-subtype upper-cut-neg-ℝ q)
-      ( neg-ℚ)
-      ( λ q (-r<q , in-lower-q) →
-        ( ( tr (le-ℚ (neg-ℚ q)) (neg-neg-ℚ r) (neg-le-ℚ (neg-ℚ r) q -r<q)) ,
-          ( tr (is-in-lower-cut-ℝ x) (inv (neg-neg-ℚ q)) in-lower-q)))
-      ( forward-implication (is-rounded-lower-cut-ℝ x (neg-ℚ r)) in-neg-upper)
-  pr2 (is-rounded-upper-cut-neg-ℝ r) exists-q =
-    backward-implication
-      ( is-rounded-lower-cut-ℝ x (neg-ℚ r))
-      ( map-exists
-        ( λ q → le-ℚ (neg-ℚ r) q × is-in-lower-cut-ℝ x q)
-        ( neg-ℚ)
-        ( λ q (q<r , in-neg-upper-q) → (neg-le-ℚ q r q<r , in-neg-upper-q))
-        ( exists-q))
+  upper-cut-neg-ℝ = cut-upper-ℝ upper-real-neg-ℝ
 
   is-disjoint-cut-neg-ℝ :
     (q : ℚ) →
@@ -135,13 +85,11 @@ module _
 
   neg-ℝ : ℝ l
   neg-ℝ =
-    real-dedekind-cut
-      ( lower-cut-neg-ℝ)
-      ( upper-cut-neg-ℝ)
-      ( (is-inhabited-lower-cut-neg-ℝ , is-inhabited-upper-cut-neg-ℝ) ,
-        ( is-rounded-lower-cut-neg-ℝ , is-rounded-upper-cut-neg-ℝ) ,
-          is-disjoint-cut-neg-ℝ ,
-          is-located-lower-upper-cut-neg-ℝ)
+    real-lower-upper-ℝ
+      ( lower-real-neg-ℝ)
+      ( upper-real-neg-ℝ)
+      ( is-disjoint-cut-neg-ℝ)
+      ( is-located-lower-upper-cut-neg-ℝ)
 ```
 
 ## Properties
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 4c9fca0bf0..1bd7e02d97 100644
--- a/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/rational-lower-dedekind-real-numbers.lagda.md
@@ -24,8 +24,9 @@ open import real-numbers.lower-dedekind-real-numbers
 
 ## Idea
 
-There is a canonical mapping from the [rational numbers](elementary-number-theory.rational-numbers.md)
-to the [lower Dedekind real numbers](real-numbers.lower-dedekind-real-numbers.md).
+There is a canonical mapping from the
+[rational numbers](elementary-number-theory.rational-numbers.md) to the
+[lower Dedekind real numbers](real-numbers.lower-dedekind-real-numbers.md).
 
 ## Definition
 
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index 55558129b1..d3829df34b 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -36,6 +36,10 @@ 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.upper-dedekind-real-numbers
+open import real-numbers.rational-lower-dedekind-real-numbers
+open import real-numbers.rational-upper-dedekind-real-numbers
 ```
 
 </details>
@@ -56,28 +60,13 @@ the [image](foundation.images.md) of this embedding
 is-dedekind-cut-le-ℚ :
   (x : ℚ) →
   is-dedekind-cut
-    (λ (q : ℚ) → le-ℚ-Prop q x)
-    (λ (r : ℚ) → le-ℚ-Prop x r)
+    (cut-lower-real-ℚ x)
+    (cut-upper-real-ℚ x)
 is-dedekind-cut-le-ℚ x =
-  ( exists-lesser-ℚ x , exists-greater-ℚ x) ,
-  ( ( λ (q : ℚ) →
-      dense-le-ℚ q x ,
-      elim-exists
-        ( le-ℚ-Prop q x)
-        ( λ r (H , H') → transitive-le-ℚ q r x H' H)) ,
-    ( λ (r : ℚ) →
-      α x r ∘ dense-le-ℚ x r ,
-      elim-exists
-        ( le-ℚ-Prop x r)
-        ( λ q (H , H') → transitive-le-ℚ x q r H H'))) ,
-  ( λ (q : ℚ) (H , H') → asymmetric-le-ℚ q x H H') ,
-  ( located-le-ℚ x)
-  where
-    α :
-      (a b : ℚ) →
-      exists ℚ (λ r → le-ℚ-Prop a r ∧ le-ℚ-Prop r b) →
-      exists ℚ (λ r → le-ℚ-Prop r b ∧ le-ℚ-Prop a r)
-    α a b = map-tot-exists (λ r (p , q) → (q , p))
+  is-lower-dedekind-cut-lower-ℝ (lower-real-ℚ x) ,
+  is-upper-dedekind-cut-upper-ℝ (upper-real-ℚ x) ,
+  (λ q (H , K) → asymmetric-le-ℚ q x H K) ,
+  located-le-ℚ x
 ```
 
 ### The canonical map from `ℚ` to `ℝ`
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 c6e1c589de..9761287d50 100644
--- a/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/rational-upper-dedekind-real-numbers.lagda.md
@@ -24,8 +24,9 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-There is a canonical mapping from the [rational numbers](elementary-number-theory.rational-numbers.md)
-to the [upper Dedekind real numbers](real-numbers.upper-dedekind-real-numbers.md).
+There is a canonical mapping from the
+[rational numbers](elementary-number-theory.rational-numbers.md) to the
+[upper Dedekind real numbers](real-numbers.upper-dedekind-real-numbers.md).
 
 ## Definition
 
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index c9dbf0f8f4..264bf07633 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -75,6 +75,9 @@ module _
   is-in-cut-upper-ℝ : ℚ → UU l
   is-in-cut-upper-ℝ = is-in-subtype cut-upper-ℝ
 
+  is-upper-dedekind-cut-upper-ℝ : is-upper-dedekind-cut cut-upper-ℝ
+  is-upper-dedekind-cut-upper-ℝ = pr2 x
+
   is-inhabited-cut-upper-ℝ : exists ℚ cut-upper-ℝ
   is-inhabited-cut-upper-ℝ = pr1 (pr2 x)
 

From 70f181c77d39b461ed5a3b79376f9f685fb70508 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 08:58:26 -0800
Subject: [PATCH 36/73] make pre-commit

---
 .../arithmetically-located-dedekind-cuts.lagda.md     |  6 +++---
 src/real-numbers/inequality-real-numbers.lagda.md     |  6 +++---
 ...egation-lower-upper-dedekind-real-numbers.lagda.md | 11 ++++++-----
 src/real-numbers/negation-real-numbers.lagda.md       |  4 ++--
 src/real-numbers/rational-real-numbers.lagda.md       |  2 +-
 5 files changed, 15 insertions(+), 14 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 31d707215b..06174ce77c 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -33,9 +33,9 @@ open import foundation.universe-levels
 
 open import group-theory.abelian-groups
 
+open import real-numbers.dedekind-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
 open import real-numbers.upper-dedekind-real-numbers
-open import real-numbers.dedekind-real-numbers
 ```
 
 </details>
@@ -48,8 +48,8 @@ is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if for
 any positive [rational number](elementary-number-theory.rational-numbers.md)
 `ε : ℚ`, there exist `p, q : ℚ` such that `0 < q - p < ε`, `p ∈ L`, and `q ∈ U`.
 Intuitively, when `L , U` represent the cuts of a real number `x`, `p` and `q`
-are rational approximations of `x` to within `ε`. This follows parts of
-Section 11 in {{#cite BauerTaylor2009}}.
+are rational approximations of `x` to within `ε`. This follows parts of Section
+11 in {{#cite BauerTaylor2009}}.
 
 ## Definitions
 
diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index 83978ab7da..579ef75be4 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -27,14 +27,14 @@ open import order-theory.large-preorders
 open import order-theory.posets
 open import order-theory.preorders
 
+open import real-numbers.dedekind-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.upper-dedekind-real-numbers
-open import real-numbers.negation-real-numbers
 open import real-numbers.negation-lower-upper-dedekind-real-numbers
-open import real-numbers.dedekind-real-numbers
+open import real-numbers.negation-real-numbers
 open import real-numbers.rational-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
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 3f6366d7ac..25081b1abe 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
@@ -29,17 +29,17 @@ open import foundation.universe-levels
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.lower-dedekind-real-numbers
-open import real-numbers.upper-dedekind-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>
 
 ## Idea
 
-The negation of a lower Dedekind real is an upper Dedekind real containing
-the negations of elements in the lower cut, and vice versa.
+The negation of a lower Dedekind real is an upper Dedekind real containing the
+negations of elements in the lower cut, and vice versa.
 
 ## Definition
 
@@ -126,7 +126,7 @@ module _
             ( λ x → le-ℚ x (neg-ℚ p))
             ( neg-neg-ℚ q)
             ( neg-le-ℚ p (neg-ℚ q) p<-q) ,
-          tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)) x<p )
+          tr (is-in-cut-upper-ℝ x) (inv (neg-neg-ℚ p)) x<p)
         ( forward-implication (is-rounded-cut-upper-ℝ x (neg-ℚ q)) x<-q)
     pr2 (is-rounded-cut-neg-upper-ℝ q) =
       elim-exists
@@ -186,7 +186,8 @@ abstract
 ### The negation of a rational projected to a lower real is the projection of its negation as an upper real
 
 ```agda
-neg-lower-real-ℚ : (q : ℚ) → neg-lower-ℝ (lower-real-ℚ q) ＝ upper-real-ℚ (neg-ℚ q)
+neg-lower-real-ℚ :
+  (q : ℚ) → neg-lower-ℝ (lower-real-ℚ q) ＝ upper-real-ℚ (neg-ℚ q)
 neg-lower-real-ℚ q =
   eq-eq-cut-upper-ℝ
     ( neg-lower-ℝ (lower-real-ℚ q))
diff --git a/src/real-numbers/negation-real-numbers.lagda.md b/src/real-numbers/negation-real-numbers.lagda.md
index b4995b61fe..bfb4895472 100644
--- a/src/real-numbers/negation-real-numbers.lagda.md
+++ b/src/real-numbers/negation-real-numbers.lagda.md
@@ -33,12 +33,12 @@ open import foundation.universe-levels
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.dedekind-real-numbers
-open import real-numbers.rational-real-numbers
 open import real-numbers.lower-dedekind-real-numbers
-open import real-numbers.upper-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.upper-dedekind-real-numbers
 ```
 
 </details>
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index d3829df34b..69ec475e2a 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -37,9 +37,9 @@ 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.upper-dedekind-real-numbers
 open import real-numbers.rational-lower-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>

From 926877950f77ce8f137452df3e56bbc22ad573bc Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 09:08:37 -0800
Subject: [PATCH 37/73] a -> an

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

diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index 264bf07633..01625b8046 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -25,7 +25,7 @@ open import foundation.universe-levels
 
 ## Idea
 
-A upper
+An upper
 {{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
 consists of a [subtype](foundation-core.subtypes.md) `U` of
 [the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,

From 1b9d81b2b52a68c6093186f8b12facffc089ed53 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 09:10:52 -0800
Subject: [PATCH 38/73] Rename some things

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

diff --git a/src/real-numbers/inequality-real-numbers.lagda.md b/src/real-numbers/inequality-real-numbers.lagda.md
index 579ef75be4..098feec9e9 100644
--- a/src/real-numbers/inequality-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-real-numbers.lagda.md
@@ -62,7 +62,7 @@ module _
 
 ## Properties
 
-### Equivalence with reversed containment of upper cuts
+### Equivalence with inequality on upper cuts
 
 ```agda
 module _

From f75209b769526ce9b5c6f17f50b75c823ba3481f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 17:08:28 -0800
Subject: [PATCH 39/73] Some more theorems

---
 ...uality-lower-dedekind-real-numbers.lagda.md | 18 ++++++++++++++++++
 ...uality-upper-dedekind-real-numbers.lagda.md | 18 ++++++++++++++++++
 ...-lower-upper-dedekind-real-numbers.lagda.md |  2 ++
 3 files changed, 38 insertions(+)

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 9cd1a90e4c..7a0a1d476b 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -14,7 +14,9 @@ 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.existential-quantification
 open import foundation.logical-equivalences
+open import foundation.negation
 open import foundation.powersets
 open import foundation.propositions
 open import foundation.subtypes
@@ -74,6 +76,22 @@ antisymmetric-leq-Large-Poset lower-ℝ-Large-Poset x y x≤y y≤x =
     ( antisymmetric-leq-subtype (cut-lower-ℝ x) (cut-lower-ℝ y) x≤y y≤x)
 ```
 
+### If a rational is in a lower Dedekind cut, its projections is less than or equal to the corresponding lower real
+
+```agda
+module _
+  {l : Level}
+  (x : lower-ℝ l)
+  (q : ℚ)
+  where
+
+  leq-is-in-cut-lower-real-ℚ : is-in-cut-lower-ℝ x q → leq-lower-ℝ (lower-real-ℚ q) x
+  leq-is-in-cut-lower-real-ℚ q∈L p p<q =
+    backward-implication
+      ( is-rounded-cut-lower-ℝ x p)
+      ( intro-exists q (p<q , q∈L))
+```
+
 ### The canonical map from the rational numbers to the lower reals preserves inequality
 
 ```agda
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 dc92a54484..ec83e2080c 100644
--- a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
@@ -14,6 +14,7 @@ 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.existential-quantification
 open import foundation.logical-equivalences
 open import foundation.powersets
 open import foundation.propositions
@@ -74,6 +75,23 @@ antisymmetric-leq-Large-Poset upper-ℝ-Large-Poset x y x≤y y≤x =
     ( antisymmetric-leq-subtype (cut-upper-ℝ x) (cut-upper-ℝ y) y≤x x≤y)
 ```
 
+### If a rational is in an upper Dedekind cut, the corresponding upper real is less than or equal to the rational's projection
+
+```agda
+module _
+  {l : Level}
+  (x : upper-ℝ l)
+  (q : ℚ)
+  where
+
+  leq-is-in-cut-upper-real-ℚ : is-in-cut-upper-ℝ x q → leq-upper-ℝ x (upper-real-ℚ q)
+  leq-is-in-cut-upper-real-ℚ q∈L p x<p =
+    backward-implication
+      ( is-rounded-cut-upper-ℝ x p)
+      ( intro-exists q (x<p , q∈L))
+```
+
+
 ### The canonical map from the rational numbers to the upper reals preserves inequality
 
 ```agda
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 25081b1abe..3af658ba6e 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
@@ -32,6 +32,8 @@ 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
 open import real-numbers.upper-dedekind-real-numbers
+open import real-numbers.inequality-lower-dedekind-real-numbers
+open import real-numbers.inequality-upper-dedekind-real-numbers
 ```
 
 </details>

From 098d4e88a48b3e1b87da820e5af988c4adc43399 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 17:20:47 -0800
Subject: [PATCH 40/73] Finish gaps

---
 .../inequality-rational-numbers.lagda.md      | 256 +++++++++---------
 .../rational-numbers.lagda.md                 |  17 +-
 ...thmetically-located-dedekind-cuts.lagda.md |   5 +-
 ...ality-lower-dedekind-real-numbers.lagda.md |   3 +-
 ...ality-upper-dedekind-real-numbers.lagda.md |   4 +-
 ...lower-upper-dedekind-real-numbers.lagda.md |   4 +-
 6 files changed, 152 insertions(+), 137 deletions(-)

diff --git a/src/elementary-number-theory/inequality-rational-numbers.lagda.md b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
index a945128fad..12ddd6f34a 100644
--- a/src/elementary-number-theory/inequality-rational-numbers.lagda.md
+++ b/src/elementary-number-theory/inequality-rational-numbers.lagda.md
@@ -103,33 +103,35 @@ refl-leq-ℚ x =
 ### Inequality on the rational numbers is antisymmetric
 
 ```agda
-antisymmetric-leq-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℚ y x → x ＝ y
-antisymmetric-leq-ℚ x y H H' =
-  ( inv (is-retraction-rational-fraction-ℚ x)) ∙
-  ( eq-ℚ-sim-fraction-ℤ
-    ( fraction-ℚ x)
-    ( fraction-ℚ y)
-    ( is-sim-antisymmetric-leq-fraction-ℤ
+abstract
+  antisymmetric-leq-ℚ : (x y : ℚ) → leq-ℚ x y → leq-ℚ y x → x ＝ y
+  antisymmetric-leq-ℚ x y H H' =
+    ( inv (is-retraction-rational-fraction-ℚ x)) ∙
+    ( eq-ℚ-sim-fraction-ℤ
       ( fraction-ℚ x)
       ( fraction-ℚ y)
-      ( H)
-      ( H'))) ∙
-  ( is-retraction-rational-fraction-ℚ y)
+      ( is-sim-antisymmetric-leq-fraction-ℤ
+        ( fraction-ℚ x)
+        ( fraction-ℚ y)
+        ( H)
+        ( H'))) ∙
+    ( is-retraction-rational-fraction-ℚ y)
 ```
 
 ### Inequality on the rational numbers is linear
 
 ```agda
-linear-leq-ℚ : (x y : ℚ) → (leq-ℚ x y) + (leq-ℚ y x)
-linear-leq-ℚ x y =
-  map-coproduct
-    ( id)
-    ( is-nonnegative-eq-ℤ
-      (distributive-neg-diff-ℤ
-        ( numerator-ℚ y *ℤ denominator-ℚ x)
-        ( numerator-ℚ x *ℤ denominator-ℚ y)))
-    ( decide-is-nonnegative-is-nonnegative-neg-ℤ
-      { cross-mul-diff-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)})
+abstract
+  linear-leq-ℚ : (x y : ℚ) → (leq-ℚ x y) + (leq-ℚ y x)
+  linear-leq-ℚ x y =
+    map-coproduct
+      ( id)
+      ( is-nonnegative-eq-ℤ
+        (distributive-neg-diff-ℤ
+          ( numerator-ℚ y *ℤ denominator-ℚ x)
+          ( numerator-ℚ x *ℤ denominator-ℚ y)))
+      ( decide-is-nonnegative-is-nonnegative-neg-ℤ
+        { cross-mul-diff-fraction-ℤ (fraction-ℚ x) (fraction-ℚ y)})
 ```
 
 ### Inequality on the rational numbers is transitive
@@ -139,12 +141,13 @@ module _
   (x y z : ℚ)
   where
 
-  transitive-leq-ℚ : leq-ℚ y z → leq-ℚ x y → leq-ℚ x z
-  transitive-leq-ℚ =
-    transitive-leq-fraction-ℤ
-      ( fraction-ℚ x)
-      ( fraction-ℚ y)
-      ( fraction-ℚ z)
+  abstract
+    transitive-leq-ℚ : leq-ℚ y z → leq-ℚ x y → leq-ℚ x z
+    transitive-leq-ℚ =
+      transitive-leq-fraction-ℤ
+        ( fraction-ℚ x)
+        ( fraction-ℚ y)
+        ( fraction-ℚ z)
 ```
 
 ### The partially ordered set of rational numbers ordered by inequality
@@ -165,43 +168,45 @@ module _
   (p q : fraction-ℤ)
   where
 
-  preserves-leq-rational-fraction-ℤ :
-    leq-fraction-ℤ p q → leq-ℚ (rational-fraction-ℤ p) (rational-fraction-ℤ q)
-  preserves-leq-rational-fraction-ℤ =
-    preserves-leq-sim-fraction-ℤ
-      ( p)
-      ( q)
-      ( reduce-fraction-ℤ p)
-      ( reduce-fraction-ℤ q)
-      ( sim-reduced-fraction-ℤ p)
-      ( sim-reduced-fraction-ℤ q)
+  abstract
+    preserves-leq-rational-fraction-ℤ :
+      leq-fraction-ℤ p q → leq-ℚ (rational-fraction-ℤ p) (rational-fraction-ℤ q)
+    preserves-leq-rational-fraction-ℤ =
+      preserves-leq-sim-fraction-ℤ
+        ( p)
+        ( q)
+        ( reduce-fraction-ℤ p)
+        ( reduce-fraction-ℤ q)
+        ( sim-reduced-fraction-ℤ p)
+        ( sim-reduced-fraction-ℤ q)
 
 module _
   (x : ℚ) (p : fraction-ℤ)
   where
 
-  preserves-leq-right-rational-fraction-ℤ :
-    leq-fraction-ℤ (fraction-ℚ x) p → leq-ℚ x (rational-fraction-ℤ p)
-  preserves-leq-right-rational-fraction-ℤ H =
-    concatenate-leq-sim-fraction-ℤ
-      ( fraction-ℚ x)
-      ( p)
-      ( fraction-ℚ ( rational-fraction-ℤ p))
-      ( H)
-      ( sim-reduced-fraction-ℤ p)
-
-  reflects-leq-right-rational-fraction-ℤ :
-    leq-ℚ x (rational-fraction-ℤ p) → leq-fraction-ℤ (fraction-ℚ x) p
-  reflects-leq-right-rational-fraction-ℤ H =
-    concatenate-leq-sim-fraction-ℤ
-      ( fraction-ℚ x)
-      ( reduce-fraction-ℤ p)
-      ( p)
-      ( H)
-      ( symmetric-sim-fraction-ℤ
+  abstract
+    preserves-leq-right-rational-fraction-ℤ :
+      leq-fraction-ℤ (fraction-ℚ x) p → leq-ℚ x (rational-fraction-ℤ p)
+    preserves-leq-right-rational-fraction-ℤ H =
+      concatenate-leq-sim-fraction-ℤ
+        ( fraction-ℚ x)
         ( p)
+        ( fraction-ℚ ( rational-fraction-ℤ p))
+        ( H)
+        ( sim-reduced-fraction-ℤ p)
+
+    reflects-leq-right-rational-fraction-ℤ :
+      leq-ℚ x (rational-fraction-ℤ p) → leq-fraction-ℤ (fraction-ℚ x) p
+    reflects-leq-right-rational-fraction-ℤ H =
+      concatenate-leq-sim-fraction-ℤ
+        ( fraction-ℚ x)
         ( reduce-fraction-ℤ p)
-        ( sim-reduced-fraction-ℤ p))
+        ( p)
+        ( H)
+        ( symmetric-sim-fraction-ℤ
+          ( p)
+          ( reduce-fraction-ℤ p)
+          ( sim-reduced-fraction-ℤ p))
 
   iff-leq-right-rational-fraction-ℤ :
     leq-fraction-ℤ (fraction-ℚ x) p ↔ leq-ℚ x (rational-fraction-ℤ p)
@@ -209,26 +214,27 @@ module _
     preserves-leq-right-rational-fraction-ℤ
   pr2 iff-leq-right-rational-fraction-ℤ = reflects-leq-right-rational-fraction-ℤ
 
-  preserves-leq-left-rational-fraction-ℤ :
-    leq-fraction-ℤ p (fraction-ℚ x) → leq-ℚ (rational-fraction-ℤ p) x
-  preserves-leq-left-rational-fraction-ℤ =
-    concatenate-sim-leq-fraction-ℤ
-      ( fraction-ℚ ( rational-fraction-ℤ p))
-      ( p)
-      ( fraction-ℚ x)
-      ( symmetric-sim-fraction-ℤ
-        ( p)
+  abstract
+    preserves-leq-left-rational-fraction-ℤ :
+      leq-fraction-ℤ p (fraction-ℚ x) → leq-ℚ (rational-fraction-ℤ p) x
+    preserves-leq-left-rational-fraction-ℤ =
+      concatenate-sim-leq-fraction-ℤ
         ( fraction-ℚ ( rational-fraction-ℤ p))
-        ( sim-reduced-fraction-ℤ p))
-
-  reflects-leq-left-rational-fraction-ℤ :
-    leq-ℚ (rational-fraction-ℤ p) x → leq-fraction-ℤ p (fraction-ℚ x)
-  reflects-leq-left-rational-fraction-ℤ =
-    concatenate-sim-leq-fraction-ℤ
-      ( p)
-      ( reduce-fraction-ℤ p)
-      ( fraction-ℚ x)
-      ( sim-reduced-fraction-ℤ p)
+        ( p)
+        ( fraction-ℚ x)
+        ( symmetric-sim-fraction-ℤ
+          ( p)
+          ( fraction-ℚ ( rational-fraction-ℤ p))
+          ( sim-reduced-fraction-ℤ p))
+
+    reflects-leq-left-rational-fraction-ℤ :
+      leq-ℚ (rational-fraction-ℤ p) x → leq-fraction-ℤ p (fraction-ℚ x)
+    reflects-leq-left-rational-fraction-ℤ =
+      concatenate-sim-leq-fraction-ℤ
+        ( p)
+        ( reduce-fraction-ℤ p)
+        ( fraction-ℚ x)
+        ( sim-reduced-fraction-ℤ p)
 
   iff-leq-left-rational-fraction-ℤ :
     leq-fraction-ℤ p (fraction-ℚ x) ↔ leq-ℚ (rational-fraction-ℤ p) x
@@ -243,26 +249,29 @@ module _
   (x y : ℚ)
   where
 
-  iff-translate-diff-leq-zero-ℚ : leq-ℚ zero-ℚ (y -ℚ x) ↔ leq-ℚ x y
-  iff-translate-diff-leq-zero-ℚ =
-    logical-equivalence-reasoning
-      leq-ℚ zero-ℚ (y -ℚ x)
-      ↔ leq-fraction-ℤ
-        ( zero-fraction-ℤ)
-        ( add-fraction-ℤ (fraction-ℚ y) (neg-fraction-ℤ (fraction-ℚ x)))
-        by
-          inv-iff
-            ( iff-leq-right-rational-fraction-ℤ
-              ( zero-ℚ)
-              ( add-fraction-ℤ (fraction-ℚ y) (neg-fraction-ℤ (fraction-ℚ x))))
-      ↔ leq-ℚ x y
-        by
-          inv-tr
-            ( _↔ leq-ℚ x y)
-            ( eq-translate-diff-leq-zero-fraction-ℤ
-              ( fraction-ℚ x)
-              ( fraction-ℚ y))
-            ( id-iff)
+  abstract
+    iff-translate-diff-leq-zero-ℚ : leq-ℚ zero-ℚ (y -ℚ x) ↔ leq-ℚ x y
+    iff-translate-diff-leq-zero-ℚ =
+      logical-equivalence-reasoning
+        leq-ℚ zero-ℚ (y -ℚ x)
+        ↔ leq-fraction-ℤ
+          ( zero-fraction-ℤ)
+          ( add-fraction-ℤ (fraction-ℚ y) (neg-fraction-ℤ (fraction-ℚ x)))
+          by
+            inv-iff
+              ( iff-leq-right-rational-fraction-ℤ
+                ( zero-ℚ)
+                ( add-fraction-ℤ
+                  ( fraction-ℚ y)
+                  ( neg-fraction-ℤ (fraction-ℚ x))))
+        ↔ leq-ℚ x y
+          by
+            inv-tr
+              ( _↔ leq-ℚ x y)
+              ( eq-translate-diff-leq-zero-fraction-ℤ
+                ( fraction-ℚ x)
+                ( fraction-ℚ y))
+              ( id-iff)
 ```
 
 ### Inequality on the rational numbers is invariant by translation
@@ -272,34 +281,35 @@ module _
   (z x y : ℚ)
   where
 
-  iff-translate-left-leq-ℚ : leq-ℚ (z +ℚ x) (z +ℚ y) ↔ leq-ℚ x y
-  iff-translate-left-leq-ℚ =
-    logical-equivalence-reasoning
-      leq-ℚ (z +ℚ x) (z +ℚ y)
-      ↔ leq-ℚ zero-ℚ ((z +ℚ y) -ℚ (z +ℚ x))
-        by (inv-iff (iff-translate-diff-leq-zero-ℚ (z +ℚ x) (z +ℚ y)))
-      ↔ leq-ℚ zero-ℚ (y -ℚ x)
-        by
-          ( inv-tr
-            ( _↔ leq-ℚ zero-ℚ (y -ℚ x))
-            ( ap (leq-ℚ zero-ℚ) (left-translation-diff-ℚ y x z))
-            ( id-iff))
-      ↔ leq-ℚ x y
-        by (iff-translate-diff-leq-zero-ℚ x y)
-
-  iff-translate-right-leq-ℚ : leq-ℚ (x +ℚ z) (y +ℚ z) ↔ leq-ℚ x y
-  iff-translate-right-leq-ℚ =
-    logical-equivalence-reasoning
-      leq-ℚ (x +ℚ z) (y +ℚ z)
-      ↔ leq-ℚ zero-ℚ ((y +ℚ z) -ℚ (x +ℚ z))
-        by (inv-iff (iff-translate-diff-leq-zero-ℚ (x +ℚ z) (y +ℚ z)))
-      ↔ leq-ℚ zero-ℚ (y -ℚ x)
-        by
-          ( inv-tr
-            ( _↔ leq-ℚ zero-ℚ (y -ℚ x))
-            ( ap (leq-ℚ zero-ℚ) (right-translation-diff-ℚ y x z))
-            ( id-iff))
-      ↔ leq-ℚ x y by (iff-translate-diff-leq-zero-ℚ x y)
+  abstract
+    iff-translate-left-leq-ℚ : leq-ℚ (z +ℚ x) (z +ℚ y) ↔ leq-ℚ x y
+    iff-translate-left-leq-ℚ =
+      logical-equivalence-reasoning
+        leq-ℚ (z +ℚ x) (z +ℚ y)
+        ↔ leq-ℚ zero-ℚ ((z +ℚ y) -ℚ (z +ℚ x))
+          by (inv-iff (iff-translate-diff-leq-zero-ℚ (z +ℚ x) (z +ℚ y)))
+        ↔ leq-ℚ zero-ℚ (y -ℚ x)
+          by
+            ( inv-tr
+              ( _↔ leq-ℚ zero-ℚ (y -ℚ x))
+              ( ap (leq-ℚ zero-ℚ) (left-translation-diff-ℚ y x z))
+              ( id-iff))
+        ↔ leq-ℚ x y
+          by (iff-translate-diff-leq-zero-ℚ x y)
+
+    iff-translate-right-leq-ℚ : leq-ℚ (x +ℚ z) (y +ℚ z) ↔ leq-ℚ x y
+    iff-translate-right-leq-ℚ =
+      logical-equivalence-reasoning
+        leq-ℚ (x +ℚ z) (y +ℚ z)
+        ↔ leq-ℚ zero-ℚ ((y +ℚ z) -ℚ (x +ℚ z))
+          by (inv-iff (iff-translate-diff-leq-zero-ℚ (x +ℚ z) (y +ℚ z)))
+        ↔ leq-ℚ zero-ℚ (y -ℚ x)
+          by
+            ( inv-tr
+              ( _↔ leq-ℚ zero-ℚ (y -ℚ x))
+              ( ap (leq-ℚ zero-ℚ) (right-translation-diff-ℚ y x z))
+              ( id-iff))
+        ↔ leq-ℚ x y by (iff-translate-diff-leq-zero-ℚ x y)
 
   preserves-leq-left-add-ℚ : leq-ℚ x y → leq-ℚ (x +ℚ z) (y +ℚ z)
   preserves-leq-left-add-ℚ = backward-implication iff-translate-right-leq-ℚ
diff --git a/src/elementary-number-theory/rational-numbers.lagda.md b/src/elementary-number-theory/rational-numbers.lagda.md
index f4b630199a..461cb3854d 100644
--- a/src/elementary-number-theory/rational-numbers.lagda.md
+++ b/src/elementary-number-theory/rational-numbers.lagda.md
@@ -155,14 +155,15 @@ mediant-ℚ x y =
 ### The rational images of two similar integer fractions are equal
 
 ```agda
-eq-ℚ-sim-fraction-ℤ :
-  (x y : fraction-ℤ) → (H : sim-fraction-ℤ x y) →
-  rational-fraction-ℤ x ＝ rational-fraction-ℤ y
-eq-ℚ-sim-fraction-ℤ x y H =
-  eq-pair-Σ'
-    ( pair
-      ( unique-reduce-fraction-ℤ x y H)
-      ( eq-is-prop (is-prop-is-reduced-fraction-ℤ (reduce-fraction-ℤ y))))
+abstract
+  eq-ℚ-sim-fraction-ℤ :
+    (x y : fraction-ℤ) → (H : sim-fraction-ℤ x y) →
+    rational-fraction-ℤ x ＝ rational-fraction-ℤ y
+  eq-ℚ-sim-fraction-ℤ x y H =
+    eq-pair-Σ'
+      ( pair
+        ( unique-reduce-fraction-ℤ x y H)
+        ( eq-is-prop (is-prop-is-reduced-fraction-ℤ (reduce-fraction-ℤ y))))
 ```
 
 ### The type of rationals is a set
diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 06174ce77c..3df7d7cd6d 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -106,7 +106,10 @@ module _
                     ( p' +ℚ (q -ℚ p))
                     ( q)
                     ( q'<p'+q-p)
-                    {! binary-tr leq-ℚ ? ? (preserves-le-left-add-ℚ q (p' -ℚ p) zero-ℚ ?)  !})
+                  ( tr
+                    ( leq-ℚ (p' +ℚ (q -ℚ p)))
+                    ( is-identity-right-conjugation-Ab abelian-group-add-ℚ p q)
+                    ( 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))
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 7a0a1d476b..7a73363b9c 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -85,7 +85,8 @@ module _
   (q : ℚ)
   where
 
-  leq-is-in-cut-lower-real-ℚ : is-in-cut-lower-ℝ x q → leq-lower-ℝ (lower-real-ℚ q) x
+  leq-is-in-cut-lower-real-ℚ :
+    is-in-cut-lower-ℝ x q → leq-lower-ℝ (lower-real-ℚ q) x
   leq-is-in-cut-lower-real-ℚ q∈L p p<q =
     backward-implication
       ( is-rounded-cut-lower-ℝ x p)
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 ec83e2080c..45cdddb26d 100644
--- a/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-upper-dedekind-real-numbers.lagda.md
@@ -84,14 +84,14 @@ module _
   (q : ℚ)
   where
 
-  leq-is-in-cut-upper-real-ℚ : is-in-cut-upper-ℝ x q → leq-upper-ℝ x (upper-real-ℚ q)
+  leq-is-in-cut-upper-real-ℚ :
+    is-in-cut-upper-ℝ x q → leq-upper-ℝ x (upper-real-ℚ q)
   leq-is-in-cut-upper-real-ℚ q∈L p x<p =
     backward-implication
       ( is-rounded-cut-upper-ℝ x p)
       ( intro-exists q (x<p , q∈L))
 ```
 
-
 ### The canonical map from the rational numbers to the upper reals preserves inequality
 
 ```agda
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 3af658ba6e..5c6f4d84f5 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
@@ -28,12 +28,12 @@ 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
 open import real-numbers.upper-dedekind-real-numbers
-open import real-numbers.inequality-lower-dedekind-real-numbers
-open import real-numbers.inequality-upper-dedekind-real-numbers
 ```
 
 </details>

From b3b72b244d870facf903ebbd95993c18881cde0b Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 19:28:57 -0800
Subject: [PATCH 41/73] Progress

---
 ...thmetically-located-dedekind-cuts.lagda.md |  10 +-
 .../dedekind-real-numbers.lagda.md            | 222 +++++++++++-------
 .../lower-dedekind-real-numbers.lagda.md      |  32 ++-
 .../upper-dedekind-real-numbers.lagda.md      |  32 ++-
 4 files changed, 199 insertions(+), 97 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 3df7d7cd6d..f4b8c30560 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -60,8 +60,8 @@ module _
   {l : Level} (L : lower-ℝ l) (U : upper-ℝ l)
   where
 
-  is-arithmetically-located-pair-of-cuts-ℚ : UU l
-  is-arithmetically-located-pair-of-cuts-ℚ =
+  arithmetically-located-lower-upper-ℝ : UU l
+  arithmetically-located-lower-upper-ℝ =
     (ε⁺ : ℚ⁺) →
     exists
       ( ℚ × ℚ)
@@ -79,13 +79,13 @@ rational numbers, it is also located.
 
 ```agda
 module _
-  {l : Level} (L : lower-ℝ l) (U : upper-ℝ l)
+  {l : Level} (x : lower-ℝ l) (y : upper-ℝ l)
   where
 
   abstract
     is-located-is-arithmetically-located-pair-of-cuts-ℚ :
-      is-arithmetically-located-pair-of-cuts-ℚ L U →
-      is-located-cut (cut-lower-ℝ L) (cut-upper-ℝ U)
+      arithmetically-located-upper-real x y →
+      is-located-cut (cut-lower-ℝ x) (cut-upper-ℝ y)
     is-located-is-arithmetically-located-pair-of-cuts-ℚ
       arithmetically-located p q p<q =
       elim-exists
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index f8515d6123..d4ec0c3e69 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -11,6 +11,7 @@ open import elementary-number-theory.inequality-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
@@ -48,88 +49,81 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-A
-{{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
-consists of a [pair](foundation.dependent-pair-types.md) `(L , U)` of a
-[lower Dedekind cut](real-numbers.lower-dedekind-real-numbers) and an
-[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers) that also satisfy
+A Dedekind real number
+consists of a [pair](foundation.dependent-pair-types.md) `(lx , uy)` of a
+[lower Dedekind real](real-numbers.lower-dedekind-real-numbers) and an
+[upper Dedekind real](real-numbers.upper-dedekind-real-numbers) that also satisfy
 the following conditions:
 
-1. _Disjointness_. `L` and `U` are disjoint subsets of `ℚ`.
-2. _Locatedness_. If `q < r` then `q ∈ L` or `r ∈ U`.
+1. _Disjointness_. The cuts of `lx` and `uy` are disjoint subsets of `ℚ`.
+2. _Locatedness_. If `q < r` then `q` is in the cut of `lx` or `r` is in the cut
+    of `uy`.
 
-The type of {{#concept "Dedekind real numbers" Agda=ℝ}} is the type of all
-Dedekind cuts. The Dedekind real numbers will be taken as the standard
+The Dedekind real numbers will be taken as the standard
 definition of the real numbers in the agda-unimath library.
 
 ## Definition
 
-### Dedekind cuts
+### Dedekind real numbers
 
 ```agda
 module _
-  {l1 l2 : Level} (L : subtype l1 ℚ) (U : subtype l2 ℚ)
+  {l1 l2 : Level} (lx : lower-ℝ l1) (uy : upper-ℝ l2)
   where
 
-  is-disjoint-cut-Prop : Prop (l1 ⊔ l2)
-  is-disjoint-cut-Prop = ∀' ℚ (λ q → ¬' (L q ∧ U q))
+  is-disjoint-prop-lower-upper-ℝ : Prop (l1 ⊔ l2)
+  is-disjoint-prop-lower-upper-ℝ =
+    ∀' ℚ (λ q → ¬' (cut-lower-ℝ lx q ∧ cut-upper-ℝ uy q))
 
-  is-disjoint-cut : UU (l1 ⊔ l2)
-  is-disjoint-cut = type-Prop is-disjoint-cut-Prop
+  is-disjoint-lower-upper-ℝ : UU (l1 ⊔ l2)
+  is-disjoint-lower-upper-ℝ = type-Prop is-disjoint-prop-lower-upper-ℝ
 
-  is-located-cut-Prop : Prop (l1 ⊔ l2)
-  is-located-cut-Prop = ∀' ℚ (λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (L q ∨ U r)))
+  is-located-prop-lower-upper-ℝ : Prop (l1 ⊔ l2)
+  is-located-prop-lower-upper-ℝ = ∀'
+    ( ℚ)
+    ( λ q → ∀' ℚ (λ r → le-ℚ-Prop q r ⇒ (cut-lower-ℝ lx q ∨ cut-upper-ℝ uy r)))
 
-  is-located-cut : UU (l1 ⊔ l2)
-  is-located-cut = type-Prop is-located-cut-Prop
+  is-located-lower-upper-ℝ : UU (l1 ⊔ l2)
+  is-located-lower-upper-ℝ = type-Prop is-located-prop-lower-upper-ℝ
 
-  is-dedekind-cut-Prop : Prop (l1 ⊔ l2)
-  is-dedekind-cut-Prop =
-    is-lower-dedekind-cut-Prop L ∧
-    is-upper-dedekind-cut-Prop U ∧
-    is-disjoint-cut-Prop ∧
-    is-located-cut-Prop
+  is-dedekind-prop-lower-upper-ℝ : Prop (l1 ⊔ l2)
+  is-dedekind-prop-lower-upper-ℝ =
+    is-disjoint-prop-lower-upper-ℝ ∧ is-located-prop-lower-upper-ℝ
 
-  is-dedekind-cut : UU (l1 ⊔ l2)
-  is-dedekind-cut = type-Prop is-dedekind-cut-Prop
-
-  is-prop-is-dedekind-cut : is-prop is-dedekind-cut
-  is-prop-is-dedekind-cut = is-prop-type-Prop is-dedekind-cut-Prop
+  is-dedekind-lower-upper-ℝ : UU (l1 ⊔ l2)
+  is-dedekind-lower-upper-ℝ = type-Prop is-dedekind-prop-lower-upper-ℝ
 ```
 
 ### The Dedekind real numbers
 
 ```agda
 ℝ : (l : Level) → UU (lsuc l)
-ℝ l = Σ (subtype l ℚ) (λ L → Σ (subtype l ℚ) (is-dedekind-cut L))
-
-real-dedekind-cut : {l : Level} (L U : subtype l ℚ) → is-dedekind-cut L U → ℝ l
-real-dedekind-cut L U H = L , U , H
+ℝ l = Σ (lower-ℝ l) (λ lx → Σ (upper-ℝ l) (is-dedekind-lower-upper-ℝ lx))
 
 real-lower-upper-ℝ :
   {l : Level} →
-  (L : lower-ℝ l) (U : upper-ℝ l) →
-  is-disjoint-cut (cut-lower-ℝ L) (cut-upper-ℝ U) →
-  is-located-cut (cut-lower-ℝ L) (cut-upper-ℝ U) →
+  (lx : lower-ℝ l) (uy : upper-ℝ l) →
+  is-disjoint-lower-upper-ℝ lx uy →
+  is-located-lower-upper-ℝ lx uy →
   ℝ l
-real-lower-upper-ℝ (L , lower-dedekind-L) (U , lower-dedekind-U) H K =
-  L , U , lower-dedekind-L , lower-dedekind-U , H , K
+real-lower-upper-ℝ lx uy H K =
+  lx , uy , H , K
 
 module _
   {l : Level} (x : ℝ l)
   where
 
-  lower-cut-ℝ : subtype l ℚ
-  lower-cut-ℝ = pr1 x
-
-  upper-cut-ℝ : subtype l ℚ
-  upper-cut-ℝ = pr1 (pr2 x)
-
   lower-real-ℝ : lower-ℝ l
-  lower-real-ℝ = lower-cut-ℝ , pr1 (pr2 (pr2 x))
+  lower-real-ℝ = pr1 x
 
   upper-real-ℝ : upper-ℝ l
-  upper-real-ℝ = upper-cut-ℝ , pr1 (pr2 (pr2 (pr2 x)))
+  upper-real-ℝ = pr1 (pr2 x)
+
+  lower-cut-ℝ : subtype l ℚ
+  lower-cut-ℝ = cut-lower-ℝ lower-real-ℝ
+
+  upper-cut-ℝ : subtype l ℚ
+  upper-cut-ℝ = cut-upper-ℝ upper-real-ℝ
 
   is-in-lower-cut-ℝ : ℚ → UU l
   is-in-lower-cut-ℝ = is-in-subtype lower-cut-ℝ
@@ -137,8 +131,8 @@ module _
   is-in-upper-cut-ℝ : ℚ → UU l
   is-in-upper-cut-ℝ = is-in-subtype upper-cut-ℝ
 
-  is-dedekind-cut-cut-ℝ : is-dedekind-cut lower-cut-ℝ upper-cut-ℝ
-  is-dedekind-cut-cut-ℝ = pr2 (pr2 x)
+  is-dedekind-ℝ : is-dedekind-lower-upper-ℝ lower-real-ℝ upper-real-ℝ
+  is-dedekind-ℝ = pr2 (pr2 x)
 
   is-inhabited-lower-cut-ℝ : exists ℚ lower-cut-ℝ
   is-inhabited-lower-cut-ℝ = is-inhabited-cut-lower-ℝ lower-real-ℝ
@@ -159,12 +153,12 @@ module _
   is-rounded-upper-cut-ℝ = is-rounded-cut-upper-ℝ upper-real-ℝ
 
   is-disjoint-cut-ℝ : (q : ℚ) → ¬ (is-in-lower-cut-ℝ q × is-in-upper-cut-ℝ q)
-  is-disjoint-cut-ℝ = pr1 (pr2 (pr2 (pr2 (pr2 x))))
+  is-disjoint-cut-ℝ = pr1 (pr2 (pr2 x))
 
   is-located-lower-upper-cut-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 (pr2 (pr2 x))))
+  is-located-lower-upper-cut-ℝ = pr2 (pr2 (pr2 x))
 
   cut-ℝ : subtype l ℚ
   cut-ℝ q =
@@ -182,18 +176,19 @@ module _
 ### The Dedekind real numbers form a set
 
 ```agda
+
 abstract
   is-set-ℝ : (l : Level) → is-set (ℝ l)
   is-set-ℝ l =
     is-set-Σ
-      ( is-set-function-type (is-trunc-Truncated-Type neg-one-𝕋))
+      ( is-set-lower-ℝ l)
       ( λ x →
         is-set-Σ
-          ( is-set-function-type (is-trunc-Truncated-Type neg-one-𝕋))
+          ( is-set-upper-ℝ l)
           ( λ y →
             ( is-set-is-prop
               ( is-prop-type-Prop
-                ( is-dedekind-cut-Prop x y)))))
+                ( is-dedekind-prop-lower-upper-ℝ x y)))))
 
 ℝ-Set : (l : Level) → Set (lsuc l)
 ℝ-Set l = ℝ l , is-set-ℝ l
@@ -212,53 +207,25 @@ module _
     le-ℚ p q →
     is-in-lower-cut-ℝ x q →
     is-in-lower-cut-ℝ x p
-  le-lower-cut-ℝ H H' =
-    ind-trunc-Prop
-      ( λ s → lower-cut-ℝ x p)
-      ( rec-coproduct
-          ( id)
-          ( λ I → ex-falso (is-disjoint-cut-ℝ x q (H' , I))))
-      ( is-located-lower-upper-cut-ℝ x p q H)
+  le-lower-cut-ℝ = le-cut-lower-ℝ (lower-real-ℝ x) p q
 
   leq-lower-cut-ℝ :
     leq-ℚ p q →
     is-in-lower-cut-ℝ x q →
     is-in-lower-cut-ℝ x p
-  leq-lower-cut-ℝ H H' =
-    rec-coproduct
-      ( λ s → le-lower-cut-ℝ s H')
-      ( λ I →
-        tr
-          ( is-in-lower-cut-ℝ x)
-          ( antisymmetric-leq-ℚ q p I H)
-          ( H'))
-      ( decide-le-leq-ℚ p q)
+  leq-lower-cut-ℝ = leq-cut-lower-ℝ (lower-real-ℝ x) p q
 
   le-upper-cut-ℝ :
     le-ℚ p q →
     is-in-upper-cut-ℝ x p →
     is-in-upper-cut-ℝ x q
-  le-upper-cut-ℝ H H' =
-    ind-trunc-Prop
-      ( λ s → upper-cut-ℝ x q)
-      ( rec-coproduct
-        ( λ I → ex-falso (is-disjoint-cut-ℝ x p ( I , H')))
-        ( id))
-      ( is-located-lower-upper-cut-ℝ x p q H)
+  le-upper-cut-ℝ = le-cut-upper-ℝ (upper-real-ℝ x) p q
 
   leq-upper-cut-ℝ :
     leq-ℚ p q →
     is-in-upper-cut-ℝ x p →
     is-in-upper-cut-ℝ x q
-  leq-upper-cut-ℝ H H' =
-    rec-coproduct
-      ( λ s → le-upper-cut-ℝ s H')
-      ( λ I →
-        tr
-          ( is-in-upper-cut-ℝ x)
-          ( antisymmetric-leq-ℚ p q H I)
-          ( H'))
-      ( decide-le-leq-ℚ p q)
+  leq-upper-cut-ℝ = leq-cut-upper-ℝ (upper-real-ℝ x) p q
 ```
 
 ### Elements of the lower cut are lower bounds of the upper cut
@@ -448,10 +415,85 @@ module _
           ( H)))
 ```
 
-### The map from a real number to its lower cut is an embedding
+### The map from a real number to its lower real is an embedding
+
+```agda
+module _
+  {l : Level}
+  (lx : lower-ℝ l)
+  where
+
+  has-upper-real-Prop : Prop (lsuc l)
+  pr1 has-upper-real-Prop = Σ (upper-ℝ l) (is-dedekind-lower-upper-ℝ lx)
+  pr2 has-upper-real-Prop =
+    ( is-prop-all-elements-equal)
+    ( λ uy uy' →
+      eq-type-subtype
+        ( is-dedekind-prop-lower-upper-ℝ lx)
+        ( eq-eq-cut-upper-ℝ
+          ( pr1 uy)
+          ( pr1 uy')
+          ( eq-upper-cut-eq-lower-cut-ℝ (lx , uy) (lx , uy') refl)))
+
+is-emb-lower-real : {l : Level} → is-emb (lower-real-ℝ {l})
+is-emb-lower-real = is-emb-inclusion-subtype has-upper-real-Prop
+```
+
+### The map from a real number to its upper real is an embedding
 
 ```agda
 module _
+  {l : Level}
+  (uy : upper-ℝ l)
+  where
+
+  has-lower-real-Prop : Prop (lsuc l)
+  pr1 has-lower-real-Prop =
+    Σ (lower-ℝ l) (λ lx → is-dedekind-lower-upper-ℝ lx uy)
+  pr2 has-lower-real-Prop =
+    ( is-prop-all-elements-equal)
+    ( λ lx lx' →
+      eq-type-subtype
+        ( λ l → is-dedekind-prop-lower-upper-ℝ l uy)
+        ( eq-eq-cut-lower-ℝ
+          ( pr1 lx)
+          ( pr1 lx')
+          ( eq-lower-cut-eq-upper-cut-ℝ
+            ( pr1 lx , uy , pr2 lx)
+            ( pr1 lx' , uy , pr2 lx')
+            ( refl))))
+
+is-emb-upper-real : {l : Level} → is-emb (upper-real-ℝ {l})
+pr1 (pr1 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) ux=uy =
+  ap
+    ( λ (uz , lz , Hz) → (lz , uz , Hz))
+    ( pr1
+      ( pr1
+        ( is-emb-inclusion-subtype
+          ( has-lower-real-Prop)
+          ( ux , lx , Hx)
+          ( uy , ly , Hy)))
+      ( ux=uy))
+pr2 (pr1 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) ux=uy =
+  {! (pr2 (pr1 (is-emb-inclusion-subtype has-lower-real-Prop (ux , lx , Hx) (uy , ly , Hy))) ux=uy) !}
+pr1 (pr2 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) ux=uy =
+  ap
+    ( λ (uz , lz , Hz) → (lz , uz , Hz))
+    ( pr1
+      (pr2
+        (is-emb-inclusion-subtype
+          ( has-lower-real-Prop)
+          ( ux , lx , Hx)
+          ( uy , ly , Hy)))
+      ux=uy)
+pr2 (pr2 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) x=y =
+  {! pr2 (pr2 (is-emb-inclusion-subtype has-lower-real-Prop (ux , lx , Hx) (uy , ly , Hy))) (ap (λ (lz , uz , Hz) → (uz , lz , Hz)) x=y) !}
+```
+
+### The map from a real number to its lower cut is an embedding
+
+```agda
+{- module _
   {l : Level} (L : subtype l ℚ)
   where
 
@@ -469,13 +511,13 @@ module _
               ( refl))))
 
 is-emb-lower-cut : {l : Level} → is-emb (lower-cut-ℝ {l})
-is-emb-lower-cut = is-emb-inclusion-subtype has-upper-cut-Prop
+is-emb-lower-cut = is-emb-inclusion-subtype has-upper-cut-Prop -}
 ```
 
 ### Two real numbers with the same lower/upper cut are equal
 
 ```agda
-module _
+{- module _
   {l : Level} (x y : ℝ l)
   where
 
@@ -483,7 +525,7 @@ module _
   eq-eq-lower-cut-ℝ = eq-type-subtype has-upper-cut-Prop
 
   eq-eq-upper-cut-ℝ : upper-cut-ℝ x ＝ upper-cut-ℝ y → x ＝ y
-  eq-eq-upper-cut-ℝ = eq-eq-lower-cut-ℝ ∘ (eq-lower-cut-eq-upper-cut-ℝ x y)
+  eq-eq-upper-cut-ℝ = eq-eq-lower-cut-ℝ ∘ (eq-lower-cut-eq-upper-cut-ℝ x y) -}
 ```
 
 ## References
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 6bc94cd66d..948f6e60bd 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -7,16 +7,22 @@ module real-numbers.lower-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequality-rational-numbers
 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.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositions
+open import foundation.sets
 open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.truncated-types
+open import foundation.truncation-levels
 open import foundation.universal-quantification
 open import foundation.universe-levels
 ```
@@ -89,7 +95,18 @@ module _
 
 ## Properties
 
-### Lower Dedekind cuts are closed under the standard ordering on the rationals
+### The lower Dedekind reals form a set
+
+```agda
+abstract
+  is-set-lower-ℝ : (l : Level) → is-set (lower-ℝ l)
+  is-set-lower-ℝ l =
+    is-set-Σ
+      ( is-set-function-type (is-trunc-Truncated-Type neg-one-𝕋))
+      ( λ q → is-set-is-prop (is-prop-type-Prop (is-lower-dedekind-cut-Prop q)))
+```
+
+### Lower Dedekind cuts are closed under strict inequality on the rationals
 
 ```agda
 module _
@@ -103,6 +120,19 @@ module _
       ( intro-exists q (p<q , q<x))
 ```
 
+### Lower Dedekind cuts are closed under inequality on the rationals
+
+```agda
+module _
+  {l : Level} (x : lower-ℝ l) (p q : ℚ)
+  where
+
+  leq-cut-lower-ℝ : leq-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
+  leq-cut-lower-ℝ p≤q q<x with decide-le-leq-ℚ p q
+  ... | inl p<q = le-cut-lower-ℝ x p q p<q q<x
+  ... | inr q≤p = tr (is-in-cut-lower-ℝ x) (antisymmetric-leq-ℚ q p q≤p p≤q) q<x
+```
+
 ### Two lower real numbers with the same cut are equal
 
 ```agda
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index 01625b8046..69d21848f4 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -7,16 +7,22 @@ module real-numbers.upper-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.inequality-rational-numbers
 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.existential-quantification
 open import foundation.identity-types
 open import foundation.logical-equivalences
 open import foundation.propositions
+open import foundation.sets
 open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.truncated-types
+open import foundation.truncation-levels
 open import foundation.universal-quantification
 open import foundation.universe-levels
 ```
@@ -89,7 +95,18 @@ module _
 
 ## Properties
 
-### Upper Dedekind cuts are closed under the standard ordering on the rationals
+### The upper Dedekind reals form a set
+
+```agda
+abstract
+  is-set-upper-ℝ : (l : Level) → is-set (upper-ℝ l)
+  is-set-upper-ℝ l =
+    is-set-Σ
+      ( is-set-function-type (is-trunc-Truncated-Type neg-one-𝕋))
+      ( λ q → is-set-is-prop (is-prop-type-Prop (is-upper-dedekind-cut-Prop q)))
+```
+
+### Upper Dedekind cuts are closed under strict inequality on the rationals
 
 ```agda
 module _
@@ -103,6 +120,19 @@ module _
       ( intro-exists p (p<q , p<x))
 ```
 
+### Upper Dedekind cuts are closed under inequality on the rationals
+
+```agda
+module _
+  {l : Level} (x : upper-ℝ l) (p q : ℚ)
+  where
+
+  leq-cut-upper-ℝ : leq-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
+  leq-cut-upper-ℝ p≤q x<p with decide-le-leq-ℚ p q
+  ... | inl p<q = le-cut-upper-ℝ x p q p<q x<p
+  ... | inr q≤p = tr (is-in-cut-upper-ℝ x) (antisymmetric-leq-ℚ p q p≤q q≤p) x<p
+```
+
 ### Two upper real numbers with the same cut are equal
 
 ```agda

From 1f9d73a7860aa92fe678acf2683ac5783726d6d8 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Mon, 10 Feb 2025 19:45:56 -0800
Subject: [PATCH 42/73] Recover all the previous results

---
 .../dedekind-real-numbers.lagda.md            | 104 +++++-------------
 1 file changed, 27 insertions(+), 77 deletions(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index d4ec0c3e69..c973a7b3cc 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -21,6 +21,7 @@ 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.functoriality-cartesian-product-types
@@ -397,6 +398,14 @@ module _
           ( upper-cut-ℝ y)
           ( H)))
 
+  eq-lower-real-eq-upper-real-ℝ :
+    upper-real-ℝ x ＝ upper-real-ℝ y → lower-real-ℝ x ＝ lower-real-ℝ y
+  eq-lower-real-eq-upper-real-ℝ ux=uy =
+    eq-eq-cut-lower-ℝ
+      ( lower-real-ℝ x)
+      ( lower-real-ℝ y)
+      ( eq-lower-cut-eq-upper-cut-ℝ (ap cut-upper-ℝ ux=uy))
+
   eq-upper-cut-eq-lower-cut-ℝ :
     lower-cut-ℝ x ＝ lower-cut-ℝ y → upper-cut-ℝ x ＝ upper-cut-ℝ y
   eq-upper-cut-eq-lower-cut-ℝ H =
@@ -413,6 +422,14 @@ module _
           ( lower-cut-ℝ x)
           ( lower-cut-ℝ y)
           ( H)))
+
+  eq-upper-real-eq-lower-real-ℝ :
+    lower-real-ℝ x ＝ lower-real-ℝ y → upper-real-ℝ x ＝ upper-real-ℝ y
+  eq-upper-real-eq-lower-real-ℝ lx=ly =
+    eq-eq-cut-upper-ℝ
+      ( upper-real-ℝ x)
+      ( upper-real-ℝ y)
+      ( eq-upper-cut-eq-lower-cut-ℝ (ap cut-lower-ℝ lx=ly))
 ```
 
 ### The map from a real number to its lower real is an embedding
@@ -439,93 +456,26 @@ is-emb-lower-real : {l : Level} → is-emb (lower-real-ℝ {l})
 is-emb-lower-real = is-emb-inclusion-subtype has-upper-real-Prop
 ```
 
-### The map from a real number to its upper real is an embedding
+### Two real numbers with the same lower/upper real are equal
 
 ```agda
 module _
-  {l : Level}
-  (uy : upper-ℝ l)
-  where
-
-  has-lower-real-Prop : Prop (lsuc l)
-  pr1 has-lower-real-Prop =
-    Σ (lower-ℝ l) (λ lx → is-dedekind-lower-upper-ℝ lx uy)
-  pr2 has-lower-real-Prop =
-    ( is-prop-all-elements-equal)
-    ( λ lx lx' →
-      eq-type-subtype
-        ( λ l → is-dedekind-prop-lower-upper-ℝ l uy)
-        ( eq-eq-cut-lower-ℝ
-          ( pr1 lx)
-          ( pr1 lx')
-          ( eq-lower-cut-eq-upper-cut-ℝ
-            ( pr1 lx , uy , pr2 lx)
-            ( pr1 lx' , uy , pr2 lx')
-            ( refl))))
-
-is-emb-upper-real : {l : Level} → is-emb (upper-real-ℝ {l})
-pr1 (pr1 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) ux=uy =
-  ap
-    ( λ (uz , lz , Hz) → (lz , uz , Hz))
-    ( pr1
-      ( pr1
-        ( is-emb-inclusion-subtype
-          ( has-lower-real-Prop)
-          ( ux , lx , Hx)
-          ( uy , ly , Hy)))
-      ( ux=uy))
-pr2 (pr1 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) ux=uy =
-  {! (pr2 (pr1 (is-emb-inclusion-subtype has-lower-real-Prop (ux , lx , Hx) (uy , ly , Hy))) ux=uy) !}
-pr1 (pr2 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) ux=uy =
-  ap
-    ( λ (uz , lz , Hz) → (lz , uz , Hz))
-    ( pr1
-      (pr2
-        (is-emb-inclusion-subtype
-          ( has-lower-real-Prop)
-          ( ux , lx , Hx)
-          ( uy , ly , Hy)))
-      ux=uy)
-pr2 (pr2 (is-emb-upper-real (lx , ux , Hx) (ly , uy , Hy))) x=y =
-  {! pr2 (pr2 (is-emb-inclusion-subtype has-lower-real-Prop (ux , lx , Hx) (uy , ly , Hy))) (ap (λ (lz , uz , Hz) → (uz , lz , Hz)) x=y) !}
-```
-
-### The map from a real number to its lower cut is an embedding
-
-```agda
-{- module _
-  {l : Level} (L : subtype l ℚ)
+  {l : Level} (x y : ℝ l)
   where
 
-  has-upper-cut-Prop : Prop (lsuc l)
-  has-upper-cut-Prop =
-    pair
-      ( Σ (subtype l ℚ) (is-dedekind-cut L))
-      ( is-prop-all-elements-equal
-        ( λ U U' →
-          eq-type-subtype
-            ( is-dedekind-cut-Prop L)
-            ( eq-upper-cut-eq-lower-cut-ℝ
-              ( pair L U)
-              ( pair L U')
-              ( refl))))
-
-is-emb-lower-cut : {l : Level} → is-emb (lower-cut-ℝ {l})
-is-emb-lower-cut = is-emb-inclusion-subtype has-upper-cut-Prop -}
-```
-
-### Two real numbers with the same lower/upper cut are equal
+  eq-eq-lower-real-ℝ : lower-real-ℝ x ＝ lower-real-ℝ y → x ＝ y
+  eq-eq-lower-real-ℝ = eq-type-subtype has-upper-real-Prop
 
-```agda
-{- module _
-  {l : Level} (x y : ℝ l)
-  where
+  eq-eq-upper-real-ℝ : upper-real-ℝ x ＝ upper-real-ℝ y → x ＝ y
+  eq-eq-upper-real-ℝ = eq-eq-lower-real-ℝ ∘ (eq-lower-real-eq-upper-real-ℝ x y)
 
   eq-eq-lower-cut-ℝ : lower-cut-ℝ x ＝ lower-cut-ℝ y → x ＝ y
-  eq-eq-lower-cut-ℝ = eq-type-subtype has-upper-cut-Prop
+  eq-eq-lower-cut-ℝ lcx=lcy =
+    eq-eq-lower-real-ℝ
+      ( eq-eq-cut-lower-ℝ (lower-real-ℝ x) (lower-real-ℝ y) lcx=lcy)
 
   eq-eq-upper-cut-ℝ : upper-cut-ℝ x ＝ upper-cut-ℝ y → x ＝ y
-  eq-eq-upper-cut-ℝ = eq-eq-lower-cut-ℝ ∘ (eq-lower-cut-eq-upper-cut-ℝ x y) -}
+  eq-eq-upper-cut-ℝ = eq-eq-lower-cut-ℝ ∘ (eq-lower-cut-eq-upper-cut-ℝ x y)
 ```
 
 ## References

From 263f367ecf3f72d4c45766fcacf734c7e7eceda7 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Feb 2025 15:40:17 -0800
Subject: [PATCH 43/73] make pre-commit

---
 ...ithmetically-located-dedekind-cuts.lagda.md | 14 +++++++-------
 .../dedekind-real-numbers.lagda.md             | 16 +++++++---------
 .../rational-real-numbers.lagda.md             | 18 ++++++------------
 3 files changed, 20 insertions(+), 28 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index f4b8c30560..ee15d2d9c1 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -83,22 +83,22 @@ module _
   where
 
   abstract
-    is-located-is-arithmetically-located-pair-of-cuts-ℚ :
-      arithmetically-located-upper-real x y →
-      is-located-cut (cut-lower-ℝ x) (cut-upper-ℝ y)
-    is-located-is-arithmetically-located-pair-of-cuts-ℚ
+    is-located-is-arithmetically-located-lower-upper-ℝ :
+      arithmetically-located-lower-upper-ℝ x y →
+      is-located-lower-upper-ℝ x y
+    is-located-is-arithmetically-located-lower-upper-ℝ
       arithmetically-located p q p<q =
       elim-exists
-        ( cut-lower-ℝ L p ∨ cut-upper-ℝ U q)
+        ( cut-lower-ℝ x p ∨ cut-upper-ℝ y q)
         ( λ (p' , q') (q'<p'+q-p , p'∈L , q'∈U) →
           rec-coproduct
             ( λ p<p' →
               inl-disjunction
-                ( le-cut-lower-ℝ L p p' p<p' p'∈L))
+                ( le-cut-lower-ℝ x p p' p<p' p'∈L))
             ( λ p'≤p →
               inr-disjunction
                 (le-cut-upper-ℝ
-                  ( U)
+                  ( y)
                   ( q')
                   ( q)
                   ( concatenate-le-leq-ℚ
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index c973a7b3cc..037d51b24b 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -50,18 +50,17 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-A Dedekind real number
-consists of a [pair](foundation.dependent-pair-types.md) `(lx , uy)` of a
-[lower Dedekind real](real-numbers.lower-dedekind-real-numbers) and an
-[upper Dedekind real](real-numbers.upper-dedekind-real-numbers) that also satisfy
-the following conditions:
+A Dedekind real number consists of a [pair](foundation.dependent-pair-types.md)
+`(lx , uy)` of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers)
+and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers) that also
+satisfy the following conditions:
 
 1. _Disjointness_. The cuts of `lx` and `uy` are disjoint subsets of `ℚ`.
 2. _Locatedness_. If `q < r` then `q` is in the cut of `lx` or `r` is in the cut
-    of `uy`.
+   of `uy`.
 
-The Dedekind real numbers will be taken as the standard
-definition of the real numbers in the agda-unimath library.
+The Dedekind real numbers will be taken as the standard definition of the real
+numbers in the agda-unimath library.
 
 ## Definition
 
@@ -177,7 +176,6 @@ module _
 ### The Dedekind real numbers form a set
 
 ```agda
-
 abstract
   is-set-ℝ : (l : Level) → is-set (ℝ l)
   is-set-ℝ l =
diff --git a/src/real-numbers/rational-real-numbers.lagda.md b/src/real-numbers/rational-real-numbers.lagda.md
index 69ec475e2a..c6c037d04f 100644
--- a/src/real-numbers/rational-real-numbers.lagda.md
+++ b/src/real-numbers/rational-real-numbers.lagda.md
@@ -57,14 +57,12 @@ the [image](foundation.images.md) of this embedding
 ### Strict inequality on rationals gives Dedekind cuts
 
 ```agda
-is-dedekind-cut-le-ℚ :
+is-dedekind-lower-upper-real-ℚ :
   (x : ℚ) →
-  is-dedekind-cut
-    (cut-lower-real-ℚ x)
-    (cut-upper-real-ℚ x)
-is-dedekind-cut-le-ℚ x =
-  is-lower-dedekind-cut-lower-ℝ (lower-real-ℚ x) ,
-  is-upper-dedekind-cut-upper-ℝ (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
 ```
@@ -73,11 +71,7 @@ is-dedekind-cut-le-ℚ x =
 
 ```agda
 real-ℚ : ℚ → ℝ lzero
-real-ℚ x =
-  real-dedekind-cut
-    ( λ (q : ℚ) → le-ℚ-Prop q x)
-    ( λ (r : ℚ) → le-ℚ-Prop x r)
-    ( is-dedekind-cut-le-ℚ x)
+real-ℚ x = lower-real-ℚ x , upper-real-ℚ x , is-dedekind-lower-upper-real-ℚ x
 ```
 
 ### The property of being a rational real number

From f2355703e42f3a0cd2b6c1b27609329e35d5fee9 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Feb 2025 15:47:49 -0800
Subject: [PATCH 44/73] Remove empty bibliography

---
 .../inequality-lower-dedekind-real-numbers.lagda.md           | 4 ----
 1 file changed, 4 deletions(-)

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 7a73363b9c..26abd2fb42 100644
--- a/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/inequality-lower-dedekind-real-numbers.lagda.md
@@ -111,7 +111,3 @@ iff-leq-lower-real-ℚ :
 pr1 (iff-leq-lower-real-ℚ p q) = preserves-leq-lower-real-ℚ p q
 pr2 (iff-leq-lower-real-ℚ p q) = reflects-leq-lower-real-ℚ p q
 ```
-
-## References
-
-{{#bibliography}}

From 827111e46a69879d9d18e743bfc0e3de068f73b7 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Feb 2025 15:58:20 -0800
Subject: [PATCH 45/73] Review changes

---
 .../lower-dedekind-real-numbers.lagda.md      | 28 ++++++++++++-------
 .../upper-dedekind-real-numbers.lagda.md      | 28 ++++++++++++-------
 2 files changed, 36 insertions(+), 20 deletions(-)

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index 948f6e60bd..db4cf96ad6 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -15,8 +15,10 @@ open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 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.subtypes
@@ -31,9 +33,8 @@ open import foundation.universe-levels
 
 ## Idea
 
-A lower
-{{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
-consists of a [subtype](foundation-core.subtypes.md) `L` of
+A {{#concept "lower Dedekind cut" Agda=is-lower-dedekind-cut}} consists of a
+[subtype](foundation-core.subtypes.md) `L` of
 [the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
 satisfying the following two conditions:
 
@@ -42,8 +43,8 @@ satisfying the following two conditions:
    [if and only if](foundation.logical-equivalences.md) there
    [exists](foundation.existential-quantification.md) `q < r` such that `r ∈ L`.
 
-The type of all lower Dedekind real numbers is the type of all lower Dedekind
-cuts.
+The {{#concept "lower Dedekind real numbers" Agda=lower-ℝ}} is the type of all
+lower Dedekind cuts.
 
 ## Definition
 
@@ -113,8 +114,9 @@ module _
   {l : Level} (x : lower-ℝ l) (p q : ℚ)
   where
 
-  le-cut-lower-ℝ : le-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
-  le-cut-lower-ℝ p<q q<x =
+  is-in-cut-le-ℚ-lower-ℝ :
+    le-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
+  is-in-cut-le-ℚ-lower-ℝ p<q q<x =
     backward-implication
       ( is-rounded-cut-lower-ℝ x p)
       ( intro-exists q (p<q , q<x))
@@ -127,9 +129,10 @@ module _
   {l : Level} (x : lower-ℝ l) (p q : ℚ)
   where
 
-  leq-cut-lower-ℝ : leq-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
-  leq-cut-lower-ℝ p≤q q<x with decide-le-leq-ℚ p q
-  ... | inl p<q = le-cut-lower-ℝ x p q p<q q<x
+  is-in-cut-leq-ℚ-lower-ℝ :
+    leq-ℚ p q → is-in-cut-lower-ℝ x q → is-in-cut-lower-ℝ x p
+  is-in-cut-leq-ℚ-lower-ℝ p≤q q<x with decide-le-leq-ℚ p q
+  ... | inl p<q = is-in-cut-le-ℚ-lower-ℝ x p q p<q q<x
   ... | inr q≤p = tr (is-in-cut-lower-ℝ x) (antisymmetric-leq-ℚ q p q≤p p≤q) q<x
 ```
 
@@ -142,6 +145,11 @@ module _
 
   eq-eq-cut-lower-ℝ : cut-lower-ℝ x ＝ cut-lower-ℝ y → x ＝ y
   eq-eq-cut-lower-ℝ = eq-type-subtype is-lower-dedekind-cut-Prop
+
+  eq-sim-cut-lower-ℝ : sim-subtype (cut-lower-ℝ x) (cut-lower-ℝ y) → x ＝ y
+  eq-sim-cut-lower-ℝ =
+    eq-eq-cut-lower-ℝ ∘
+    antisymmetric-sim-subtype (cut-lower-ℝ x) (cut-lower-ℝ y)
 ```
 
 ## See also
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index 69d21848f4..83d2328e84 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -15,8 +15,10 @@ open import foundation.conjunction
 open import foundation.coproduct-types
 open import foundation.dependent-pair-types
 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.subtypes
@@ -31,9 +33,8 @@ open import foundation.universe-levels
 
 ## Idea
 
-An upper
-{{#concept "Dedekind cut" Agda=is-dedekind-cut WD="dedekind cut" WDID=Q851333}}
-consists of a [subtype](foundation-core.subtypes.md) `U` of
+An {{#concept "upper Dedekind cut" Agda=is-upper-dedekind-cut}} consists of a
+[subtype](foundation-core.subtypes.md) `U` of
 [the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
 satisfying the following two conditions:
 
@@ -42,8 +43,8 @@ satisfying the following two conditions:
    [if and only if](foundation.logical-equivalences.md) there
    [exists](foundation.existential-quantification.md) `p < q` such that `p ∈ U`.
 
-The type of all upper Dedekind real numbers is the type of all upper Dedekind
-cuts.
+The {{#concept "upper Dedekind real numbers" Agda=upper-ℝ}} is the type of all
+upper Dedekind cuts.
 
 ## Definition
 
@@ -113,8 +114,9 @@ module _
   {l : Level} (x : upper-ℝ l) (p q : ℚ)
   where
 
-  le-cut-upper-ℝ : le-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
-  le-cut-upper-ℝ p<q p<x =
+  is-in-cut-le-ℚ-upper-ℝ :
+    le-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
+  is-in-cut-le-ℚ-upper-ℝ p<q p<x =
     backward-implication
       ( is-rounded-cut-upper-ℝ x q)
       ( intro-exists p (p<q , p<x))
@@ -127,9 +129,10 @@ module _
   {l : Level} (x : upper-ℝ l) (p q : ℚ)
   where
 
-  leq-cut-upper-ℝ : leq-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
-  leq-cut-upper-ℝ p≤q x<p with decide-le-leq-ℚ p q
-  ... | inl p<q = le-cut-upper-ℝ x p q p<q x<p
+  is-in-cut-leq-ℚ-upper-ℝ :
+    leq-ℚ p q → is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x q
+  is-in-cut-leq-ℚ-upper-ℝ p≤q x<p with decide-le-leq-ℚ p q
+  ... | inl p<q = is-in-cut-le-ℚ-upper-ℝ x p q p<q x<p
   ... | inr q≤p = tr (is-in-cut-upper-ℝ x) (antisymmetric-leq-ℚ p q p≤q q≤p) x<p
 ```
 
@@ -142,6 +145,11 @@ module _
 
   eq-eq-cut-upper-ℝ : cut-upper-ℝ x ＝ cut-upper-ℝ y → x ＝ y
   eq-eq-cut-upper-ℝ = eq-type-subtype is-upper-dedekind-cut-Prop
+
+  eq-sim-cut-upper-ℝ : sim-subtype (cut-upper-ℝ x) (cut-upper-ℝ y) → x ＝ y
+  eq-sim-cut-upper-ℝ =
+    eq-eq-cut-upper-ℝ ∘
+    antisymmetric-sim-subtype (cut-upper-ℝ x) (cut-upper-ℝ y)
 ```
 
 ## See also

From f734874211fb5bcd7d6cd7d9918c44527c7c8c20 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Feb 2025 16:15:25 -0800
Subject: [PATCH 46/73] Fix names

---
 .../arithmetically-located-dedekind-cuts.lagda.md         | 4 ++--
 src/real-numbers/dedekind-real-numbers.lagda.md           | 8 ++++----
 2 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index ee15d2d9c1..38e39bfb9e 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -94,10 +94,10 @@ module _
           rec-coproduct
             ( λ p<p' →
               inl-disjunction
-                ( le-cut-lower-ℝ x p p' p<p' p'∈L))
+                ( is-in-cut-le-ℚ-lower-ℝ x p p' p<p' p'∈L))
             ( λ p'≤p →
               inr-disjunction
-                (le-cut-upper-ℝ
+                ( is-in-cut-le-ℚ-upper-ℝ
                   ( y)
                   ( q')
                   ( q)
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 037d51b24b..76134fd985 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -206,25 +206,25 @@ module _
     le-ℚ p q →
     is-in-lower-cut-ℝ x q →
     is-in-lower-cut-ℝ x p
-  le-lower-cut-ℝ = le-cut-lower-ℝ (lower-real-ℝ x) p q
+  le-lower-cut-ℝ = is-in-cut-le-ℚ-lower-ℝ (lower-real-ℝ x) p q
 
   leq-lower-cut-ℝ :
     leq-ℚ p q →
     is-in-lower-cut-ℝ x q →
     is-in-lower-cut-ℝ x p
-  leq-lower-cut-ℝ = leq-cut-lower-ℝ (lower-real-ℝ x) p q
+  leq-lower-cut-ℝ = is-in-cut-leq-ℚ-lower-ℝ (lower-real-ℝ x) p q
 
   le-upper-cut-ℝ :
     le-ℚ p q →
     is-in-upper-cut-ℝ x p →
     is-in-upper-cut-ℝ x q
-  le-upper-cut-ℝ = le-cut-upper-ℝ (upper-real-ℝ x) p q
+  le-upper-cut-ℝ = is-in-cut-le-ℚ-upper-ℝ (upper-real-ℝ x) p q
 
   leq-upper-cut-ℝ :
     leq-ℚ p q →
     is-in-upper-cut-ℝ x p →
     is-in-upper-cut-ℝ x q
-  leq-upper-cut-ℝ = leq-cut-upper-ℝ (upper-real-ℝ x) p q
+  leq-upper-cut-ℝ = is-in-cut-leq-ℚ-upper-ℝ (upper-real-ℝ x) p q
 ```
 
 ### Elements of the lower cut are lower bounds of the upper cut

From 53e54d8b312861efccf3634ceaa32d58446e520e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Tue, 11 Feb 2025 21:07:48 -0800
Subject: [PATCH 47/73] Fix links

---
 src/real-numbers/dedekind-real-numbers.lagda.md | 5 +++--
 1 file changed, 3 insertions(+), 2 deletions(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 76134fd985..3e6695ebce 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -51,8 +51,9 @@ open import real-numbers.upper-dedekind-real-numbers
 ## Idea
 
 A Dedekind real number consists of a [pair](foundation.dependent-pair-types.md)
-`(lx , uy)` of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers)
-and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers) that also
+`(lx , uy)` of a
+[lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md) and an
+[upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md) that also
 satisfy the following conditions:
 
 1. _Disjointness_. The cuts of `lx` and `uy` are disjoint subsets of `ℚ`.

From 71479cac22d2f14bac981f0d66592b74e1d85048 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 11:20:08 -0800
Subject: [PATCH 48/73] Fix naming

---
 .../arithmetically-located-dedekind-cuts.lagda.md             | 4 ++--
 1 file changed, 2 insertions(+), 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 38e39bfb9e..fe8f273ea0 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -42,8 +42,8 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Definition
 
-A pair of a [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md)
-`L` and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`
+A pair of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md)
+`L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md) `U`
 is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if for
 any positive [rational number](elementary-number-theory.rational-numbers.md)
 `ε : ℚ`, there exist `p, q : ℚ` such that `0 < q - p < ε`, `p ∈ L`, and `q ∈ U`.

From 9f12eee4548ad0a47f816c940993fbbd0b9b948e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 11:31:14 -0800
Subject: [PATCH 49/73] Progress on upper addition

---
 ...ition-upper-dedekind-real-numbers.lagda.md | 208 ++++++++++++++++++
 1 file changed, 208 insertions(+)
 create mode 100644 src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md

diff --git a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
new file mode 100644
index 0000000000..bc9d4da02a
--- /dev/null
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -0,0 +1,208 @@
+# Addition of upper Dedekind real numbers
+
+```agda
+{-# OPTIONS --lossy-unification #-}
+
+module real-numbers.addition-upper-dedekind-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.dependent-pair-types
+open import foundation.existential-quantification
+open import foundation.identity-types
+open import foundation.logical-equivalences
+open import foundation.subtypes
+open import foundation.transport-along-identifications
+open import foundation.universe-levels
+
+open import logic.functoriality-existential-quantification
+
+open import group-theory.abelian-groups
+open import group-theory.groups
+open import group-theory.minkowski-multiplication-commutative-monoids
+
+open import real-numbers.upper-dedekind-real-numbers
+```
+
+</details>
+
+## Idea
+
+The sum of two [upper Dedekind real numbers](real-numbers.upper-dedekind-real-numbers.md)
+is the Minkowski sum of their cuts.
+
+```agda
+module _
+  {l1 l2 : Level}
+  (x : upper-ℝ l1)
+  (y : upper-ℝ l2)
+  where
+
+  cut-add-upper-ℝ : subtype (l1 ⊔ l2) ℚ
+  cut-add-upper-ℝ =
+    minkowski-mul-Commutative-Monoid
+      ( commutative-monoid-add-ℚ)
+      ( cut-upper-ℝ x)
+      ( cut-upper-ℝ y)
+
+  is-in-cut-add-upper-ℝ : ℚ → UU (l1 ⊔ l2)
+  is-in-cut-add-upper-ℝ = is-in-subtype cut-add-upper-ℝ
+
+  is-inhabited-cut-add-upper-ℝ : exists ℚ cut-add-upper-ℝ
+  is-inhabited-cut-add-upper-ℝ =
+    minkowski-mul-inhabited-is-inhabited-Commutative-Monoid
+      ( commutative-monoid-add-ℚ)
+      ( cut-upper-ℝ x)
+      ( cut-upper-ℝ y)
+      ( is-inhabited-cut-upper-ℝ x)
+      ( is-inhabited-cut-upper-ℝ y)
+
+  abstract
+    is-rounded-cut-add-upper-ℝ :
+      (q : ℚ) →
+      is-in-cut-add-upper-ℝ q ↔
+      exists ℚ (λ p → le-ℚ-Prop p q ∧ cut-add-upper-ℝ p)
+    pr1 (is-rounded-cut-add-upper-ℝ q) =
+      elim-exists
+        ( ∃ ℚ (λ p → le-ℚ-Prop p q ∧ cut-add-upper-ℝ p))
+        ( λ (ux , uy) (x<ux , y<uy , q=ux+uy) →
+          map-binary-exists
+            ( λ p → le-ℚ p q × is-in-cut-add-upper-ℝ p)
+            ( add-ℚ)
+            ( λ ux' uy' (ux'<ux , x<ux') (uy'<uy , y<uy') →
+              tr
+                ( le-ℚ (ux' +ℚ uy') )
+                ( inv (q=ux+uy))
+                ( preserves-le-add-ℚ {ux'} {ux} {uy'} {uy} ux'<ux uy'<uy) ,
+              intro-exists (ux' , uy') (x<ux' , y<uy' , refl))
+            ( forward-implication (is-rounded-cut-upper-ℝ x ux) x<ux)
+            ( forward-implication (is-rounded-cut-upper-ℝ y uy) y<uy))
+    pr2 (is-rounded-cut-add-upper-ℝ q) = {!   !}
+    {-
+    pr2 (is-rounded-cut-add-upper-ℝ q) =
+      elim-exists
+        ( cut-add-upper-ℝ q)
+        ( λ r (q<r , r<x+y) →
+          elim-exists
+            ( cut-add-upper-ℝ q)
+            ( λ (rx , ry) (rx<x , ry<y , r=rx+ry) →
+              let
+                r-q⁺ : ℚ⁺
+                r-q⁺ = positive-diff-le-ℚ q r q<r
+                ε⁺ : ℚ⁺
+                ε⁺ = mediant-zero-ℚ⁺ r-q⁺
+                ε : ℚ
+                ε = rational-ℚ⁺ ε⁺
+                ε<r-q : le-ℚ ε (r -ℚ q)
+                ε<r-q = le-mediant-zero-ℚ⁺ r-q⁺
+              in
+              intro-exists
+                ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
+                ( le-cut-upper-ℝ
+                    ( x)
+                    ( rx -ℚ ε)
+                    ( rx)
+                    ( le-diff-rational-ℚ⁺ rx ε⁺)
+                    ( rx<x) ,
+                  le-cut-upper-ℝ
+                    ( y)
+                    ( q -ℚ (rx -ℚ ε))
+                    ( ry)
+                    ( binary-tr
+                        ( le-ℚ)
+                        ( equational-reasoning
+                          (q -ℚ rx) +ℚ ε
+                          ＝ q +ℚ (neg-ℚ rx +ℚ ε)
+                            by associative-add-ℚ q (neg-ℚ rx) ε
+                          ＝ q +ℚ (ε -ℚ rx)
+                            by ap (q +ℚ_) (commutative-add-ℚ (neg-ℚ rx) ε)
+                          ＝ q -ℚ (rx -ℚ ε)
+                            by ap (q +ℚ_) (inv (distributive-neg-diff-ℚ rx ε)))
+                        ( equational-reasoning
+                          (q -ℚ rx) +ℚ (r -ℚ q)
+                          ＝ (q -ℚ rx) +ℚ (neg-ℚ q +ℚ r)
+                            by
+                              ap ((q -ℚ rx) +ℚ_) (commutative-add-ℚ r (neg-ℚ q))
+                          ＝ (q -ℚ q) +ℚ (neg-ℚ rx +ℚ r)
+                            by
+                              interchange-law-add-add-ℚ q (neg-ℚ rx) (neg-ℚ q) r
+                          ＝ zero-ℚ +ℚ (neg-ℚ rx +ℚ r)
+                            by
+                              ap
+                                ( _+ℚ (neg-ℚ rx +ℚ r))
+                                ( right-inverse-law-add-ℚ q)
+                          ＝ neg-ℚ rx +ℚ r by left-unit-law-add-ℚ (neg-ℚ rx +ℚ r)
+                          ＝ neg-ℚ rx +ℚ (rx +ℚ ry) by ap (neg-ℚ rx +ℚ_) r=rx+ry
+                          ＝ ry
+                            by is-retraction-left-div-Group group-add-ℚ rx ry)
+                        ( preserves-le-right-add-ℚ
+                          ( q -ℚ rx)
+                          ( ε)
+                          ( r -ℚ q)
+                          ( ε<r-q)))
+                    ( ry<y) ,
+                  inv
+                    ( is-identity-right-conjugation-Ab
+                      ( abelian-group-add-ℚ)
+                      ( rx -ℚ ε)
+                      ( q))))
+            ( r<x+y))-}
+
+  add-upper-ℝ : upper-ℝ (l1 ⊔ l2)
+  pr1 add-upper-ℝ = cut-add-upper-ℝ
+  pr1 (pr2 add-upper-ℝ) = is-inhabited-cut-add-upper-ℝ
+  pr2 (pr2 add-upper-ℝ) = is-rounded-cut-add-upper-ℝ
+```
+
+## Properties
+
+### Addition of upper Dedekind real numbers is commutative
+
+```agda
+module _
+  {l1 l2 : Level} (x : upper-ℝ l1) (y : upper-ℝ l2)
+  where
+
+  commutative-add-upper-ℝ : add-upper-ℝ x y ＝ add-upper-ℝ y x
+  commutative-add-upper-ℝ =
+    eq-eq-cut-upper-ℝ
+      ( add-upper-ℝ x y)
+      ( add-upper-ℝ y x)
+      ( commutative-minkowski-mul-Commutative-Monoid
+        ( commutative-monoid-add-ℚ)
+        ( cut-upper-ℝ x)
+        ( cut-upper-ℝ y))
+```
+
+### Addition of upper Dedekind real numbers is associative
+
+```agda
+module _
+  {l1 l2 l3 : Level} (x : upper-ℝ l1) (y : upper-ℝ l2) (z : upper-ℝ l3)
+  where
+
+  associative-add-upper-ℝ :
+    add-upper-ℝ (add-upper-ℝ x y) z ＝ add-upper-ℝ x (add-upper-ℝ y z)
+  associative-add-upper-ℝ =
+    eq-eq-cut-upper-ℝ
+      ( add-upper-ℝ (add-upper-ℝ x y) z)
+      ( add-upper-ℝ x (add-upper-ℝ y z))
+      ( associative-minkowski-mul-Commutative-Monoid
+        ( commutative-monoid-add-ℚ)
+        ( cut-upper-ℝ x)
+        ( cut-upper-ℝ y)
+        ( cut-upper-ℝ z))
+```

From e53543547c419196f42eed070551b537a68d302f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 11:31:45 -0800
Subject: [PATCH 50/73] make pre-commit

---
 .../arithmetically-located-dedekind-cuts.lagda.md           | 6 +++---
 1 file changed, 3 insertions(+), 3 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index fe8f273ea0..405a77a7e2 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -43,9 +43,9 @@ open import real-numbers.upper-dedekind-real-numbers
 ## Definition
 
 A pair of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md)
-`L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md) `U`
-is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if for
-any positive [rational number](elementary-number-theory.rational-numbers.md)
+`L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md)
+`U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if
+for any positive [rational number](elementary-number-theory.rational-numbers.md)
 `ε : ℚ`, there exist `p, q : ℚ` such that `0 < q - p < ε`, `p ∈ L`, and `q ∈ U`.
 Intuitively, when `L , U` represent the cuts of a real number `x`, `p` and `q`
 are rational approximations of `x` to within `ε`. This follows parts of Section

From 74a4defe05d5e9546041eaf5e2253dba830ee7db Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 12:15:35 -0800
Subject: [PATCH 51/73] make pre-commit

---
 src/real-numbers.lagda.md                     |   2 +
 ...ition-lower-dedekind-real-numbers.lagda.md |  15 ++-
 ...ition-upper-dedekind-real-numbers.lagda.md | 117 ++++++++----------
 3 files changed, 66 insertions(+), 68 deletions(-)

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 1cd86ceaa3..ce1bcaf270 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -5,6 +5,8 @@
 ```agda
 module real-numbers where
 
+open import real-numbers.addition-lower-dedekind-real-numbers public
+open import real-numbers.addition-upper-dedekind-real-numbers public
 open import real-numbers.arithmetically-located-dedekind-cuts public
 open import real-numbers.dedekind-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
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 c45c75d99b..0073594d5b 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -28,12 +28,12 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import group-theory.abelian-groups
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 
+open import logic.functoriality-existential-quantification
+
 open import real-numbers.lower-dedekind-real-numbers
 ```
 
@@ -41,8 +41,11 @@ open import real-numbers.lower-dedekind-real-numbers
 
 ## Idea
 
-The sum of two [lower Dedekind real numbers](real-numbers.lower-dedekind-real-numbers.md)
-is the Minkowski sum of their cuts.
+The sum of two
+[lower Dedekind real numbers](real-numbers.lower-dedekind-real-numbers.md) is
+the
+[Minkowski sum](group-theory.minkowski-multiplication-commutative-monoids.md) of
+their cuts.
 
 ```agda
 module _
@@ -109,13 +112,13 @@ module _
               in
               intro-exists
                 ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
-                ( le-cut-lower-ℝ
+                ( is-in-cut-le-ℚ-lower-ℝ
                     ( x)
                     ( rx -ℚ ε)
                     ( rx)
                     ( le-diff-rational-ℚ⁺ rx ε⁺)
                     ( rx<x) ,
-                  le-cut-lower-ℝ
+                  is-in-cut-le-ℚ-lower-ℝ
                     ( y)
                     ( q -ℚ (rx -ℚ ε))
                     ( ry)
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 bc9d4da02a..5e8a0a37ee 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -28,12 +28,12 @@ open import foundation.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
 
-open import logic.functoriality-existential-quantification
-
 open import group-theory.abelian-groups
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 
+open import logic.functoriality-existential-quantification
+
 open import real-numbers.upper-dedekind-real-numbers
 ```
 
@@ -41,8 +41,11 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-The sum of two [upper Dedekind real numbers](real-numbers.upper-dedekind-real-numbers.md)
-is the Minkowski sum of their cuts.
+The sum of two
+[upper Dedekind real numbers](real-numbers.upper-dedekind-real-numbers.md) is
+the
+[Minkowski sum](group-theory.minkowski-multiplication-commutative-monoids.md) of
+their cuts.
 
 ```agda
 module _
@@ -84,82 +87,72 @@ module _
             ( add-ℚ)
             ( λ ux' uy' (ux'<ux , x<ux') (uy'<uy , y<uy') →
               tr
-                ( le-ℚ (ux' +ℚ uy') )
+                ( le-ℚ (ux' +ℚ uy'))
                 ( inv (q=ux+uy))
                 ( preserves-le-add-ℚ {ux'} {ux} {uy'} {uy} ux'<ux uy'<uy) ,
               intro-exists (ux' , uy') (x<ux' , y<uy' , refl))
             ( forward-implication (is-rounded-cut-upper-ℝ x ux) x<ux)
             ( forward-implication (is-rounded-cut-upper-ℝ y uy) y<uy))
-    pr2 (is-rounded-cut-add-upper-ℝ q) = {!   !}
-    {-
     pr2 (is-rounded-cut-add-upper-ℝ q) =
       elim-exists
         ( cut-add-upper-ℝ q)
-        ( λ r (q<r , r<x+y) →
+        ( λ p (p<q , x+y<r) →
           elim-exists
             ( cut-add-upper-ℝ q)
-            ( λ (rx , ry) (rx<x , ry<y , r=rx+ry) →
+            ( λ (px , py) (x<px , y<py , p=px+py) →
               let
-                r-q⁺ : ℚ⁺
-                r-q⁺ = positive-diff-le-ℚ q r q<r
-                ε⁺ : ℚ⁺
-                ε⁺ = mediant-zero-ℚ⁺ r-q⁺
-                ε : ℚ
+                q-p⁺ = positive-diff-le-ℚ p q p<q
+                ε⁺ = mediant-zero-ℚ⁺ q-p⁺
                 ε = rational-ℚ⁺ ε⁺
-                ε<r-q : le-ℚ ε (r -ℚ q)
-                ε<r-q = le-mediant-zero-ℚ⁺ r-q⁺
+                ε<q-p = le-mediant-zero-ℚ⁺ q-p⁺
               in
-              intro-exists
-                ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
-                ( le-cut-upper-ℝ
+                intro-exists
+                  ( px +ℚ ε , q -ℚ (px +ℚ ε))
+                  ( is-in-cut-le-ℚ-upper-ℝ
                     ( x)
-                    ( rx -ℚ ε)
-                    ( rx)
-                    ( le-diff-rational-ℚ⁺ rx ε⁺)
-                    ( rx<x) ,
-                  le-cut-upper-ℝ
-                    ( y)
-                    ( q -ℚ (rx -ℚ ε))
-                    ( ry)
-                    ( binary-tr
+                    ( px)
+                    ( px +ℚ ε)
+                    ( le-right-add-rational-ℚ⁺ px ε⁺)
+                    ( x<px) ,
+                    is-in-cut-le-ℚ-upper-ℝ
+                      ( y)
+                      ( py)
+                      ( q -ℚ (px +ℚ ε))
+                      ( binary-tr
                         ( le-ℚ)
                         ( equational-reasoning
-                          (q -ℚ rx) +ℚ ε
-                          ＝ q +ℚ (neg-ℚ rx +ℚ ε)
-                            by associative-add-ℚ q (neg-ℚ rx) ε
-                          ＝ q +ℚ (ε -ℚ rx)
-                            by ap (q +ℚ_) (commutative-add-ℚ (neg-ℚ rx) ε)
-                          ＝ q -ℚ (rx -ℚ ε)
-                            by ap (q +ℚ_) (inv (distributive-neg-diff-ℚ rx ε)))
+                            (q -ℚ px) -ℚ (q -ℚ p)
+                            ＝ neg-ℚ (q -ℚ p) +ℚ (q -ℚ px)
+                              by commutative-add-ℚ (q -ℚ px) (neg-ℚ (q -ℚ p))
+                            ＝ (p -ℚ q) +ℚ (q -ℚ px)
+                              by
+                                ap (_+ℚ (q -ℚ px)) (distributive-neg-diff-ℚ q p)
+                            ＝ p -ℚ px by triangle-diff-ℚ p q px
+                            ＝ (px +ℚ py) -ℚ px by ap (_-ℚ px) p=px+py
+                            ＝ py
+                              by
+                                is-identity-left-conjugation-Ab
+                                  ( abelian-group-add-ℚ)
+                                  ( px)
+                                  ( py))
                         ( equational-reasoning
-                          (q -ℚ rx) +ℚ (r -ℚ q)
-                          ＝ (q -ℚ rx) +ℚ (neg-ℚ q +ℚ r)
-                            by
-                              ap ((q -ℚ rx) +ℚ_) (commutative-add-ℚ r (neg-ℚ q))
-                          ＝ (q -ℚ q) +ℚ (neg-ℚ rx +ℚ r)
-                            by
-                              interchange-law-add-add-ℚ q (neg-ℚ rx) (neg-ℚ q) r
-                          ＝ zero-ℚ +ℚ (neg-ℚ rx +ℚ r)
-                            by
-                              ap
-                                ( _+ℚ (neg-ℚ rx +ℚ r))
-                                ( right-inverse-law-add-ℚ q)
-                          ＝ neg-ℚ rx +ℚ r by left-unit-law-add-ℚ (neg-ℚ rx +ℚ r)
-                          ＝ neg-ℚ rx +ℚ (rx +ℚ ry) by ap (neg-ℚ rx +ℚ_) r=rx+ry
-                          ＝ ry
-                            by is-retraction-left-div-Group group-add-ℚ rx ry)
+                          (q -ℚ px) -ℚ ε
+                          ＝ q +ℚ (neg-ℚ px +ℚ neg-ℚ ε)
+                            by associative-add-ℚ q (neg-ℚ px) (neg-ℚ ε)
+                          ＝ q -ℚ (px +ℚ ε)
+                            by ap (q +ℚ_) (inv (distributive-neg-add-ℚ px ε)))
                         ( preserves-le-right-add-ℚ
-                          ( q -ℚ rx)
-                          ( ε)
-                          ( r -ℚ q)
-                          ( ε<r-q)))
-                    ( ry<y) ,
-                  inv
-                    ( is-identity-right-conjugation-Ab
-                      ( abelian-group-add-ℚ)
-                      ( rx -ℚ ε)
-                      ( q))))
-            ( r<x+y))-}
+                          ( q -ℚ px)
+                          ( neg-ℚ (q -ℚ p))
+                          ( neg-ℚ ε)
+                          ( neg-le-ℚ ε (q -ℚ p) ε<q-p)))
+                      ( y<py) ,
+                    inv
+                      ( is-identity-right-conjugation-Ab
+                        ( abelian-group-add-ℚ)
+                        ( px +ℚ ε)
+                        ( q))))
+            ( x+y<r))
 
   add-upper-ℝ : upper-ℝ (l1 ⊔ l2)
   pr1 add-upper-ℝ = cut-add-upper-ℝ

From 68ab24923d7be3bf10f6220b0398408bd7bd3826 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 22:14:37 -0800
Subject: [PATCH 52/73] Rewrite in terms of propositional truncation

---
 .../existential-quantification.lagda.md       | 42 -------------
 .../propositional-truncations.lagda.md        | 59 +++++++++++++++++++
 .../strict-inequality-real-numbers.lagda.md   | 11 ++--
 3 files changed, 65 insertions(+), 47 deletions(-)

diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md
index 04483c575f..a15d2258a9 100644
--- a/src/foundation/existential-quantification.lagda.md
+++ b/src/foundation/existential-quantification.lagda.md
@@ -331,48 +331,6 @@ module _
     eq-distributive-product-exists-structure P
 ```
 
-## Existential quantification `do` syntax
-
-When eliminating a chain of existential quantifications, which may be
-interdependent,
-[Agda's `do` syntax](https://agda.readthedocs.io/en/stable/language/syntactic-sugar.html#do-notation)
-can eliminate many levels of nesting.
-
-For example, suppose we have
-
-```text
-R : Prop l
-
-exists-a-p : exists A P
-a-p-implies-b-q-exists : (a : A) → P a → exists B Q
-a-b-p-q-implies-r : (a : A) (b : B) → P a → Q b → R
-```
-
-To deduce `R`, we could write
-
-```text
-do
-  a , pa ← exists-a-p
-  b , qb ← a-p-implies-b-q-exists a pa
-  a-b-p-q-implies-r a b pa qb
-where open do-syntax-elim-exists R
-```
-
-instead of a nested chain of `elim-exists`.
-
-```agda
-module do-syntax-elim-exists
-  {l : Level}
-  (P : Prop l)
-  where
-
-  _>>=_ :
-    {l1 l2 : Level} {A : UU l1} {B : A -> UU l2} →
-    exists-structure A B →
-    (Σ A B -> type-Prop P) -> type-Prop P
-  x >>= f = elim-exists P (ev-pair f) x
-```
-
 ## See also
 
 - Existential quantification is the indexed counterpart to
diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 1181cc2572..5f2cfd7bb5 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -446,6 +446,65 @@ module _
     is-equiv-map-inv-trunc-Prop-diagonal-coproduct
 ```
 
+## `do` syntax for propositional truncation
+
+[Agda's `do` syntax](https://agda.readthedocs.io/en/v2.7.0/language/syntactic-sugar.html#do-notation)
+is a handy tool to avoid deeply nesting calls to the same lambda-based function.
+For example, consider a case where you are trying to prove a proposition,
+`motive : Prop l`, from a chain of truncated propositions `P` and `Q`, where
+showing `Q` requires having a value of `type-Prop P`:
+
+```text
+rec-trunc-Prop
+  ( motive)
+  ( λ q →
+    rec-trunc-Prop
+      ( motive)
+      ( λ q → derive-motive-from p q)
+      ( truncated-prop-Q p))
+  ( truncated-prop-P)
+```
+
+The tower of indentation, with many layers of indentation in the innermost
+derivation, is painful; there are many duplicated lines of
+`rec-trunc-Prop motive`. Agda's `do` syntax offers us an alternative.
+
+```agda
+module do-syntax-trunc-Prop {l : Level} (motive : Prop l) where
+  _>>=_ :
+    {l : Level} {A : UU l} →
+    type-trunc-Prop A → (A → type-Prop motive) →
+    type-Prop motive
+  trunc-prop-a >>= k = rec-trunc-Prop motive k trunc-prop-a
+```
+
+This allows us to rewrite the deeply nested chain above
+
+```text
+do
+  p ← truncated-prop-P
+  q ← truncated-prop-Q p
+  derive-motive-from p q
+where open do-syntax-trunc-Prop motive
+```
+
+Note that the last line, `where open do-syntax-trunc-Prop motive`, is required
+to make this syntax possible.
+
+In general, you can read the `p ← truncated-prop-P` as an _instruction_ saying,
+"Get the value `p` out of its truncation `truncated-prop-Q`." We obviously
+cannot get values out of truncations in general, but conceptually, we can do it
+in the service of proving a proposition, and that's what we're doing here. The
+following line `q ← truncated-prop-Q p` says, "Get the value `q` out of the
+truncation `truncated-prop-Q p`" -- noticing that we can make use of `p` in that
+line. Finally, `derive-motive-from p q` must give us a value of type
+`type-Prop motive` from `p` and `q`. The result of the entire `do` block is the
+value of type `type-Prop motive`, which is internally constructed with an
+appropriate chain of `rec-trunc-Prop` from the intermediate steps.
+
+While we had each line depend on all the values from the lines above it, this is
+not at all required.
+
 ## Table of files about propositional logic
 
 The following table gives an overview of basic constructions in propositional
diff --git a/src/real-numbers/strict-inequality-real-numbers.lagda.md b/src/real-numbers/strict-inequality-real-numbers.lagda.md
index e6c6041f4b..1cf4481302 100644
--- a/src/real-numbers/strict-inequality-real-numbers.lagda.md
+++ b/src/real-numbers/strict-inequality-real-numbers.lagda.md
@@ -24,6 +24,7 @@ open import foundation.functoriality-cartesian-product-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
@@ -105,7 +106,7 @@ module _
             ( q)
             ( le-lower-upper-cut-ℝ y p q p<y y<q))
         ( decide-le-leq-ℚ p q)
-    where open do-syntax-elim-exists empty-Prop
+    where open do-syntax-trunc-Prop empty-Prop
 ```
 
 ### Strict inequality on the reals is transitive
@@ -127,7 +128,7 @@ module _
         ( p)
         ( x<p ,
           le-lower-cut-ℝ z p q (le-lower-upper-cut-ℝ y p q p<y y<q) q<z)
-    where open do-syntax-elim-exists (le-ℝ-Prop x z)
+    where open do-syntax-trunc-Prop (le-ℝ-Prop x z)
 ```
 
 ### The canonical map from rationals to reals preserves and reflects strict inequality
@@ -196,7 +197,7 @@ module _
       q , x<q ← is-inhabited-upper-cut-ℝ x
       r , r<q , x<r ← forward-implication (is-rounded-upper-cut-ℝ x q) x<q
       intro-exists (real-ℚ q) (intro-exists r (x<r , r<q))
-    where open do-syntax-elim-exists (∃ (ℝ lzero) (le-ℝ-Prop x))
+    where open do-syntax-trunc-Prop (∃ (ℝ lzero) (le-ℝ-Prop x))
 ```
 
 ### Negation reverses the strict ordering of real numbers
@@ -216,7 +217,7 @@ module _
         (neg-ℚ p)
         ( tr (is-in-lower-cut-ℝ y) (inv (neg-neg-ℚ p)) p<y ,
           tr (is-in-upper-cut-ℝ x) (inv (neg-neg-ℚ p)) x<p)
-    where open do-syntax-elim-exists (le-ℝ-Prop (neg-ℝ y) (neg-ℝ x))
+    where open do-syntax-trunc-Prop (le-ℝ-Prop (neg-ℝ y) (neg-ℝ x))
 ```
 
 ### If `x` is less than `y`, then `y` is not less than or equal to `x`
@@ -339,7 +340,7 @@ module _
         ( intro-exists p (x<p , p<q) , intro-exists r (q<r , r<y))
     where
       open
-        do-syntax-elim-exists
+        do-syntax-trunc-Prop
           ( ∃ (ℝ lzero) (λ z → le-ℝ-Prop x z ∧ le-ℝ-Prop z y))
 ```
 

From df87847f80c8d715c321f4ffa6c10920a0d72bc2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 22:22:58 -0800
Subject: [PATCH 53/73] More words

---
 src/foundation/propositional-truncations.lagda.md | 9 ++++++---
 1 file changed, 6 insertions(+), 3 deletions(-)

diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 5f2cfd7bb5..87986c98ab 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -466,8 +466,11 @@ rec-trunc-Prop
 ```
 
 The tower of indentation, with many layers of indentation in the innermost
-derivation, is painful; there are many duplicated lines of
-`rec-trunc-Prop motive`. Agda's `do` syntax offers us an alternative.
+derivation, is a little awkward even at two levels, let alone more.  In
+particular, we have the many duplicated lines of
+`rec-trunc-Prop motive`, and the increasing distance between the
+`rec-trunc-Prop` and the `trunc-Prop` being recursed on.
+Agda's `do` syntax offers us an alternative.
 
 ```agda
 module do-syntax-trunc-Prop {l : Level} (motive : Prop l) where
@@ -478,7 +481,7 @@ module do-syntax-trunc-Prop {l : Level} (motive : Prop l) where
   trunc-prop-a >>= k = rec-trunc-Prop motive k trunc-prop-a
 ```
 
-This allows us to rewrite the deeply nested chain above
+This allows us to rewrite the nested chain above as
 
 ```text
 do

From e510c3d10b7d37a49fa96bab3302b37e69a184a2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 22:23:29 -0800
Subject: [PATCH 54/73] make pre-commit

---
 src/foundation/propositional-truncations.lagda.md | 9 ++++-----
 1 file changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 87986c98ab..5c6123e515 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -466,11 +466,10 @@ rec-trunc-Prop
 ```
 
 The tower of indentation, with many layers of indentation in the innermost
-derivation, is a little awkward even at two levels, let alone more.  In
-particular, we have the many duplicated lines of
-`rec-trunc-Prop motive`, and the increasing distance between the
-`rec-trunc-Prop` and the `trunc-Prop` being recursed on.
-Agda's `do` syntax offers us an alternative.
+derivation, is a little awkward even at two levels, let alone more. In
+particular, we have the many duplicated lines of `rec-trunc-Prop motive`, and
+the increasing distance between the `rec-trunc-Prop` and the `trunc-Prop` being
+recursed on. Agda's `do` syntax offers us an alternative.
 
 ```agda
 module do-syntax-trunc-Prop {l : Level} (motive : Prop l) where

From b4e7234f8a96af9dff8fa8c28abe883de6253c62 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Wed, 12 Feb 2025 22:25:44 -0800
Subject: [PATCH 55/73] Reorganize

---
 .../propositional-truncations.lagda.md        | 32 +++++++++----------
 1 file changed, 16 insertions(+), 16 deletions(-)

diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 5c6123e515..1a38564da6 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -490,22 +490,22 @@ do
 where open do-syntax-trunc-Prop motive
 ```
 
-Note that the last line, `where open do-syntax-trunc-Prop motive`, is required
-to make this syntax possible.
-
-In general, you can read the `p ← truncated-prop-P` as an _instruction_ saying,
-"Get the value `p` out of its truncation `truncated-prop-Q`." We obviously
-cannot get values out of truncations in general, but conceptually, we can do it
-in the service of proving a proposition, and that's what we're doing here. The
-following line `q ← truncated-prop-Q p` says, "Get the value `q` out of the
-truncation `truncated-prop-Q p`" -- noticing that we can make use of `p` in that
-line. Finally, `derive-motive-from p q` must give us a value of type
-`type-Prop motive` from `p` and `q`. The result of the entire `do` block is the
-value of type `type-Prop motive`, which is internally constructed with an
-appropriate chain of `rec-trunc-Prop` from the intermediate steps.
-
-While we had each line depend on all the values from the lines above it, this is
-not at all required.
+Going through each line:
+
+1. You can read the `p ← truncated-prop-P` as an _instruction_ saying, "Get the
+   value `p` out of its truncation `truncated-prop-Q`." We obviously cannot get
+   values out of truncations in general, but conceptually, we can do it in the
+   service of proving a proposition, and that's what we're doing here.
+2. `q ← truncated-prop-Q p` says, "Get the value `q` out of the truncation
+   `truncated-prop-Q p`" -- noticing that we can make use of `p` in that line.
+3. `derive-motive-from p q` must give us a value of type `type-Prop motive` from
+   `p` and `q`.
+4. `where open do-syntax-trunc-Prop motive` is required to allow us to use the
+   `do` syntax.
+
+The result of the entire `do` block is the value of type `type-Prop motive`,
+which is internally constructed with an appropriate chain of `rec-trunc-Prop`
+from the intermediate steps.
 
 ## Table of files about propositional logic
 

From 6cd9e60a6d17c1639fd6d03945c76bd8f9f33ea7 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 14:38:04 -0800
Subject: [PATCH 56/73] Apply suggestions from code review
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Co-authored-by: Vojtěch Štěpančík <vojtechstepancik@outlook.com>
---
 src/foundation/propositional-truncations.lagda.md | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 1a38564da6..6d4d6c259e 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -446,7 +446,7 @@ module _
     is-equiv-map-inv-trunc-Prop-diagonal-coproduct
 ```
 
-## `do` syntax for propositional truncation
+## `do` syntax for propositional truncation { #do-syntax }
 
 [Agda's `do` syntax](https://agda.readthedocs.io/en/v2.7.0/language/syntactic-sugar.html#do-notation)
 is a handy tool to avoid deeply nesting calls to the same lambda-based function.

From 45f47a805d65d39b85f5859043e23e8d5a9f97e2 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 14:59:08 -0800
Subject: [PATCH 57/73] Formatting

---
 .../existential-quantification.lagda.md       |  5 ++
 .../propositional-truncations.lagda.md        | 59 ++++++++++++-------
 2 files changed, 43 insertions(+), 21 deletions(-)

diff --git a/src/foundation/existential-quantification.lagda.md b/src/foundation/existential-quantification.lagda.md
index a15d2258a9..07e29f4bb7 100644
--- a/src/foundation/existential-quantification.lagda.md
+++ b/src/foundation/existential-quantification.lagda.md
@@ -202,6 +202,11 @@ module _
         ( elim-exists Q)
 ```
 
+Note that since existential quantification is implemented using propositional
+truncation, the associated
+[do syntax](foundation.propositional-truncations.md#do-syntax) can be used to
+simplify deeply nested chains of `elim-exists`.
+
 ### The existential quantification satisfies the universal property of existential quantification
 
 ```agda
diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index 1a38564da6..5f59a9bafc 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -446,23 +446,23 @@ module _
     is-equiv-map-inv-trunc-Prop-diagonal-coproduct
 ```
 
-## `do` syntax for propositional truncation
+## `do` syntax for propositional truncation {#do-syntax}
 
 [Agda's `do` syntax](https://agda.readthedocs.io/en/v2.7.0/language/syntactic-sugar.html#do-notation)
 is a handy tool to avoid deeply nesting calls to the same lambda-based function.
 For example, consider a case where you are trying to prove a proposition,
-`motive : Prop l`, from a chain of truncated propositions `P` and `Q`, where
-showing `Q` requires having a value of `type-Prop P`:
+`motive : Prop l`, from witnesses of propositional truncations of types `P` and
+`Q`:
 
 ```text
 rec-trunc-Prop
   ( motive)
-  ( λ q →
+  ( λ (p : P) →
     rec-trunc-Prop
       ( motive)
-      ( λ q → derive-motive-from p q)
-      ( truncated-prop-Q p))
-  ( truncated-prop-P)
+      ( λ (q : Q) → witness-motive-P-Q p q)
+      ( witness-truncated-prop-Q p))
+  ( witness-truncated-prop-P)
 ```
 
 The tower of indentation, with many layers of indentation in the innermost
@@ -484,23 +484,40 @@ This allows us to rewrite the nested chain above as
 
 ```text
 do
-  p ← truncated-prop-P
-  q ← truncated-prop-Q p
-  derive-motive-from p q
+  p ← witness-truncated-prop-P
+  q ← witness-truncated-prop-Q p
+  witness-motive-P-Q p q
 where open do-syntax-trunc-Prop motive
 ```
 
-Going through each line:
-
-1. You can read the `p ← truncated-prop-P` as an _instruction_ saying, "Get the
-   value `p` out of its truncation `truncated-prop-Q`." We obviously cannot get
-   values out of truncations in general, but conceptually, we can do it in the
-   service of proving a proposition, and that's what we're doing here.
-2. `q ← truncated-prop-Q p` says, "Get the value `q` out of the truncation
-   `truncated-prop-Q p`" -- noticing that we can make use of `p` in that line.
-3. `derive-motive-from p q` must give us a value of type `type-Prop motive` from
-   `p` and `q`.
-4. `where open do-syntax-trunc-Prop motive` is required to allow us to use the
+Since Agda's `do` syntax desugars to calls to `>>=`, this is simply syntactic
+sugar for
+
+```text
+witness-truncated-prop-P >>=
+  ( λ p → witness-truncated-prop-Q p >>=
+    ( λ q → witness-motive-P-Q p q))
+```
+
+which, inlining the definition of `>>=`, becomes exactly the chain of
+`rec-trunc-Prop` used above.
+
+To read the `do` syntax, it may help to go through each line:
+
+1. `do` simply indicates that we will be using Agda's syntactic sugar for the
+   `>>=` function defined in the `do-syntax-trunc-Prop` module.
+1. You can read the `p ← witness-truncated-prop-P` as an _instruction_ saying,
+   "Get the value `p` out of the witness of `trunc-Prop P`." We cannot extract
+   elements out of witnesses of propositionally truncated types, but since we're
+   eliminating into a proposition, the universal property of the truncation
+   allows us to lift a map from the untruncated type to a map from its
+   truncation.
+1. `q ← witness-truncated-prop-Q p` says, "Get the value `q` out of the witness
+   for the truncation `witness-truncated-prop-Q p`" -- noticing that we can make
+   use of `p : P` in that line.
+1. `witness-motive-P-Q p q` must give us a witness of `motive` -- that is, a
+   value of type `type-Prop motive` -- from `p` and `q`.
+1. `where open do-syntax-trunc-Prop motive` is required to allow us to use the
    `do` syntax.
 
 The result of the entire `do` block is the value of type `type-Prop motive`,

From 23a33d354e3d0c034e5cec8180b9c058a04d6db9 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 15:07:01 -0800
Subject: [PATCH 58/73] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 ...rithmetically-located-dedekind-cuts.lagda.md |  2 +-
 src/real-numbers/dedekind-real-numbers.lagda.md | 17 ++++++++---------
 2 files changed, 9 insertions(+), 10 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 405a77a7e2..aaba62504f 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -46,7 +46,7 @@ A pair of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md)
 `L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md)
 `U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if
 for any positive [rational number](elementary-number-theory.rational-numbers.md)
-`ε : ℚ`, there exist `p, q : ℚ` such that `0 < q - p < ε`, `p ∈ L`, and `q ∈ U`.
+`ε : ℚ`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that `0 < q - p < ε`.
 Intuitively, when `L , U` represent the cuts of a real number `x`, `p` and `q`
 are rational approximations of `x` to within `ε`. This follows parts of Section
 11 in {{#cite BauerTaylor2009}}.
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 3e6695ebce..e68152a823 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -50,22 +50,21 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-A Dedekind real number consists of a [pair](foundation.dependent-pair-types.md)
-`(lx , uy)` of a
-[lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md) and an
-[upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md) that also
-satisfy the following conditions:
+A {{#concept "Dedekind real number" WD="real number" WDID=Q12916 Agda=ℝ}} `x` consists of a [pair](foundation.dependent-pair-types.md)
+ of _cuts_ `(L , U)` of [rational numbers](elementary-number-theory.rational-numbers.md), a
+[lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md) `L` and an
+[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`. These cuts present lower and upper bounds on the Dedekind real number, and must satisfy the conditions
 
-1. _Disjointness_. The cuts of `lx` and `uy` are disjoint subsets of `ℚ`.
-2. _Locatedness_. If `q < r` then `q` is in the cut of `lx` or `r` is in the cut
-   of `uy`.
+1. _Disjointness_. The cuts `lx` and `uy` are disjoint subsets of `ℚ`.
+2. _Locatedness_. If `q < r` then `q` is in the cut `L` or `r` is in the cut `U`.
 
+The {{#concept "collection of all Dedekind real numbers" WD="set of real numbers" WDID=Q1174982 Agda=ℝ}} is denoted `ℝ`.
 The Dedekind real numbers will be taken as the standard definition of the real
 numbers in the agda-unimath library.
 
 ## Definition
 
-### Dedekind real numbers
+### Dedekind cuts
 
 ```agda
 module _

From f2c3f7646486d0f5b5894418ea4baa08847a3f65 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 15:13:49 -0800
Subject: [PATCH 59/73] renaming

---
 src/real-numbers/dedekind-real-numbers.lagda.md | 13 +++++++------
 1 file changed, 7 insertions(+), 6 deletions(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index e68152a823..4c2e2f5a10 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -438,10 +438,11 @@ module _
   (lx : lower-ℝ l)
   where
 
-  has-upper-real-Prop : Prop (lsuc l)
-  pr1 has-upper-real-Prop = Σ (upper-ℝ l) (is-dedekind-lower-upper-ℝ lx)
-  pr2 has-upper-real-Prop =
-    ( is-prop-all-elements-equal)
+  is-dedekind-cut-lower-ℝ-Prop : Prop (lsuc l)
+  pr1 is-dedekind-cut-lower-ℝ-Prop =
+    Σ (upper-ℝ l) (is-dedekind-lower-upper-ℝ lx)
+  pr2 is-dedekind-cut-lower-ℝ-Prop =
+    is-prop-all-elements-equal
     ( λ uy uy' →
       eq-type-subtype
         ( is-dedekind-prop-lower-upper-ℝ lx)
@@ -451,7 +452,7 @@ module _
           ( eq-upper-cut-eq-lower-cut-ℝ (lx , uy) (lx , uy') refl)))
 
 is-emb-lower-real : {l : Level} → is-emb (lower-real-ℝ {l})
-is-emb-lower-real = is-emb-inclusion-subtype has-upper-real-Prop
+is-emb-lower-real = is-emb-inclusion-subtype is-dedekind-cut-lower-ℝ-Prop
 ```
 
 ### Two real numbers with the same lower/upper real are equal
@@ -462,7 +463,7 @@ module _
   where
 
   eq-eq-lower-real-ℝ : lower-real-ℝ x ＝ lower-real-ℝ y → x ＝ y
-  eq-eq-lower-real-ℝ = eq-type-subtype has-upper-real-Prop
+  eq-eq-lower-real-ℝ = eq-type-subtype is-dedekind-cut-lower-ℝ-Prop
 
   eq-eq-upper-real-ℝ : upper-real-ℝ x ＝ upper-real-ℝ y → x ＝ y
   eq-eq-upper-real-ℝ = eq-eq-lower-real-ℝ ∘ (eq-lower-real-eq-upper-real-ℝ x y)

From 36731631bde357cfda2527326f45a3d388ab945f Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 15:58:03 -0800
Subject: [PATCH 60/73] make pre-commit

---
 ...thmetically-located-dedekind-cuts.lagda.md |  8 ++++----
 .../dedekind-real-numbers.lagda.md            | 19 ++++++++++++-------
 2 files changed, 16 insertions(+), 11 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index aaba62504f..f290bbe754 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -46,10 +46,10 @@ A pair of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md)
 `L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md)
 `U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if
 for any positive [rational number](elementary-number-theory.rational-numbers.md)
-`ε : ℚ`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that `0 < q - p < ε`.
-Intuitively, when `L , U` represent the cuts of a real number `x`, `p` and `q`
-are rational approximations of `x` to within `ε`. This follows parts of Section
-11 in {{#cite BauerTaylor2009}}.
+`ε : ℚ`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that
+`0 < q - p < ε`. Intuitively, when `L , U` represent the cuts of a real number
+`x`, `p` and `q` are rational approximations of `x` to within `ε`. This follows
+parts of Section 11 in {{#cite BauerTaylor2009}}.
 
 ## Definitions
 
diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 4c2e2f5a10..3b6c9ce2bc 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -50,17 +50,22 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-A {{#concept "Dedekind real number" WD="real number" WDID=Q12916 Agda=ℝ}} `x` consists of a [pair](foundation.dependent-pair-types.md)
- of _cuts_ `(L , U)` of [rational numbers](elementary-number-theory.rational-numbers.md), a
+A {{#concept "Dedekind real number" WD="real number" WDID=Q12916 Agda=ℝ}} `x`
+consists of a [pair](foundation.dependent-pair-types.md) of _cuts_ `(L , U)` of
+[rational numbers](elementary-number-theory.rational-numbers.md), a
 [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md) `L` and an
-[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`. These cuts present lower and upper bounds on the Dedekind real number, and must satisfy the conditions
+[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`. These
+cuts present lower and upper bounds on the Dedekind real number, and must
+satisfy the conditions
 
 1. _Disjointness_. The cuts `lx` and `uy` are disjoint subsets of `ℚ`.
-2. _Locatedness_. If `q < r` then `q` is in the cut `L` or `r` is in the cut `U`.
+2. _Locatedness_. If `q < r` then `q` is in the cut `L` or `r` is in the cut
+   `U`.
 
-The {{#concept "collection of all Dedekind real numbers" WD="set of real numbers" WDID=Q1174982 Agda=ℝ}} is denoted `ℝ`.
-The Dedekind real numbers will be taken as the standard definition of the real
-numbers in the agda-unimath library.
+The
+{{#concept "collection of all Dedekind real numbers" WD="set of real numbers" WDID=Q1174982 Agda=ℝ}}
+is denoted `ℝ`. The Dedekind real numbers will be taken as the standard
+definition of the real numbers in the agda-unimath library.
 
 ## Definition
 

From 324595aba9740dec0caaae65e69922c4fb08631e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:02:06 -0800
Subject: [PATCH 61/73] Respond to review comments

---
 src/real-numbers/lower-dedekind-real-numbers.lagda.md | 8 ++++----
 src/real-numbers/upper-dedekind-real-numbers.lagda.md | 8 ++++----
 2 files changed, 8 insertions(+), 8 deletions(-)

diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index db4cf96ad6..ddae49559d 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -33,10 +33,10 @@ open import foundation.universe-levels
 
 ## Idea
 
-A {{#concept "lower Dedekind cut" Agda=is-lower-dedekind-cut}} consists of a
-[subtype](foundation-core.subtypes.md) `L` of
-[the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
-satisfying the following two conditions:
+A [subtype](foundation-core.subtypes.md) `L` of
+[the rational numbers](elementary-number-theory.rational-numbers.md) is a
+{{#concept "lower Dedekind cut" Agda=is-lower-dedekind-cut}} if it satisfies the
+following two conditions:
 
 1. _Inhabitedness_. `L` is [inhabited](foundation.inhabited-subtypes.md).
 2. _Roundedness_. A rational number `q` is in `L`
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index 83d2328e84..da99075a67 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -33,10 +33,10 @@ open import foundation.universe-levels
 
 ## Idea
 
-An {{#concept "upper Dedekind cut" Agda=is-upper-dedekind-cut}} consists of a
-[subtype](foundation-core.subtypes.md) `U` of
-[the rational numbers](elementary-number-theory.rational-numbers.md) `ℚ`,
-satisfying the following two conditions:
+A [subtype](foundation-core.subtypes.md) `U` of
+[the rational numbers](elementary-number-theory.rational-numbers.md) is an
+{{#concept "upper Dedekind cut" Agda=is-upper-dedekind-cut}} if it satisfies the
+following two conditions:
 
 1. _Inhabitedness_. `U` is [inhabited](foundation.inhabited-subtypes.md).
 2. _Roundedness_. A rational number `q` is in `U`

From 10a0a3612d92e4893433fcbdedf8d7a8d23d58c1 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:06:06 -0800
Subject: [PATCH 62/73] Inline explanations of lower and upper cuts

---
 src/real-numbers/dedekind-real-numbers.lagda.md | 11 +++++++++--
 1 file changed, 9 insertions(+), 2 deletions(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 3b6c9ce2bc..912f8f9676 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -54,8 +54,15 @@ A {{#concept "Dedekind real number" WD="real number" WDID=Q12916 Agda=ℝ}} `x`
 consists of a [pair](foundation.dependent-pair-types.md) of _cuts_ `(L , U)` of
 [rational numbers](elementary-number-theory.rational-numbers.md), a
 [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md) `L` and an
-[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`. These
-cuts present lower and upper bounds on the Dedekind real number, and must
+[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`. A lower
+Dedekind cut `L` is a subtype of the rational numbers that is
+[inhabited](foundation.inhabited-subtypes.md) and rounded, meaning that `q ∈ L`
+[if and only if](foundation.logical-equivalences.md) there exists `r`, with
+`q < r` and `r ∈ L`. An upper Dedekind cut is a subtype of the rational numbers
+that is inhabited and rounded "in the other direction": `q ∈ U` if and only if
+there is a `p < q` where `p ∈ U`.
+
+These cuts present lower and upper bounds on the Dedekind real number, and must
 satisfy the conditions
 
 1. _Disjointness_. The cuts `lx` and `uy` are disjoint subsets of `ℚ`.

From 6430f86be2f1172423b78799be52cbdfc11075c5 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:07:55 -0800
Subject: [PATCH 63/73] Reword

---
 .../dedekind-real-numbers.lagda.md            | 19 ++++++++++++-------
 1 file changed, 12 insertions(+), 7 deletions(-)

diff --git a/src/real-numbers/dedekind-real-numbers.lagda.md b/src/real-numbers/dedekind-real-numbers.lagda.md
index 912f8f9676..3f83f4b510 100644
--- a/src/real-numbers/dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/dedekind-real-numbers.lagda.md
@@ -54,13 +54,18 @@ A {{#concept "Dedekind real number" WD="real number" WDID=Q12916 Agda=ℝ}} `x`
 consists of a [pair](foundation.dependent-pair-types.md) of _cuts_ `(L , U)` of
 [rational numbers](elementary-number-theory.rational-numbers.md), a
 [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md) `L` and an
-[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`. A lower
-Dedekind cut `L` is a subtype of the rational numbers that is
-[inhabited](foundation.inhabited-subtypes.md) and rounded, meaning that `q ∈ L`
-[if and only if](foundation.logical-equivalences.md) there exists `r`, with
-`q < r` and `r ∈ L`. An upper Dedekind cut is a subtype of the rational numbers
-that is inhabited and rounded "in the other direction": `q ∈ U` if and only if
-there is a `p < q` where `p ∈ U`.
+[upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`.
+
+A lower Dedekind cut `L` is a subtype of the rational numbers that is
+
+1. [Inhabited](foundation.inhabited-subtypes.md), meaning it has at least one
+   element.
+2. Rounded, meaning that `q ∈ L`
+   [if and only if](foundation.logical-equivalences.md) there exists `r`, with
+   `q < r` and `r ∈ L`.
+
+An upper Dedekind cut is analogous, but must be rounded "in the other
+direction": `q ∈ U` if and only if there is a `p < q` where `p ∈ U`.
 
 These cuts present lower and upper bounds on the Dedekind real number, and must
 satisfy the conditions

From 69430a2f07180d25ca98d501cbfb8d678b8f1c54 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:19:20 -0800
Subject: [PATCH 64/73] Update
 src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md

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

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index f290bbe754..4826d54cc4 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -44,7 +44,7 @@ open import real-numbers.upper-dedekind-real-numbers
 
 A pair of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md)
 `L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md)
-`U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut"}} if
+`U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut" Agda=arithmetically-located-lower-upper-ℝ}} if
 for any positive [rational number](elementary-number-theory.rational-numbers.md)
 `ε : ℚ`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that
 `0 < q - p < ε`. Intuitively, when `L , U` represent the cuts of a real number

From 8c1ad1108ac8a1b05fc13089c18e82f6cf961a7e Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:21:08 -0800
Subject: [PATCH 65/73] Progress

---
 .../arithmetically-located-dedekind-cuts.lagda.md  | 14 +++++++-------
 1 file changed, 7 insertions(+), 7 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 4826d54cc4..abd7034a5e 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -42,18 +42,19 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Definition
 
-A pair of a [lower Dedekind real](real-numbers.lower-dedekind-real-numbers.md)
-`L` and an [upper Dedekind real](real-numbers.upper-dedekind-real-numbers.md)
+A pair of a [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md)
+`L` and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md)
 `U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut" Agda=arithmetically-located-lower-upper-ℝ}} if
-for any positive [rational number](elementary-number-theory.rational-numbers.md)
-`ε : ℚ`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that
+for any [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational number](elementary-number-theory.rational-numbers.md)
+`ε : ℚ⁺`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that
 `0 < q - p < ε`. Intuitively, when `L , U` represent the cuts of a real number
 `x`, `p` and `q` are rational approximations of `x` to within `ε`. This follows
 parts of Section 11 in {{#cite BauerTaylor2009}}.
 
 ## Definitions
 
-### The predicate of being arithmetically located on pairs of subtypes of rational numbers
+### The predicate of being arithmetically located on pairs of lower and upper Dedekind real numbers
 
 ```agda
 module _
@@ -74,8 +75,7 @@ module _
 
 ### Arithmetically located cuts are located
 
-If a cut is arithmetically located and closed under strict inequality on the
-rational numbers, it is also located.
+If a pair of lower and upper Dedekind cuts is arithmetically located, it is also located.
 
 ```agda
 module _

From 9f4197ca964c339bb510ff2b7849c203b1e2246c Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:21:52 -0800
Subject: [PATCH 66/73] Review comments

---
 .../arithmetically-located-dedekind-cuts.lagda.md             | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index abd7034a5e..5b589d3f7e 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -49,7 +49,9 @@ for any [positive](elementary-number-theory.positive-rational-numbers.md)
 [rational number](elementary-number-theory.rational-numbers.md)
 `ε : ℚ⁺`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that
 `0 < q - p < ε`. Intuitively, when `L , U` represent the cuts of a real number
-`x`, `p` and `q` are rational approximations of `x` to within `ε`. This follows
+`x`, `p` and `q` are rational approximations of `x` to within `ε`.
+
+This follows
 parts of Section 11 in {{#cite BauerTaylor2009}}.
 
 ## Definitions

From d6da82420b1046f383d342b4128edd2bafe5e501 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:24:05 -0800
Subject: [PATCH 67/73] reword negation

---
 ...thmetically-located-dedekind-cuts.lagda.md | 21 ++++++++++---------
 ...lower-upper-dedekind-real-numbers.lagda.md |  6 ++++--
 2 files changed, 15 insertions(+), 12 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index 5b589d3f7e..dc0ce73fc2 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -43,16 +43,16 @@ open import real-numbers.upper-dedekind-real-numbers
 ## Definition
 
 A pair of a [lower Dedekind cut](real-numbers.lower-dedekind-real-numbers.md)
-`L` and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md)
-`U` is {{#concept "arithmetically located" Disambiguation="Dedekind cut" Agda=arithmetically-located-lower-upper-ℝ}} if
-for any [positive](elementary-number-theory.positive-rational-numbers.md)
-[rational number](elementary-number-theory.rational-numbers.md)
-`ε : ℚ⁺`, there exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that
-`0 < q - p < ε`. Intuitively, when `L , U` represent the cuts of a real number
-`x`, `p` and `q` are rational approximations of `x` to within `ε`.
+`L` and an [upper Dedekind cut](real-numbers.upper-dedekind-real-numbers.md) `U`
+is
+{{#concept "arithmetically located" Disambiguation="Dedekind cut" Agda=arithmetically-located-lower-upper-ℝ}}
+if for any [positive](elementary-number-theory.positive-rational-numbers.md)
+[rational number](elementary-number-theory.rational-numbers.md) `ε : ℚ⁺`, there
+exists `p, q : ℚ` where `p ∈ L` and `q ∈ U`, such that `0 < q - p < ε`.
+Intuitively, when `L , U` represent the cuts of a real number `x`, `p` and `q`
+are rational approximations of `x` to within `ε`.
 
-This follows
-parts of Section 11 in {{#cite BauerTaylor2009}}.
+This follows parts of Section 11 in {{#cite BauerTaylor2009}}.
 
 ## Definitions
 
@@ -77,7 +77,8 @@ module _
 
 ### Arithmetically located cuts are located
 
-If a pair of lower and upper Dedekind cuts is arithmetically located, it is also located.
+If a pair of lower and upper Dedekind cuts is arithmetically located, it is also
+located.
 
 ```agda
 module _
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 5c6f4d84f5..b9847a632c 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
@@ -40,8 +40,10 @@ open import real-numbers.upper-dedekind-real-numbers
 
 ## Idea
 
-The negation of a lower Dedekind real is an upper Dedekind real containing the
-negations of elements in the lower cut, and vice versa.
+The negation of a
+[lower Dedekind real number](real-numbers.lower-dedekind-real-numbers.md) is an
+[upper Dedekind real number](real-numbers.upper-dedekind-real-numbers.md) whose
+cut contains negations of elements in the lower cut, and vice versa.
 
 ## Definition
 

From 0a2691ca5cddb2441a32e7c73cf5133cdec1bb03 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:24:49 -0800
Subject: [PATCH 68/73] Apply suggestions from code review

Co-authored-by: Fredrik Bakke <fredrbak@gmail.com>
---
 .../arithmetically-located-dedekind-cuts.lagda.md             | 4 ++--
 1 file changed, 2 insertions(+), 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 dc0ce73fc2..c3d4d59716 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -60,7 +60,7 @@ This follows parts of Section 11 in {{#cite BauerTaylor2009}}.
 
 ```agda
 module _
-  {l : Level} (L : lower-ℝ l) (U : upper-ℝ l)
+  {l1 l2 : Level} (L : lower-ℝ l1) (U : upper-ℝ l2)
   where
 
   arithmetically-located-lower-upper-ℝ : UU l
@@ -82,7 +82,7 @@ located.
 
 ```agda
 module _
-  {l : Level} (x : lower-ℝ l) (y : upper-ℝ l)
+  {l1 l2 : Level} (x : lower-ℝ l1) (y : upper-ℝ l2)
   where
 
   abstract

From f462be5d7d0007edf17871bde17dcd75435cb29d Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:32:12 -0800
Subject: [PATCH 69/73] Fix compile

---
 .../arithmetically-located-dedekind-cuts.lagda.md         | 8 ++++----
 1 file changed, 4 insertions(+), 4 deletions(-)

diff --git a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
index c3d4d59716..fa5295f6a6 100644
--- a/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
+++ b/src/real-numbers/arithmetically-located-dedekind-cuts.lagda.md
@@ -60,17 +60,17 @@ This follows parts of Section 11 in {{#cite BauerTaylor2009}}.
 
 ```agda
 module _
-  {l1 l2 : Level} (L : lower-ℝ l1) (U : upper-ℝ l2)
+  {l1 l2 : Level} (x : lower-ℝ l1) (y : upper-ℝ l2)
   where
 
-  arithmetically-located-lower-upper-ℝ : UU l
+  arithmetically-located-lower-upper-ℝ : UU (l1 ⊔ l2)
   arithmetically-located-lower-upper-ℝ =
     (ε⁺ : ℚ⁺) →
     exists
       ( ℚ × ℚ)
       ( λ (p , q) → le-ℚ-Prop q (p +ℚ rational-ℚ⁺ ε⁺) ∧
-        cut-lower-ℝ L p ∧
-        cut-upper-ℝ U q)
+        cut-lower-ℝ x p ∧
+        cut-upper-ℝ y q)
 ```
 
 ## Properties

From 5978e1a566b16e416be7efcee630e639ebadc241 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Thu, 13 Feb 2025 16:44:22 -0800
Subject: [PATCH 70/73] Review comments

---
 src/foundation/propositional-truncations.lagda.md | 15 +++++++--------
 1 file changed, 7 insertions(+), 8 deletions(-)

diff --git a/src/foundation/propositional-truncations.lagda.md b/src/foundation/propositional-truncations.lagda.md
index f067b32f78..8580a7744a 100644
--- a/src/foundation/propositional-truncations.lagda.md
+++ b/src/foundation/propositional-truncations.lagda.md
@@ -490,8 +490,7 @@ do
 where open do-syntax-trunc-Prop motive
 ```
 
-Since Agda's `do` syntax desugars to calls to `>>=`, this is simply syntactic
-sugar for
+Since Agda's `do` syntax desugars to calls to `>>=`, this is syntactic sugar for
 
 ```text
 witness-truncated-prop-P >>=
@@ -504,8 +503,8 @@ which, inlining the definition of `>>=`, becomes exactly the chain of
 
 To read the `do` syntax, it may help to go through each line:
 
-1. `do` simply indicates that we will be using Agda's syntactic sugar for the
-   `>>=` function defined in the `do-syntax-trunc-Prop` module.
+1. `do` indicates that we will be using Agda's syntactic sugar for the `>>=`
+   function defined in the `do-syntax-trunc-Prop` module.
 1. You can read the `p ← witness-truncated-prop-P` as an _instruction_ saying,
    "Get the value `p` out of the witness of `trunc-Prop P`." We cannot extract
    elements out of witnesses of propositionally truncated types, but since we're
@@ -513,10 +512,10 @@ To read the `do` syntax, it may help to go through each line:
    allows us to lift a map from the untruncated type to a map from its
    truncation.
 1. `q ← witness-truncated-prop-Q p` says, "Get the value `q` out of the witness
-   for the truncation `witness-truncated-prop-Q p`" -- noticing that we can make
-   use of `p : P` in that line.
-1. `witness-motive-P-Q p q` must give us a witness of `motive` -- that is, a
-   value of type `type-Prop motive` -- from `p` and `q`.
+   for the truncation `witness-truncated-prop-Q p`" --- noticing that we can
+   make use of `p : P` in that line.
+1. `witness-motive-P-Q p q` must give us a witness of `motive` --- that is, a
+   value of type `type-Prop motive` --- from `p` and `q`.
 1. `where open do-syntax-trunc-Prop motive` is required to allow us to use the
    `do` syntax.
 

From e981a15b8285ee4130b013f3f0a46c0bb542f730 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 14 Feb 2025 11:50:03 -0800
Subject: [PATCH 71/73] Progress

---
 .../addition-lower-dedekind-real-numbers.lagda.md           | 6 ++++++
 1 file changed, 6 insertions(+)

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 0073594d5b..36113fa021 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -207,3 +207,9 @@ module _
         ( cut-lower-ℝ y)
         ( cut-lower-ℝ z))
 ```
+
+### Unit laws for the addition of lower Dedekind real numbers
+
+```agda
+
+```

From d90e459a320fef8aa2e97962d8a5b781d5d78159 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 14 Feb 2025 13:32:26 -0800
Subject: [PATCH 72/73] Unit laws and simplifications

---
 ...ition-lower-dedekind-real-numbers.lagda.md | 194 ++++++++++--------
 ...ition-upper-dedekind-real-numbers.lagda.md | 180 +++++++++-------
 .../lower-dedekind-real-numbers.lagda.md      |  17 ++
 .../upper-dedekind-real-numbers.lagda.md      |  19 ++
 4 files changed, 250 insertions(+), 160 deletions(-)

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 36113fa021..652c87812c 100644
--- a/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-lower-dedekind-real-numbers.lagda.md
@@ -24,6 +24,7 @@ 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.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -35,6 +36,7 @@ open import group-theory.minkowski-multiplication-commutative-monoids
 open import logic.functoriality-existential-quantification
 
 open import real-numbers.lower-dedekind-real-numbers
+open import real-numbers.rational-lower-dedekind-real-numbers
 ```
 
 </details>
@@ -78,89 +80,67 @@ module _
       (q : ℚ) →
       is-in-cut-add-lower-ℝ q ↔
       exists ℚ (λ r → le-ℚ-Prop q r ∧ cut-add-lower-ℝ r)
-    pr1 (is-rounded-cut-add-lower-ℝ q) =
-      elim-exists
-        ( ∃ ℚ (λ r → le-ℚ-Prop q r ∧ cut-add-lower-ℝ r))
-        ( λ (lx , ly) (lx<x , ly<y , q=lx+ly) →
-          map-binary-exists
-            ( λ r → le-ℚ q r × is-in-cut-add-lower-ℝ r)
-            ( add-ℚ)
-            ( λ lx' ly' (lx<lx' , lx'<x) (ly<ly' , ly'<y) →
-              tr
-                ( λ p → le-ℚ p (lx' +ℚ ly'))
-                ( inv q=lx+ly)
-                ( preserves-le-add-ℚ {lx} {lx'} {ly} {ly'} lx<lx' ly<ly') ,
-              intro-exists (lx' , ly') (lx'<x , ly'<y , refl))
-            ( forward-implication (is-rounded-cut-lower-ℝ x lx) lx<x)
-            ( forward-implication (is-rounded-cut-lower-ℝ y ly) ly<y))
-    pr2 (is-rounded-cut-add-lower-ℝ q) =
-      elim-exists
-        ( cut-add-lower-ℝ q)
-        ( λ r (q<r , r<x+y) →
-          elim-exists
-            ( cut-add-lower-ℝ q)
-            ( λ (rx , ry) (rx<x , ry<y , r=rx+ry) →
-              let
-                r-q⁺ : ℚ⁺
-                r-q⁺ = positive-diff-le-ℚ q r q<r
-                ε⁺ : ℚ⁺
-                ε⁺ = mediant-zero-ℚ⁺ r-q⁺
-                ε : ℚ
-                ε = rational-ℚ⁺ ε⁺
-                ε<r-q : le-ℚ ε (r -ℚ q)
-                ε<r-q = le-mediant-zero-ℚ⁺ r-q⁺
-              in
-              intro-exists
-                ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
-                ( is-in-cut-le-ℚ-lower-ℝ
-                    ( x)
-                    ( rx -ℚ ε)
-                    ( rx)
-                    ( le-diff-rational-ℚ⁺ rx ε⁺)
-                    ( rx<x) ,
-                  is-in-cut-le-ℚ-lower-ℝ
-                    ( y)
-                    ( q -ℚ (rx -ℚ ε))
-                    ( ry)
-                    ( binary-tr
-                        ( le-ℚ)
-                        ( equational-reasoning
-                          (q -ℚ rx) +ℚ ε
-                          ＝ q +ℚ (neg-ℚ rx +ℚ ε)
-                            by associative-add-ℚ q (neg-ℚ rx) ε
-                          ＝ q +ℚ (ε -ℚ rx)
-                            by ap (q +ℚ_) (commutative-add-ℚ (neg-ℚ rx) ε)
-                          ＝ q -ℚ (rx -ℚ ε)
-                            by ap (q +ℚ_) (inv (distributive-neg-diff-ℚ rx ε)))
-                        ( equational-reasoning
-                          (q -ℚ rx) +ℚ (r -ℚ q)
-                          ＝ (q -ℚ rx) +ℚ (neg-ℚ q +ℚ r)
-                            by
-                              ap ((q -ℚ rx) +ℚ_) (commutative-add-ℚ r (neg-ℚ q))
-                          ＝ (q -ℚ q) +ℚ (neg-ℚ rx +ℚ r)
-                            by
-                              interchange-law-add-add-ℚ q (neg-ℚ rx) (neg-ℚ q) r
-                          ＝ zero-ℚ +ℚ (neg-ℚ rx +ℚ r)
-                            by
-                              ap
-                                ( _+ℚ (neg-ℚ rx +ℚ r))
-                                ( right-inverse-law-add-ℚ q)
-                          ＝ neg-ℚ rx +ℚ r by left-unit-law-add-ℚ (neg-ℚ rx +ℚ r)
-                          ＝ neg-ℚ rx +ℚ (rx +ℚ ry) by ap (neg-ℚ rx +ℚ_) r=rx+ry
-                          ＝ ry
-                            by is-retraction-left-div-Group group-add-ℚ rx ry)
-                        ( preserves-le-right-add-ℚ
-                          ( q -ℚ rx)
-                          ( ε)
-                          ( r -ℚ q)
-                          ( ε<r-q)))
-                    ( ry<y) ,
-                  inv
-                    ( is-identity-right-conjugation-Ab
-                      ( abelian-group-add-ℚ)
-                      ( rx -ℚ ε)
-                      ( q))))
-            ( r<x+y))
+    pr1 (is-rounded-cut-add-lower-ℝ q) q<x+y =
+      do
+        ((lx , ly) , (lx<x , ly<y , q=lx+ly)) ← q<x+y
+        (lx' , lx<lx' , lx'<x) ←
+          forward-implication (is-rounded-cut-lower-ℝ x lx) lx<x
+        (ly' , ly<ly' , ly'<y) ←
+          forward-implication (is-rounded-cut-lower-ℝ y ly) ly<y
+        intro-exists
+          ( lx' +ℚ ly')
+          ( tr
+              ( λ p → le-ℚ p (lx' +ℚ ly'))
+              ( inv q=lx+ly)
+              ( preserves-le-add-ℚ {lx} {lx'} {ly} {ly'} lx<lx' ly<ly') ,
+            intro-exists (lx' , ly') (lx'<x , ly'<y , refl))
+      where
+        open
+          do-syntax-trunc-Prop (∃ ℚ (λ r → le-ℚ-Prop q r ∧ cut-add-lower-ℝ r))
+    pr2 (is-rounded-cut-add-lower-ℝ q) exists-r =
+      do
+        (r , q<r , r<x+y) ← exists-r
+        ((rx , ry) , (rx<x , ry<y , r=rx+ry)) ← r<x+y
+        let
+          r-q⁺ = positive-diff-le-ℚ q r q<r
+          ε⁺ = mediant-zero-ℚ⁺ r-q⁺
+          ε = rational-ℚ⁺ ε⁺
+          ε<r-q = le-mediant-zero-ℚ⁺ r-q⁺
+        intro-exists
+          ( rx -ℚ ε , q -ℚ (rx -ℚ ε))
+          ( is-in-cut-diff-rational-ℚ⁺-lower-ℝ x rx ε⁺ rx<x ,
+            is-in-cut-le-ℚ-lower-ℝ
+              ( y)
+              ( q -ℚ (rx -ℚ ε))
+              ( ry)
+              ( binary-tr
+                ( le-ℚ)
+                ( equational-reasoning
+                  (q -ℚ rx) +ℚ ε
+                  ＝ q +ℚ (neg-ℚ rx +ℚ ε)
+                    by associative-add-ℚ q (neg-ℚ rx) ε
+                  ＝ q +ℚ (ε -ℚ rx)
+                    by ap (q +ℚ_) (commutative-add-ℚ (neg-ℚ rx) ε)
+                  ＝ q -ℚ (rx -ℚ ε)
+                    by ap (q +ℚ_) (inv (distributive-neg-diff-ℚ rx ε)))
+                ( equational-reasoning
+                  (q -ℚ rx) +ℚ (r -ℚ q)
+                  ＝ (q -ℚ rx) -ℚ (q -ℚ r)
+                    by ap ((q -ℚ rx) +ℚ_) (inv (distributive-neg-diff-ℚ q r))
+                  ＝ neg-ℚ rx -ℚ neg-ℚ r
+                    by left-translation-diff-ℚ (neg-ℚ rx) (neg-ℚ r) q
+                  ＝ neg-ℚ rx +ℚ (rx +ℚ ry)
+                    by ap (neg-ℚ rx +ℚ_) (neg-neg-ℚ r ∙ r=rx+ry)
+                  ＝ ry
+                    by is-retraction-left-div-Group group-add-ℚ rx ry)
+                ( preserves-le-right-add-ℚ (q -ℚ rx) ε (r -ℚ q) ε<r-q))
+              ( ry<y) ,
+            inv
+              ( is-identity-right-conjugation-Ab
+                ( abelian-group-add-ℚ)
+                ( rx -ℚ ε)
+                ( q)))
+      where open do-syntax-trunc-Prop (cut-add-lower-ℝ q)
 
   add-lower-ℝ : lower-ℝ (l1 ⊔ l2)
   pr1 add-lower-ℝ = cut-add-lower-ℝ
@@ -211,5 +191,53 @@ module _
 ### Unit laws for the addition of lower Dedekind real numbers
 
 ```agda
+module _
+  {l : Level} (x : lower-ℝ l)
+  where
 
+  abstract
+    right-unit-law-add-lower-ℝ : add-lower-ℝ x (lower-real-ℚ zero-ℚ) ＝ x
+    right-unit-law-add-lower-ℝ =
+      eq-sim-cut-lower-ℝ
+        ( add-lower-ℝ x (lower-real-ℚ zero-ℚ))
+        ( x)
+        ( (λ r →
+            elim-exists
+              ( cut-lower-ℝ x r)
+              ( λ (p , q) (p<x , q<0 , r=p+q) →
+                inv-tr
+                  ( is-in-cut-lower-ℝ x)
+                  ( r=p+q ∙ ap (p +ℚ_) (inv (neg-neg-ℚ q)))
+                  ( is-in-cut-diff-rational-ℚ⁺-lower-ℝ
+                    ( x)
+                    ( p)
+                    ( neg-ℚ q ,
+                      is-positive-le-zero-ℚ (neg-ℚ q) (neg-le-ℚ q zero-ℚ q<0))
+                    ( p<x)))) ,
+          (λ p p<x →
+            elim-exists
+              ( cut-add-lower-ℝ x (lower-real-ℚ zero-ℚ) p)
+              ( λ q (p<q , q<x) →
+                intro-exists
+                  ( q , p -ℚ q)
+                  ( q<x ,
+                    tr
+                      ( λ r → le-ℚ r zero-ℚ)
+                      ( distributive-neg-diff-ℚ q p)
+                      ( neg-le-ℚ
+                        ( zero-ℚ)
+                        ( q -ℚ p)
+                        ( backward-implication
+                          ( iff-translate-diff-le-zero-ℚ p q)
+                          ( p<q))) ,
+                    inv
+                      ( is-identity-right-conjugation-Ab
+                        ( abelian-group-add-ℚ)
+                        ( q)
+                        ( p))))
+              ( forward-implication (is-rounded-cut-lower-ℝ x p) p<x)))
+
+  left-unit-law-add-lower-ℝ : add-lower-ℝ (lower-real-ℚ zero-ℚ) x ＝ x
+  left-unit-law-add-lower-ℝ =
+    commutative-add-lower-ℝ (lower-real-ℚ zero-ℚ) x ∙ right-unit-law-add-lower-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 5e8a0a37ee..0941ed1a99 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -24,6 +24,7 @@ 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.subtypes
 open import foundation.transport-along-identifications
 open import foundation.universe-levels
@@ -32,9 +33,8 @@ open import group-theory.abelian-groups
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 
-open import logic.functoriality-existential-quantification
-
 open import real-numbers.upper-dedekind-real-numbers
+open import real-numbers.rational-upper-dedekind-real-numbers
 ```
 
 </details>
@@ -78,81 +78,67 @@ module _
       (q : ℚ) →
       is-in-cut-add-upper-ℝ q ↔
       exists ℚ (λ p → le-ℚ-Prop p q ∧ cut-add-upper-ℝ p)
-    pr1 (is-rounded-cut-add-upper-ℝ q) =
-      elim-exists
-        ( ∃ ℚ (λ p → le-ℚ-Prop p q ∧ cut-add-upper-ℝ p))
-        ( λ (ux , uy) (x<ux , y<uy , q=ux+uy) →
-          map-binary-exists
-            ( λ p → le-ℚ p q × is-in-cut-add-upper-ℝ p)
-            ( add-ℚ)
-            ( λ ux' uy' (ux'<ux , x<ux') (uy'<uy , y<uy') →
-              tr
-                ( le-ℚ (ux' +ℚ uy'))
-                ( inv (q=ux+uy))
-                ( preserves-le-add-ℚ {ux'} {ux} {uy'} {uy} ux'<ux uy'<uy) ,
-              intro-exists (ux' , uy') (x<ux' , y<uy' , refl))
-            ( forward-implication (is-rounded-cut-upper-ℝ x ux) x<ux)
-            ( forward-implication (is-rounded-cut-upper-ℝ y uy) y<uy))
-    pr2 (is-rounded-cut-add-upper-ℝ q) =
-      elim-exists
-        ( cut-add-upper-ℝ q)
-        ( λ p (p<q , x+y<r) →
-          elim-exists
-            ( cut-add-upper-ℝ q)
-            ( λ (px , py) (x<px , y<py , p=px+py) →
-              let
-                q-p⁺ = positive-diff-le-ℚ p q p<q
-                ε⁺ = mediant-zero-ℚ⁺ q-p⁺
-                ε = rational-ℚ⁺ ε⁺
-                ε<q-p = le-mediant-zero-ℚ⁺ q-p⁺
-              in
-                intro-exists
-                  ( px +ℚ ε , q -ℚ (px +ℚ ε))
-                  ( is-in-cut-le-ℚ-upper-ℝ
-                    ( x)
-                    ( px)
-                    ( px +ℚ ε)
-                    ( le-right-add-rational-ℚ⁺ px ε⁺)
-                    ( x<px) ,
-                    is-in-cut-le-ℚ-upper-ℝ
-                      ( y)
-                      ( py)
-                      ( q -ℚ (px +ℚ ε))
-                      ( binary-tr
-                        ( le-ℚ)
-                        ( equational-reasoning
-                            (q -ℚ px) -ℚ (q -ℚ p)
-                            ＝ neg-ℚ (q -ℚ p) +ℚ (q -ℚ px)
-                              by commutative-add-ℚ (q -ℚ px) (neg-ℚ (q -ℚ p))
-                            ＝ (p -ℚ q) +ℚ (q -ℚ px)
-                              by
-                                ap (_+ℚ (q -ℚ px)) (distributive-neg-diff-ℚ q p)
-                            ＝ p -ℚ px by triangle-diff-ℚ p q px
-                            ＝ (px +ℚ py) -ℚ px by ap (_-ℚ px) p=px+py
-                            ＝ py
-                              by
-                                is-identity-left-conjugation-Ab
-                                  ( abelian-group-add-ℚ)
-                                  ( px)
-                                  ( py))
-                        ( equational-reasoning
-                          (q -ℚ px) -ℚ ε
-                          ＝ q +ℚ (neg-ℚ px +ℚ neg-ℚ ε)
-                            by associative-add-ℚ q (neg-ℚ px) (neg-ℚ ε)
-                          ＝ q -ℚ (px +ℚ ε)
-                            by ap (q +ℚ_) (inv (distributive-neg-add-ℚ px ε)))
-                        ( preserves-le-right-add-ℚ
-                          ( q -ℚ px)
-                          ( neg-ℚ (q -ℚ p))
-                          ( neg-ℚ ε)
-                          ( neg-le-ℚ ε (q -ℚ p) ε<q-p)))
-                      ( y<py) ,
-                    inv
-                      ( is-identity-right-conjugation-Ab
-                        ( abelian-group-add-ℚ)
-                        ( px +ℚ ε)
-                        ( q))))
-            ( x+y<r))
+    pr1 (is-rounded-cut-add-upper-ℝ q) q<x+y =
+      do
+        ((ux , uy) , (x<ux , y<uy , q=ux+uy)) ← q<x+y
+        (ux' , ux'<ux , x<ux') ←
+          forward-implication (is-rounded-cut-upper-ℝ x ux) x<ux
+        (uy' , uy'<uy , y<uy') ←
+          forward-implication (is-rounded-cut-upper-ℝ y uy) y<uy
+        intro-exists
+          ( ux' +ℚ uy')
+          ( tr
+              ( le-ℚ (ux' +ℚ uy'))
+              ( inv (q=ux+uy))
+              ( preserves-le-add-ℚ {ux'} {ux} {uy'} {uy} ux'<ux uy'<uy) ,
+            intro-exists (ux' , uy') (x<ux' , y<uy' , refl))
+      where
+        open
+          do-syntax-trunc-Prop (∃ ℚ (λ p → le-ℚ-Prop p q ∧ cut-add-upper-ℝ p))
+    pr2 (is-rounded-cut-add-upper-ℝ q) exists-p =
+      do
+        (p , p<q , x+y<p) ← exists-p
+        ((px , py) , (x<px , y<py , p=px+py)) ← x+y<p
+        let
+          q-p⁺ = positive-diff-le-ℚ p q p<q
+          ε⁺ = mediant-zero-ℚ⁺ q-p⁺
+          ε = rational-ℚ⁺ ε⁺
+          ε<q-p = le-mediant-zero-ℚ⁺ q-p⁺
+        intro-exists
+          ( px +ℚ ε , q -ℚ (px +ℚ ε))
+          ( is-in-cut-add-rational-ℚ⁺-upper-ℝ x px ε⁺ x<px ,
+            is-in-cut-le-ℚ-upper-ℝ
+              ( y)
+              ( py)
+              ( q -ℚ (px +ℚ ε))
+              ( binary-tr
+                ( le-ℚ)
+                ( equational-reasoning
+                    (q -ℚ px) -ℚ (q -ℚ p)
+                    ＝ neg-ℚ px -ℚ neg-ℚ p
+                      by left-translation-diff-ℚ (neg-ℚ px) (neg-ℚ p) q
+                    ＝ neg-ℚ px +ℚ (px +ℚ py)
+                      by ap (neg-ℚ px +ℚ_) (neg-neg-ℚ p ∙ p=px+py)
+                    ＝ py
+                      by is-retraction-left-div-Group group-add-ℚ px py)
+                ( equational-reasoning
+                  (q -ℚ px) -ℚ ε
+                  ＝ q +ℚ (neg-ℚ px +ℚ neg-ℚ ε)
+                    by associative-add-ℚ q (neg-ℚ px) (neg-ℚ ε)
+                  ＝ q -ℚ (px +ℚ ε)
+                    by ap (q +ℚ_) (inv (distributive-neg-add-ℚ px ε)))
+                ( preserves-le-right-add-ℚ
+                  ( q -ℚ px)
+                  ( neg-ℚ (q -ℚ p))
+                  ( neg-ℚ ε)
+                  ( neg-le-ℚ ε (q -ℚ p) ε<q-p)))
+              ( y<py) ,
+            inv
+              ( is-identity-right-conjugation-Ab
+                ( abelian-group-add-ℚ)
+                ( px +ℚ ε)
+                ( q)))
+      where open do-syntax-trunc-Prop (cut-add-upper-ℝ q)
 
   add-upper-ℝ : upper-ℝ (l1 ⊔ l2)
   pr1 add-upper-ℝ = cut-add-upper-ℝ
@@ -199,3 +185,43 @@ module _
         ( cut-upper-ℝ y)
         ( cut-upper-ℝ z))
 ```
+
+### Unit laws for addition of upper Dedekind real numbers
+
+```agda
+module _
+  {l : Level} (x : upper-ℝ l)
+  where
+
+  right-unit-law-add-upper-ℝ : add-upper-ℝ x (upper-real-ℚ zero-ℚ) ＝ x
+  right-unit-law-add-upper-ℝ =
+    eq-sim-cut-upper-ℝ
+      ( add-upper-ℝ x (upper-real-ℚ zero-ℚ))
+      ( x)
+      ( (λ r →
+          elim-exists
+            ( cut-upper-ℝ x r)
+            ( λ (p , q) (x<p , 0<q , r=p+q) →
+              inv-tr
+                ( is-in-cut-upper-ℝ x)
+                ( r=p+q)
+                ( is-in-cut-add-rational-ℚ⁺-upper-ℝ
+                  ( x)
+                  ( p)
+                  ( q , is-positive-le-zero-ℚ q 0<q)
+                  ( x<p)))) ,
+        ( λ q x<q →
+          elim-exists
+            ( cut-add-upper-ℝ x (upper-real-ℚ zero-ℚ) q)
+            ( λ r (r<q , x<r) →
+              intro-exists
+                ( r , q -ℚ r)
+                ( x<r ,
+                  backward-implication (iff-translate-diff-le-zero-ℚ r q) r<q ,
+                  inv
+                    ( is-identity-right-conjugation-Ab
+                      ( abelian-group-add-ℚ)
+                      ( r)
+                      ( q))))
+            ( forward-implication (is-rounded-cut-upper-ℝ x q) x<q)))
+```
diff --git a/src/real-numbers/lower-dedekind-real-numbers.lagda.md b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
index ddae49559d..71f60cde97 100644
--- a/src/real-numbers/lower-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/lower-dedekind-real-numbers.lagda.md
@@ -1,13 +1,17 @@
 # Lower Dedekind real numbers
 
 ```agda
+{-# OPTIONS --lossy-unification #-}
+
 module real-numbers.lower-dedekind-real-numbers where
 ```
 
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.difference-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-rational-numbers
 
@@ -122,6 +126,19 @@ module _
       ( intro-exists q (p<q , q<x))
 ```
 
+### Lower Dedekind cuts are closed under subtraction by positive rational numbers
+
+```agda
+module _
+  {l : Level} (x : lower-ℝ l) (p : ℚ) (d : ℚ⁺)
+  where
+
+  is-in-cut-diff-rational-ℚ⁺-lower-ℝ :
+    is-in-cut-lower-ℝ x p → is-in-cut-lower-ℝ x (p -ℚ rational-ℚ⁺ d)
+  is-in-cut-diff-rational-ℚ⁺-lower-ℝ =
+    is-in-cut-le-ℚ-lower-ℝ x (p -ℚ rational-ℚ⁺ d) p (le-diff-rational-ℚ⁺ p d)
+```
+
 ### Lower Dedekind cuts are closed under inequality on the rationals
 
 ```agda
diff --git a/src/real-numbers/upper-dedekind-real-numbers.lagda.md b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
index da99075a67..6eccf28a9c 100644
--- a/src/real-numbers/upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/upper-dedekind-real-numbers.lagda.md
@@ -7,7 +7,9 @@ module real-numbers.upper-dedekind-real-numbers where
 <details><summary>Imports</summary>
 
 ```agda
+open import elementary-number-theory.addition-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-rational-numbers
 
@@ -122,6 +124,23 @@ module _
       ( intro-exists p (p<q , p<x))
 ```
 
+### Upper Dedekind cuts are closed under the addition of positive rational numbers
+
+```agda
+module _
+  {l : Level} (x : upper-ℝ l) (p : ℚ) (d : ℚ⁺)
+  where
+
+  is-in-cut-add-rational-ℚ⁺-upper-ℝ :
+    is-in-cut-upper-ℝ x p → is-in-cut-upper-ℝ x (p +ℚ rational-ℚ⁺ d)
+  is-in-cut-add-rational-ℚ⁺-upper-ℝ =
+    is-in-cut-le-ℚ-upper-ℝ
+      ( x)
+      ( p)
+      ( p +ℚ rational-ℚ⁺ d)
+      ( le-right-add-rational-ℚ⁺ p d)
+```
+
 ### Upper Dedekind cuts are closed under inequality on the rationals
 
 ```agda

From 1d4965babf9796463e46d779d5fa6ec9e9114ff9 Mon Sep 17 00:00:00 2001
From: Louis Wasserman <wasserman.louis@gmail.com>
Date: Fri, 14 Feb 2025 13:38:01 -0800
Subject: [PATCH 73/73] make pre-commit

---
 src/real-numbers.lagda.md                                      | 2 +-
 src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md | 2 +-
 2 files changed, 2 insertions(+), 2 deletions(-)

diff --git a/src/real-numbers.lagda.md b/src/real-numbers.lagda.md
index 840e17a3da..24d0f66b34 100644
--- a/src/real-numbers.lagda.md
+++ b/src/real-numbers.lagda.md
@@ -5,9 +5,9 @@
 ```agda
 module real-numbers where
 
-open import real-numbers.apartness-real-numbers public
 open import real-numbers.addition-lower-dedekind-real-numbers public
 open import real-numbers.addition-upper-dedekind-real-numbers public
+open import real-numbers.apartness-real-numbers public
 open import real-numbers.arithmetically-located-dedekind-cuts public
 open import real-numbers.dedekind-real-numbers public
 open import real-numbers.inequality-lower-dedekind-real-numbers public
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 0941ed1a99..33dd0ded7d 100644
--- a/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
+++ b/src/real-numbers/addition-upper-dedekind-real-numbers.lagda.md
@@ -33,8 +33,8 @@ open import group-theory.abelian-groups
 open import group-theory.groups
 open import group-theory.minkowski-multiplication-commutative-monoids
 
-open import real-numbers.upper-dedekind-real-numbers
 open import real-numbers.rational-upper-dedekind-real-numbers
+open import real-numbers.upper-dedekind-real-numbers
 ```
 
 </details>