[Add] Consequences of associativity for Semigroups (#2688)

* Adds reasonig combinator for semigroup

* Adds reasonig combinator for semigroup

* Adds reasonig combinator for semigroup

* Adds reasonig combinator for semigroup

* Add some more missing reasoning combinators

* add module Extends

* rename SemiGroup to Semigroup

* fix-whitespace

* Update CHANGELOG.md

Co-authored-by: jamesmckinna <[email protected]>

* New names

* improve syntx

* fix-whitespace

* Name changes

* Names change

* Space

* Proof of assoc with PUshes and Pulles

* Proof of assoc with PUshes and Pulles

* white space

* Update src/Algebra/Properties/Semigroup/Reasoning.agda

Co-authored-by: jamesmckinna <[email protected]>

* Reasoning to Semigroup and explicit variables

* fix bug

* space

* Update src/Algebra/Properties/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update CHANGELOG.md

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

* Update src/Algebra/Properties/Semigroup.agda

Co-authored-by: jamesmckinna <[email protected]>

* variables

* update CHANGELOG

* Explicit varaibles

* remove white space

* update Changelog and minor improvments

---------

Co-authored-by: jamesmckinna <[email protected]>

Author: jmougeot

CHANGELOG.md

@@ -239,6 +239,29 @@ Additions to existing modules
   ∣ˡ-preorder   : Preorder a ℓ _
   ```
 
+* In `Algebra.Properties.Semigroup` adding consequences for associativity for semigroups
+
+```
+  uv≈w⇒xu∙v≈xw          : ∀ x → (x ∙ u) ∙ v ≈ x ∙ w
+  uv≈w⇒u∙vx≈wx          : ∀ x → u ∙ (v ∙ x) ≈ w ∙ x
+  uv≈w⇒u[vx∙y]≈w∙xy     : ∀ x y → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ (x ∙ y)
+  uv≈w⇒x[uv∙y]≈x∙wy     : ∀ x y → x ∙ (u ∙ (v ∙ y)) ≈ x ∙ (w ∙ y)
+  uv≈w⇒[x∙yu]v≈x∙yw     : ∀ x y → (x ∙ (y ∙ u)) ∙ v ≈ x ∙ (y ∙ w)
+  uv≈w⇒[xu∙v]y≈x∙wy     : ∀ x y → ((x ∙ u) ∙ v) ∙ y ≈ x ∙ (w ∙ y)
+  uv≈w⇒[xy∙u]v≈x∙yw     : ∀ x y → ((x ∙ y) ∙ u) ∙ v ≈ x ∙ (y ∙ w)
+  uv≈w⇒xu∙vy≈x∙wy       : ∀ x y → (x ∙ u) ∙ (v ∙ y) ≈ x ∙ (w ∙ y)
+  uv≈w⇒xy≈z⇒u[vx∙y]≈wz  : ∀ z → x ∙ y ≈ z → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ z
+  uv≈w⇒x∙wy≈x∙[u∙vy]    : x ∙ (w ∙ y) ≈ x ∙ (u ∙ (v ∙ y))
+  [uv∙w]x≈u[vw∙x]       : ((u ∙ v) ∙ w) ∙ x ≈ u ∙ ((v ∙ w) ∙ x)
+  [uv∙w]x≈u[v∙wx]       : ((u ∙ v) ∙ w) ∙ x ≈ u ∙ (v ∙ (w ∙ x))
+  [u∙vw]x≈uv∙wx         : (u ∙ (v ∙ w)) ∙ x ≈ (u ∙ v) ∙ (w ∙ x)
+  [u∙vw]x≈u[v∙wx]       : (u ∙ (v ∙ w)) ∙ x ≈ u ∙ (v ∙ (w ∙ x))
+  uv∙wx≈u[vw∙x]         : (u ∙ v) ∙ (w ∙ x) ≈ u ∙ ((v ∙ w) ∙ x)
+  uv≈wx⇒yu∙v≈yw∙x       : ∀ y → (y ∙ u) ∙ v ≈ (y ∙ w) ∙ x
+  uv≈wx⇒u∙vy≈w∙xy       : ∀ y → u ∙ (v ∙ y) ≈ w ∙ (x ∙ y)
+  uv≈wx⇒yu∙vz≈yw∙xz     : ∀ y z → (y ∙ u) ∙ (v ∙ z) ≈ (y ∙ w) ∙ (x ∙ z)
+```
+
 * In `Algebra.Properties.Semigroup.Divisibility`:
   ```agda
   ∣ˡ-trans     : Transitive _∣ˡ_

src/Algebra/Properties/Semigroup.agda

@@ -10,9 +10,16 @@ open import Algebra using (Semigroup)
 
 module Algebra.Properties.Semigroup {a ℓ} (S : Semigroup a ℓ) where
 
+open import Data.Product.Base using (_,_)
+
 open Semigroup S
+  using (Carrier; _∙_; _≈_; setoid; trans ; refl; sym; assoc; ∙-cong; ∙-congˡ; ∙-congʳ)
 open import Algebra.Definitions _≈_
-open import Data.Product.Base using (_,_)
+  using (Alternative; LeftAlternative; RightAlternative; Flexible)
+
+private
+  variable
+    u v w x y z : Carrier
 
 x∙yz≈xy∙z : ∀ x y z → x ∙ (y ∙ z) ≈ (x ∙ y) ∙ z
 x∙yz≈xy∙z x y z = sym (assoc x y z)
@@ -28,3 +35,62 @@ alternative = alternativeˡ , alternativeʳ
 
 flexible : Flexible _∙_
 flexible x y = assoc x y x
+
+module _ (uv≈w : u ∙ v ≈ w) where
+  uv≈w⇒xu∙v≈xw : ∀ x → (x ∙ u) ∙ v ≈ x ∙ w
+  uv≈w⇒xu∙v≈xw x = trans (assoc x u v) (∙-congˡ uv≈w)
+
+  uv≈w⇒u∙vx≈wx : ∀ x → u ∙ (v ∙ x) ≈ w ∙ x
+  uv≈w⇒u∙vx≈wx x = trans (sym (assoc u v x)) (∙-congʳ uv≈w)
+
+  uv≈w⇒u[vx∙y]≈w∙xy : ∀ x y → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ (x ∙ y)
+  uv≈w⇒u[vx∙y]≈w∙xy x y = trans (∙-congˡ (assoc v x y)) (uv≈w⇒u∙vx≈wx (x ∙ y))
+
+  uv≈w⇒x[uv∙y]≈x∙wy : ∀ x y → x ∙ (u ∙ (v ∙ y)) ≈ x ∙ (w ∙ y)
+  uv≈w⇒x[uv∙y]≈x∙wy x y = ∙-congˡ (uv≈w⇒u∙vx≈wx y)
+
+  uv≈w⇒[x∙yu]v≈x∙yw : ∀ x y → (x ∙ (y ∙ u)) ∙ v ≈ x ∙ (y ∙ w)
+  uv≈w⇒[x∙yu]v≈x∙yw x y = trans (assoc x (y ∙ u) v) (∙-congˡ (uv≈w⇒xu∙v≈xw y))
+
+  uv≈w⇒[xu∙v]y≈x∙wy : ∀ x y → ((x ∙ u) ∙ v) ∙ y ≈ x ∙ (w ∙ y)
+  uv≈w⇒[xu∙v]y≈x∙wy x y = trans (∙-congʳ (uv≈w⇒xu∙v≈xw x)) (assoc x w y)
+
+  uv≈w⇒[xy∙u]v≈x∙yw : ∀ x y → ((x ∙ y) ∙ u) ∙ v ≈ x ∙ (y ∙ w)
+  uv≈w⇒[xy∙u]v≈x∙yw x y = trans (∙-congʳ (assoc x y u)) (uv≈w⇒[x∙yu]v≈x∙yw x y )
+
+module _ (uv≈w : u ∙ v ≈ w) where
+
+  uv≈w⇒xu∙vy≈x∙wy : ∀ x y → (x ∙ u) ∙ (v ∙ y) ≈ x ∙ (w ∙ y)
+  uv≈w⇒xu∙vy≈x∙wy x y = uv≈w⇒xu∙v≈xw (uv≈w⇒u∙vx≈wx uv≈w y) x
+
+  uv≈w⇒xy≈z⇒u[vx∙y]≈wz : ∀ x y → x ∙ y ≈ z → u ∙ ((v ∙ x) ∙ y) ≈ w ∙ z
+  uv≈w⇒xy≈z⇒u[vx∙y]≈wz x y xy≈z = trans (∙-congˡ (uv≈w⇒xu∙v≈xw xy≈z v)) (uv≈w⇒u∙vx≈wx uv≈w _)
+
+  uv≈w⇒x∙wy≈x∙[u∙vy] : x ∙ (w ∙ y) ≈ x ∙ (u ∙ (v ∙ y))
+  uv≈w⇒x∙wy≈x∙[u∙vy] = sym (uv≈w⇒x[uv∙y]≈x∙wy uv≈w _ _)
+
+module _ u v w x where
+  [uv∙w]x≈u[vw∙x] : ((u ∙ v) ∙ w) ∙ x ≈ u ∙ ((v ∙ w) ∙ x)
+  [uv∙w]x≈u[vw∙x] = uv≈w⇒[xu∙v]y≈x∙wy refl u x
+
+  [uv∙w]x≈u[v∙wx] : ((u ∙ v) ∙ w) ∙ x ≈ u ∙ (v ∙ (w ∙ x))
+  [uv∙w]x≈u[v∙wx] = uv≈w⇒[xy∙u]v≈x∙yw refl u v
+
+  [u∙vw]x≈uv∙wx : (u ∙ (v ∙ w)) ∙ x ≈ (u ∙ v) ∙ (w ∙ x)
+  [u∙vw]x≈uv∙wx = trans (sym (∙-congʳ (assoc u v w))) (assoc (u ∙ v) w x)
+
+  [u∙vw]x≈u[v∙wx] : (u ∙ (v ∙ w)) ∙ x ≈ u ∙ (v ∙ (w ∙ x))
+  [u∙vw]x≈u[v∙wx] = uv≈w⇒[x∙yu]v≈x∙yw refl u v
+
+  uv∙wx≈u[vw∙x] : (u ∙ v) ∙ (w ∙ x) ≈ u ∙ ((v ∙ w) ∙ x)
+  uv∙wx≈u[vw∙x] = uv≈w⇒xu∙vy≈x∙wy refl u x
+
+module _ (uv≈wx : u ∙ v ≈ w ∙ x) where
+  uv≈wx⇒yu∙v≈yw∙x : ∀ y → (y ∙ u) ∙ v ≈ (y ∙ w) ∙ x
+  uv≈wx⇒yu∙v≈yw∙x y = trans (uv≈w⇒xu∙v≈xw uv≈wx y) (sym (assoc y w x))
+
+  uv≈wx⇒u∙vy≈w∙xy : ∀ y → u ∙ (v ∙ y) ≈ w ∙ (x ∙ y)
+  uv≈wx⇒u∙vy≈w∙xy y = trans (uv≈w⇒u∙vx≈wx uv≈wx y) (assoc w x y)
+
+  uv≈wx⇒yu∙vz≈yw∙xz : ∀ y z → (y ∙ u) ∙ (v ∙ z) ≈ (y ∙ w) ∙ (x ∙ z)
+  uv≈wx⇒yu∙vz≈yw∙xz y z = trans (uv≈w⇒xu∙v≈xw (uv≈wx⇒u∙vy≈w∙xy z) y) (sym (assoc y w (x ∙ z)))