Version 2.0-dev

The library has been tested using Agda 2.6.3.

Highlights

A golden testing library in Test.Golden. This allows you to run a set of tests and make sure their output matches an expected golden value. The test runner has many options: filtering tests by name, dumping the list of failures to a file, timing the runs, coloured output, etc. Cf. the comments in Test.Golden and the standard library's own tests in tests/ for documentation on how to use the library.
A new tactic cong! available from Tactic.Cong which automatically infers the argument to cong for you via anti-unification.
Improved the solve tactic in Tactic.RingSolver to work in a much wider range of situations.
Added ⌊log₂_⌋ and ⌈log₂_⌉ on Natural Numbers.
A massive refactoring of the unindexed Functor/Applicative/Monad hierarchy and the MonadReader / MonadState type classes. These are now usable with instance arguments as demonstrated in the tests/monad examples.

Bug-fixes

The combinators _⟶-∘_, _↣-∘_, _↠-∘_, _⤖-∘_, _⇔-∘_, _↩-∘_, _↪-∘_, _↔-∘_ in Function.Construct.Composition had their arguments in the wrong order. They have been flipped so they can actually be used as a composition operator.

The following operators were missing a fixity declaration, which has now been fixed -

infixr  5 _∷_                                              (Codata.Guarded.Stream)
infix   4 _[_]                                             (Codata.Guarded.Stream)
infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Binary.Pointwise)
infix   4 _≈∞_                                             (Codata.Guarded.Stream.Relation.Binary.Pointwise)
infixr  5 _∷_                                              (Codata.Musical.Colist)
infix   4 _≈_                                              (Codata.Musical.Conat)
infixr  5 _∷_                                              (Codata.Musical.Colist.Bisimilarity)
infixr  5 _∷_                                              (Codata.Musical.Colist.Relation.Unary.All)
infixr  5 _∷_                                              (Codata.Sized.Colist)
infixr  5 _∷_                                              (Codata.Sized.Covec)
infixr  5 _∷_                                              (Codata.Sized.Cowriter)
infixl  1 _>>=_                                            (Codata.Sized.Cowriter)
infixr  5 _∷_                                              (Codata.Sized.Stream)
infixr  5 _∷_                                              (Codata.Sized.Colist.Bisimilarity)
infix   4 _ℕ≤?_                                            (Codata.Sized.Conat.Properties)
infixr  5 _∷_                                              (Codata.Sized.Covec.Bisimilarity)
infixr  5 _∷_                                              (Codata.Sized.Cowriter.Bisimilarity)
infixr  5 _∷_                                              (Codata.Sized.Stream.Bisimilarity)
infixr  8 _⇒_ _⊸_                                          (Data.Container.Core)
infixr -1 _<$>_ _<*>_                                      (Data.Container.FreeMonad)
infixl  1 _>>=_                                            (Data.Container.FreeMonad)
infix   5 _▷_                                              (Data.Container.Indexed)
infix   4 _≈_                                              (Data.Float.Base)
infixl  7 _⊓′_                                             (Data.Nat.Base)
infixl  6 _⊔′_                                             (Data.Nat.Base)
infixr  8 _^_                                              (Data.Nat.Base)
infix   4 _!≢0 _!*_!≢0                                     (Data.Nat.Properties)
infix   4 _≃?_                                             (Data.Rational.Unnormalised.Properties)
infix   4 _≈ₖᵥ_                                            (Data.Tree.AVL.Map.Membership.Propositional)
infix   4 _<_                                              (Induction.WellFounded)
infix  -1 _$ⁿ_                                             (Data.Vec.N-ary)
infix   4 _≋_                                              (Data.Vec.Functional.Relation.Binary.Equality.Setoid)
infix   4 _≟_                                              (Reflection.AST.Definition)
infix   4 _≡ᵇ_                                             (Reflection.AST.Literal)
infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Meta)
infix   4 _≈?_ _≟_ _≈_                                     (Reflection.AST.Name)
infix   4 _≟-Telescope_                                    (Reflection.AST.Term)
infix   4 _≟_                                              (Reflection.AST.Argument.Information)
infix   4 _≟_                                              (Reflection.AST.Argument.Modality)
infix   4 _≟_                                              (Reflection.AST.Argument.Quantity)
infix   4 _≟_                                              (Reflection.AST.Argument.Relevance)
infix   4 _≟_                                              (Reflection.AST.Argument.Visibility)
infixr  8 _^_                                              (Function.Endomorphism.Propositional)
infixr  8 _^_                                              (Function.Endomorphism.Setoid)
infix   4 _≃_                                              (Function.HalfAdjointEquivalence)
infix   4 _≈_ _≈ᵢ_ _≤_                                     (Function.Metric.Bundles)
infixl  6 _∙_                                              (Function.Metric.Bundles)
infix   4 _≈_                                              (Function.Metric.Nat.Bundles)
infix   3 _←_ _↢_                                          (Function.Related)
infix   4 _ℕ<_ _ℕ≤infinity _ℕ≤_                            (Codata.Sized.Conat)
infix   6 _ℕ+_ _+ℕ_                                        (Codata.Sized.Conat)
infixl  4 _+ _*                                            (Data.List.Kleene.Base)
infixr  4 _++++_ _+++*_ _*+++_ _*++*_                      (Data.List.Kleene.Base)
infix   4 _[_]* _[_]+                                      (Data.List.Kleene.Base)
infix   4 _≢∈_                                             (Data.List.Membership.Propositional)
infixr  5 _`∷_                                             (Data.List.Reflection)
infix   4 _≡?_                                             (Data.List.Relation.Binary.Equality.DecPropositional)
infixr  5 _++ᵖ_                                            (Data.List.Relation.Binary.Prefix.Heterogeneous)
infixr  5 _++ˢ_                                            (Data.List.Relation.Binary.Suffix.Heterogeneous)
infixr  5 _++_ _++[]                                       (Data.List.Relation.Ternary.Appending.Propositional)
infixr  5 _∷=_                                             (Data.List.Relation.Unary.Any)
infixr  5 _++_                                             (Data.List.Ternary.Appending)
infixr  2 _×-⇔_ _×-↣_ _×-↞_ _×-↠_ _×-↔_ _×-cong_           (Data.Product.Function.NonDependent.Propositional)
infixr  2 _×-⟶_                                           (Data.Product.Function.NonDependent.Setoid)
infixr  2 _×-equivalence_ _×-injection_ _×-left-inverse_   (Data.Product.Function.NonDependent.Setoid)
infixr  2 _×-surjection_ _×-inverse_                       (Data.Product.Function.NonDependent.Setoid)
infixr  1 _⊎-⇔_ _⊎-↣_ _⊎-↞_ _⊎-↠_ _⊎-↔_ _⊎-cong_           (Data.Sum.Function.Propositional)
infixr  1 _⊎-⟶_                                           (Data.Sum.Function.Setoid)
infixr  1 _⊎-equivalence_ _⊎-injection_ _⊎-left-inverse_   (Data.Sum.Function.Setoid)
infixr  1 _⊎-surjection_ _⊎-inverse_                       (Data.Sum.Function.Setoid)
infix   8 _⁻¹                                              (Data.Parity.Base)
infixr  5 _`∷_                                             (Data.Vec.Reflection)
infixr  5 _∷=_                                             (Data.Vec.Membership.Setoid)
infix   4 _≟H_ _≟N_                                        (Algebra.Solver.Ring)
infix   4 _≈_                                              (Relation.Binary.Bundles)
infixl  6 _∩_                                              (Relation.Binary.Construct.Intersection)
infix   4 _<₋_                                             (Relation.Binary.Construct.Add.Infimum.Strict)
infix   4 _≈∙_                                             (Relation.Binary.Construct.Add.Point.Equality)
infix   4 _≤⁺_ _≤⊤⁺                                        (Relation.Binary.Construct.Add.Supremum.NonStrict)
infixr  5 _∷ʳ_                                             (Relation.Binary.Construct.Closure.Transitive)
infixr  5 _∷_                                              (Codata.Guarded.Stream.Relation.Unary.All)
infixr  5 _∷_                                              (Foreign.Haskell.List.NonEmpty)
infix   4 _≈_                                              (Function.Metric.Rational.Bundles)
infixl  6 _ℕ+_                                             (Level.Literals)
infixr  6 _∪_                                              (Relation.Binary.Construct.Union)
infixl  6 _+ℤ_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
infix   4 _≉_ _≈ᵢ_ _≤ᵢ_                                    (Relation.Binary.Indexed.Homogeneous.Bundles)
infixr  5 _∷ᴹ_ _∷⁻¹ᴹ_                                      (Text.Regex.Search)
infixr  4 _,_                                              (Data.Refinement)
infixr  4 _,_                                              (Data.Container.Relation.Binary.Pointwise)
infixr  4 _,_                                              (Data.Tree.AVL.Value)
infixr  4 _,_                                              (Foreign.Haskell.Pair)
infixr  4 _,_                                              (Reflection.AnnotatedAST)
infixr  4 _,_                                              (Reflection.AST.Traversal)
infixl  6.5 _P′_ _P_ _C′_ _C_                              (Data.Nat.Combinatorics.Base)
infixl  1 _>>=-cong_ _≡->>=-cong_                          (Effect.Monad.Partiality)
infixl  1 _?>=′_                                           (Effect.Monad.Predicate)
infixl  1 _>>=-cong_ _>>=-congP_                           (Effect.Monad.Partiality.All)
infix   4 _∈FV_                                            (Reflection.AST.DeBruijn)
infixr  9 _;_                                              (Relation.Binary.Construct.Composition)
infixl  6 _+²_                                             (Relation.Binary.HeterogeneousEquality.Quotients.Examples)
infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Heterogeneous.Construct.At)
infixr -1 _atₛ_                                             (Relation.Binary.Indexed.Homogeneous.Construct.At)
infix   4 _∈_ _∉_                                          (Relation.Unary.Indexed)
infixr  9 _⍮_                                              (Relation.Unary.PredicateTransformer)
infix   8 ∼_                                               (Relation.Unary.PredicateTransformer)
infix   2 _×?_ _⊙?_                                        (Relation.Unary.Properties)
infix   10 _~?                                             (Relation.Unary.Properties)
infixr  1 _⊎?_                                             (Relation.Unary.Properties)
infixr  7 _∩?_                                             (Relation.Unary.Properties)
infixr  6 _∪?_                                             (Relation.Unary.Properties)
infixl  6 _`⊜_                                             (Tactic.RingSolver)
infix   8 ⊝_                                               (Tactic.RingSolver.Core.Expression)
infix   4 _∈ᴿ?_ _∉ᴿ?_ _∈?ε _∈?[_] _∈?[^_]                  (Text.Regex.Properties)
infix   4 _∈?_ _∉?_                                        (Text.Regex.Derivative.Brzozowski)
infix   4 _∈_ _∉_ _∈?_ _∉?_                                (Text.Regex.String.Unsafe)

In System.Exit, the ExitFailure constructor is now carrying an integer rather than a natural. The previous binding was incorrectly assuming that all exit codes where non-negative.
In /-monoˡ-≤ in Data.Nat.DivMod the parameter o was implicit but not inferrable. It has been changed to be explicit.
In +-distrib-/-∣ʳ in Data.Nat.DivMod the parameter m was implicit but not inferrable, while n is explicit but inferrable. They have been changed.
In Function.Definitions the definitions of Surjection, Inverseˡ, Inverseʳ were not being re-exported correctly and therefore had an unsolved meta-variable whenever this module was explicitly parameterised. This has been fixed.
Add module Algebra.Module that re-exports the contents of Algebra.Module.(Definitions/Structures/Bundles)
Various module-like bundles in Algebra.Module.Bundles were missing a fixity declaration for _*ᵣ_. This has been fixed.
In Algebra.Definitions.RawSemiring the record prime add p∤1 : p ∤ 1# to the field.
In Data.List.Relation.Ternary.Appending.Setoid we re-export specialised versions of the constructors using the setoid's carrier set. Prior to that, the constructor were re-exported in their full generality which would lead to unsolved meta variables at their use sites.
In Data.Container.FreeMonad, we give a direct definition of _⋆_ as an inductive type rather than encoding it as an instance of μ. This ensures Agda notices that C ⋆ X is strictly positive in X which in turn allows us to use the free monad when defining auxiliary (co)inductive types (cf. the Tap example in README.Data.Container.FreeMonad).
In Data.Maybe.Base the fixity declaration of _<∣>_ was missing. This has been fixed.

Non-backwards compatible changes

Removed deprecated names

All modules and names that were deprecated in v1.2 and before have been removed.

Moved `Codata` modules to `Codata.Sized`

Due to the change in Agda 2.6.2 where sized types are no longer compatible with the --safe flag, it has become clear that a third variant of codata will be needed using coinductive records. Therefore all existing modules in Codata which used sized types have been moved inside a new folder named Sized, e.g. Codata.Stream has become Codata.Sized.Stream.

Moved `Category` modules to `Effect`

As observed by Wen Kokke in Issue #1636, it no longer really makes sense to group the modules which correspond to the variety of concepts of (effectful) type constructor arising in functional programming (esp. in haskell) such as Monad, Applicative, Functor, etc, under Category.*, as this obstructs the importing of the agda-categories development into the Standard Library, and moreover needlessly restricts the applicability of categorical concepts to this (highly specific) mode of use. Correspondingly, client modules grouped under *.Categorical.* which exploit such structure for effectful programming have been renamed *.Effectful, with the originals being deprecated.

Improvements to pretty printing and regexes

In Text.Pretty, Doc is now a record rather than a type alias. This helps Agda reconstruct the width parameter when the module is opened without it being applied. In particular this allows users to write width-generic pretty printers and only pick a width when calling the renderer by using this import style:

open import Text.Pretty using (Doc; render)
--                      ^-- no width parameter for Doc & render
open module Pretty {w} = Text.Pretty w hiding (Doc; render)
--                 ^-- implicit width parameter for the combinators

pretty : Doc w
pretty = ? -- you can use the combinators here and there won't be any
           -- issues related to the fact that `w` cannot be reconstructed
           -- anymore

main = do
  -- you can now use the same pretty with different widths:
  putStrLn $ render 40 pretty
  putStrLn $ render 80 pretty

In Text.Regex.Search the searchND function finding infix matches has been tweaked to make sure the returned solution is a local maximum in terms of length. It used to be that we would make the match start as late as possible and finish as early as possible. It's now the reverse.

So [a-zA-Z]+.agdai? run on "the path _build/Main.agdai corresponds to" will return "Main.agdai" when it used to be happy to just return "n.agda".

Refactoring of algebraic lattice hierarchy

In order to improve modularity and consistency with Relation.Binary.Lattice, the structures & bundles for Semilattice, Lattice, DistributiveLattice & BooleanAlgebra have been moved out of the Algebra modules and into their own hierarchy in Algebra.Lattice.
All submodules, (e.g. Algebra.Properties.Semilattice or Algebra.Morphism.Lattice) have been moved to the corresponding place under Algebra.Lattice (e.g. Algebra.Lattice.Properties.Semilattice or Algebra.Lattice.Morphism.Lattice). See the Deprecated modules section below for full details.

Changed definition of IsDistributiveLattice and IsBooleanAlgebra so that they are no longer right-biased which hinders compositionality. More concretely, IsDistributiveLattice now has fields:

∨-distrib-∧ : ∨ DistributesOver ∧
∧-distrib-∨ : ∧ DistributesOver ∨

instead of

∨-distribʳ-∧ : ∨ DistributesOverʳ ∧

and IsBooleanAlgebra now has fields:

∨-complement : Inverse ⊤ ¬ ∨
∧-complement : Inverse ⊥ ¬ ∧

instead of:

∨-complementʳ : RightInverse ⊤ ¬ ∨
∧-complementʳ : RightInverse ⊥ ¬ ∧

To allow construction of these structures via their old form, smart constructors have been added to a new module Algebra.Lattice.Structures.Biased, which are by re-exported automatically by Algebra.Lattice. For example, if before you wrote:
```
∧-∨-isDistributiveLattice = record
  { isLattice    = ∧-∨-isLattice
  ; ∨-distribʳ-∧ = ∨-distribʳ-∧
  }
```
you can use the smart constructor isDistributiveLatticeʳʲᵐ to write:
```
∧-∨-isDistributiveLattice = isDistributiveLatticeʳʲᵐ (record
  { isLattice    = ∧-∨-isLattice
  ; ∨-distribʳ-∧ = ∨-distribʳ-∧
  })
```
without having to prove full distributivity.
Added new IsBoundedSemilattice/BoundedSemilattice records.
Added new aliases Is(Meet/Join)(Bounded)Semilattice for Is(Bounded)Semilattice which can be used to indicate meet/join-ness of the original structures.

Removal of the old function hierarchy

The switch to the new function hierarchy is complete and the following modules have been completely switched over to use the new definitions:
```
Data.Sum.Function.Setoid
Data.Sum.Function.Propositional
```
Additionally the following proofs now use the new definitions instead of the old ones:
- Algebra.Lattice.Properties.BooleanAlgebra
- Algebra.Properties.CommutativeMonoid.Sum
- Algebra.Properties.Lattice
  - replace-equality
- Data.Fin.Properties
  - ∀-cons-⇔
  - ⊎⇔∃
- Data.Fin.Subset.Properties
  - out⊆-⇔
  - in⊆in-⇔
  - out⊂in-⇔
  - out⊂out-⇔
  - in⊂in-⇔
  - x∈⁅y⁆⇔x≡y
  - ∩⇔×
  - ∪⇔⊎
  - ∃-Subset-[]-⇔
  - ∃-Subset-∷-⇔
- Data.List.Countdown
  - empty
- Data.List.Fresh.Relation.Unary.Any
  - ⊎⇔Any
- Data.List.NonEmpty
- Data.List.Relation.Binary.Lex
  - []<[]-⇔
  - ∷<∷-⇔
- Data.List.Relation.Binary.Sublist.Heterogeneous.Properties
  - ∷⁻¹
  - ∷ʳ⁻¹
  - Sublist-[x]-bijection
- Data.List.Relation.Binary.Sublist.Setoid.Properties
  - ∷⁻¹
  - ∷ʳ⁻¹
  - [x]⊆xs⤖x∈xs
- Data.List.Relation.Unary.Grouped.Properties
- Data.Maybe.Relation.Binary.Connected
  - just-equivalence
- Data.Maybe.Relation.Binary.Pointwise
  - just-equivalence
- Data.Maybe.Relation.Unary.All
  - just-equivalence
- Data.Maybe.Relation.Unary.Any
  - just-equivalence
- Data.Nat.Divisibility
  - m%n≡0⇔n∣m
- Data.Nat.Properties
  - eq?
- Data.Vec.N-ary
  - uncurry-∀ⁿ
  - uncurry-∃ⁿ
- Data.Vec.Relation.Binary.Lex.Core
  - P⇔[]<[]
  - ∷<∷-⇔
- Data.Vec.Relation.Binary.Pointwise.Extensional
  - equivalent
  - Pointwise-≡↔≡
- Data.Vec.Relation.Binary.Pointwise.Inductive
  - Pointwise-≡↔≡
- Effect.Monad.Partiality
  - correct
- Relation.Binary.Construct.Closure.Reflexive.Properties
  - ⊎⇔Refl
- Relation.Binary.Construct.Closure.Transitive
  - equivalent
- Relation.Binary.Reflection
  - solve₁
- Relation.Nullary.Decidable
  - map

Changes to the new function hierarchy

The names of the fields in Function.Bundles have been changed from f, g, cong₁ and cong₂ to to, from, to-cong, from-cong.
The module Function.Definitions no longer has two equalities as module arguments, as they did not interact as intended with the re-exports from Function.Definitions.(Core1/Core2). The latter have been removed and their definitions folded into Function.Definitions.
In Function.Definitions the types of Surjective, Injective and Surjective have been changed from:
```
Surjective f = ∀ y → ∃ λ x → f x ≈₂ y
Inverseˡ f g = ∀ y → f (g y) ≈₂ y
Inverseʳ f g = ∀ x → g (f x) ≈₁ x
```
to:
```
Surjective f = ∀ y → ∃ λ x → ∀ {z} → z ≈₁ x → f z ≈₂ y
Inverseˡ f g = ∀ {x y} → y ≈₁ g x → f y ≈₂ x
Inverseʳ f g = ∀ {x y} → y ≈₂ f x → g y ≈₁ x
```
This is for several reasons: i) the new definitions compose much more easily, ii) Agda can better infer the equalities used.

To ease backwards compatibility:
- the old definitions have been moved to the new names StrictlySurjective, StrictlyInverseˡ and StrictlyInverseʳ.
- The records in Function.Structures and Function.Bundles export proofs of these under the names strictlySurjective, strictlyInverseˡ and strictlyInverseʳ,
- Conversion functions have been added in both directions to Function.Consequences(.Propositional).

Proofs of non-zeroness/positivity/negativity as instance arguments

Many numeric operations in the library require their arguments to be non-zero, and various proofs require their arguments to be non-zero/positive/negative etc. As discussed on the mailing list, the previous way of constructing and passing round these proofs was extremely clunky and lead to messy and difficult to read code. We have therefore changed every occurrence where we need a proof of non-zeroness/positivity/etc. to take it as an irrelevant instance argument. See the mailing list for a fuller explanation of the motivation and implementation.
For example, whereas the type signature of division used to be:
```
_/_ : (dividend divisor : ℕ) {≢0 : False (divisor ≟ 0)} → ℕ
```
it is now:
```
_/_ : (dividend divisor : ℕ) .{{_ : NonZero divisor}} → ℕ
```
which means that as long as an instance of NonZero n is in scope then you can write m / n without having to explicitly provide a proof, as instance search will fill it in for you. The full list of such operations changed is as follows:
- In Data.Nat.DivMod: _/_, _%_, _div_, _mod_
- In Data.Nat.Pseudorandom.LCG: Generator
- In Data.Integer.DivMod: _divℕ_, _div_, _modℕ_, _mod_
- In Data.Rational: mkℚ+, normalize, _/_, 1/_
- In Data.Rational.Unnormalised: _/_, 1/_, _÷_
At the moment, there are 4 different ways such instance arguments can be provided, listed in order of convenience and clarity:
1. Automatic basic instances - the standard library provides instances based on the constructors of each numeric type in Data.X.Base. For example, Data.Nat.Base constrains an instance of NonZero (suc n) for any n and Data.Integer.Base contains an instance of NonNegative (+ n) for any n. Consequently, if the argument is of the required form, these instances will always be filled in by instance search automatically, e.g.
```
0/n≡0 : 0 / suc n ≡ 0
```
2. Take the instance as an argument - You can provide the instance argument as a parameter to your function and Agda's instance search will automatically use it in the correct place without you having to explicitly pass it, e.g.
```
0/n≡0 : .{{_ : NonZero n}} → 0 / n ≡ 0
```
3. Define the instance locally - You can define an instance argument in scope (e.g. in a where clause) and Agda's instance search will again find it automatically, e.g.
```
instance
  n≢0 : NonZero n
  n≢0 = ...

0/n≡0 : 0 / n ≡ 0
```
4. Pass the instance argument explicitly - Finally, if all else fails you can pass the instance argument explicitly into the function using {{ }}, e.g.
```
0/n≡0 : ∀ n (n≢0 : NonZero n) → ((0 / n) {{n≢0}}) ≡ 0
```
  Suitable constructors for NonZero/Positive etc. can be found in Data.X.Base.
A full list of proofs that have changed to use instance arguments is available at the end of this file. Notable changes to proofs are now discussed below.
Previously one of the hacks used in proofs was to explicitly refer to everything in the correct form, e.g. if the argument n had to be non-zero then you would refer to the argument as suc n everywhere instead of n, e.g.
```
n/n≡1 : ∀ n → suc n / suc n ≡ 1
```
This made the proofs extremely difficult to use if your term wasn't in the right form. After being updated to use instance arguments instead, the proof above becomes:
```
n/n≡1 : ∀ n {{_ : NonZero n}} → n / n ≡ 1
```
However, note that this means that if you passed in the value x to these proofs before, then you will now have to pass in suc x. The proofs for which the arguments have changed form in this way are highlighted in the list at the bottom of the file.
Finally, the definition of _≢0 has been removed from Data.Rational.Unnormalised.Base and the following proofs about it have been removed from Data.Rational.Unnormalised.Properties:
```
p≄0⇒∣↥p∣≢0 : ∀ p → p ≠ 0ℚᵘ → ℤ.∣ (↥ p) ∣ ≢0
∣↥p∣≢0⇒p≄0 : ∀ p → ℤ.∣ (↥ p) ∣ ≢0 → p ≠ 0ℚᵘ
```

Change in reduction behaviour of rationals

Currently arithmetic expressions involving rationals (both normalised and unnormalised) undergo disastrous exponential normalisation. For example, p + q would often be normalised by Agda to (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q). While the normalised form of p + q + r + s + t + u + v would be ~700 lines long. This behaviour often chokes both type-checking and the display of the expressions in the IDE.
To avoid this expansion and make non-trivial reasoning about rationals actually feasible:
1. the records ℚᵘ and ℚ have both had the no-eta-equality flag enabled
2. definition of arithmetic operations have trivial pattern matching added to prevent them reducing, e.g.
```
p + q = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
```
  has been changed to
```
p@record{} + q@record{} = (↥ p ℤ.* ↧ q ℤ.+ ↥ q ℤ.* ↧ p) / (↧ₙ p ℕ.* ↧ₙ q)
```
As a consequence of this, some proofs that relied on this reduction behaviour or on eta-equality may no longer go through. There are several ways to fix this:
1. The principled way is to not rely on this reduction behaviour in the first place. The Properties files for rational numbers have been greatly expanded in v1.7 and v2.0, and we believe most proofs should be able to be built up from existing proofs contained within these files.
2. Alternatively, annotating any rational arguments to a proof with either @record{} or @(mkℚ _ _ _) should restore the old reduction behaviour for any terms involving those parameters.
3. Finally, if the above approaches are not viable then you may be forced to explicitly use cong combined with a lemma that proves the old reduction behaviour.

Change to the definition of `LeftCancellative` and `RightCancellative`

The definitions of the types for cancellativity in Algebra.Definitions previously made some of their arguments implicit. This was under the assumption that the operators were defined by pattern matching on the left argument so that Agda could always infer the argument on the RHS.
Although many of the operators defined in the library follow this convention, this is not always true and cannot be assumed in user's code.
Therefore the definitions have been changed as follows to make all their arguments explicit:
- LeftCancellative _•_
  - From: ∀ x {y z} → (x • y) ≈ (x • z) → y ≈ z
  - To: ∀ x y z → (x • y) ≈ (x • z) → y ≈ z
- RightCancellative _•_
  - From: ∀ {x} y z → (y • x) ≈ (z • x) → y ≈ z
  - To: ∀ x y z → (y • x) ≈ (z • x) → y ≈ z
- AlmostLeftCancellative e _•_
  - From: ∀ {x} y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z
  - To: ∀ x y z → ¬ x ≈ e → (x • y) ≈ (x • z) → y ≈ z
- AlmostRightCancellative e _•_
  - From: ∀ {x} y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z
  - To: ∀ x y z → ¬ x ≈ e → (y • x) ≈ (z • x) → y ≈ z
Correspondingly some proofs of the above types will need additional arguments passed explicitly. Instances can easily be fixed by adding additional underscores, e.g.
- ∙-cancelˡ x to ∙-cancelˡ x _ _
- ∙-cancelʳ y z to ∙-cancelʳ _ y z

Change in the definition of `Prime`

The definition of Prime in Data.Nat.Primality was:
```
Prime 0             = ⊥
Prime 1             = ⊥
Prime (suc (suc n)) = (i : Fin n) → 2 + toℕ i ∤ 2 + n
```
which was very hard to reason about as not only did it involve conversion to and from the Fin type, it also required that the divisor was of the form 2 + toℕ i, which has exactly the same problem as the suc n hack described above used for non-zeroness.

To make it easier to use, reason about and read, the definition has been changed to:

Prime 0 = ⊥
Prime 1 = ⊥
Prime n = ∀ {d} → 2 ≤ d → d < n → d ∤ n

Change to the definition of `Induction.WellFounded.WfRec` (issue #2083)

Previously, the following definition was adopted
```
WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
WfRec _<_ P x = ∀ y → y < x → P y
```
with the consequence that all arguments involving about accesibility and wellfoundedness proofs were polluted by almost-always-inferrable explicit arguments for the y position. The definition has now been changed to make that argument implicit, as
```
WfRec : Rel A r → ∀ {ℓ} → RecStruct A ℓ _
WfRec _<_ P x = ∀ {y} → y < x → P y
```

Change in the definition of `_≤″_` (issue #1919)

The definition of _≤″_ in Data.Nat.Base was previously:
```
record _≤″_ (m n : ℕ) : Set where
  constructor less-than-or-equal
  field
    {k}   : ℕ
    proof : m + k ≡ n
```
which introduced a spurious additional definition, when this is in fact, modulo field names and implicit/explicit qualifiers, equivalent to the definition of left- divisibility, _∣ˡ_ for the RawMagma structure of _+_. Since the addition of raw bundles to Data.X.Base, this definition can now be made directly. Knock-on consequences include the need to retain the old constructor name, now introduced as a pattern synonym, and introduction of (a function equivalent to) the former field name/projection function proof as ≤″-proof in Data.Nat.Properties.

Accordingly, the definition has been changed to:

_≤″_ : (m n : ℕ)  → Set
_≤″_ = _∣ˡ_ +-rawMagma

pattern less-than-or-equal {k} prf = k , prf

Renaming of `Reflection` modules

Under the Reflection module, there were various impending name clashes between the core AST as exposed by Agda and the annotated AST defined in the library.

While the content of the modules remain the same, the modules themselves have therefore been renamed as follows:

Reflection.Annotated              ↦ Reflection.AnnotatedAST
Reflection.Annotated.Free         ↦ Reflection.AnnotatedAST.Free

Reflection.Abstraction            ↦ Reflection.AST.Abstraction
Reflection.Argument               ↦ Reflection.AST.Argument
Reflection.Argument.Information   ↦ Reflection.AST.Argument.Information
Reflection.Argument.Quantity      ↦ Reflection.AST.Argument.Quantity
Reflection.Argument.Relevance     ↦ Reflection.AST.Argument.Relevance
Reflection.Argument.Modality      ↦ Reflection.AST.Argument.Modality
Reflection.Argument.Visibility    ↦ Reflection.AST.Argument.Visibility
Reflection.DeBruijn               ↦ Reflection.AST.DeBruijn
Reflection.Definition             ↦ Reflection.AST.Definition
Reflection.Instances              ↦ Reflection.AST.Instances
Reflection.Literal                ↦ Reflection.AST.Literal
Reflection.Meta                   ↦ Reflection.AST.Meta
Reflection.Name                   ↦ Reflection.AST.Name
Reflection.Pattern                ↦ Reflection.AST.Pattern
Reflection.Show                   ↦ Reflection.AST.Show
Reflection.Traversal              ↦ Reflection.AST.Traversal
Reflection.Universe               ↦ Reflection.AST.Universe

Reflection.TypeChecking.Monad             ↦ Reflection.TCM
Reflection.TypeChecking.Monad.Categorical ↦ Reflection.TCM.Categorical
Reflection.TypeChecking.Monad.Format      ↦ Reflection.TCM.Format
Reflection.TypeChecking.Monad.Syntax      ↦ Reflection.TCM.Instances
Reflection.TypeChecking.Monad.Instances   ↦ Reflection.TCM.Syntax

A new module Reflection.AST that re-exports the contents of the submodules has been added.

Implementation of division and modulus for `ℤ`

The previous implementations of _divℕ_, _div_, _modℕ_, _mod_ in Data.Integer.DivMod internally converted to the unary Fin datatype resulting in poor performance. The implementation has been updated to use the corresponding operations from Data.Nat.DivMod which are efficiently implemented using the Haskell backend.

Strict functions

The module Strict has been deprecated in favour of Function.Strict and the definitions of strict application, _$!_ and _$!′_, have been moved from Function.Base to Function.Strict.
The contents of Function.Strict is now re-exported by Function.

Changes to ring structures

Several ring-like structures now have the multiplicative structure defined by its laws rather than as a substructure, to avoid repeated proofs that the underlying relation is an equivalence. These are:
- IsNearSemiring
- IsSemiringWithoutOne
- IsSemiringWithoutAnnihilatingZero
- IsRing
To aid with migration, structures matching the old style ones have been added to Algebra.Structures.Biased, with conversion functions:
- IsNearSemiring* and isNearSemiring*
- IsSemiringWithoutOne* and isSemiringWithoutOne*
- IsSemiringWithoutAnnihilatingZero* and isSemiringWithoutAnnihilatingZero*
- IsRing* and isRing*

Proof-irrelevant empty type

The definition of ⊥ has been changed to
```
private
  data Empty : Set where

⊥ : Set
⊥ = Irrelevant Empty
```
in order to make ⊥ proof irrelevant. Any two proofs of ⊥ or of a negated statements are now judgmentally equal to each other.

Consequently we have modified the following definitions:

In Relation.Nullary.Decidable.Core, the type of dec-no has changed

dec-no : (p? : Dec P) → ¬ P → ∃ λ ¬p′ → p? ≡ no ¬p′
  ↦ dec-no : (p? : Dec P) (¬p : ¬ P) → p? ≡ no ¬p

In Relation.Binary.PropositionalEquality, the type of ≢-≟-identity has changed

≢-≟-identity : x ≢ y → ∃ λ ¬eq → x ≟ y ≡ no ¬eq
  ↦ ≢-≟-identity : (x≢y : x ≢ y) → x ≟ y ≡ no x≢y

Reorganisation of the `Relation.Nullary` hierarchy

It was very difficult to use the Relation.Nullary modules, as Relation.Nullary contained the basic definitions of negation, decidability etc., and the operations and proofs were smeared over Relation.Nullary.(Negation/Product/Sum/Implication etc.).
In order to fix this:
- the definition of Dec and recompute have been moved to Relation.Nullary.Decidable.Core
- the definition of Reflects has been moved to Relation.Nullary.Reflects
- the definition of ¬_ has been moved to Relation.Nullary.Negation.Core
Backwards compatibility has been maintained, as Relation.Nullary still re-exports these publicly.
The modules:
```
Relation.Nullary.Product
Relation.Nullary.Sum
Relation.Nullary.Implication
```
have been deprecated and their contents moved to Relation.Nullary.(Negation/Reflects/Decidable) however all their contents is re-exported by Relation.Nullary which is the easiest way to access it now.
In order to facilitate this reorganisation the following breaking moves have occurred:
- ¬? has been moved from Relation.Nullary.Negation.Core to Relation.Nullary.Decidable.Core
- ¬-reflects has been moved from Relation.Nullary.Negation.Core to Relation.Nullary.Reflects.
- decidable-stable, excluded-middle and ¬-drop-Dec have been moved from Relation.Nullary.Negation to Relation.Nullary.Decidable.
- fromDec and toDec have been moved from Data.Sum.Base to Data.Sum.

Refactoring of the unindexed Functor/Applicative/Monad hierarchy

The unindexed versions are not defined in terms of the named versions anymore
The RawApplicative and RawMonad type classes have been relaxed so that the underlying functors do not need their domain and codomain to live at the same Set level. This is needed for level-increasing functors like IO : Set l → Set (suc l).
RawApplicative is now RawFunctor + pure + _<*>_ and RawMonad is now RawApplicative + _>>=_ and so return is not used anywhere anymore. This fixes the conflict with case_return_of (#356) This reorganisation means in particular that the functor/applicative of a monad are not computed using _>>=_. This may break proofs.
When F : Set f → Set f we moreover have a definable join/μ operator join : (M : RawMonad F) → F (F A) → F A.
We now have RawEmpty and RawChoice respectively packing empty : M A and (<|>) : M A → M A → M A. RawApplicativeZero, RawAlternative, RawMonadZero, RawMonadPlus are all defined in terms of these.
MonadT T now returns a MonadTd record that packs both a proof that the Monad M transformed by T is a monad and that we can lift a computation M A to a transformed computation T M A.
The monad transformer are not mere aliases anymore, they are record-wrapped which allows constraints such as MonadIO (StateT S (ReaderT R IO)) to be discharged by instance arguments.
The mtl-style type classes (MonadState, MonadReader) do not contain a proof that the underlying functor is a Monad anymore. This ensures we do not have conflicting Monad M instances from a pair of MonadState S M & MonadReader R M constraints.
MonadState S M is now defined in terms of
```
gets : (S → A) → M A
modify : (S → S) → M ⊤
```
with get and put defined as derived notions. This is needed because MonadState S M does not pack a Monad M instance anymore and so we cannot define modify f as get >>= λ s → put (f s).
MonadWriter 𝕎 M is defined similarly:
```
writer : W × A → M A
listen : M A → M (W × A)
pass   : M ((W → W) × A) → M A
```
with tell defined as a derived notion. Note that 𝕎 is a RawMonoid, not a Set and W is the carrier of the monoid.

New modules:

Algebra.Construct.Initial
Algebra.Construct.Terminal
Data.List.Effectful.Transformer
Data.List.NonEmpty.Effectful.Transformer
Data.Maybe.Effectful.Transformer
Data.Sum.Effectful.Left.Transformer
Data.Sum.Effectful.Right.Transformer
Data.Vec.Effectful.Transformer
Effect.Empty
Effect.Choice
Effect.Monad.Error.Transformer
Effect.Monad.Identity
Effect.Monad.IO
Effect.Monad.IO.Instances
Effect.Monad.Reader.Indexed
Effect.Monad.Reader.Instances
Effect.Monad.Reader.Transformer
Effect.Monad.Reader.Transformer.Base
Effect.Monad.State.Indexed
Effect.Monad.State.Instances
Effect.Monad.State.Transformer
Effect.Monad.State.Transformer.Base
Effect.Monad.Writer
Effect.Monad.Writer.Indexed
Effect.Monad.Writer.Instances
Effect.Monad.Writer.Transformer
Effect.Monad.Writer.Transformer.Base
IO.Effectful
IO.Instances

(Issue #2096) Introduction of flipped relation symbol for `Relation.Binary.Bundles.Preorder`

Previously, the relation symbol _∼_ was (notationally) symmetric, so that its converse relation could only be discussed semantically in terms of flip _∼_ in Relation.Binary.Properties.Preorder, Relation.Binary.Construct.Flip.{Ord|EqAndOrd}
Now, the symbol _∼_ has been renamed to a new symbol _≲_, with _≳_ introduced as a definition in Relation.Binary.Bundles.Preorder whose properties in Relation.Binary.Properties.Preorder now refer to it. Partial backwards compatible has been achieved by redeclaring a deprecated version of the old name in the record. Therefore, only declarations of PartialOrder records will need their field names updating.
NB (issues #1214 #2098) the corresponding situation regarding the flipped relation symbols _≥_, _>_ (and their negated versions!) has not (yet) been addressed.

Standardisation of `insertAt`/`updateAt`/`removeAt`

Previously, the names and argument order of index-based insertion, update and removal functions for various types of lists and vectors were inconsistent.
To fix this the names have all been standardised to insertAt/updateAt/removeAt.
Correspondingly the following changes have occurred:

In Data.List.Base the following have been added:

insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A
removeAt : (xs : List A) → Fin (length xs) → List A

and the following has been deprecated

_─_ ↦ removeAt

In Data.Vec.Base:

insert ↦ insertAt
remove ↦ removeAt

updateAt : Fin n → (A → A) → Vec A n → Vec A n
  ↦
updateAt : Vec A n → Fin n → (A → A) → Vec A n

In Data.Vec.Functional:

remove : Fin (suc n) → Vector A (suc n) → Vector A n
  ↦
removeAt : Vector A (suc n) → Fin (suc n) → Vector A n

updateAt : Fin n → (A → A) → Vector A n → Vector A n
  ↦
updateAt : Vector A n → Fin n → (A → A) → Vector A n

The old names (and the names of all proofs about these functions) have been deprecated appropriately.

Changes to triple reasoning interface

The module Relation.Binary.Reasoning.Base.Triple now takes an extra proof that the strict relation is irreflexive.
This allows the following new proof combinator:
```
begin-contradiction : (r : x IsRelatedTo x) → {s : True (IsStrict? r)} → A
```
that takes a proof that a value is strictly less than itself and then applies the principle of explosion.

Specialised versions of this combinator are available in the following derived modules:

Data.Nat.Properties
Data.Nat.Binary.Properties
Data.Integer.Properties
Data.Rational.Unnormalised.Properties
Data.Rational.Properties
Data.Vec.Relation.Binary.Lex.Strict
Data.Vec.Relation.Binary.Lex.NonStrict
Relation.Binary.Reasoning.StrictPartialOrder
Relation.Binary.Reasoning.PartialOrder

Other

In accordance with changes to the flags in Agda 2.6.3, all modules that previously used the --without-K flag now use the --cubical-compatible flag instead.
To avoid large indices that are by default no longer allowed in Agda 2.6.4, universe levels have been increased in the following definitions:
- Data.Star.Decoration.DecoratedWith
- Data.Star.Pointer.Pointer
- Reflection.AnnotatedAST.Typeₐ
- Reflection.AnnotatedAST.AnnotationFun
The first two arguments of m≡n⇒m-n≡0 (now i≡j⇒i-j≡0) in Data.Integer.Base have been made implicit.
The relations _≤_ _≥_ _<_ _>_ in Data.Fin.Base have been generalised so they can now relate Fin terms with different indices. Should be mostly backwards compatible, but very occasionally when proving properties about the orderings themselves the second index must be provided explicitly.
The argument xs in xs≮[] in Data.{List|Vec}.Relation.Binary.Lex.Strict introduced in PRs #1648 and #1672 has now been made implicit.
Issue #2075 (Johannes Waldmann): wellfoundedness of the lexicographic ordering on products, defined in Data.Product.Relation.Binary.Lex.Strict, no longer requires the assumption of symmetry for the first equality relation _≈₁_, leading to a new lemma Induction.WellFounded.Acc-resp-flip-≈, and refactoring of the previous proof Induction.WellFounded.Acc-resp-≈.
The operation SymClosure on relations in Relation.Binary.Construct.Closure.Symmetric has been reimplemented as a data type SymClosure _⟶_ a b that is parameterized by the input relation _⟶_ (as well as the elements a and b of the domain) so that _⟶_ can be inferred, which it could not from the previous implementation using the sum type a ⟶ b ⊎ b ⟶ a.
In Algebra.Morphism.Structures, IsNearSemiringHomomorphism, IsSemiringHomomorphism, and IsRingHomomorphism have been redesigned to build up from IsMonoidHomomorphism, IsNearSemiringHomomorphism, and IsSemiringHomomorphism respectively, adding a single property at each step. This means that they no longer need to have two separate proofs of IsRelHomomorphism. Similarly, IsLatticeHomomorphism is now built as IsRelHomomorphism along with proofs that _∧_ and _∨_ are homomorphic.

Also, ⁻¹-homo in IsRingHomomorphism has been renamed to -‿homo.
Moved definition of _>>=_ under Data.Vec.Base to its submodule CartesianBind in order to keep another new definition of _>>=_, located in DiagonalBind which is also a submodule of Data.Vec.Base.
The functions split, flatten and flatten-split have been removed from Data.List.NonEmpty. In their place groupSeqs and ungroupSeqs have been added to Data.List.NonEmpty.Base which morally perform the same operations but without computing the accompanying proofs. The proofs can be found in Data.List.NonEmpty.Properties under the names groupSeqs-groups and ungroupSeqs and groupSeqs.
In Data.List.Relation.Unary.Grouped.Properties the proofs map⁺ and map⁻ have had their preconditions weakened so the equivalences no longer require congruence proofs.
The constructors +0 and +[1+_] from Data.Integer.Base are no longer exported by Data.Rational.Base. You will have to open Data.Integer(.Base) directly to use them.
The names of the (in)equational reasoning combinators defined in the internal modules Data.Rational(.Unnormalised).Properties.≤-Reasoning have been renamed (issue #1437) to conform with the defined setoid equality _≃_ on Rationals:
```
step-≈  ↦  step-≃
step-≈˘ ↦  step-≃˘
```
with corresponding associated syntax:
```
_≈⟨_⟩_  ↦  _≃⟨_⟩_
_≈˘⟨_⟩_ ↦  _≃˘⟨_⟩_
```
NB. It is not possible to rename or deprecate syntax declarations, so Agda will only issue a "Could not parse the application begin ... when scope checking" warning if the old combinators are used.
The types of the proofs pos⇒1/pos/1/pos⇒pos and neg⇒1/neg/1/neg⇒neg in Data.Rational(.Unnormalised).Properties have been switched, as the previous naming scheme didn't correctly generalise to e.g. pos+pos⇒pos. For example the types of pos⇒1/pos/1/pos⇒pos were:
```
pos⇒1/pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
1/pos⇒pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
```
but are now:
```
pos⇒1/pos : ∀ p .{{_ : Positive p}} → Positive (1/ p)
1/pos⇒pos : ∀ p .{{_ : NonZero p}} .{{Positive (1/ p)}} → Positive p
```
Opₗ and Opᵣ have moved from Algebra.Core to Algebra.Module.Core.
In Data.Nat.GeneralisedArithmetic, the s and z arguments to the following functions have been made explicit:
- fold-+
- fold-k
- fold-*
- fold-pull
In Data.Fin.Properties:
- the i argument to opposite-suc has been made explicit;
- pigeonhole has been strengthened: wlog, we return a proof that i < j rather than a mere i ≢ j.
In Data.Sum.Base the definitions fromDec and toDec have been moved to Data.Sum.
In Data.Vec.Base: the definitions init and last have been changed from the initLast view-derived implementation to direct recursive definitions.
In Codata.Guarded.Stream the following functions have been modified to have simpler definitions:
- cycle
- interleave⁺
- cantor Furthermore, the direction of interleaving of cantor has changed. Precisely, suppose pair is the cantor pairing function, then lookup (pair i j) (cantor xss) according to the old definition corresponds to lookup (pair j i) (cantor xss) according to the new definition. For a concrete example see the one included at the end of the module.
Removed m/n/o≡m/[n*o] from Data.Nat.Divisibility and added a more general m/n/o≡m/[n*o] to Data.Nat.DivMod that doesn't require n * o ∣ m.

Updated lookup functions (and variants) to match the conventions adopted in the List module i.e. lookup takes its container first and the index (whose type may depend on the container value) second. This affects modules:

Codata.Guarded.Stream
Codata.Guarded.Stream.Relation.Binary.Pointwise
Codata.Musical.Colist.Base
Codata.Musical.Colist.Relation.Unary.Any.Properties
Codata.Musical.Covec
Codata.Musical.Stream
Codata.Sized.Colist
Codata.Sized.Covec
Codata.Sized.Stream
Data.Vec.Relation.Unary.All
Data.Star.Environment
Data.Star.Pointer
Data.Star.Vec
Data.Trie
Data.Trie.NonEmpty
Data.Tree.AVL
Data.Tree.AVL.Indexed
Data.Tree.AVL.Map
Data.Tree.AVL.NonEmpty
Data.Vec.Recursive
Tactic.RingSolver
Tactic.RingSolver.Core.NatSet

Moved & renamed from Data.Vec.Relation.Unary.All to Data.Vec.Relation.Unary.All.Properties:
```
lookup ↦ lookup⁺
tabulate ↦ lookup⁻
```
Renamed in Data.Vec.Relation.Unary.Linked.Properties and Codata.Guarded.Stream.Relation.Binary.Pointwise:
```
lookup ↦ lookup⁺
```

Added the following new definitions to Data.Vec.Relation.Unary.All:

lookupAny : All P xs → (i : Any Q xs) → (P ∩ Q) (Any.lookup i)
lookupWith : ∀[ P ⇒ Q ⇒ R ] → All P xs → (i : Any Q xs) → R (Any.lookup i)
lookup : All P xs → (∀ {x} → x ∈ₚ xs → P x)
lookupₛ : P Respects _≈_ → All P xs → (∀ {x} → x ∈ xs → P x)

excluded-middle in Relation.Nullary.Decidable.Core has been renamed to ¬¬-excluded-middle.
iterate and replicate in Data.Vec.Base and Data.Vec.Functional now take the length of vector, n, as an explicit rather than an implicit argument.
```
iterate : (A → A) → A → ∀ n → Vec A n
replicate : ∀ n → A → Vec A n
```

Major improvements

Improvements to ring solver tactic

The ring solver tactic has been greatly improved. In particular:
1. When the solver is used for concrete ring types, e.g. ℤ, the equality can now use all the ring operations defined natively for that type, rather than having to use the operations defined in AlmostCommutativeRing. For example previously you could not use Data.Integer.Base._*_ but instead had to use AlmostCommutativeRing._*_.
2. The solver now supports use of the subtraction operator _-_ whenever it is defined immediately in terms of _+_ and -_. This is the case for Data.Integer and Data.Rational.

Moved `_%_` and `_/_` operators to `Data.Nat.Base`

Previously the division and modulus operators were defined in Data.Nat.DivMod which in turn meant that using them required importing Data.Nat.Properties which is a very heavy dependency.
To fix this, these operators have been moved to Data.Nat.Base. The properties for them still live in Data.Nat.DivMod (which also publicly re-exports them to provide backwards compatibility).
Beneficiaries of this change include Data.Rational.Unnormalised.Base whose dependencies are now significantly smaller.

Moved raw bundles from Data.X.Properties to Data.X.Base

As mentioned by MatthewDaggitt in Issue #1755, Raw bundles defined in Data.X.Properties should be defined in Data.X.Base as they don't require any properties.
- Moved raw bundles From Data.Nat.Properties to Data.Nat.Base
- Moved raw bundles From Data.Nat.Binary.Properties to Data.Nat.Binary.Base
- Moved raw bundles From Data.Rational.Properties to Data.Rational.Base
- Moved raw bundles From Data.Rational.Unnormalised.Properties to Data.Rational.Unnormalised.Base

Moved the definition of RawX from `Algebra.X.Bundles` to `Algebra.X.Bundles.Raw`

A new module Algebra.Bundles.Raw containing the definitions of the raw bundles can be imported at much lower cost from Data.X.Base. The following definitions have been moved:
- RawMagma
- RawMonoid
- RawGroup
- RawNearSemiring
- RawSemiring
- RawRingWithoutOne
- RawRing
- RawQuasigroup
- RawLoop
- RawKleeneAlgebra
A new module Algebra.Lattice.Bundles.Raw is also introduced.
- RawLattice has been moved from Algebra.Lattice.Bundles to this new module.
In Relation.Binary.Reasoning.Base.Triple, added a new parameter <-irrefl : Irreflexive _≈_ _<_

Deprecated modules

Moving `Relation.Binary.Construct.(Converse/Flip)`

The following files have been moved:

Relation.Binary.Construct.Converse               ↦ Relation.Binary.Construct.Flip.EqAndOrd
Relation.Binary.Construct.Flip                   ↦ Relation.Binary.Construct.Flip.Ord

Deprecation of old function hierarchy

The module Function.Related has been deprecated in favour of Function.Related.Propositional whose code uses the new function hierarchy. This also opens up the possibility of a more general Function.Related.Setoid at a later date. Several of the names have been changed in this process to bring them into line with the camelcase naming convention used in the rest of the library:
```
reverse-implication ↦ reverseImplication
reverse-injection   ↦ reverseInjection
left-inverse        ↦ leftInverse

Symmetric-kind      ↦ SymmetricKind
Forward-kind        ↦ ForwardKind
Backward-kind       ↦ BackwardKind
Equivalence-kind    ↦ EquivalenceKind
```

Moving `Algebra.Lattice` files

As discussed above the following files have been moved:

Algebra.Properties.Semilattice               ↦ Algebra.Lattice.Properties.Semilattice
Algebra.Properties.Lattice                   ↦ Algebra.Lattice.Properties.Lattice
Algebra.Properties.DistributiveLattice       ↦ Algebra.Lattice.Properties.DistributiveLattice
Algebra.Properties.BooleanAlgebra            ↦ Algebra.Lattice.Properties.BooleanAlgebra
Algebra.Properties.BooleanAlgebra.Expression ↦ Algebra.Lattice.Properties.BooleanAlgebra.Expression
Algebra.Morphism.LatticeMonomorphism         ↦ Algebra.Lattice.Morphism.LatticeMonomorphism

Moving `.Catgeorical.` files

As discussed above the following files have been moved:

Codata.Sized.Colist.Categorical            ↦ Codata.Sized.Colist.Effectful
Codata.Sized.Covec.Categorical             ↦ Codata.Sized.Covec.Effectful
Codata.Sized.Delay.Categorical             ↦ Codata.Sized.Delay.Effectful
Codata.Sized.Stream.Categorical            ↦ Codata.Sized.Stream.Effectful
Data.List.Categorical                      ↦ Data.List.Effectful
Data.List.Categorical.Transformer          ↦ Data.List.Effectful.Transformer
Data.List.NonEmpty.Categorical             ↦ Data.List.NonEmpty.Effectful
Data.List.NonEmpty.Categorical.Transformer ↦ Data.List.NonEmpty.Effectful.Transformer
Data.Maybe.Categorical                     ↦ Data.Maybe.Effectful
Data.Maybe.Categorical.Transformer         ↦ Data.Maybe.Effectful.Transformer
Data.Product.Categorical.Examples          ↦ Data.Product.Effectful.Examples
Data.Product.Categorical.Left              ↦ Data.Product.Effectful.Left
Data.Product.Categorical.Left.Base         ↦ Data.Product.Effectful.Left.Base
Data.Product.Categorical.Right             ↦ Data.Product.Effectful.Right
Data.Product.Categorical.Right.Base        ↦ Data.Product.Effectful.Right.Base
Data.Sum.Categorical.Examples              ↦ Data.Sum.Effectful.Examples
Data.Sum.Categorical.Left                  ↦ Data.Sum.Effectful.Left
Data.Sum.Categorical.Left.Transformer      ↦ Data.Sum.Effectful.Left.Transformer
Data.Sum.Categorical.Right                 ↦ Data.Sum.Effectful.Right
Data.Sum.Categorical.Right.Transformer     ↦ Data.Sum.Effectful.Right.Transformer
Data.These.Categorical.Examples            ↦ Data.These.Effectful.Examples
Data.These.Categorical.Left                ↦ Data.These.Effectful.Left
Data.These.Categorical.Left.Base           ↦ Data.These.Effectful.Left.Base
Data.These.Categorical.Right               ↦ Data.These.Effectful.Right
Data.These.Categorical.Right.Base          ↦ Data.These.Effectful.Right.Base
Data.Vec.Categorical                       ↦ Data.Vec.Effectful
Data.Vec.Categorical.Transformer           ↦ Data.Vec.Effectful.Transformer
Data.Vec.Recursive.Categorical             ↦ Data.Vec.Recursive.Effectful
Function.Identity.Categorical              ↦ Function.Identity.Effectful
IO.Categorical                             ↦ IO.Effectful
Reflection.TCM.Categorical                 ↦ Reflection.TCM.Effectful

Moving `Relation.Binary.Properties.XLattice` files

The following files have been moved:

Relation.Binary.Properties.BoundedJoinSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedJoinSemilattice.agda
Relation.Binary.Properties.BoundedLattice.agda          ↦ Relation.Binary.Lattice.Properties.BoundedLattice.agda
Relation.Binary.Properties.BoundedMeetSemilattice.agda  ↦ Relation.Binary.Lattice.Properties.BoundedMeetSemilattice.agda
Relation.Binary.Properties.DistributiveLattice.agda     ↦ Relation.Binary.Lattice.Properties.DistributiveLattice.agda
Relation.Binary.Properties.HeytingAlgebra.agda         ↦ Relation.Binary.Lattice.Properties.HeytingAlgebra.agda
Relation.Binary.Properties.JoinSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.JoinSemilattice.agda
Relation.Binary.Properties.Lattice.agda                 ↦ Relation.Binary.Lattice.Properties.Lattice.agda
Relation.Binary.Properties.MeetSemilattice.agda         ↦ Relation.Binary.Lattice.Properties.MeetSemilattice.agda

Deprecation of `Data.Nat.Properties.Core`

The module Data.Nat.Properties.Core has been deprecated, and its one entry moved to Data.Nat.Properties

Deprecation of `Data.Fin.Substitution.Example`

The module Data.Fin.Substitution.Example has been deprecated, and moved to README.Data.Fin.Substitution.UntypedLambda

Deprecation of `Data.Product.Function.Dependent.Setoid.WithK`

This module has been deprecated, as none of its contents actually depended on axiom K. The contents has been moved to Data.Product.Function.Dependent.Setoid.

Deprecated names

In Algebra.Construct.Zero:

rawMagma   ↦  Algebra.Construct.Terminal.rawMagma
magma      ↦  Algebra.Construct.Terminal.magma
semigroup  ↦  Algebra.Construct.Terminal.semigroup
band       ↦  Algebra.Construct.Terminal.band

In Codata.Guarded.Stream.Properties:

map-identity  ↦  map-id
map-fusion    ↦  map-∘
drop-fusion   ↦  drop-drop

In Codata.Sized.Colist.Properties:

map-identity      ↦  map-id
map-map-fusion    ↦  map-∘
drop-drop-fusion  ↦  drop-drop

In Codata.Sized.Covec.Properties:

map-identity   ↦  map-id
map-map-fusion  ↦  map-∘

In Codata.Sized.Delay.Properties:

map-identity      ↦  map-id
map-map-fusion    ↦  map-∘
map-unfold-fusion  ↦  map-unfold

In Codata.Sized.M.Properties:
```
map-compose  ↦  map-∘
```

In Codata.Sized.Stream.Properties:

map-identity   ↦  map-id
map-map-fusion  ↦  map-∘

In Data.Bool.Properties (Issue #2046):
```
push-function-into-if ↦ if-float
```
In Data.Container.Related:
```
_∼[_]_  ↦  _≲[_]_
```
In Data.Fin.Base: two new, hopefully more memorable, names ↑ˡ ↑ʳ for the 'left', resp. 'right' injection of a Fin m into a 'larger' type, Fin (m + n), resp. Fin (n + m), with argument order to reflect the position of the Fin m argument.
```
inject+  ↦  flip _↑ˡ_
raise    ↦  _↑ʳ_
```
Issue #1726: the relation _≺_ and its single constructor _≻toℕ_ have been deprecated in favour of their extensional equivalent _<_ but omitting the inversion principle which pattern matching on _≻toℕ_ would achieve; this instead is proxied by the property Data.Fin.Properties.toℕ<.
In Data.Fin.Induction:
```
≺-Rec
≺-wellFounded
≺-recBuilder
≺-rec
```
As with Issue #1726 above: the deprecation of relation _≺_ means that these definitions associated with wf-recursion are deprecated in favour of their _<_ counterparts. But it's not quite sensible to say that these definitions should be renamed to anything, least of all those counterparts... the type confusion would be intolerable.

In Data.Fin.Properties:

toℕ-raise        ↦ toℕ-↑ʳ
toℕ-inject+      ↦ toℕ-↑ˡ
splitAt-inject+  ↦ splitAt-↑ˡ m i n
splitAt-raise    ↦ splitAt-↑ʳ
Fin0↔⊥           ↦ 0↔⊥
eq?              ↦ inj⇒≟

Likewise under issue #1726: the properties ≺⇒<′ and <′⇒≺ have been deprecated in favour of their proxy counterparts <⇒<′ and <′⇒<.

In Data.Fin.Permutation.Components:

reverse            ↦ Data.Fin.Base.opposite
reverse-prop       ↦ Data.Fin.Properties.opposite-prop
reverse-involutive ↦ Data.Fin.Properties.opposite-involutive
reverse-suc        ↦ Data.Fin.Properties.opposite-suc

In Data.Integer.DivMod the operator names have been renamed to be consistent with those in Data.Nat.DivMod:
```
_divℕ_  ↦ _/ℕ_
_div_   ↦ _/_
_modℕ_  ↦ _%ℕ_
_mod_   ↦ _%_
```

In Data.Integer.Properties references to variables in names have been made consistent so that m, n always refer to naturals and i and j always refer to integers:

≤-steps        ↦  i≤j⇒i≤k+j
≤-step         ↦  i≤j⇒i≤1+j

≤-steps-neg    ↦  i≤j⇒i-k≤j
≤-step-neg     ↦  i≤j⇒pred[i]≤j

n≮n            ↦  i≮i
∣n∣≡0⇒n≡0       ↦  ∣i∣≡0⇒i≡0
∣-n∣≡∣n∣         ↦  ∣-i∣≡∣i∣
0≤n⇒+∣n∣≡n      ↦  0≤i⇒+∣i∣≡i
+∣n∣≡n⇒0≤n      ↦  +∣i∣≡i⇒0≤i
+∣n∣≡n⊎+∣n∣≡-n   ↦  +∣i∣≡i⊎+∣i∣≡-i
∣m+n∣≤∣m∣+∣n∣     ↦  ∣i+j∣≤∣i∣+∣j∣
∣m-n∣≤∣m∣+∣n∣     ↦  ∣i-j∣≤∣i∣+∣j∣
signₙ◃∣n∣≡n     ↦  signᵢ◃∣i∣≡i
◃-≡            ↦  ◃-cong
∣m-n∣≡∣n-m∣      ↦  ∣i-j∣≡∣j-i∣
m≡n⇒m-n≡0      ↦  i≡j⇒i-j≡0
m-n≡0⇒m≡n      ↦  i-j≡0⇒i≡j
m≤n⇒m-n≤0      ↦  i≤j⇒i-j≤0
m-n≤0⇒m≤n      ↦  i-j≤0⇒i≤j
m≤n⇒0≤n-m      ↦  i≤j⇒0≤j-i
0≤n-m⇒m≤n      ↦  0≤i-j⇒j≤i
n≤1+n          ↦  i≤suc[i]
n≢1+n          ↦  i≢suc[i]
m≤pred[n]⇒m<n  ↦  i≤pred[j]⇒i<j
m<n⇒m≤pred[n]  ↦  i<j⇒i≤pred[j]
-1*n≡-n        ↦  -1*i≡-i
m*n≡0⇒m≡0∨n≡0  ↦  i*j≡0⇒i≡0∨j≡0
∣m*n∣≡∣m∣*∣n∣  ↦  ∣i*j∣≡∣i∣*∣j∣
m≤m+n          ↦  i≤i+j
n≤m+n          ↦  i≤j+i
m-n≤m          ↦  i≤i-j

+-pos-monoʳ-≤    ↦  +-monoʳ-≤
+-neg-monoʳ-≤    ↦  +-monoʳ-≤
*-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
*-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
*-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
*-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
*-cancelˡ-<-neg  ↦  *-cancelˡ-<-nonPos
*-cancelʳ-<-neg  ↦  *-cancelʳ-<-nonPos

^-semigroup-morphism ↦ ^-isMagmaHomomorphism
^-monoid-morphism    ↦ ^-isMonoidHomomorphism

pos-distrib-* ↦ pos-*
pos-+-commute ↦ pos-+
abs-*-commute ↦ abs-*

+-isAbelianGroup ↦ +-0-isAbelianGroup

In Data.List.Base:
```
_─_  ↦  removeAt
```

In Data.List.Properties:

map-id₂         ↦  map-id-local
map-cong₂       ↦  map-cong-local
map-compose     ↦  map-∘

map-++-commute       ↦  map-++
sum-++-commute       ↦  sum-++
reverse-map-commute  ↦  reverse-map
reverse-++-commute   ↦  reverse-++

zipWith-identityˡ  ↦  zipWith-zeroˡ
zipWith-identityʳ  ↦  zipWith-zeroʳ

ʳ++-++  ↦  ++-ʳ++

take++drop  ↦  take++drop≡id

length-─  ↦  length-removeAt
map-─     ↦  map-removeAt

In Data.List.NonEmpty.Properties:

map-compose     ↦  map-∘

map-++⁺-commute ↦  map-++⁺

In Data.List.Relation.Unary.All.Properties:

updateAt-id-relative      ↦  updateAt-id-local
updateAt-compose-relative ↦  updateAt-updateAt-local
updateAt-compose          ↦  updateAt-updateAt
updateAt-cong-relative    ↦  updateAt-cong-local

In Data.List.Zipper.Properties:

toList-reverse-commute ↦  toList-reverse
toList-ˡ++-commute     ↦  toList-ˡ++
toList-++ʳ-commute     ↦  toList-++ʳ
toList-map-commute    ↦  toList-map
toList-foldr-commute  ↦  toList-foldr

In Data.Maybe.Properties:

map-id₂     ↦  map-id-local
map-cong₂   ↦  map-cong-local

map-compose    ↦  map-∘

map-<∣>-commute ↦  map-<∣>

In Data.List.Relation.Binary.Subset.Propositional.Properties:
```
map-with-∈⁺  ↦  mapWith∈⁺
```

In Data.List.Relation.Unary.Any.Properties:

map-with-∈⁺  ↦  mapWith∈⁺
map-with-∈⁻  ↦  mapWith∈⁻
map-with-∈↔  ↦  mapWith∈↔

In Data.List.Relation.Unary.All.Properties:
```
gmap  ↦  gmap⁺
```

In Data.Nat.Properties:

suc[pred[n]]≡n  ↦  suc-pred
≤-step          ↦  m≤n⇒m≤1+n
≤-stepsˡ        ↦  m≤n⇒m≤o+n
≤-stepsʳ        ↦  m≤n⇒m≤n+o
<-step          ↦  m<n⇒m<1+n

<-transʳ        ↦  ≤-<-trans
<-transˡ        ↦  <-≤-trans

In Data.Rational.Unnormalised.Base:

_≠_  ↦  _≄_
+-rawMonoid ↦ +-0-rawMonoid
*-rawMonoid ↦ *-1-rawMonoid

In Data.Rational.Unnormalised.Properties:

↥[p/q]≡p         ↦  ↥[n/d]≡n
↧[p/q]≡q         ↦  ↧[n/d]≡d
*-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
*-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
≤-steps          ↦  p≤q⇒p≤r+q
*-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
*-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
*-cancelˡ-<-pos  ↦  *-cancelˡ-<-nonNeg
*-cancelʳ-<-pos  ↦  *-cancelʳ-<-nonNeg

positive⇒nonNegative  ↦ pos⇒nonNeg
negative⇒nonPositive  ↦ neg⇒nonPos
negative<positive     ↦ neg<pos

In Data.Rational.Base:

+-rawMonoid ↦ +-0-rawMonoid
*-rawMonoid ↦ *-1-rawMonoid

In Data.Rational.Properties:

*-monoʳ-≤-neg    ↦  *-monoʳ-≤-nonPos
*-monoˡ-≤-neg    ↦  *-monoˡ-≤-nonPos
*-monoʳ-≤-pos    ↦  *-monoʳ-≤-nonNeg
*-monoˡ-≤-pos    ↦  *-monoˡ-≤-nonNeg
*-cancelˡ-<-pos  ↦  *-cancelˡ-<-nonNeg
*-cancelʳ-<-pos  ↦  *-cancelʳ-<-nonNeg
*-cancelˡ-<-neg  ↦  *-cancelˡ-<-nonPos
*-cancelʳ-<-neg  ↦  *-cancelʳ-<-nonPos

negative<positive     ↦ neg<pos

In Data.Rational.Unnormalised.Base:

+-rawMonoid ↦ +-0-rawMonoid
*-rawMonoid ↦ *-1-rawMonoid

In Data.Rational.Unnormalised.Properties:
```
≤-steps  ↦  p≤q⇒p≤r+q
```

In Data.Sum.Properties:

[,]-∘-distr      ↦  [,]-∘
[,]-map-commute  ↦  [,]-map
map-commute      ↦  map-map
map₁₂-commute    ↦  map₁₂-map₂₁

In Data.Tree.AVL:
```
_∈?_ ↦ member
```
In Data.Tree.AVL.IndexedMap:
```
_∈?_ ↦ member
```
In Data.Tree.AVL.Map:
```
_∈?_ ↦ member
```
In Data.Tree.AVL.Sets:
```
_∈?_ ↦ member
```

In Data.Tree.Binary.Zipper.Properties:

toTree-#nodes-commute   ↦  toTree-#nodes
toTree-#leaves-commute  ↦  toTree-#leaves
toTree-map-commute      ↦  toTree-map
toTree-foldr-commute    ↦  toTree-foldr
toTree-⟪⟫ˡ-commute      ↦  toTree--⟪⟫ˡ
toTree-⟪⟫ʳ-commute      ↦  toTree-⟪⟫ʳ

In Data.Tree.Rose.Properties:
```
map-compose     ↦  map-∘
```

In Data.Vec.Base:

remove  ↦  removeAt
insert  ↦  insertAt

In Data.Vec.Properties:

take-distr-zipWith ↦  take-zipWith
take-distr-map     ↦  take-map
drop-distr-zipWith ↦  drop-zipWith
drop-distr-map     ↦  drop-map

updateAt-id-relative      ↦  updateAt-id-local
updateAt-compose-relative ↦  updateAt-updateAt-local
updateAt-compose          ↦  updateAt-updateAt
updateAt-cong-relative    ↦  updateAt-cong-local

[]%=-compose    ↦  []%=-∘

[]≔-++-inject+  ↦ []≔-++-↑ˡ
[]≔-++-raise    ↦ []≔-++-↑ʳ
idIsFold        ↦ id-is-foldr
sum-++-commute  ↦ sum-++

take-drop-id  ↦  take++drop≡id

map-insert       ↦  map-insertAt

insert-lookup    ↦  insertAt-lookup
insert-punchIn   ↦  insertAt-punchIn
remove-PunchOut  ↦  removeAt-punchOut
remove-insert    ↦  removeAt-insertAt
insert-remove    ↦  insertAt-removeAt

lookup-inject≤-take ↦ lookup-take-inject≤

and the type of the proof zipWith-comm has been generalised from:

zipWith-comm : ∀ {f : A → A → B} (comm : ∀ x y → f x y ≡ f y x) (xs ys : Vec A n) → zipWith f xs ys ≡ zipWith f ys xs

to

zipWith-comm : ∀ {f : A → B → C} {g : B → A → C}  (comm : ∀ x y → f x y ≡ g y x) (xs : Vec A n) ys → zipWith f xs ys ≡ zipWith g ys xs

In Data.Vec.Functional.Properties:

updateAt-id-relative      ↦  updateAt-id-local
updateAt-compose-relative ↦  updateAt-updateAt-local
updateAt-compose          ↦  updateAt-updateAt
updateAt-cong-relative    ↦  updateAt-cong-local

map-updateAt              ↦  map-updateAt-local

insert-lookup             ↦  insertAt-lookup
insert-punchIn            ↦  insertAt-punchIn
remove-punchOut           ↦  removeAt-punchOut
remove-insert             ↦  removeAt-insertAt
insert-remove             ↦  insertAt-removeAt

NB. This last one is complicated by the addition of a 'global' property map-updateAt

In Data.Vec.Recursive.Properties:
```
append-cons-commute  ↦  append-cons
```
In Data.Vec.Relation.Binary.Equality.Setoid:
```
map-++-commute ↦  map-++
```

In Function.Construct.Composition:

_∘-⟶_   ↦   _⟶-∘_
_∘-↣_   ↦   _↣-∘_
_∘-↠_   ↦   _↠-∘_
_∘-⤖_  ↦   _⤖-∘_
_∘-⇔_   ↦   _⇔-∘_
_∘-↩_   ↦   _↩-∘_
_∘-↪_   ↦   _↪-∘_
_∘-↔_   ↦   _↔-∘_

In Function.Construct.Identity:

id-⟶   ↦   ⟶-id
id-↣   ↦   ↣-id
id-↠   ↦   ↠-id
id-⤖  ↦   ⤖-id
id-⇔   ↦   ⇔-id
id-↩   ↦   ↩-id
id-↪   ↦   ↪-id
id-↔   ↦   ↔-id

In Function.Construct.Symmetry:

sym-⤖   ↦   ⤖-sym
sym-⇔   ↦   ⇔-sym
sym-↩   ↦   ↩-sym
sym-↪   ↦   ↪-sym
sym-↔   ↦   ↔-sym

In Foreign.Haskell.Either and Foreign.Haskell.Pair:

toForeign   ↦ Foreign.Haskell.Coerce.coerce
fromForeign ↦ Foreign.Haskell.Coerce.coerce

In Relation.Binary.Bundles.Preorder:
```
_∼_  ↦  _≲_
```
In Relation.Binary.Indexed.Heterogeneous.Bundles.Preorder:
```
_∼_  ↦  _≲_
```

In Relation.Binary.Properties.Preorder:

invIsPreorder ↦ converse-isPreorder
invPreorder   ↦ converse-preorder

In Relation.Nullary.Decidable.Core:

excluded-middle  ↦  ¬¬-excluded-middle

Renamed Data.Erased to Data.Irrelevant

This fixes the fact we had picked the wrong name originally. The erased modality corresponds to @0 whereas the irrelevance one corresponds to ..

New modules

Algebraic structures when freely adding an identity element:
```
Algebra.Construct.Add.Identity
```
Operations for module-like algebraic structures:
```
Algebra.Module.Core
```

Constructive algebraic structures with apartness relations:

Algebra.Apartness
Algebra.Apartness.Bundles
Algebra.Apartness.Structures
Algebra.Apartness.Properties.CommutativeHeytingAlgebra
Relation.Binary.Properties.ApartnessRelation

Algebraic structures obtained by flipping their binary operations:
```
Algebra.Construct.Flip.Op
```

Morphisms between module-like algebraic structures:

Algebra.Module.Morphism.Construct.Composition
Algebra.Module.Morphism.Construct.Identity
Algebra.Module.Morphism.Definitions
Algebra.Module.Morphism.Structures
Algebra.Module.Properties

Identity morphisms and composition of morphisms between algebraic structures:
```
Algebra.Morphism.Construct.Composition
Algebra.Morphism.Construct.Identity
```

Ordered algebraic structures (pomonoids, posemigroups, etc.)

Algebra.Ordered
Algebra.Ordered.Bundles
Algebra.Ordered.Structures

'Optimised' tail-recursive exponentiation properties:

Algebra.Properties.Semiring.Exp.TailRecursiveOptimised

A definition of infinite streams using coinductive records:

Codata.Guarded.Stream
Codata.Guarded.Stream.Properties
Codata.Guarded.Stream.Relation.Binary.Pointwise
Codata.Guarded.Stream.Relation.Unary.All
Codata.Guarded.Stream.Relation.Unary.Any

A small library for function arguments with default values:
```
Data.Default
```
A small library defining a structurally recursive view of Fin n:
```
Data.Fin.Relation.Unary.Top
```
A small library for a non-empty fresh list:
```
Data.List.Fresh.NonEmpty
```
A small library defining a structurally inductive view of lists:
```
Data.List.Relation.Unary.Sufficient
```

Combinations and permutations for ℕ.

Data.Nat.Combinatorics
Data.Nat.Combinatorics.Base
Data.Nat.Combinatorics.Spec

A small library defining parity and its algebra:

Data.Parity
Data.Parity.Base
Data.Parity.Instances
Data.Parity.Properties

New base module for Data.Product containing only the basic definitions.
```
Data.Product.Base
```
Reflection utilities for some specific types:
```
Data.List.Reflection
Data.Vec.Reflection
```
The All predicate over non-empty lists:
```
Data.List.NonEmpty.Relation.Unary.All
```
Show module for unnormalised rationals:
```
Data.Rational.Unnormalised.Show
```

Membership relations for maps and sets

Data.Tree.AVL.Map.Membership.Propositional
Data.Tree.AVL.Map.Membership.Propositional.Properties
Data.Tree.AVL.Sets.Membership
Data.Tree.AVL.Sets.Membership.Properties

Properties of bijections:

Function.Consequences
Function.Properties.Bijection
Function.Properties.RightInverse
Function.Properties.Surjection

In order to improve modularity, the contents of Relation.Binary.Lattice has been split out into the standard:
```
Relation.Binary.Lattice.Definitions
Relation.Binary.Lattice.Structures
Relation.Binary.Lattice.Bundles
```
All contents is re-exported by Relation.Binary.Lattice as before.
Algebraic properties of _∩_ and _∪_ for predicates
```
Relation.Unary.Algebra
```
Both versions of equality on predications are equivalences
```
Relation.Unary.Relation.Binary.Equality
```
The subset relations on predicates define an order
```
Relation.Unary.Relation.Binary.Subset
```

Polymorphic versions of some unary relations and their properties

Relation.Unary.Polymorphic
Relation.Unary.Polymorphic.Properties

Alpha equality over reflected terms
```
Reflection.AST.AlphaEquality
```
cong! tactic for deriving arguments to cong
```
Tactic.Cong
```

Various system types and primitives:

System.Clock.Primitive
System.Clock
System.Console.ANSI
System.Directory.Primitive
System.Directory
System.FilePath.Posix.Primitive
System.FilePath.Posix
System.Process.Primitive
System.Process

A golden testing library with test pools, an options parser, a runner:
```
Test.Golden
```
New interfaces for using Haskell datatypes:
```
Foreign.Haskell.List.NonEmpty
```

Added new module Algebra.Properties.RingWithoutOne:

-‿distribˡ-* : ∀ x y → - (x * y) ≈ - x * y
-‿distribʳ-* : ∀ x y → - (x * y) ≈ x * - y

An implementation of M-types with --guardedness flag:
```
Codata.Guarded.M
```

Port of Linked to Vec:

Data.Vec.Relation.Unary.Linked
Data.Vec.Relation.Unary.Linked.Properties

Added Logarithm base 2 on Natural Numbers:

Data.Nat.Logarithm.Core
Data.Nat.Logarithm

Proofs of some axioms of linearity:

Algebra.Module.Morphism.ModuleHomomorphism
Algebra.Module.Properties

Properties of MoufangLoop
```
Algebra.Properties.MoufangLoop
```
Properties of Quasigroup
```
Algebra.Properties.Quasigroup
```
Properties of MiddleBolLoop
```
Algebra.Properties.MiddleBolLoop
```
Properties of Loop
```
Algebra.Properties.Loop
```

Properties of (Commutative)Semiring: the Binomial Theorem

Algebra.Properties.CommutativeSemiring.Binomial
Algebra.Properties.Semiring.Binomial

Some n-ary functions manipulating lists
```
Data.List.Nary.NonDependent
```
Properties of KleeneAlgebra
```
Algebra.Properties.KleeneAlgebra
```
Relations on indexed sets
```
Function.Indexed.Bundles
```
Combinators for propositional equational reasoning on vectors with different indices
```
Data.Vec.Relation.Binary.Equality.Cast
```

Additions to existing modules

Added new proof to Data.Maybe.Properties
```
  <∣>-idem : Idempotent _<∣>_
```
The module Algebra now publicly re-exports the contents of Algebra.Structures.Biased.

Added new definitions to Algebra.Bundles:

record UnitalMagma c ℓ : Set (suc (c ⊔ ℓ))
record InvertibleMagma c ℓ : Set (suc (c ⊔ ℓ))
record InvertibleUnitalMagma c ℓ : Set (suc (c ⊔ ℓ))
record RawQuasiGroup c ℓ : Set (suc (c ⊔ ℓ))
record Quasigroup c ℓ : Set (suc (c ⊔ ℓ))
record RawLoop  c ℓ : Set (suc (c ⊔ ℓ))
record Loop  c ℓ : Set (suc (c ⊔ ℓ))
record RingWithoutOne c ℓ : Set (suc (c ⊔ ℓ))
record IdempotentSemiring c ℓ : Set (suc (c ⊔ ℓ))
record KleeneAlgebra c ℓ : Set (suc (c ⊔ ℓ))
record RawRingWithoutOne c ℓ : Set (suc (c ⊔ ℓ))
record Quasiring c ℓ : Set (suc (c ⊔ ℓ)) where
record Nearring c ℓ : Set (suc (c ⊔ ℓ)) where
record IdempotentMagma c ℓ : Set (suc (c ⊔ ℓ))
record AlternateMagma c ℓ : Set (suc (c ⊔ ℓ))
record FlexibleMagma c ℓ : Set (suc (c ⊔ ℓ))
record MedialMagma c ℓ : Set (suc (c ⊔ ℓ))
record SemimedialMagma c ℓ : Set (suc (c ⊔ ℓ))
record LeftBolLoop c ℓ : Set (suc (c ⊔ ℓ))
record RightBolLoop c ℓ : Set (suc (c ⊔ ℓ))
record MoufangLoop c ℓ : Set (suc (c ⊔ ℓ))
record NonAssociativeRing c ℓ : Set (suc (c ⊔ ℓ))
record MiddleBolLoop c ℓ : Set (suc (c ⊔ ℓ))

and the existing record Lattice now provides

∨-commutativeSemigroup : CommutativeSemigroup c ℓ
∧-commutativeSemigroup : CommutativeSemigroup c ℓ

and their corresponding algebraic subbundles.

Added new proofs to Algebra.Consequences.Base:

reflexive+selfInverse⇒involutive : Reflexive _≈_ →
                                   SelfInverse _≈_ f →
                                   Involutive _≈_ f

Added new proofs to Algebra.Consequences.Propositional:

comm+assoc⇒middleFour     : Commutative _•_ →
                            Associative _•_ →
                            _•_ MiddleFourExchange _•_
identity+middleFour⇒assoc : Identity e _•_ →
                            _•_ MiddleFourExchange _•_ →
                            Associative _•_
identity+middleFour⇒comm  : Identity e _+_ →
                            _•_ MiddleFourExchange _+_ →
                            Commutative _•_

Added new proofs to Algebra.Consequences.Setoid:

comm+assoc⇒middleFour     : Congruent₂ _•_ → Commutative _•_ → Associative _•_ →
                            _•_ MiddleFourExchange _•_
identity+middleFour⇒assoc : Congruent₂ _•_ → Identity e _•_ →
                            _•_ MiddleFourExchange _•_ →
                            Associative _•_
identity+middleFour⇒comm  : Congruent₂ _•_ → Identity e _+_ →
                            _•_ MiddleFourExchange _+_ →
                            Commutative _•_

involutive⇒surjective  : Involutive f  → Surjective f
selfInverse⇒involutive : SelfInverse f → Involutive f
selfInverse⇒congruent  : SelfInverse f → Congruent f
selfInverse⇒inverseᵇ   : SelfInverse f → Inverseᵇ f f
selfInverse⇒surjective : SelfInverse f → Surjective f
selfInverse⇒injective  : SelfInverse f → Injective f
selfInverse⇒bijective  : SelfInverse f → Bijective f

comm+idˡ⇒id              : Commutative _•_ → LeftIdentity  e _•_ → Identity e _•_
comm+idʳ⇒id              : Commutative _•_ → RightIdentity e _•_ → Identity e _•_
comm+zeˡ⇒ze              : Commutative _•_ → LeftZero      e _•_ → Zero     e _•_
comm+zeʳ⇒ze              : Commutative _•_ → RightZero     e _•_ → Zero     e _•_
comm+invˡ⇒inv            : Commutative _•_ → LeftInverse  e _⁻¹ _•_ → Inverse e _⁻¹ _•_
comm+invʳ⇒inv            : Commutative _•_ → RightInverse e _⁻¹ _•_ → Inverse e _⁻¹ _•_
comm+distrˡ⇒distr        : Commutative _•_ → _•_ DistributesOverˡ _◦_ → _•_ DistributesOver _◦_
comm+distrʳ⇒distr        : Commutative _•_ → _•_ DistributesOverʳ _◦_ → _•_ DistributesOver _◦_
distrib+absorbs⇒distribˡ : Associative _•_ → Commutative _◦_ → _•_ Absorbs _◦_ → _◦_ Absorbs _•_ → _◦_ DistributesOver _•_ → _•_ DistributesOverˡ _◦_

Added new functions to Algebra.Construct.DirectProduct:

rawSemiring : RawSemiring a ℓ₁ → RawSemiring b ℓ₂ → RawSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
rawRing : RawRing a ℓ₁ → RawRing b ℓ₂ → RawRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
semiringWithoutAnnihilatingZero : SemiringWithoutAnnihilatingZero a ℓ₁ →
                                  SemiringWithoutAnnihilatingZero b ℓ₂ →
                                  SemiringWithoutAnnihilatingZero (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
semiring : Semiring a ℓ₁ → Semiring b ℓ₂ → Semiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
commutativeSemiring : CommutativeSemiring a ℓ₁ → CommutativeSemiring b ℓ₂ →
                      CommutativeSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
ring : Ring a ℓ₁ → Ring b ℓ₂ → Ring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
commutativeRing : CommutativeRing a ℓ₁ → CommutativeRing b ℓ₂ →
                  CommutativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
rawQuasigroup : RawQuasigroup a ℓ₁ → RawQuasigroup b ℓ₂ →
                RawQuasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
rawLoop : RawLoop a ℓ₁ → RawLoop b ℓ₂ → RawLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
unitalMagma : UnitalMagma a ℓ₁ → UnitalMagma b ℓ₂ →
              UnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
invertibleMagma : InvertibleMagma a ℓ₁ → InvertibleMagma b ℓ₂ →
                  InvertibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
invertibleUnitalMagma : InvertibleUnitalMagma a ℓ₁ → InvertibleUnitalMagma b ℓ₂ →
                        InvertibleUnitalMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
quasigroup : Quasigroup a ℓ₁ → Quasigroup b ℓ₂ → Quasigroup (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
loop : Loop a ℓ₁ → Loop b ℓ₂ → Loop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
idempotentSemiring : IdempotentSemiring a ℓ₁ → IdempotentSemiring b ℓ₂ →
                     IdempotentSemiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
idempotentMagma : IdempotentMagma a ℓ₁ → IdempotentMagma b ℓ₂ →
                  IdempotentMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
alternativeMagma : AlternativeMagma a ℓ₁ → AlternativeMagma b ℓ₂ →
                   AlternativeMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
flexibleMagma : FlexibleMagma a ℓ₁ → FlexibleMagma b ℓ₂ →
                FlexibleMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
medialMagma : MedialMagma a ℓ₁ → MedialMagma b ℓ₂ → MedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
semimedialMagma : SemimedialMagma a ℓ₁ → SemimedialMagma b ℓ₂ →
                  SemimedialMagma (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
kleeneAlgebra : KleeneAlgebra a ℓ₁ → KleeneAlgebra b ℓ₂ → KleeneAlgebra (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
leftBolLoop : LeftBolLoop a ℓ₁ → LeftBolLoop b ℓ₂ → LeftBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
rightBolLoop : RightBolLoop a ℓ₁ → RightBolLoop b ℓ₂ → RightBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
middleBolLoop : MiddleBolLoop a ℓ₁ → MiddleBolLoop b ℓ₂ → MiddleBolLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
moufangLoop : MoufangLoop a ℓ₁ → MoufangLoop b ℓ₂ → MoufangLoop (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
rawRingWithoutOne : RawRingWithoutOne a ℓ₁ → RawRingWithoutOne b ℓ₂ → RawRingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
ringWithoutOne : RingWithoutOne a ℓ₁ → RingWithoutOne b ℓ₂ → RingWithoutOne (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
nonAssociativeRing : NonAssociativeRing a ℓ₁ → NonAssociativeRing b ℓ₂ → NonAssociativeRing (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
quasiring : Quasiring a ℓ₁ → Quasiring b ℓ₂ → Quasiring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)
nearring : Nearring a ℓ₁ → Nearring b ℓ₂ → Nearring (a ⊔ b) (ℓ₁ ⊔ ℓ₂)


* Added new definition to `Algebra.Definitions`:
 ```agda
 _MiddleFourExchange_ : Op₂ A → Op₂ A → Set _

 SelfInverse : Op₁ A → Set _

 LeftDividesˡ  : Op₂ A → Op₂ A → Set _
 LeftDividesʳ  : Op₂ A → Op₂ A → Set _
 RightDividesˡ : Op₂ A → Op₂ A → Set _
 RightDividesʳ : Op₂ A → Op₂ A → Set _
 LeftDivides   : Op₂ A → Op₂ A → Set _
 RightDivides  : Op₂ A → Op₂ A → Set _

 LeftInvertible  e _∙_ x = ∃[ x⁻¹ ] (x⁻¹ ∙ x) ≈ e
 RightInvertible e _∙_ x = ∃[ x⁻¹ ] (x ∙ x⁻¹) ≈ e
 Invertible      e _∙_ x = ∃[ x⁻¹ ] ((x⁻¹ ∙ x) ≈ e) × ((x ∙ x⁻¹) ≈ e)
 StarRightExpansive e _+_ _∙_ _⁻* = ∀ x → (e + (x ∙ (x ⁻*))) ≈ (x ⁻*)
 StarLeftExpansive e _+_ _∙_ _⁻* = ∀ x →  (e + ((x ⁻*) ∙ x)) ≈ (x ⁻*)
 StarExpansive e _+_ _∙_ _* = (StarLeftExpansive e _+_ _∙_ _*) × (StarRightExpansive e _+_ _∙_ _*)
 StarLeftDestructive _+_ _∙_ _* = ∀ a b x → (b + (a ∙ x)) ≈ x → ((a *) ∙ b) ≈ x
 StarRightDestructive _+_ _∙_ _* = ∀ a b x → (b + (x ∙ a)) ≈ x → (b ∙ (a *)) ≈ x
 StarDestructive _+_ _∙_ _* = (StarLeftDestructive _+_ _∙_ _*) × (StarRightDestructive _+_ _∙_ _*)
 LeftAlternative _∙_ = ∀ x y  →  ((x ∙ x) ∙ y) ≈ (x ∙ (y ∙ y))
 RightAlternative _∙_ = ∀ x y → (x ∙ (y ∙ y)) ≈ ((x ∙ y) ∙ y)
 Alternative _∙_ = (LeftAlternative _∙_ ) × (RightAlternative _∙_)
 Flexible _∙_ = ∀ x y → ((x ∙ y) ∙ x) ≈ (x ∙ (y ∙ x))
 Medial _∙_ = ∀ x y u z → ((x ∙ y) ∙ (u ∙ z)) ≈ ((x ∙ u) ∙ (y ∙ z))
 LeftSemimedial _∙_ = ∀ x y z → ((x ∙ x) ∙ (y ∙ z)) ≈ ((x ∙ y) ∙ (x ∙ z))
 RightSemimedial _∙_ = ∀ x y z → ((y ∙ z) ∙ (x ∙ x)) ≈ ((y ∙ x) ∙ (z ∙ x))
 Semimedial _∙_ = (LeftSemimedial _∙_) × (RightSemimedial _∙_)
 LeftBol _∙_ = ∀ x y z → (x ∙ (y ∙ (x ∙ z))) ≈ ((x ∙ (y ∙ z)) ∙ z )
 RightBol _∙_ = ∀ x y z → (((z ∙ x) ∙ y) ∙ x) ≈ (z ∙ ((x ∙ y) ∙ x))
 MiddleBol _∙_ _\\_ _//_ = ∀ x y z → (x ∙ ((y ∙ z) \\ x)) ≈ ((x // z) ∙ (y \\ x))

Added new functions to Algebra.Definitions.RawSemiring:

_^[_]*_ : A → ℕ → A → A
_^ᵗ_     : A → ℕ → A

Algebra.Properties.Magma.Divisibility now re-exports operations _∣ˡ_, _∤ˡ_, _∣ʳ_, _∤ʳ_ from Algebra.Definitions.Magma.

Added new proofs to Algebra.Properties.CommutativeSemigroup:

interchange : Interchangable _∙_ _∙_
xy∙xx≈x∙yxx : ∀ x y → (x ∙ y) ∙ (x ∙ x) ≈ x ∙ (y ∙ (x ∙ x))
leftSemimedial : LeftSemimedial _∙_
rightSemimedial : RightSemimedial _∙_
middleSemimedial : ∀ x y z → (x ∙ y) ∙ (z ∙ x) ≈ (x ∙ z) ∙ (y ∙ x)
semimedial : Semimedial _∙_

Added new proof to Algebra.Properties.Monoid.Mult:

×-congˡ : ∀ {x} → (_× x) Preserves _≡_ ⟶ _≈_

Added new proof to Algebra.Properties.Monoid.Sum:

sum-init-last : (t : Vector _ (suc n)) → sum t ≈ sum (init t) + last t

Added new proofs to Algebra.Properties.Semigroup:

leftAlternative : LeftAlternative _∙_
rightAlternative : RightAlternative _∙_
alternative : Alternative _∙_
flexible : Flexible _∙_

Added new proofs to Algebra.Properties.Semiring.Exp:

^-congʳ               : (x ^_) Preserves _≡_ ⟶ _≈_
y*x^m*y^n≈x^m*y^[n+1] : (x * y ≈ y * x) → y * (x ^ m * y ^ n) ≈ x ^ m * y ^ suc n

Added new proofs to Algebra.Properties.Semiring.Mult:

1×-identityʳ : 1 × x ≈ x
×-comm-*    : x * (n × y) ≈ n × (x * y)
×-assoc-*   : (n × x) * y ≈ n × (x * y)

Added new proofs to Algebra.Properties.Ring:

-1*x≈-x : ∀ x → - 1# * x ≈ - x
x+x≈x⇒x≈0 : ∀ x → x + x ≈ x → x ≈ 0#
x[y-z]≈xy-xz : ∀ x y z → x * (y - z) ≈ x * y - x * z
[y-z]x≈yx-zx : ∀ x y z → (y - z) * x ≈ (y * x) - (z * x)

Added new definitions to Algebra.Structures:

record IsUnitalMagma (_∙_ : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
record IsInvertibleMagma (_∙_ : Op₂ A) (ε : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
record IsInvertibleUnitalMagma (_∙_ : Op₂ A) (ε : A) (⁻¹ : Op₁ A) : Set (a ⊔ ℓ)
record IsQuasigroup (∙ \\ // : Op₂ A) : Set (a ⊔ ℓ)
record IsLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
record IsRingWithoutOne (+ * : Op₂ A) (-_ : Op₁ A) (0# : A) : Set (a ⊔ ℓ)
record IsIdempotentSemiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ)
record IsKleeneAlgebra (+ * : Op₂ A) (⋆ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
record IsQuasiring (+ * : Op₂ A) (0# 1# : A) : Set (a ⊔ ℓ) where
record IsNearring (+ * : Op₂ A) (0# 1# : A) (_⁻¹ : Op₁ A) : Set (a ⊔ ℓ) where
record IsIdempotentMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
record IsAlternativeMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
record IsFlexibleMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
record IsMedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
record IsSemimedialMagma (∙ : Op₂ A) : Set (a ⊔ ℓ)
record IsLeftBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
record IsRightBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
record IsMoufangLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)
record IsNonAssociativeRing (+ * : Op₂ A) (-_ : Op₁ A) (0# 1# : A) : Set (a ⊔ ℓ)
record IsMiddleBolLoop (∙ \\ // : Op₂ A) (ε : A) : Set (a ⊔ ℓ)

and the existing record IsLattice now provides

∨-isCommutativeSemigroup : IsCommutativeSemigroup ∨
∧-isCommutativeSemigroup : IsCommutativeSemigroup ∧

and their corresponding algebraic substructures.

Added new records to Algebra.Morphism.Structures:

record IsQuasigroupHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsQuasigroupMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsQuasigroupIsomorphism  (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
record IsLoopHomomorphism       (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsLoopMonomorphism       (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsLoopIsomorphism        (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
record IsRingWithoutOneHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsRingWithoutOneMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsRingWithoutOneIsoMorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)
record IsKleeneAlgebraHomomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsKleeneAlgebraMonomorphism (⟦_⟧ : A → B) : Set (a ⊔ ℓ₁ ⊔ ℓ₂)
record IsKleeneAlgebraIsomorphism (⟦_⟧ : A → B) : Set (a ⊔ b ⊔ ℓ₁ ⊔ ℓ₂)

Added new proofs in Data.Bool.Properties:

<-wellFounded : WellFounded _<_
∨-conicalˡ : LeftConical false _∨_
∨-conicalʳ : RightConical false _∨_
∨-conical  : Conical false _∨_
∧-conicalˡ : LeftConical true _∧_
∧-conicalʳ : RightConical true _∧_
∧-conical  : Conical true _∧_

true-xor            : true xor x ≡ not x
xor-same            : x xor x ≡ false
not-distribˡ-xor    : not (x xor y) ≡ (not x) xor y
not-distribʳ-xor    : not (x xor y) ≡ x xor (not y)
xor-assoc           : Associative _xor_
xor-comm            : Commutative _xor_
xor-identityˡ       : LeftIdentity false _xor_
xor-identityʳ       : RightIdentity false _xor_
xor-identity        : Identity false _xor_
xor-inverseˡ        : LeftInverse true not _xor_
xor-inverseʳ        : RightInverse true not _xor_
xor-inverse         : Inverse true not _xor_
∧-distribˡ-xor      : _∧_ DistributesOverˡ _xor_
∧-distribʳ-xor      : _∧_ DistributesOverʳ _xor_
∧-distrib-xor       : _∧_ DistributesOver _xor_
xor-annihilates-not : (not x) xor (not y) ≡ x xor y

Added new functions in Data.Fin.Base:

finToFun  : Fin (m ^ n) → (Fin n → Fin m)
funToFin  : (Fin m → Fin n) → Fin (n ^ m)
quotient  : Fin (m * n) → Fin m
remainder : Fin (m * n) → Fin n

Added new proofs in Data.Fin.Induction: every (strict) partial order is well-founded and Noetherian.

spo-wellFounded : ∀ {r} {_⊏_ : Rel (Fin n) r} → IsStrictPartialOrder _≈_ _⊏_ → WellFounded _⊏_
spo-noetherian  : ∀ {r} {_⊏_ : Rel (Fin n) r} → IsStrictPartialOrder _≈_ _⊏_ → WellFounded (flip _⊏_)

Added new definitions and proofs in Data.Fin.Permutation:

insert         : Fin (suc m) → Fin (suc n) → Permutation m n → Permutation (suc m) (suc n)
insert-punchIn : insert i j π ⟨$⟩ʳ punchIn i k ≡ punchIn j (π ⟨$⟩ʳ k)
insert-remove  : insert i (π ⟨$⟩ʳ i) (remove i π) ≈ π
remove-insert  : remove i (insert i j π) ≈ π

In Data.Fin.Properties: the proof that an injection from a type A into Fin n induces a decision procedure for _≡_ on A has been generalized to other equivalences over A (i.e. to arbitrary setoids), and renamed from eq? to the more descriptive inj⇒≟ and inj⇒decSetoid.

Added new proofs in Data.Fin.Properties:

1↔⊤                : Fin 1 ↔ ⊤

0≢1+n              : zero ≢ suc i

↑ˡ-injective       : i ↑ˡ n ≡ j ↑ˡ n → i ≡ j
↑ʳ-injective       : n ↑ʳ i ≡ n ↑ʳ j → i ≡ j
finTofun-funToFin  : funToFin ∘ finToFun ≗ id
funTofin-funToFun  : finToFun (funToFin f) ≗ f
^↔→                : Extensionality _ _ → Fin (m ^ n) ↔ (Fin n → Fin m)

toℕ-mono-<         : i < j → toℕ i ℕ.< toℕ j
toℕ-mono-≤         : i ≤ j → toℕ i ℕ.≤ toℕ j
toℕ-cancel-≤       : toℕ i ℕ.≤ toℕ j → i ≤ j
toℕ-cancel-<       : toℕ i ℕ.< toℕ j → i < j

splitAt⁻¹-↑ˡ       : splitAt m {n} i ≡ inj₁ j → j ↑ˡ n ≡ i
splitAt⁻¹-↑ʳ       : splitAt m {n} i ≡ inj₂ j → m ↑ʳ j ≡ i

toℕ-combine        : toℕ (combine i j) ≡ k ℕ.* toℕ i ℕ.+ toℕ j
combine-injectiveˡ : combine i j ≡ combine k l → i ≡ k
combine-injectiveʳ : combine i j ≡ combine k l → j ≡ l
combine-injective  : combine i j ≡ combine k l → i ≡ k × j ≡ l
combine-surjective : ∀ i → ∃₂ λ j k → combine j k ≡ i
combine-monoˡ-<    : i < j → combine i k < combine j l

ℕ-ℕ≡toℕ‿ℕ-         : n ℕ-ℕ i ≡ toℕ (n ℕ- i)

lower₁-injective   : lower₁ i n≢i ≡ lower₁ j n≢j → i ≡ j
pinch-injective    : suc i ≢ j → suc i ≢ k → pinch i j ≡ pinch i k → j ≡ k

i<1+i              : i < suc i

injective⇒≤               : ∀ {f : Fin m → Fin n} → Injective f → m ℕ.≤ n
<⇒notInjective            : ∀ {f : Fin m → Fin n} → n ℕ.< m → ¬ (Injective f)
ℕ→Fin-notInjective        : ∀ (f : ℕ → Fin n) → ¬ (Injective f)
cantor-schröder-bernstein : ∀ {f : Fin m → Fin n} {g : Fin n → Fin m} → Injective f → Injective g → m ≡ n

cast-is-id    : cast eq k ≡ k
subst-is-cast : subst Fin eq k ≡ cast eq k
cast-trans    : cast eq₂ (cast eq₁ k) ≡ cast (trans eq₁ eq₂) k

fromℕ≢inject₁      : {i : Fin n} → fromℕ n ≢ inject₁ i

inject≤-trans      : inject≤ (inject≤ i m≤n) n≤o ≡ inject≤ i (≤-trans m≤n n≤o)
inject≤-irrelevant : inject≤ i m≤n ≡ inject≤ i m≤n′

Changed the fixity of Data.Fin.Substitution.TermSubst._/Var_.
```
infix 8 ↦ infixl 8
```

Added new lemmas in Data.Fin.Substitution.Lemmas.TermLemmas:

map-var≡ : {ρ₁ : Sub Fin m n} {ρ₂ : Sub T m n} {f : Fin m → Fin n} →
           (∀ x → lookup ρ₁ x ≡ f x) →
           (∀ x → lookup ρ₂ x ≡ T.var (f x)) →
           map T.var ρ₁ ≡ ρ₂
wk≡wk : map T.var VarSubst.wk ≡ T.wk {n = n}
id≡id : map T.var VarSubst.id ≡ T.id {n = n}
sub≡sub : {x : Fin n} → map T.var (VarSubst.sub x) ≡ T.sub (T.var x)
↑≡↑ : {ρ : Sub Fin m n} → map T.var (ρ VarSubst.↑) ≡ map T.var ρ T.↑
/Var≡/ : {ρ : Sub Fin m n} {t} → t /Var ρ ≡ t T./ map T.var ρ
sub-renaming-commutes : {ρ : Sub T m n} →
  t /Var VarSubst.sub x T./ ρ ≡ t T./ ρ T.↑ T./ T.sub (lookup ρ x)
sub-commutes-with-renaming : {ρ : Sub Fin m n} →
  t T./ T.sub t′ /Var ρ ≡ t /Var ρ VarSubst.↑ T./ T.sub (t′ /Var ρ)

Added new functions in Data.Integer.Base:

_^_ : ℤ → ℕ → ℤ

+-0-rawGroup  : Rawgroup 0ℓ 0ℓ

*-rawMagma    : RawMagma 0ℓ 0ℓ
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ


* Added new proofs in `Data.Integer.Properties`:
 ```agda
 sign-cong′ : s₁ ◃ n₁ ≡ s₂ ◃ n₂ → s₁ ≡ s₂ ⊎ (n₁ ≡ 0 × n₂ ≡ 0)
 ≤-⊖        : m ℕ.≤ n → n ⊖ m ≡ + (n ∸ m)
 ∣⊖∣-≤      : m ℕ.≤ n → ∣ m ⊖ n ∣ ≡ n ∸ m
 ∣-∣-≤      : i ≤ j → + ∣ i - j ∣ ≡ j - i

 i^n≡0⇒i≡0      : i ^ n ≡ 0ℤ → i ≡ 0ℤ
 ^-identityʳ    : i ^ 1 ≡ i
 ^-zeroˡ        : 1 ^ n ≡ 1
 ^-*-assoc      : (i ^ m) ^ n ≡ i ^ (m ℕ.* n)
 ^-distribˡ-+-* : i ^ (m ℕ.+ n) ≡ i ^ m * i ^ n

 ^-isMagmaHomomorphism  : IsMagmaHomomorphism  ℕ.+-rawMagma    *-rawMagma    (i ^_)
 ^-isMonoidHomomorphism : IsMonoidHomomorphism ℕ.+-0-rawMonoid *-1-rawMonoid (i ^_)

Added new proofs in Data.Integer.GCD:

gcd-assoc : Associative gcd
gcd-zeroˡ : LeftZero 1ℤ gcd
gcd-zeroʳ : RightZero 1ℤ gcd
gcd-zero  : Zero 1ℤ gcd

Added new functions and definitions to Data.List.Base:

takeWhileᵇ   : (A → Bool) → List A → List A
dropWhileᵇ   : (A → Bool) → List A → List A
filterᵇ      : (A → Bool) → List A → List A
partitionᵇ   : (A → Bool) → List A → List A × List A
spanᵇ        : (A → Bool) → List A → List A × List A
breakᵇ       : (A → Bool) → List A → List A × List A
linesByᵇ     : (A → Bool) → List A → List (List A)
wordsByᵇ     : (A → Bool) → List A → List (List A)
derunᵇ       : (A → A → Bool) → List A → List A
deduplicateᵇ : (A → A → Bool) → List A → List A

findᵇ        : (A → Bool) → List A -> Maybe A
findIndexᵇ   : (A → Bool) → (xs : List A) → Maybe $ Fin (length xs)
findIndicesᵇ : (A → Bool) → (xs : List A) → List $ Fin (length xs)
find         : Decidable P → List A → Maybe A
findIndex    : Decidable P → (xs : List A) → Maybe $ Fin (length xs)
findIndices  : Decidable P → (xs : List A) → List $ Fin (length xs)

catMaybes       : List (Maybe A) → List A
ap              : List (A → B) → List A → List B
++-rawMagma     : Set a → RawMagma a _
++-[]-rawMonoid : Set a → RawMonoid a _

iterate : (A → A) → A → ℕ → List A
insertAt : (xs : List A) → Fin (suc (length xs)) → A → List A
updateAt : (xs : List A) → Fin (length xs) → (A → A) → List A

Added new proofs in Data.List.Relation.Binary.Lex.Strict:
```
xs≮[] : ¬ xs < []
```

Added new proofs to Data.List.Relation.Binary.Permutation.Propositional.Properties:

Any-resp-[σ⁻¹∘σ] : (σ : xs ↭ ys) → (ix : Any P xs) → Any-resp-↭ (trans σ (↭-sym σ)) ix ≡ ix
∈-resp-[σ⁻¹∘σ]   : (σ : xs ↭ ys) → (ix : x ∈ xs)   → ∈-resp-↭   (trans σ (↭-sym σ)) ix ≡ ix

Added new functions in Data.List.Relation.Unary.All:

decide :  Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]

Added new functions in Data.List.Fresh.Relation.Unary.All:

decide :  Π[ P ∪ Q ] → Π[ All {R = R} P ∪ Any Q ]

Added new proofs to Data.List.Membership.Propositional.Properties:

mapWith∈-id  : mapWith∈ xs (λ {x} _ → x) ≡ xs
map-mapWith∈ : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)

Added new proofs to Data.List.Membership.Setoid.Properties:

mapWith∈-id  : mapWith∈ xs (λ {x} _ → x) ≡ xs
map-mapWith∈ : map g (mapWith∈ xs f) ≡ mapWith∈ xs (g ∘′ f)
index-injective : index x₁∈xs ≡ index x₂∈xs → x₁ ≈ x₂

Add new proofs in Data.List.Properties:

∈⇒∣product : n ∈ ns → n ∣ product ns
∷ʳ-++ : xs ∷ʳ a ++ ys ≡ xs ++ a ∷ ys

concatMap-cong : f ≗ g → concatMap f ≗ concatMap g
concatMap-pure : concatMap [_] ≗ id
concatMap-map  : concatMap g (map f xs) ≡ concatMap (g ∘′ f) xs
map-concatMap  : map f ∘′ concatMap g ≗ concatMap (map f ∘′ g)

length-isMagmaHomomorphism  : (A : Set a) → IsMagmaHomomorphism (++-rawMagma A) +-rawMagma length
length-isMonoidHomomorphism : (A : Set a) → IsMonoidHomomorphism (++-[]-rawMonoid A) +-0-rawMonoid length

take-map : take n (map f xs) ≡ map f (take n xs)
drop-map : drop n (map f xs) ≡ map f (drop n xs)
head-map : head (map f xs) ≡ Maybe.map f (head xs)

take-suc               : take (suc m) xs ≡ take m xs ∷ʳ lookup xs i
take-suc-tabulate      : take (suc m) (tabulate f) ≡ take m (tabulate f) ∷ʳ f i
drop-take-suc          : drop m (take (suc m) xs) ≡ [ lookup xs i ]
drop-take-suc-tabulate : drop m (take (suc m) (tabulate f)) ≡ [ f i ]

take-all : n ≥ length xs → take n xs ≡ xs
drop-all : n ≥ length xs → drop n xs ≡ []

take-[] : take m [] ≡ []
drop-[] : drop m [] ≡ []

drop-drop : drop n (drop m xs) ≡ drop (m + n) xs

lookup-replicate  : lookup (replicate n x) i ≡ x
map-replicate     : map f (replicate n x) ≡ replicate n (f x)
zipWith-replicate : zipWith _⊕_ (replicate n x) (replicate n y) ≡ replicate n (x ⊕ y)

length-iterate : length (iterate f x n) ≡ n
iterate-id     : iterate id x n ≡ replicate n x
lookup-iterate : lookup (iterate f x n) (cast (sym (length-iterate f x n)) i) ≡ ℕ.iterate f x (toℕ i)

length-insertAt   : length (insertAt xs i v) ≡ suc (length xs)
length-removeAt′  : length xs ≡ suc (length (removeAt xs k))
removeAt-insertAt : removeAt (insertAt xs i v) ((cast (sym (length-insertAt xs i v)) i)) ≡ xs
insertAt-removeAt : insertAt (removeAt xs i) (cast (sym (lengthAt-removeAt xs i)) i) (lookup xs i) ≡ xs

Added new patterns and definitions to Data.Nat.Base:

pattern z<s {n}         = s≤s (z≤n {n})
pattern s<s {m} {n} m<n = s≤s {m} {n} m<n

pattern <′-base          = ≤′-refl
pattern <′-step {n} m<′n = ≤′-step {n} m<′n

_⊔′_ : ℕ → ℕ → ℕ
_⊓′_ : ℕ → ℕ → ℕ
∣_-_∣′ : ℕ → ℕ → ℕ
_! : ℕ → ℕ

parity : ℕ → Parity

+-rawMagma          : RawMagma 0ℓ 0ℓ
+-0-rawMonoid       : RawMonoid 0ℓ 0ℓ
*-rawMagma          : RawMagma 0ℓ 0ℓ
*-1-rawMonoid       : RawMonoid 0ℓ 0ℓ
+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
+-*-rawSemiring     : RawSemiring 0ℓ 0ℓ

Added a new proof to Data.Nat.Binary.Properties:

suc-injective : Injective _≡_ _≡_ suc
toℕ-inverseˡ  : Inverseˡ _≡_ _≡_ toℕ fromℕ
toℕ-inverseʳ  : Inverseʳ _≡_ _≡_ toℕ fromℕ
toℕ-inverseᵇ  : Inverseᵇ _≡_ _≡_ toℕ fromℕ

Added a new pattern synonym to Data.Nat.Divisibility.Core:
```
pattern divides-refl q = divides q refl
```

Added new definitions and proofs to Data.Nat.Primality:

Composite        : ℕ → Set
composite?       : Decidable composite
composite⇒¬prime : Composite n → ¬ Prime n
¬composite⇒prime : 2 ≤ n → ¬ Composite n → Prime n
prime⇒¬composite : Prime n → ¬ Composite n
¬prime⇒composite : 2 ≤ n → ¬ Prime n → Composite n
euclidsLemma     : Prime p → p ∣ m * n → p ∣ m ⊎ p ∣ n

Added new proofs in Data.Nat.Properties:

nonZero?  : Decidable NonZero
n≮0       : n ≮ 0
n+1+m≢m   : n + suc m ≢ m
m*n≡0⇒m≡0 : .{{_ : NonZero n}} → m * n ≡ 0 → m ≡ 0
n>0⇒n≢0   : n > 0 → n ≢ 0
m^n≢0     : .{{_ : NonZero m}} → NonZero (m ^ n)
m*n≢0     : .{{_ : NonZero m}} .{{_ : NonZero n}} → NonZero (m * n)
m≤n⇒n∸m≤n : m ≤ n → n ∸ m ≤ n

≤-pred        : suc m ≤ suc n → m ≤ n
s<s-injective : ∀ {p q : m < n} → s<s p ≡ s<s q → p ≡ q
<-pred        : suc m < suc n → m < n
<-step        : m < n → m < 1 + n
m<1+n⇒m<n∨m≡n : m < suc n → m < n ⊎ m ≡ n

z<′s : zero <′ suc n
s<′s : m <′ n → suc m <′ suc n
<⇒<′ : m < n → m <′ n
<′⇒< : m <′ n → m < n

≤″-proof : (le : m ≤″ n) → let less-than-or-equal {k} _ = le in m + k ≡ n

m+n≤p⇒m≤p∸n         : m + n ≤ p → m ≤ p ∸ n
m≤p∸n⇒m+n≤p         : n ≤ p → m ≤ p ∸ n → m + n ≤ p

1≤n!    : 1 ≤ n !
_!≢0    : NonZero (n !)
_!*_!≢0 : NonZero (m ! * n !)

anyUpTo? : ∀ (P? : U.Decidable P) (v : ℕ) → Dec (∃ λ n → n < v × P n)
allUpTo? : ∀ (P? : U.Decidable P) (v : ℕ) → Dec (∀ {n} → n < v → P n)

n≤1⇒n≡0∨n≡1 : n ≤ 1 → n ≡ 0 ⊎ n ≡ 1

m^n>0 : ∀ m .{{_ : NonZero m}} n → m ^ n > 0

^-monoˡ-≤ : ∀ n → (_^ n) Preserves _≤_ ⟶ _≤_
^-monoʳ-≤ : ∀ m .{{_ : NonZero m}} → (m ^_) Preserves _≤_ ⟶ _≤_
^-monoˡ-< : ∀ n .{{_ : NonZero n}} → (_^ n) Preserves _<_ ⟶ _<_
^-monoʳ-< : ∀ m → 1 < m → (m ^_) Preserves _<_ ⟶ _<_

n≡⌊n+n/2⌋ : n ≡ ⌊ n + n /2⌋
n≡⌈n+n/2⌉ : n ≡ ⌈ n + n /2⌉

m<n⇒m<n*o : .{{_ : NonZero o}} → m < n → m < n * o
m<n⇒m<o*n : .{{_ : NonZero o}} → m < n → m < o * n
∸-monoˡ-< : m < o → n ≤ m → m ∸ n < o ∸ n

m≤n⇒∣m-n∣≡n∸m : m ≤ n → ∣ m - n ∣ ≡ n ∸ m

⊔≡⊔′ : m ⊔ n ≡ m ⊔′ n
⊓≡⊓′ : m ⊓ n ≡ m ⊓′ n
∣-∣≡∣-∣′ : ∣ m - n ∣ ≡ ∣ m - n ∣′

Re-exported additional functions from Data.Nat:

nonZero? : Decidable NonZero
eq? : A ↣ ℕ → DecidableEquality A
≤-<-connex : Connex _≤_ _<_
≥->-connex : Connex _≥_ _>_
<-≤-connex : Connex _<_ _≤_
>-≥-connex : Connex _>_ _≥_
<-cmp : Trichotomous _≡_ _<_
anyUpTo? : (P? : Decidable P) → ∀ v → Dec (∃ λ n → n < v × P n)
allUpTo? : (P? : Decidable P) → ∀ v → Dec (∀ {n} → n < v → P n)

Added new proofs in Data.Nat.Combinatorics:

[n-k]*[n-k-1]!≡[n-k]!   : k < n → (n ∸ k) * (n ∸ suc k) ! ≡ (n ∸ k) !
[n-k]*d[k+1]≡[k+1]*d[k] : k < n → (n ∸ k) * ((suc k) ! * (n ∸ suc k) !) ≡ (suc k) * (k ! * (n ∸ k) !)
k![n∸k]!∣n!              : k ≤ n → k ! * (n ∸ k) ! ∣ n !
nP1≡n                   : n P 1 ≡ n
nC1≡n                   : n C 1 ≡ n
nCk+nC[k+1]≡[n+1]C[k+1] : n C k + n C (suc k) ≡ suc n C suc k

Added new proofs in Data.Nat.DivMod:

m%n≤n           : .{{_ : NonZero n}} → m % n ≤ n
m*n/m!≡n/[m∸1]! : .{{_ : NonZero m}} → m * n / m ! ≡ n / (pred m) !

%-congˡ             : .⦃ _ : NonZero o ⦄ → m ≡ n → m % o ≡ n % o
%-congʳ             : .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ≡ n → o % m ≡ o % n
m≤n⇒[n∸m]%m≡n%m     : .⦃ _ : NonZero m ⦄ → m ≤ n → (n ∸ m) % m ≡ n % m
m*n≤o⇒[o∸m*n]%n≡o%n : .⦃ _ : NonZero n ⦄ → m * n ≤ o → (o ∸ m * n) % n ≡ o % n
m∣n⇒o%n%m≡o%m       : .⦃ _ : NonZero m ⦄ .⦃ _ : NonZero n ⦄ → m ∣ n → o % n % m ≡ o % m
m<n⇒m%n≡m           : .⦃ _ : NonZero n ⦄ → m < n → m % n ≡ m
m*n/o*n≡m/o         : .⦃ _ : NonZero o ⦄ ⦃ _ : NonZero (o * n) ⦄ → m * n / (o * n) ≡ m / o
m<n*o⇒m/o<n         : .⦃ _ : NonZero o ⦄ → m < n * o → m / o < n
[m∸n]/n≡m/n∸1       : .⦃ _ : NonZero n ⦄ → (m ∸ n) / n ≡ pred (m / n)
[m∸n*o]/o≡m/o∸n     : .⦃ _ : NonZero o ⦄ → (m ∸ n * o) / o ≡ m / o ∸ n
m/n/o≡m/[n*o]       : .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄ .⦃ _ : NonZero (n * o) ⦄ → m / n / o ≡ m / (n * o)
m%[n*o]/o≡m/o%n     : .⦃ _ : NonZero n ⦄ .⦃ _ : NonZero o ⦄ ⦃ _ : NonZero (n * o) ⦄ → m % (n * o) / o ≡ m / o % n
m%n*o≡m*o%[n*o]     : .⦃ _ : NonZero n ⦄ ⦃ _ : NonZero (n * o) ⦄ → m % n * o ≡ m * o % (n * o)

[m*n+o]%[p*n]≡[m*n]%[p*n]+o : ⦃ _ : NonZero (p * n) ⦄ → o < n → (m * n + o) % (p * n) ≡ (m * n) % (p * n) + o

Added new proofs in Data.Nat.Divisibility:

n∣m*n*o       : n ∣ m * n * o
m*n∣⇒m∣       : m * n ∣ i → m ∣ i
m*n∣⇒n∣       : m * n ∣ i → n ∣ i
m≤n⇒m!∣n!     : m ≤ n → m ! ∣ n !
m/n/o≡m/[n*o] : .{{NonZero n}} .{{NonZero o}} → n * o ∣ m → (m / n) / o ≡ m / (n * o)

Added new proofs in Data.Nat.GCD:

gcd-assoc     : Associative gcd
gcd-identityˡ : LeftIdentity 0 gcd
gcd-identityʳ : RightIdentity 0 gcd
gcd-identity  : Identity 0 gcd
gcd-zeroˡ     : LeftZero 1 gcd
gcd-zeroʳ     : RightZero 1 gcd
gcd-zero      : Zero 1 gcd

Added new patterns in Data.Nat.Reflection:

pattern `ℕ     = def (quote ℕ) []
pattern `zero  = con (quote ℕ.zero) []
pattern `suc x = con (quote ℕ.suc) (x ⟨∷⟩ [])

Added new proofs in Data.Parity.Properties:

suc-homo-⁻¹ : (parity (suc n)) ⁻¹ ≡ parity n
+-homo-+    : parity (m ℕ.+ n) ≡ parity m ℙ.+ parity n
*-homo-*    : parity (m ℕ.* n) ≡ parity m ℙ.* parity n
parity-isMagmaHomomorphism : IsMagmaHomomorphism ℕ.+-rawMagma ℙ.+-rawMagma parity
parity-isMonoidHomomorphism : IsMonoidHomomorphism ℕ.+-0-rawMonoid ℙ.+-0-rawMonoid parity
parity-isNearSemiringHomomorphism : IsNearSemiringHomomorphism ℕ.+-*-rawNearSemiring ℙ.+-*-rawNearSemiring parity
parity-isSemiringHomomorphism : IsSemiringHomomorphism ℕ.+-*-rawSemiring ℙ.+-*-rawSemiring parity

Added new rounding functions in Data.Rational.Base:

floor ceiling truncate round ⌊_⌋ ⌈_⌉ [_] : ℚ → ℤ
fracPart : ℚ → ℚ

Added new definitions and proofs in Data.Rational.Properties:

↥ᵘ-toℚᵘ : ↥ᵘ (toℚᵘ p) ≡ ↥ p
↧ᵘ-toℚᵘ : ↧ᵘ (toℚᵘ p) ≡ ↧ p

_≥?_ : Decidable _≥_
_>?_ : Decidable _>_

+-*-rawNearSemiring                 : RawNearSemiring 0ℓ 0ℓ
+-*-rawSemiring                     : RawSemiring 0ℓ 0ℓ
toℚᵘ-isNearSemiringHomomorphism-+-* : IsNearSemiringHomomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ
toℚᵘ-isNearSemiringMonomorphism-+-* : IsNearSemiringMonomorphism +-*-rawNearSemiring ℚᵘ.+-*-rawNearSemiring toℚᵘ
toℚᵘ-isSemiringHomomorphism-+-*     : IsSemiringHomomorphism     +-*-rawSemiring     ℚᵘ.+-*-rawSemiring     toℚᵘ
toℚᵘ-isSemiringMonomorphism-+-*     : IsSemiringMonomorphism     +-*-rawSemiring     ℚᵘ.+-*-rawSemiring     toℚᵘ

pos⇒nonZero       : .{{Positive p}} → NonZero p
neg⇒nonZero       : .{{Negative p}} → NonZero p
nonZero⇒1/nonZero : .{{_ : NonZero p}} → NonZero (1/ p)

Added new rounding functions in Data.Rational.Unnormalised.Base:

floor ceiling truncate round ⌊_⌋ ⌈_⌉ [_] : ℚᵘ → ℤ
fracPart : ℚᵘ → ℚᵘ

Added new definitions in Data.Rational.Unnormalised.Properties:

+-*-rawNearSemiring : RawNearSemiring 0ℓ 0ℓ
+-*-rawSemiring     : RawSemiring 0ℓ 0ℓ

≰⇒≥ : _≰_ ⇒ _≥_

_≥?_ : Decidable _≥_
_>?_ : Decidable _>_

*-mono-≤-nonNeg   : .{{_ : NonNegative p}} .{{_ : NonNegative r}} → p ≤ q → r ≤ s → p * r ≤ q * s
*-mono-<-nonNeg   : .{{_ : NonNegative p}} .{{_ : NonNegative r}} → p < q → r < s → p * r < q * s
1/-antimono-≤-pos : .{{_ : Positive p}}    .{{_ : Positive q}}    → p ≤ q → 1/ q ≤ 1/ p
⊓-mono-<          : _⊓_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_
⊔-mono-<          : _⊔_ Preserves₂ _<_ ⟶ _<_ ⟶ _<_

pos⇒nonZero          : .{{_ : Positive p}} → NonZero p
neg⇒nonZero          : .{{_ : Negative p}} → NonZero p
pos+pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p + q)
nonNeg+nonNeg⇒nonNeg : .{{_ : NonNegative p}} .{{_ : NonNegative q}} → NonNegative (p + q)
pos*pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p * q)
nonNeg*nonNeg⇒nonNeg : .{{_ : NonNegative p}} .{{_ : NonNegative q}} → NonNegative (p * q)
pos⊓pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊓ q)
pos⊔pos⇒pos          : .{{_ : Positive p}}    .{{_ : Positive q}}    → Positive (p ⊔ q)
1/nonZero⇒nonZero    : .{{_ : NonZero p}} → NonZero (1/ p)

Added new functions to Data.Product.Nary.NonDependent:

zipWith : (∀ k → Projₙ as k → Projₙ bs k → Projₙ cs k) →
          Product n as → Product n bs → Product n cs

Added new proof to Data.Product.Properties:

map-cong : f ≗ g → h ≗ i → map f h ≗ map g i

Added new definitions to Data.Product.Properties and Function.Related.TypeIsomorphisms for convenience:

Σ-≡,≡→≡ : (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂) → p₁ ≡ p₂
Σ-≡,≡←≡ : p₁ ≡ p₂ → (∃ λ (p : proj₁ p₁ ≡ proj₁ p₂) → subst B p (proj₂ p₁) ≡ proj₂ p₂)
×-≡,≡→≡ : (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂) → p₁ ≡ p₂
×-≡,≡←≡ : p₁ ≡ p₂ → (proj₁ p₁ ≡ proj₁ p₂ × proj₂ p₁ ≡ proj₂ p₂)

Added new proofs to Data.Product.Relation.Binary.Lex.Strict

×-respectsʳ : Transitive _≈₁_ →
              _<₁_ Respectsʳ _≈₁_ → _<₂_ Respectsʳ _≈₂_ → _<ₗₑₓ_ Respectsʳ _≋_
×-respectsˡ : Symmetric _≈₁_ → Transitive _≈₁_ →
               _<₁_ Respectsˡ _≈₁_ → _<₂_ Respectsˡ _≈₂_ → _<ₗₑₓ_ Respectsˡ _≋_
×-wellFounded' : Transitive _≈₁_ → _<₁_ Respectsʳ _≈₁_ →
                 WellFounded _<₁_ → WellFounded _<₂_ → WellFounded _<ₗₑₓ_

Added new definitions to Data.Sign.Base:

*-rawMagma : RawMagma 0ℓ 0ℓ
*-1-rawMonoid : RawMonoid 0ℓ 0ℓ
*-1-rawGroup : RawGroup 0ℓ 0ℓ

Added new proofs to Data.Sign.Properties:

*-inverse : Inverse + id _*_
*-isCommutativeSemigroup : IsCommutativeSemigroup _*_
*-isCommutativeMonoid : IsCommutativeMonoid _*_ +
*-isGroup : IsGroup _*_ + id
*-isAbelianGroup : IsAbelianGroup _*_ + id
*-commutativeSemigroup : CommutativeSemigroup 0ℓ 0ℓ
*-commutativeMonoid : CommutativeMonoid 0ℓ 0ℓ
*-group : Group 0ℓ 0ℓ
*-abelianGroup : AbelianGroup 0ℓ 0ℓ
≡-isDecEquivalence : IsDecEquivalence _≡_

Added new functions in Data.String.Base:

wordsByᵇ : (Char → Bool) → String → List String
linesByᵇ : (Char → Bool) → String → List String

Added new proofs in Data.String.Properties:

≤-isDecTotalOrder-≈ : IsDecTotalOrder _≈_ _≤_
≤-decTotalOrder-≈   :  DecTotalOrder _ _ _

Added new definitions in Data.Sum.Properties:
```
swap-↔ : (A ⊎ B) ↔ (B ⊎ A)
```

Added new proofs in Data.Sum.Properties:

map-assocˡ : (f : A → C) (g : B → D) (h : C → F) →
  map (map f g) h ∘ assocˡ ≗ assocˡ ∘ map f (map g h)
map-assocʳ : (f : A → C) (g : B → D) (h : C → F) →
  map f (map g h) ∘ assocʳ ≗ assocʳ ∘ map (map f g) h

Made Map public in Data.Tree.AVL.IndexedMap

Added new definitions in Data.Vec.Base:

truncate : m ≤ n → Vec A n → Vec A m
pad      : m ≤ n → A → Vec A m → Vec A n

FoldrOp A B = ∀ {n} → A → B n → B (suc n)
FoldlOp A B = ∀ {n} → B n → A → B (suc n)

foldr′ : (A → B → B) → B → Vec A n → B
foldl′ : (B → A → B) → B → Vec A n → B
countᵇ : (A → Bool) → Vec A n → ℕ

iterate : (A → A) → A → Vec A n

diagonal           : Vec (Vec A n) n → Vec A n
DiagonalBind._>>=_ : Vec A n → (A → Vec B n) → Vec B n

_ʳ++_              : Vec A m → Vec A n → Vec A (m + n)

cast : .(eq : m ≡ n) → Vec A m → Vec A n

Added new instance in Data.Vec.Effectful:

monad : RawMonad (λ (A : Set a) → Vec A n)

Added new proofs in Data.Vec.Properties:

padRight-refl      : padRight ≤-refl a xs ≡ xs
padRight-replicate : replicate a ≡ padRight le a (replicate a)
padRight-trans     : padRight (≤-trans m≤n n≤p) a xs ≡ padRight n≤p a (padRight m≤n a xs)

truncate-refl     : truncate ≤-refl xs ≡ xs
truncate-trans    : truncate (≤-trans m≤n n≤p) xs ≡ truncate m≤n (truncate n≤p xs)
truncate-padRight : truncate m≤n (padRight m≤n a xs) ≡ xs

map-const    : map (const x) xs ≡ replicate x
map-⊛        : map f xs ⊛ map g xs ≡ map (f ˢ g) xs
map-++       : map f (xs ++ ys) ≡ map f xs ++ map f ys
map-is-foldr : map f ≗ foldr (Vec B) (λ x ys → f x ∷ ys) []
map-∷ʳ       : map f (xs ∷ʳ x) ≡ (map f xs) ∷ʳ (f x)
map-reverse  : map f (reverse xs) ≡ reverse (map f xs)
map-ʳ++      : map f (xs ʳ++ ys) ≡ map f xs ʳ++ map f ys
map-insert   : map f (insert xs i x) ≡ insert (map f xs) i (f x)
toList-map   : toList (map f xs) ≡ List.map f (toList xs)

lookup-concat : lookup (concat xss) (combine i j) ≡ lookup (lookup xss i) j

⊛-is->>=    : fs ⊛ xs ≡ fs >>= flip map xs
lookup-⊛*   : lookup (fs ⊛* xs) (combine i j) ≡ (lookup fs i $ lookup xs j)
++-is-foldr : xs ++ ys ≡ foldr ((Vec A) ∘ (_+ n)) _∷_ ys xs
[]≔-++-↑ʳ   : (xs ++ ys) [ m ↑ʳ i ]≔ y ≡ xs ++ (ys [ i ]≔ y)
unfold-ʳ++  : xs ʳ++ ys ≡ reverse xs ++ ys

foldl-universal : ∀ (h : ∀ {c} (C : ℕ → Set c) (g : FoldlOp A C) (e : C zero) → ∀ {n} → Vec A n → C n) →
                  (∀ ... → h C g e [] ≡ e) →
                  (∀ ... → h C g e ∘ (x ∷_) ≗ h (C ∘ suc) g (g e x)) →
                  h B f e ≗ foldl B f e
foldl-fusion  : h d ≡ e → (∀ ... → h (f b x) ≡ g (h b) x) → h ∘ foldl B f d ≗ foldl C g e
foldl-∷ʳ      : foldl B f e (ys ∷ʳ y) ≡ f (foldl B f e ys) y
foldl-[]      : foldl B f e [] ≡ e
foldl-reverse : foldl B {n} f e ∘ reverse ≗ foldr B (flip f) e

foldr-[]      : foldr B f e [] ≡ e
foldr-++      : foldr B f e (xs ++ ys) ≡ foldr (B ∘ (_+ n)) f (foldr B f e ys) xs
foldr-∷ʳ      : foldr B f e (ys ∷ʳ y) ≡ foldr (B ∘ suc) f (f y e) ys
foldr-ʳ++     : foldr B f e (xs ʳ++ ys) ≡ foldl (B ∘ (_+ n)) (flip f) (foldr B f e ys) xs
foldr-reverse : foldr B f e ∘ reverse ≗ foldl B (flip f) e

∷ʳ-injective  : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys × x ≡ y
∷ʳ-injectiveˡ : xs ∷ʳ x ≡ ys ∷ʳ y → xs ≡ ys
∷ʳ-injectiveʳ : xs ∷ʳ x ≡ ys ∷ʳ y → x ≡ y

unfold-∷ʳ : cast eq (xs ∷ʳ x) ≡ xs ++ [ x ]
init-∷ʳ   : init (xs ∷ʳ x) ≡ xs
last-∷ʳ   : last (xs ∷ʳ x) ≡ x
cast-∷ʳ   : cast eq (xs ∷ʳ x) ≡ (cast (cong pred eq) xs) ∷ʳ x
++-∷ʳ     : cast eq ((xs ++ ys) ∷ʳ z) ≡ xs ++ (ys ∷ʳ z)
∷ʳ-++     : cast eq ((xs ∷ʳ a) ++ ys) ≡ xs ++ (a ∷ ys)

reverse-∷          : reverse (x ∷ xs) ≡ reverse xs ∷ʳ x
reverse-involutive : Involutive _≡_ reverse
reverse-reverse    : reverse xs ≡ ys → reverse ys ≡ xs
reverse-injective  : reverse xs ≡ reverse ys → xs ≡ ys

transpose-replicate : transpose (replicate xs) ≡ map replicate xs
toList-replicate    : toList (replicate {n = n} a) ≡ List.replicate n a

toList-++ : toList (xs ++ ys) ≡ toList xs List.++ toList ys

cast-is-id    : cast eq xs ≡ xs
subst-is-cast : subst (Vec A) eq xs ≡ cast eq xs
cast-sym      : cast eq xs ≡ ys → cast (sym eq) ys ≡ xs
cast-trans    : cast eq₂ (cast eq₁ xs) ≡ cast (trans eq₁ eq₂) xs
map-cast      : map f (cast eq xs) ≡ cast eq (map f xs)
lookup-cast   : lookup (cast eq xs) (Fin.cast eq i) ≡ lookup xs i
lookup-cast₁  : lookup (cast eq xs) i ≡ lookup xs (Fin.cast (sym eq) i)
lookup-cast₂  : lookup xs (Fin.cast eq i) ≡ lookup (cast (sym eq) xs) i
cast-reverse  : cast eq ∘ reverse ≗ reverse ∘ cast eq
cast-++ˡ      : cast (cong (_+ n) eq) (xs ++ ys) ≡ cast eq xs ++ ys
cast-++ʳ      : cast (cong (m +_) eq) (xs ++ ys) ≡ xs ++ cast eq ys

iterate-id     : iterate id x n ≡ replicate x
take-iterate   : take n (iterate f x (n + m)) ≡ iterate f x n
drop-iterate   : drop n (iterate f x n) ≡ []
lookup-iterate : lookup (iterate f x n) i ≡ ℕ.iterate f x (toℕ i)
toList-iterate : toList (iterate f x n) ≡ List.iterate f x n

zipwith-++ : zipWith f (xs ++ ys) (xs' ++ ys') ≡ zipWith f xs xs' ++ zipWith f ys ys'

++-assoc     : cast eq ((xs ++ ys) ++ zs) ≡ xs ++ (ys ++ zs)
++-identityʳ : cast eq (xs ++ []) ≡ xs
init-reverse : init ∘ reverse ≗ reverse ∘ tail
last-reverse : last ∘ reverse ≗ head
reverse-++   : cast eq (reverse (xs ++ ys)) ≡ reverse ys ++ reverse xs

toList-cast   : toList (cast eq xs) ≡ toList xs
cast-fromList : cast _ (fromList xs) ≡ fromList ys
fromList-map  : cast _ (fromList (List.map f xs)) ≡ map f (fromList xs)
fromList-++   : cast _ (fromList (xs List.++ ys)) ≡ fromList xs ++ fromList ys
fromList-reverse : cast (Listₚ.length-reverse xs) (fromList (List.reverse xs)) ≡ reverse (fromList xs)

∷-ʳ++   : cast eq ((a ∷ xs) ʳ++ ys) ≡ xs ʳ++ (a ∷ ys)
++-ʳ++  : cast eq ((xs ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ʳ++ zs)
ʳ++-ʳ++ : cast eq ((xs ʳ++ ys) ʳ++ zs) ≡ ys ʳ++ (xs ++ zs)

length-toList   : List.length (toList xs) ≡ length xs
toList-insertAt : toList (insertAt xs i v) ≡ List.insertAt (toList xs) (Fin.cast (cong suc (sym (length-toList xs))) i) v

truncate≡take       : .(eq : n ≡ m + o) → truncate m≤n xs ≡ take m (cast eq xs)
take≡truncate       : take m xs ≡ truncate (m≤m+n m n) xs
lookup-truncate     : lookup (truncate m≤n xs) i ≡ lookup xs (Fin.inject≤ i m≤n)
lookup-take-inject≤ : lookup (take m xs) i ≡ lookup xs (Fin.inject≤ i (m≤m+n m n))

Added new proofs in Data.Vec.Membership.Propositional.Properties:

index-∈-fromList⁺ : Any.index (∈-fromList⁺ v∈xs) ≡ indexₗ v∈xs

Added new proofs in Data.Vec.Functional.Properties:

map-updateAt : f ∘ g ≗ h ∘ f → map f (updateAt i g xs) ≗ updateAt i h (map f xs)

Added new proofs in Data.Vec.Relation.Binary.Lex.Strict:

xs≮[] : {xs : Vec A n} → ¬ xs < []
<-respectsˡ : IsPartialEquivalence _≈_ → _≺_ Respectsˡ _≈_ →
              ∀ {m n} → _Respectsˡ_ (_<_ {m} {n}) _≋_
<-respectsʳ : IsPartialEquivalence _≈_ → _≺_ Respectsʳ _≈_ →
              ∀ {m n} → _Respectsʳ_ (_<_ {m} {n}) _≋_
<-wellFounded : Transitive _≈_ → _≺_ Respectsʳ _≈_ → WellFounded _≺_ →
                ∀ {n} → WellFounded (_<_ {n})


* Added new functions in `Data.Vec.Relation.Unary.Any`:

lookup : Any P xs → A


* Added new functions in `Data.Vec.Relation.Unary.All`:

decide : Π[ P ∪ Q ] → Π[ All P ∪ Any Q ]


* Added vector associativity proof to  `Data.Vec.Relation.Binary.Equality.Setoid`:

++-assoc : (xs ++ ys) ++ zs ≋ xs ++ (ys ++ zs)


* Added new functions in `Effect.Monad.State`:

runState : State s a → s → a × s evalState : State s a → s → a execState : State s a → s → s


* Added new proofs in `Function.Construct.Symmetry`:

bijective : Bijective ≈₁ ≈₂ f → Symmetric ≈₂ → Transitive ≈₂ → Congruent ≈₁ ≈₂ f → Bijective ≈₂ ≈₁ f⁻¹ isBijection : IsBijection ≈₁ ≈₂ f → Congruent ≈₂ ≈₁ f⁻¹ → IsBijection ≈₂ ≈₁ f⁻¹ isBijection-≡ : IsBijection ≈₁ ≡ f → IsBijection ≡ ≈₁ f⁻¹ bijection : Bijection R S → Congruent IB.Eq₂.≈ IB.Eq₁.≈ f⁻¹ → Bijection S R bijection-≡ : Bijection R (setoid B) → Bijection (setoid B) R sym-⤖ : A ⤖ B → B ⤖ A


* Added new operations in `Function.Strict`:

!|> : (a : A) → (∀ a → B a) → B a !|>′ : A → (A → B) → B


* Added new definition to the `Surjection` module in `Function.Related.Surjection`:

f⁻ = proj₁ ∘ surjective


* Added new operations in `IO`:

Colist.forM : Colist A → (A → IO B) → IO (Colist B) Colist.forM′ : Colist A → (A → IO B) → IO ⊤

List.forM : List A → (A → IO B) → IO (List B) List.forM′ : List A → (A → IO B) → IO ⊤


* Added new operations in `IO.Base`:

lift! : IO A → IO (Lift b A) <$ : B → IO A → IO B =<< : (A → IO B) → IO A → IO B << : IO B → IO A → IO B lift′ : Prim.IO ⊤ → IO {a} ⊤

when : Bool → IO ⊤ → IO ⊤ unless : Bool → IO ⊤ → IO ⊤

whenJust : Maybe A → (A → IO ⊤) → IO ⊤ untilJust : IO (Maybe A) → IO A untilRight : (A → IO (A ⊎ B)) → A → IO B


* Added new functions in `Reflection.AST.Term`:

stripPis : Term → List (String × Arg Type) × Term prependLams : List (String × Visibility) → Term → Term prependHLams : List String → Term → Term prependVLams : List String → Term → Term


* Added new operations in `Relation.Binary.Construct.Closure.Equivalence`:

fold : IsEquivalence ∼ → ⟶ ⇒ ∼ → EqClosure ⟶ ⇒ ∼ gfold : IsEquivalence ∼ → ⟶ =[ f ]⇒ ∼ → EqClosure ⟶ =[ f ]⇒ ∼ return : ⟶ ⇒ EqClosure ⟶ join : EqClosure (EqClosure ⟶) ⇒ EqClosure ⟶ _⋆ : ⟶₁ ⇒ EqClosure ⟶₂ → EqClosure ⟶₁ ⇒ EqClosure ⟶₂


* Added new operations in `Relation.Binary.Construct.Closure.Symmetric`:

fold : Symmetric ∼ → ⟶ ⇒ ∼ → SymClosure ⟶ ⇒ ∼ gfold : Symmetric ∼ → ⟶ =[ f ]⇒ ∼ → SymClosure ⟶ =[ f ]⇒ ∼ return : ⟶ ⇒ SymClosure ⟶ join : SymClosure (SymClosure ⟶) ⇒ SymClosure ⟶ _⋆ : ⟶₁ ⇒ SymClosure ⟶₂ → SymClosure ⟶₁ ⇒ SymClosure ⟶₂


* Added new proofs to `Relation.Binary.Lattice.Properties.{Join,Meet}Semilattice`:
```agda
isPosemigroup : IsPosemigroup _≈_ _≤_ _∨_
posemigroup : Posemigroup c ℓ₁ ℓ₂
≈-dec⇒≤-dec : Decidable _≈_ → Decidable _≤_
≈-dec⇒isDecPartialOrder : Decidable _≈_ → IsDecPartialOrder _≈_ _≤_

Added new proofs to Relation.Binary.Lattice.Properties.Bounded{Join,Meet}Semilattice:

isCommutativePomonoid : IsCommutativePomonoid _≈_ _≤_ _∨_ ⊥
commutativePomonoid : CommutativePomonoid c ℓ₁ ℓ₂

Added new proofs to Relation.Binary.Properties.Poset:

≤-dec⇒≈-dec : Decidable _≤_ → Decidable _≈_
≤-dec⇒isDecPartialOrder : Decidable _≤_ → IsDecPartialOrder _≈_ _≤_

Added new proofs in Relation.Binary.Properties.StrictPartialOrder:
```
>-strictPartialOrder : StrictPartialOrder s₁ s₂ s₃
```

Added new proofs in Relation.Binary.PropositionalEquality.Properties:

subst-application′ : subst Q eq (f x p) ≡ f y (subst P eq p)
sym-cong : sym (cong f p) ≡ cong f (sym p)

Added new proofs in Relation.Binary.HeterogeneousEquality:

subst₂-removable : subst₂ _∼_ eq₁ eq₂ p ≅ p

Added new definitions in Relation.Unary:

_≐_  : Pred A ℓ₁ → Pred A ℓ₂ → Set _
_≐′_ : Pred A ℓ₁ → Pred A ℓ₂ → Set _
_∖_  : Pred A ℓ₁ → Pred A ℓ₂ → Pred A _

Added new proofs in Relation.Unary.Properties:

⊆-reflexive : _≐_ ⇒ _⊆_
⊆-antisym : Antisymmetric _≐_ _⊆_
⊆-min : Min _⊆_ ∅
⊆-max : Max _⊆_ U
⊂⇒⊆ : _⊂_ ⇒ _⊆_
⊂-trans : Trans _⊂_ _⊂_ _⊂_
⊂-⊆-trans : Trans _⊂_ _⊆_ _⊂_
⊆-⊂-trans : Trans _⊆_ _⊂_ _⊂_
⊂-respʳ-≐ : _⊂_ Respectsʳ _≐_
⊂-respˡ-≐ : _⊂_ Respectsˡ _≐_
⊂-resp-≐ : _⊂_ Respects₂ _≐_
⊂-irrefl : Irreflexive _≐_ _⊂_
⊂-antisym : Antisymmetric _≐_ _⊂_
∅-⊆′ : (P : Pred A ℓ) → ∅ ⊆′ P
⊆′-U : (P : Pred A ℓ) → P ⊆′ U
⊆′-refl : Reflexive {A = Pred A ℓ} _⊆′_
⊆′-reflexive : _≐′_ ⇒ _⊆′_
⊆′-trans : Trans _⊆′_ _⊆′_ _⊆′_
⊆′-antisym : Antisymmetric _≐′_ _⊆′_
⊆′-min : Min _⊆′_ ∅
⊆′-max : Max _⊆′_ U
⊂′⇒⊆′ : _⊂′_ ⇒ _⊆′_
⊂′-trans : Trans _⊂′_ _⊂′_ _⊂′_
⊂′-⊆′-trans : Trans _⊂′_ _⊆′_ _⊂′_
⊆′-⊂′-trans : Trans _⊆′_ _⊂′_ _⊂′_
⊂′-respʳ-≐′ : _⊂′_ Respectsʳ _≐′_
⊂′-respˡ-≐′ : _⊂′_ Respectsˡ _≐′_
⊂′-resp-≐′ : _⊂′_ Respects₂ _≐′_
⊂′-irrefl : Irreflexive _≐′_ _⊂′_
⊂′-antisym : Antisymmetric _≐′_ _⊂′_
⊆⇒⊆′ : _⊆_ ⇒ _⊆′_
⊆′⇒⊆ : _⊆′_ ⇒ _⊆_
⊂⇒⊂′ : _⊂_ ⇒ _⊂′_
⊂′⇒⊂ : _⊂′_ ⇒ _⊂_
≐-refl : Reflexive _≐_
≐-sym : Sym _≐_ _≐_
≐-trans : Trans _≐_ _≐_ _≐_
≐′-refl : Reflexive _≐′_
≐′-sym : Sym _≐′_ _≐′_
≐′-trans : Trans _≐′_ _≐′_ _≐′_
≐⇒≐′ : _≐_ ⇒ _≐′_
≐′⇒≐ : _≐′_ ⇒ _≐_

U-irrelevant : Irrelevant {A = A} U
∁-irrelevant : (P : Pred A ℓ) → Irrelevant (∁ P)

Generalised proofs in Relation.Unary.Properties:
```
⊆-trans : Trans _⊆_ _⊆_ _⊆_
```

Added new proofs in Relation.Binary.Properties.Setoid:

≈-isPreorder     : IsPreorder _≈_ _≈_
≈-isPartialOrder : IsPartialOrder _≈_ _≈_

≈-preorder : Preorder a ℓ ℓ
≈-poset    : Poset a ℓ ℓ

Added new definitions in Relation.Binary.Definitions:

RightTrans R S = Trans R S R
LeftTrans  S R = Trans S R R

Dense        _<_ = ∀ {x y} → x < y → ∃[ z ] x < z × z < y
Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)
Tight    _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)

Monotonic₁         _≤_ _⊑_ f     = f Preserves _≤_ ⟶ _⊑_
Antitonic₁         _≤_ _⊑_ f     = f Preserves (flip _≤_) ⟶ _⊑_
Monotonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ _⊑_ ⟶ _≼_
MonotonicAntitonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ _≤_ ⟶ (flip _⊑_) ⟶ _≼_
AntitonicMonotonic _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ _⊑_ ⟶ _≼_
Antitonic₂         _≤_ _⊑_ _≼_ ∙ = ∙ Preserves₂ (flip _≤_) ⟶ (flip _⊑_) ⟶ _≼_
Adjoint            _≤_ _⊑_ f g   = ∀ {x y} → (f x ⊑ y → x ≤ g y) × (x ≤ g y → f x ⊑ y)

Added new definitions in Relation.Binary.Bundles:

record DenseLinearOrder c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where
record ApartnessRelation c ℓ₁ ℓ₂ : Set (suc (c ⊔ ℓ₁ ⊔ ℓ₂)) where

Added new definitions in Relation.Binary.Structures:

record IsDenseLinearOrder (_<_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where
record IsApartnessRelation (_#_ : Rel A ℓ₂) : Set (a ⊔ ℓ ⊔ ℓ₂) where

Added new proofs to Relation.Binary.Consequences:

sym⇒¬-sym       : Symmetric _∼_ → Symmetric _≁_
cotrans⇒¬-trans : Cotransitive _∼_ → Transitive _≁_
irrefl⇒¬-refl   : Reflexive _≈_ → Irreflexive _≈_ _∼_ →  Reflexive _≁_
mono₂⇒cong₂     : Symmetric ≈₁ → ≈₁ ⇒ ≤₁ → Antisymmetric ≈₂ ≤₂ → ∀ {f} →
                  f Preserves₂ ≤₁ ⟶ ≤₁ ⟶ ≤₂ →
                  f Preserves₂ ≈₁ ⟶ ≈₁ ⟶ ≈₂

Added new proofs to Relation.Binary.Construct.Closure.Transitive:

accessible⁻ : ∀ {x} → Acc _∼⁺_ x → Acc _∼_ x
wellFounded⁻ : WellFounded _∼⁺_ → WellFounded _∼_
accessible : ∀ {x} → Acc _∼_ x → Acc _∼⁺_ x

Added new operations in Relation.Binary.PropositionalEquality.Properties:

J       : (B : (y : A) → x ≡ y → Set b) (p : x ≡ y) → B x refl → B y p
dcong   : (p : x ≡ y) → subst B p (f x) ≡ f y
dcong₂  : (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → f x₁ y₁ ≡ f x₂ y₂
dsubst₂ : (p : x₁ ≡ x₂) → subst B p y₁ ≡ y₂ → C x₁ y₁ → C x₂ y₂
ddcong₂ : (p : x₁ ≡ x₂) (q : subst B p y₁ ≡ y₂) → dsubst₂ C p q (f x₁ y₁) ≡ f x₂ y₂

isPartialOrder : IsPartialOrder _≡_ _≡_
poset          : Set a → Poset _ _ _

Added new operations in System.Exit:

isSuccess : ExitCode → Bool
isFailure : ExitCode → Bool

Added new functions in Codata.Guarded.Stream:

transpose : List (Stream A) → Stream (List A)
transpose⁺ : List⁺ (Stream A) → Stream (List⁺ A)
concat : Stream (List⁺ A) → Stream A

Added new proofs in Codata.Guarded.Stream.Properties:

cong-concat : {ass bss : Stream (List⁺.List⁺ A)} → ass ≈ bss → concat ass ≈ concat bss
map-concat : ∀ (f : A → B) ass → map f (concat ass) ≈ concat (map (List⁺.map f) ass)
lookup-transpose : ∀ n (ass : List (Stream A)) → lookup n (transpose ass) ≡ List.map (lookup n) ass
lookup-transpose⁺ : ∀ n (ass : List⁺ (Stream A)) → lookup n (transpose⁺ ass) ≡ List⁺.map (lookup n) ass

Added new corollaries in Data.List.Membership.Setoid.Properties:

∈-++⁺∘++⁻ : ∀ {v} xs {ys} (p : v ∈ xs ++ ys) → [ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ (∈-++⁻ xs p) ≡ p
∈-++⁻∘++⁺ : ∀ {v} xs {ys} (p : v ∈ xs ⊎ v ∈ ys) → ∈-++⁻ xs ([ ∈-++⁺ˡ , ∈-++⁺ʳ xs ]′ p) ≡ p
∈-++↔ : ∀ {v xs ys} → (v ∈ xs ⊎ v ∈ ys) ↔ v ∈ xs ++ ys
∈-++-comm : ∀ {v} xs ys → v ∈ xs ++ ys → v ∈ ys ++ xs
∈-++-comm∘++-comm : ∀ {v} xs {ys} (p : v ∈ xs ++ ys) → ∈-++-comm ys xs (∈-++-comm xs ys p) ≡ p
∈-++↔++ : ∀ {v} xs ys → v ∈ xs ++ ys ↔ v ∈ ys ++ xs

Exposed container combinator conversion functions from Data.Container.Combinator.Properties separate from their correctness proofs in Data.Container.Combinator:

to-id      : F.id A → ⟦ id ⟧ A
from-id    : ⟦ id ⟧ A → F.id A
to-const   : (A : Set s) → A → ⟦ const A ⟧ B
from-const : (A : Set s) → ⟦ const A ⟧ B → A
to-∘       : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A) → ⟦ C₁ ∘ C₂ ⟧ A
from-∘     : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ ∘ C₂ ⟧ A → ⟦ C₁ ⟧ (⟦ C₂ ⟧ A)
to-×       : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A → ⟦ C₁ × C₂ ⟧ A
from-×     : (C₁ : Container s₁ p₁) (C₂ : Container s₂ p₂) → ⟦ C₁ × C₂ ⟧ A → ⟦ C₁ ⟧ A P.× ⟦ C₂ ⟧ A
to-Π       : (I : Set i) (Cᵢ : I → Container s p) → (∀ i → ⟦ Cᵢ i ⟧ A) → ⟦ Π I Cᵢ ⟧ A
from-Π     : (I : Set i) (Cᵢ : I → Container s p) → ⟦ Π I Cᵢ ⟧ A → ∀ i → ⟦ Cᵢ i ⟧ A
to-⊎       : (C₁ : Container s₁ p) (C₂ : Container s₂ p) → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A → ⟦ C₁ ⊎ C₂ ⟧ A
from-⊎     : (C₁ : Container s₁ p) (C₂ : Container s₂ p) → ⟦ C₁ ⊎ C₂ ⟧ A → ⟦ C₁ ⟧ A S.⊎ ⟦ C₂ ⟧ A
to-Σ       : (I : Set i) (C : I → Container s p) → (∃ λ i → ⟦ C i ⟧ A) → ⟦ Σ I C ⟧ A
from-Σ     : (I : Set i) (C : I → Container s p) → ⟦ Σ I C ⟧ A → ∃ λ i → ⟦ C i ⟧ A

Added a non-dependent version of Function.Base.flip due to an issue noted in Pull Request #1812:
```
flip′ : (A → B → C) → (B → A → C)
```

Added new proofs to Data.List.Properties

cartesianProductWith-zeroˡ       : cartesianProductWith f [] ys ≡ []
cartesianProductWith-zeroʳ       : cartesianProductWith f xs [] ≡ []
cartesianProductWith-distribʳ-++ : cartesianProductWith f (xs ++ ys) zs ≡
                                   cartesianProductWith f xs zs ++ cartesianProductWith f ys zs
foldr-map : foldr f x (map g xs) ≡ foldr (g -⟨ f ∣) x xs
foldl-map : foldl f x (map g xs) ≡ foldl (∣ f ⟩- g) x xs

NonZero/Positive/Negative changes

This is a full list of proofs that have changed form to use irrelevant instance arguments:

In Data.Nat.Base:

≢-nonZero⁻¹ : ∀ {n} → .(NonZero n) → n ≢ 0
>-nonZero⁻¹ : ∀ {n} → .(NonZero n) → n > 0

In Data.Nat.Properties:

*-cancelʳ-≡ : ∀ m n {o} → m * suc o ≡ n * suc o → m ≡ n
*-cancelˡ-≡ : ∀ {m n} o → suc o * m ≡ suc o * n → m ≡ n
*-cancelʳ-≤ : ∀ m n o → m * suc o ≤ n * suc o → m ≤ n
*-cancelˡ-≤ : ∀ {m n} o → suc o * m ≤ suc o * n → m ≤ n
*-monoˡ-<   : ∀ n → (_* suc n) Preserves _<_ ⟶ _<_
*-monoʳ-<   : ∀ n → (suc n *_) Preserves _<_ ⟶ _<_

m≤m*n          : ∀ m {n} → 0 < n → m ≤ m * n
m≤n*m          : ∀ m {n} → 0 < n → m ≤ n * m
m<m*n          : ∀ {m n} → 0 < m → 1 < n → m < m * n
suc[pred[n]]≡n : ∀ {n} → n ≢ 0 → suc (pred n) ≡ n

In Data.Nat.Coprimality:

Bézout-coprime : ∀ {i j d} → Bézout.Identity (suc d) (i * suc d) (j * suc d) → Coprime i j

In Data.Nat.Divisibility

m%n≡0⇒n∣m : ∀ m n → m % suc n ≡ 0 → suc n ∣ m
n∣m⇒m%n≡0 : ∀ m n → suc n ∣ m → m % suc n ≡ 0
m%n≡0⇔n∣m : ∀ m n → m % suc n ≡ 0 ⇔ suc n ∣ m
∣⇒≤       : ∀ {m n} → m ∣ suc n → m ≤ suc n
>⇒∤        : ∀ {m n} → m > suc n → m ∤ suc n
*-cancelˡ-∣ : ∀ {i j} k → suc k * i ∣ suc k * j → i ∣ j

In Data.Nat.DivMod:

m≡m%n+[m/n]*n : ∀ m n → m ≡ m % suc n + (m / suc n) * suc n
m%n≡m∸m/n*n   : ∀ m n → m % suc n ≡ m ∸ (m / suc n) * suc n
n%n≡0         : ∀ n → suc n % suc n ≡ 0
m%n%n≡m%n     : ∀ m n → m % suc n % suc n ≡ m % suc n
[m+n]%n≡m%n   : ∀ m n → (m + suc n) % suc n ≡ m % suc n
[m+kn]%n≡m%n  : ∀ m k n → (m + k * (suc n)) % suc n ≡ m % suc n
m*n%n≡0       : ∀ m n → (m * suc n) % suc n ≡ 0
m%n<n         : ∀ m n → m % suc n < suc n
m%n≤m         : ∀ m n → m % suc n ≤ m
m≤n⇒m%n≡m     : ∀ {m n} → m ≤ n → m % suc n ≡ m

m/n<m         : ∀ m n {≢0} → m ≥ 1 → n ≥ 2 → (m / n) {≢0} < m

In Data.Nat.GCD

GCD-* : ∀ {m n d c} → GCD (m * suc c) (n * suc c) (d * suc c) → GCD m n d
gcd[m,n]≤n : ∀ m n → gcd m (suc n) ≤ suc n

In Data.Integer.Properties:

positive⁻¹        : ∀ {i} → Positive i → i > 0ℤ
negative⁻¹        : ∀ {i} → Negative i → i < 0ℤ
nonPositive⁻¹     : ∀ {i} → NonPositive i → i ≤ 0ℤ
nonNegative⁻¹     : ∀ {i} → NonNegative i → i ≥ 0ℤ
negative<positive : ∀ {i j} → Negative i → Positive j → i < j

sign-◃    : ∀ s n → sign (s ◃ suc n) ≡ s
sign-cong : ∀ {s₁ s₂ n₁ n₂} → s₁ ◃ suc n₁ ≡ s₂ ◃ suc n₂ → s₁ ≡ s₂
-◃<+◃     : ∀ m n → Sign.- ◃ (suc m) < Sign.+ ◃ n
m⊖1+n<m   : ∀ m n → m ⊖ suc n < + m

*-cancelʳ-≡     : ∀ i j k → k ≢ + 0 → i * k ≡ j * k → i ≡ j
*-cancelˡ-≡     : ∀ i j k → i ≢ + 0 → i * j ≡ i * k → j ≡ k
*-cancelʳ-≤-pos : ∀ m n o → m * + suc o ≤ n * + suc o → m ≤ n

≤-steps     : ∀ n → i ≤ j → i ≤ + n + j
≤-steps-neg : ∀ n → i ≤ j → i - + n ≤ j
n≤m+n       : ∀ n → i ≤ + n + i
m≤m+n       : ∀ n → i ≤ i + + n
m-n≤m       : ∀ i n → i - + n ≤ i

*-cancelʳ-≤-pos    : ∀ m n o → m * + suc o ≤ n * + suc o → m ≤ n
*-cancelˡ-≤-pos    : ∀ m j k → + suc m * j ≤ + suc m * k → j ≤ k
*-cancelˡ-≤-neg    : ∀ m {j k} → -[1+ m ] * j ≤ -[1+ m ] * k → j ≥ k
*-cancelʳ-≤-neg    : ∀ {n o} m → n * -[1+ m ] ≤ o * -[1+ m ] → n ≥ o
*-cancelˡ-<-nonNeg : ∀ n → + n * i < + n * j → i < j
*-cancelʳ-<-nonNeg : ∀ n → i * + n < j * + n → i < j
*-monoʳ-≤-nonNeg   : ∀ n → (_* + n) Preserves _≤_ ⟶ _≤_
*-monoˡ-≤-nonNeg   : ∀ n → (+ n *_) Preserves _≤_ ⟶ _≤_
*-monoˡ-≤-nonPos   : ∀ i → NonPositive i → (i *_) Preserves _≤_ ⟶ _≥_
*-monoʳ-≤-nonPos   : ∀ i → NonPositive i → (_* i) Preserves _≤_ ⟶ _≥_
*-monoˡ-<-pos      : ∀ n → (+[1+ n ] *_) Preserves _<_ ⟶ _<_
*-monoʳ-<-pos      : ∀ n → (_* +[1+ n ]) Preserves _<_ ⟶ _<_
*-monoˡ-<-neg      : ∀ n → (-[1+ n ] *_) Preserves _<_ ⟶ _>_
*-monoʳ-<-neg      : ∀ n → (_* -[1+ n ]) Preserves _<_ ⟶ _>_
*-cancelˡ-<-nonPos : ∀ n → NonPositive n → n * i < n * j → i > j
*-cancelʳ-<-nonPos : ∀ n → NonPositive n → i * n < j * n → i > j

*-distribˡ-⊓-nonNeg : ∀ m j k → + m * (j ⊓ k) ≡ (+ m * j) ⊓ (+ m * k)
*-distribʳ-⊓-nonNeg : ∀ m j k → (j ⊓ k) * + m ≡ (j * + m) ⊓ (k * + m)
*-distribˡ-⊓-nonPos : ∀ i → NonPositive i → ∀ j k → i * (j ⊓ k) ≡ (i * j) ⊔ (i * k)
*-distribʳ-⊓-nonPos : ∀ i → NonPositive i → ∀ j k → (j ⊓ k) * i ≡ (j * i) ⊔ (k * i)
*-distribˡ-⊔-nonNeg : ∀ m j k → + m * (j ⊔ k) ≡ (+ m * j) ⊔ (+ m * k)
*-distribʳ-⊔-nonNeg : ∀ m j k → (j ⊔ k) * + m ≡ (j * + m) ⊔ (k * + m)
*-distribˡ-⊔-nonPos : ∀ i → NonPositive i → ∀ j k → i * (j ⊔ k) ≡ (i * j) ⊓ (i * k)
*-distribʳ-⊔-nonPos : ∀ i → NonPositive i → ∀ j k → (j ⊔ k) * i ≡ (j * i) ⊓ (k * i)

In Data.Integer.Divisibility:

*-cancelˡ-∣ : ∀ k {i j} → k ≢ + 0 → k * i ∣ k * j → i ∣ j
*-cancelʳ-∣ : ∀ k {i j} → k ≢ + 0 → i * k ∣ j * k → i ∣ j

In Data.Integer.Divisibility.Signed:

*-cancelˡ-∣ : ∀ k {i j} → k ≢ + 0 → k * i ∣ k * j → i ∣ j
*-cancelʳ-∣ : ∀ k {i j} → k ≢ + 0 → i * k ∣ j * k → i ∣ j

In Data.Rational.Unnormalised.Properties:

positive⁻¹           : ∀ {q} → .(Positive q) → q > 0ℚᵘ
nonNegative⁻¹        : ∀ {q} → .(NonNegative q) → q ≥ 0ℚᵘ
negative⁻¹           : ∀ {q} → .(Negative q) → q < 0ℚᵘ
nonPositive⁻¹        : ∀ {q} → .(NonPositive q) → q ≤ 0ℚᵘ
positive⇒nonNegative : ∀ {p} → Positive p → NonNegative p
negative⇒nonPositive : ∀ {p} → Negative p → NonPositive p
negative<positive    : ∀ {p q} → .(Negative p) → .(Positive q) → p < q
nonNeg∧nonPos⇒0      : ∀ {p} → .(NonNegative p) → .(NonPositive p) → p ≃ 0ℚᵘ

p≤p+q   : ∀ {p q} → NonNegative q → p ≤ p + q
p≤q+p   : ∀ {p} → NonNegative p → ∀ {q} → q ≤ p + q

*-cancelʳ-≤-pos    : ∀ {r} → Positive r → ∀ {p q} → p * r ≤ q * r → p ≤ q
*-cancelˡ-≤-pos    : ∀ {r} → Positive r → ∀ {p q} → r * p ≤ r * q → p ≤ q
*-cancelʳ-≤-neg    : ∀ r → Negative r → ∀ {p q} → p * r ≤ q * r → q ≤ p
*-cancelˡ-≤-neg    : ∀ r → Negative r → ∀ {p q} → r * p ≤ r * q → q ≤ p
*-cancelˡ-<-nonNeg : ∀ {r} → NonNegative r → ∀ {p q} → r * p < r * q → p < q
*-cancelʳ-<-nonNeg : ∀ {r} → NonNegative r → ∀ {p q} → p * r < q * r → p < q
*-cancelˡ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → r * p < r * q → q < p
*-cancelʳ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → p * r < q * r → q < p
*-monoˡ-≤-nonNeg   : ∀ {r} → NonNegative r → (_* r) Preserves _≤_ ⟶ _≤_
*-monoʳ-≤-nonNeg   : ∀ {r} → NonNegative r → (r *_) Preserves _≤_ ⟶ _≤_
*-monoˡ-≤-nonPos   : ∀ r → NonPositive r → (_* r) Preserves _≤_ ⟶ _≥_
*-monoʳ-≤-nonPos   : ∀ r → NonPositive r → (r *_) Preserves _≤_ ⟶ _≥_
*-monoˡ-<-pos      : ∀ {r} → Positive r → (_* r) Preserves _<_ ⟶ _<_
*-monoʳ-<-pos      : ∀ {r} → Positive r → (r *_) Preserves _<_ ⟶ _<_
*-monoˡ-<-neg      : ∀ r → Negative r → (_* r) Preserves _<_ ⟶ _>_
*-monoʳ-<-neg      : ∀ r → Negative r → (r *_) Preserves _<_ ⟶ _>_

pos⇒1/pos : ∀ p (p>0 : Positive p) → Positive ((1/ p) {{pos⇒≢0 p p>0}})
neg⇒1/neg : ∀ p (p<0 : Negative p) → Negative ((1/ p) {{neg⇒≢0 p p<0}})

*-distribʳ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊓ (r * p)
*-distribˡ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊓ (p * r)
*-distribˡ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊔ (p * r)
*-distribʳ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊔ (r * p)
*-distribˡ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊔ r) ≃ (p * q) ⊓ (p * r)
*-distribʳ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊔ r) * p ≃ (q * p) ⊓ (r * p)
*-distribˡ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊓ r) ≃ (p * q) ⊔ (p * r)
*-distribʳ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊓ r) * p ≃ (q * p) ⊔ (r * p)

In Data.Rational.Properties:

positive⁻¹ : Positive p → p > 0ℚ
nonNegative⁻¹ : NonNegative p → p ≥ 0ℚ
negative⁻¹ : Negative p → p < 0ℚ
nonPositive⁻¹ : NonPositive p → p ≤ 0ℚ
negative<positive : Negative p → Positive q → p < q
nonNeg≢neg : ∀ p q → NonNegative p → Negative q → p ≢ q
pos⇒nonNeg : ∀ p → Positive p → NonNegative p
neg⇒nonPos : ∀ p → Negative p → NonPositive p
nonNeg∧nonZero⇒pos : ∀ p → NonNegative p → NonZero p → Positive p

*-cancelʳ-≤-pos    : ∀ r → Positive r → ∀ {p q} → p * r ≤ q * r → p ≤ q
*-cancelˡ-≤-pos    : ∀ r → Positive r → ∀ {p q} → r * p ≤ r * q → p ≤ q
*-cancelʳ-≤-neg    : ∀ r → Negative r → ∀ {p q} → p * r ≤ q * r → p ≥ q
*-cancelˡ-≤-neg    : ∀ r → Negative r → ∀ {p q} → r * p ≤ r * q → p ≥ q
*-cancelˡ-<-nonNeg : ∀ r → NonNegative r → ∀ {p q} → r * p < r * q → p < q
*-cancelʳ-<-nonNeg : ∀ r → NonNegative r → ∀ {p q} → p * r < q * r → p < q
*-cancelˡ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → r * p < r * q → p > q
*-cancelʳ-<-nonPos : ∀ r → NonPositive r → ∀ {p q} → p * r < q * r → p > q
*-monoʳ-≤-nonNeg   : ∀ r → NonNegative r → (_* r) Preserves _≤_ ⟶ _≤_
*-monoˡ-≤-nonNeg   : ∀ r → NonNegative r → (r *_) Preserves _≤_ ⟶ _≤_
*-monoʳ-≤-nonPos   : ∀ r → NonPositive r → (_* r) Preserves _≤_ ⟶ _≥_
*-monoˡ-≤-nonPos   : ∀ r → NonPositive r → (r *_) Preserves _≤_ ⟶ _≥_
*-monoˡ-<-pos      : ∀ r → Positive r → (_* r) Preserves _<_ ⟶ _<_
*-monoʳ-<-pos      : ∀ r → Positive r → (r *_) Preserves _<_ ⟶ _<_
*-monoˡ-<-neg      : ∀ r → Negative r → (_* r) Preserves _<_ ⟶ _>_
*-monoʳ-<-neg      : ∀ r → Negative r → (r *_) Preserves _<_ ⟶ _>_

*-distribˡ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊓ (p * r)
*-distribʳ-⊓-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊓ (r * p)
*-distribˡ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊔ (p * r)
*-distribʳ-⊔-nonNeg : ∀ p → NonNegative p → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊔ (r * p)
*-distribˡ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊔ r) ≡ (p * q) ⊓ (p * r)
*-distribʳ-⊔-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊔ r) * p ≡ (q * p) ⊓ (r * p)
*-distribˡ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → p * (q ⊓ r) ≡ (p * q) ⊔ (p * r)
*-distribʳ-⊓-nonPos : ∀ p → NonPositive p → ∀ q r → (q ⊓ r) * p ≡ (q * p) ⊔ (r * p)

pos⇒1/pos : ∀ p (p>0 : Positive p) → Positive ((1/ p) {{pos⇒≢0 p p>0}})
neg⇒1/neg : ∀ p (p<0 : Negative p) → Negative ((1/ p) {{neg⇒≢0 p p<0}})
1/pos⇒pos : ∀ p .{{_ : NonZero p}} → (1/p : Positive (1/ p)) → Positive p
1/neg⇒neg : ∀ p .{{_ : NonZero p}} → (1/p : Negative (1/ p)) → Negative p

In Data.List.NonEmpty.Base:
```
drop+ : ℕ → List⁺ A → List⁺ A
```
When drop+ping more than the size of the length of the list, the last element remains.

Added new proofs in Data.List.NonEmpty.Properties:

length-++⁺ : length (xs ++⁺ ys) ≡ length xs + length ys
length-++⁺-tail : length (xs ++⁺ ys) ≡ suc (length xs + length (tail ys))
++-++⁺ : (xs ++ ys) ++⁺ zs ≡ xs ++⁺ ys ++⁺ zs
++⁺-cancelˡ′ : xs ++⁺ zs ≡ ys ++⁺ zs′ → List.length xs ≡ List.length ys → zs ≡ zs′
++⁺-cancelˡ : xs ++⁺ ys ≡ xs ++⁺ zs → ys ≡ zs
drop+-++⁺ : drop+ (length xs) (xs ++⁺ ys) ≡ ys
map-++⁺-commute : map f (xs ++⁺ ys) ≡ map f xs ++⁺ map f ys
length-map : length (map f xs) ≡ length xs
map-cong : f ≗ g → map f ≗ map g
map-compose : map (g ∘ f) ≗ map g ∘ map f

Added new functions and proofs in Data.Nat.GeneralisedArithmetic:

iterate : (A → A) → A → ℕ → A
iterate-is-fold : ∀ (z : A) s m → fold z s m ≡ iterate s z m

Added new proofs to Function.Properties.Inverse:

Inverse⇒Injection : Inverse S T → Injection S T
↔⇒↣ : A ↔ B → A ↣ B

Added a new isomorphism to Data.Fin.Properties:
```
2↔Bool : Fin 2 ↔ Bool
```
Added new isomorphisms to Data.Unit.Polymorphic.Properties:
```
⊤↔⊤* : ⊤ {ℓ} ↔ ⊤*
```

Added new isomorphisms to Data.Vec.N-ary:

Vec↔N-ary : ∀ n → (Vec A n → B) ↔ N-ary n A B

Added new isomorphisms to Data.Vec.Recursive:

lift↔ : ∀ n → A ↔ B → A ^ n ↔ B ^ n
Fin[m^n]↔Fin[m]^n : ∀ m n → Fin (m ^ n) ↔ Fin m Vec.^ n

Added new functions to Function.Properties.Inverse:

↔-refl  : Reflexive _↔_
↔-sym   : Symmetric _↔_
↔-trans : Transitive _↔_

Added new isomorphisms to Function.Properties.Inverse:

↔-fun : A ↔ B → C ↔ D → (A → C) ↔ (B → D)

Added new function to Data.Fin.Properties

i≤inject₁[j]⇒i≤1+j : i ≤ inject₁ j → i ≤ suc j

Added new function to Data.Fin.Induction

<-weakInduction-startingFrom : P i →  (∀ j → P (inject₁ j) → P (suc j)) → ∀ {j} → j ≥ i → P j

In Data.List.Relation.Binary.Permutation.Setoid.Properties:

foldr-commMonoid : xs ↭ ys → foldr _∙_ ε xs ≈ foldr _∙_ ε ys

Added new proof, structure, and bundle to Data.Rational.Properties

<-dense              : Dense _<_
<-isDenseLinearOrder : IsDenseLinearOrder _≡_ _<_
<-denseLinearOrder   : DenseLinearOrder 0ℓ 0ℓ 0ℓ

Added new module to Data.Rational.Unnormalised.Properties
```
module ≃-Reasoning = SetoidReasoning ≃-setoid
```

Added new functions to Data.Rational.Unnormalised.Properties

0≠1 : 0ℚᵘ ≠ 1ℚᵘ
≃-≠-irreflexive : Irreflexive _≃_ _≠_
≠-symmetric : Symmetric _≠_
≠-cotransitive : Cotransitive _≠_
≠⇒invertible : p ≠ q → Invertible _≃_ 1ℚᵘ _*_ (p - q)

<-dense : Dense _<_

Added new structures to Data.Rational.Unnormalised.Properties

<-isDenseLinearOrder : IsDenseLinearOrder _≃_ _<_
+-*-isHeytingCommutativeRing : IsHeytingCommutativeRing _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ
+-*-isHeytingField : IsHeytingField _≃_ _≠_ _+_ _*_ -_ 0ℚᵘ 1ℚᵘ

Added new bundles to Data.Rational.Unnormalised.Properties

<-denseLinearOrder : DenseLinearOrder 0ℓ 0ℓ 0ℓ
+-*-heytingCommutativeRing : HeytingCommutativeRing 0ℓ 0ℓ 0ℓ
+-*-heytingField : HeytingField 0ℓ 0ℓ 0ℓ

Added new function to Data.Vec.Relation.Binary.Pointwise.Inductive

cong-[_]≔ : Pointwise _∼_ xs ys → Pointwise _∼_ (xs [ i ]≔ p) (ys [ i ]≔ p)

Added new function to Data.Vec.Relation.Binary.Equality.Setoid

map-[]≔ : map f (xs [ i ]≔ p) ≋ map f xs [ i ]≔ f p

Added new function to Data.List.Relation.Binary.Permutation.Propositional.Properties
```
↭-reverse : (xs : List A) → reverse xs ↭ xs
```

Added new functions to Algebra.Properties.CommutativeMonoid

invertibleˡ⇒invertibleʳ : LeftInvertible _≈_ 0# _+_ x → RightInvertible _≈_ 0# _+_ x
invertibleʳ⇒invertibleˡ : RightInvertible _≈_ 0# _+_ x → LeftInvertible _≈_ 0# _+_ x
invertibleˡ⇒invertible  : LeftInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x
invertibleʳ⇒invertible  : RightInvertible _≈_ 0# _+_ x → Invertible _≈_ 0# _+_ x

Added new functions to Algebra.Apartness.Bundles

invertibleˡ⇒# : LeftInvertible _≈_ 1# _*_ (x - y) → x # y
invertibleʳ⇒# : RightInvertible _≈_ 1# _*_ (x - y) → x # y
x#0y#0→xy#0   : x # 0# → y # 0# → x * y # 0#
#-sym         : Symmetric _#_
#-congʳ       : x ≈ y → x # z → y # z
#-congˡ       : y ≈ z → x # y → x # z

Added new proof to Data.List.Relation.Unary.All.Properties:
```
gmap⁻ : Q ∘ f ⋐ P → All Q ∘ map f ⋐ All P
```
Added new proofs to Data.List.Relation.Binary.Sublist.Setoid.Properties and Data.List.Relation.Unary.Sorted.TotalOrder.Properties.
```
⊆-mergeˡ : ∀ xs ys → xs ⊆ merge _≤?_ xs ys
⊆-mergeʳ : ∀ xs ys → ys ⊆ merge _≤?_ xs ys
```

Added new proof to Induction.WellFounded

Acc-resp-flip-≈ : _<_ Respectsʳ (flip _≈_) → (Acc _<_) Respects _≈_

acc⇒asym : Acc _<_ x → x < y → ¬ (y < x)
wf⇒asym : WellFounded _<_ → Asymmetric _<_
wf⇒irrefl : _<_ Respects₂ _≈_ → Symmetric _≈_ →
            WellFounded _<_ → Irreflexive _≈_ _<_

Added new file Relation.Binary.Reasoning.Base.Apartness

This is how to use it:

_ : a # d
_ = begin-apartness
  a ≈⟨ a≈b ⟩
  b #⟨ b#c ⟩
  c ≈⟨ c≈d ⟩
  d ∎

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