CHANGELOG.md

Version 2.0-dev

The library has been tested using Agda 2.6.2.

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.

Bug-fixes

• 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)

• In Algebra.Definitions.RawSemiring the record prime add p∤1 : p ∤ 1# to the field.

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, modules grouped under *.Categorical.* which exploited these structures for effectful programming have been renamed *.Effectful.

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.

Function hierarchy

• The switch to the new function hierarchy is complete and the following definitions now use the new definitions instead of the old ones:

  • 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.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.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.Relation.Binary.Lex.Core
    • P⇔[]<[]
    • ∷<∷-⇔
  • Data.Vec.Relation.Binary.Pointwise.Extensional
    • equivalent
    • Pointwise-≡↔≡
  • Data.Vec.Relation.Binary.Pointwise.Inductive
    • Pointwise-≡↔≡
  • Relation.Binary.Construct.Closure.Reflexive.Properties
    • ⊎⇔Refl
  • Relation.Binary.Construct.Closure.Transitive
    • equivalent
  • Relation.Nullary.Decidable
    • map

• The names of the fields in the records of the new hierarchy have been changed from f, g, cong₁, cong₂ to to, from, to-cong, from-cong.

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 constains 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 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

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 addeed.

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 conversionFunctions:
  • 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 judgementally 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

Other

• 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 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 redeisgned 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 homorphic.

  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.

• 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 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 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.

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 compatability).

• Beneficieries of this change include Data.Rational.Unnormalised.Base whose dependencies are now significantly smaller.

Deprecated modules

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 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.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

Deprecated names

• 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_
  

• 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⇒≟
  

• 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:

  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
  

• In Data.List.Properties:

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

• In Data.Nat.Properties:

  suc[pred[n]]≡n  ↦  suc-pred
  

• 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
  *-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.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.Vec.Properties:

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

  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 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.Properties.Preorder:

  invIsPreorder ↦ converse-isPreorder
  invPreorder   ↦ converse-preorder
  

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

• 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
  

• Identity morphisms and composition of morphisms between algebraic structures:

  Algebra.Morphism.Construct.Composition
  Algebra.Morphism.Construct.Identity
  

• '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 for a non-empty fresh list:

  Data.List.Fresh.NonEmpty
  

• Combinations and permutations for ℕ.

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

• 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
  

• A small library for heterogenous equational reasoning on vectors:

  Data.Vec.Properties.Heterogeneous
  

• Show module for unnormalised rationals:

  Data.Rational.Unnormalised.Show
  

• 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
  

Other minor changes

• 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 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

  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.Setoid:

  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)
  kleeneAlgebra : KleeneAlgebra a ℓ₁ → KleeneAlgebra b ℓ₂ →
                  KleeneAlgebra (a ⊔ b) (ℓ₁ ⊔ ℓ₂)

* Added new definition to `Algebra.Definitions`:
 ```agda
 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)

• Added new functions to Algebra.Definitions.RawSemiring:

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

• Added new proofs to Algebra.Properties.CommutativeSemigroup:

  interchange : Interchangable _∙_ _∙_
  

• 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 IsKleeneAlgebra (+ * : Op₂ 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

  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)

• Added new functions in Category.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 Data.Bool.Properties:

  <-wellFounded : WellFounded _<_

• 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
  
  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
  
  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
  

• Added new functions in Data.Integer.Base:

  _^_ : ℤ → ℕ → ℤ
  

• Added new proofs in Data.Integer.Properties:

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

• Added new functions in Data.List:

  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

• 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 ]
  

• Add new proofs in Data.List.Properties:

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

• 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

• 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
  
  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)

• Added new functions in Data.Nat:

  _! : ℕ → ℕ

• 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) !

• 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 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 rounding functions in Data.Rational.Base:

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

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

  +-*-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
  
  ≰⇒≥ : _≰_ ⇒ _≥_
  
  *-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 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.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

• 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
  

• 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)

• Added new instance in Data.Vec.Categorical:

  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
  
  ⊛-is->>=    : fs ⊛ xs ≡ fs >>= flip map xs
  ++-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
  
  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

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

  xs≮[] : ∀ {n} (xs : Vec A n) → ¬ xs < []

• 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 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
  

• 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:

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

• 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:

  Cotransitive _#_ = ∀ {x y} → x # y → ∀ z → (x # z) ⊎ (z # y)
  Tight    _≈_ _#_ = ∀ x y → (¬ x # y → x ≈ y) × (x ≈ y → ¬ x # y)
  

• Added new definitions in Relation.Binary.Bundles:

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

• Added new definitions in Relation.Binary.Structures:

  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 _≁_
  

• 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
  

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
  
  ≤-steps : ∀ {p q r} → NonNegative r → p ≤ q → p ≤ r + q
  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