https://wg21.link/P2214 ## advice • always use range algorithms first ■ constrained, better errors ■ projections, superior results • view adapters first • don't be evil and write inscrutable pipelines ##

Syntax 11 3.3 Implicit Arguments 13 3.4 Basic Data Types and Assertions 14 3.5 Constructors, Projections, and Matching 15 3.6 Structured Proofs 17 3.7 Computation 18 3.8 Axioms 19 4 Declarations \alpha $ or report an error if it fails. Lean supports anonymous constructor notation, anonymous projections, and various forms of match syntax, including destructuring $ \lambda $ and let. These, as well → a^n + b^n ≠ c^n def unbounded (f : N → N) : Prop := ∀ M, ∃ n, f n ≥ M ### 3.5 Constructors, Projections, and Matching Lean’s foundation, the Calculus of Inductive Constructions, supports the declaration

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private (continues allowed: -- third (cons _ (cons _ (cons x _))) = x Instead, you can use the record fields as projections: third str = str .tl .tl .hd The constructor can be used as usual in the right-hand side of definitions: their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private (continues allowed: -- third (cons _ (cons _ (cons x _))) = x Instead, you can use the record fields as projections: third str = str .tl .tl .hd The constructor can be used as usual in the right-hand side of definitions: their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private (continues allowed: -- third (cons _ (cons _ (cons x _))) = x Instead, you can use the record fields as projections: third str = str .tl .tl .hd The constructor can be used as usual in the right-hand side of definitions: their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private (continues allowed: -- third (cons _ (cons _ (cons x _))) = x Instead, you can use the record fields as projections: third str = str .tl .tl .hd The constructor can be used as usual in the right-hand side of definitions: their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private posZ : Z allowed: -- third (cons _ (cons _ (cons x _))) = x Instead, you can use the record fields as projections: third str = str .tl .tl .hd The constructor can be used as usual in the right-hand side of definitions: their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private posZ allowed: -- third (cons _ (cons _ (cons x _))) = x Instead, you can use the record fields as projections: third str = str .tl .tl .hd The constructor can be used as usual in the right-hand side of definitions: their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record

signatures, even when in an abstract block! To work around this we have to define aliases for the projections functions: -- A property about the representation of zero integers: abstract private posZ : Z their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record -> (x : Subset A P) -> P (Subset.elem x) certificate (a # p) = irrAx p Example 4. Irrelevant projections are justified by the irrelevance axiom. .unsquash' : ∀ {A} → Squash A → A unsquash' (squash

their name with a dot in the definition of the record type. Projections for irrelevant fields are only created if option --irrelevant-projections is supplied (since Agda > 2.5.4). Example 1. A record inconsistency. This might be fixed in the future. --experimental-irrelevance and --irrelevant-projections; enables potentially unsound irrelevance features (irrelevant levels, irrelevant data matching time. We can for instance prove that any pair is equal to the pairing of its first and second projections, a property commonly called eta-equality: eta : (p@(a , b) : Σ A B) → p ≡ (a , b) eta p = refl