Assertions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.5 Constructors, Projections, and Matching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 3.6 Structured Proofs type α or report an error if it fails. Lean supports anonymous constructor notation, anonymous projections, and various forms of match syntax, including destructuring λ and let. These, as well as notation 2 → 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 type

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 type

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 type

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 type

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 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 type

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 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 type -> (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 x)

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 type

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 : 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 type -> (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 x)

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 type 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