short guide contributed by a community member illustrates how to prove associativity of addition on Nat using the interactive theorem prover. First we define a module Foo.idr module Foo plusAssoc : plus the initial context. This looks as follows: *Foo> :p rhs Goal: { hole 0 }: (n : Nat) -> (m : Nat) -> (o : Nat) -> plus n (plus m o) = plus (plus n m) o ### 2.2 Application of Intros We first tactic: --Foo.rhs> intros Other goals: { hole 2 } { hole 1 } { hole 0 } Assumptions: n : Nat m : Nat o : Nat Goal: { hole 3 }: plus n (plus m o) = plus (plus n m) o ### 2.3 Induction on n Then apply

short guide contributed by a community member illustrates how to prove associativity of addition on Nat using the interactive theorem prover. First we define a module Foo.idr module Foo plusAssoc : plus the initial context. This looks as follows: *Foo> :p rhs Goal: { hole 0 }: (n : Nat) -> (m : Nat) -> (o : Nat) -> plus n (plus m o) = plus (plus n m) o ### 2.2 Application of Intros We first tactic: --Foo.rhs> intros Other goals: { hole 2 } { hole 1 } { hole 0 } Assumptions: n : Nat m : Nat o : Nat Goal: { hole 3 }: plus n (plus m o) = plus (plus n m) o ### 2.3 Induction on n Then apply

short guide contributed by a community member illustrates how to prove associativity of addition on Nat using the interactive theorem prover. First we define a module Foo.idr module Foo plusAssoc : plus the initial context. This looks as follows: *Foo> :p rhs Goal: { hole 0 }: (n : Nat) -> (m : Nat) -> (o : Nat) -> plus n (plus m o) = plus (plus n m) o ### 2.2 Application of Intros We first tactic: --Foo.rhs> intros Other goals: { hole 2 } { hole 1 } { hole 0 } Assumptions: n : Nat m : Nat o : Nat Goal: { hole 3 }: plus n (plus m o) = plus (plus n m) o ### 2.3 Induction on n Then apply

various ways, e.g. as difference of two natural numbers: module Integer where abstract = Nat × Nat 0 : 0 = 0, 0 1 : 1 = 1, 0 +_: (x y :) → (p, n) + (p' can see through the abstractions of their uncles: module Where where abstract x : Nat x = 0 y : Nat y = x where xy : x 0 xy = refl Type signatures in where blocks abstract definitions: module WherePrivate where abstract x : Nat x = proj $ _{1} $ t where T = Nat $ \times $ Nat t : T t = 0, 1 p : proj $ _{1} $ t 0

various ways, e.g. as difference of two natural numbers: module Integer where abstract = Nat × Nat 0 : 0 = 0, 0 1 : 1 = 1, 0 +_: (x y :) → (p, n) + (p' can see through the abstractions of their uncles: module Where where abstract x : Nat x = 0 y : Nat y = x where xy : x 0 xy = refl Type signatures in where blocks abstract definitions: module WherePrivate where abstract x : Nat x = proj $ _{1} $ t where T = Nat $ \times $ Nat t : T t = 0, 1 p : proj $ _{1} $ t 0

various ways, e.g. as difference of two natural numbers: module Integer where abstract = Nat × Nat 0 : 0 = 0, 0 1 : 1 = 1, 0 +_: (x y :) → (p, n) + (p' can see through the abstractions of their uncles: module Where where abstract x : Nat x = 0 y : Nat y = x where xy : x 0 xy = refl Type signatures in where blocks abstract definitions: module WherePrivate where abstract x : Nat x = proj $ _{1} $ t where T = Nat $ \times $ Nat t : T t = 0, 1 p : proj $ _{1} $ t 0

in various ways, e.g. as difference of two natural numbers: module Integer where abstract Z = Nat × Nat 0Z : Z 0Z = 0, 0 1Z : Z 1Z = 1, 0 +Z : (x y : Z) → Z (p, n) +Z (p', n') = (p + p') can see through the abstractions of their uncles: module Where where abstract x : Nat x = 0 y : Nat y = x where x ≡ y : x ≡ 0 x ≡ y = refl Type signatures in where where blocks of abstract definitions: module WherePrivate where abstract x : Nat x = proj₁ t where T = Nat × Nat t : T t = ℤ, 1 p : proj₁ t ≡ ℤ p = refl Note that if p was not

can define the type vec n of vectors of length n. This is a family of types indexed by objects in Nat (a type parameterized by natural numbers). Having dependent types, we must generalize the type of is then a function of type identity : (n : Nat) -> (Mat n n). Remark: We could, of course, just specify the identity function with the type Nat -> Nat -> Mat, where Mat is the type of matrices to define: • equality on natural numbers • properties of arithmetical operations - the type (n : Nat) -> (PrimRoot n) consisting of functions computing primitive root in modular arithmetic. Of course

can define the type vec n of vectors of length n. This is a family of types indexed by objects in Nat (a type parameterized by natural numbers). Having dependent types, we must generalize the type of is then a function of type identity : (n : Nat) -> (Mat n n). Remark: We could, of course, just specify the identity function with the type Nat -> Nat -> Mat, where Mat is the type of matrices to define: • equality on natural numbers • properties of arithmetical operations - the type (n : Nat) -> (PrimRoot n) consisting of functions computing primitive root in modular arithmetic. Of course

in various ways, e.g. as difference of two natural numbers: module Integer where abstract Z = Nat × Nat 0Z : Z 0Z = 0, 0 1Z : Z 1Z = 1, 0 +Z : (x y : Z) → Z (p, n) +Z (p', n') = (p + p') can see through the abstractions of their uncles: module Where where abstract x : Nat x = 0 y : Nat y = x where x ≡ y : x ≡ 0 x ≡ y = refl Type signatures in where where blocks of abstract definitions: module WherePrivate where abstract x : Nat x = proj₁ t where T = Nat × Nat t : T t = ℤ, 1 p : proj₁ t ≡ ℤ p = refl Note that if p was not