lean2/tests/lean/run/algebra1.lean

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import logic
definition Type1 := Type.{1}
context
variable {A : Type}
variable f : A → A → A
variable one : A
variable inv : A → A
infixl `*` := f
postfix `^-1`:100 := inv
definition is_assoc := ∀ a b c, (a*b)*c = a*b*c
definition is_id := ∀ a, a*one = a
definition is_inv := ∀ a, a*a^-1 = one
end
namespace algebra
inductive mul_struct [class] (A : Type) : Type :=
mk : (A → A → A) → mul_struct A
inductive add_struct [class] (A : Type) : Type :=
mk : (A → A → A) → add_struct A
definition mul {A : Type} [s : mul_struct A] (a b : A)
:= mul_struct.rec (fun f, f) s a b
infixl `*` := mul
definition add {A : Type} [s : add_struct A] (a b : A)
:= add_struct.rec (fun f, f) s a b
infixl `+` := add
end algebra
open algebra
inductive nat : Type :=
zero : nat,
succ : nat → nat
namespace nat
constant add : nat → nat → nat
constant mul : nat → nat → nat
definition is_mul_struct [instance] : algebra.mul_struct nat
:= algebra.mul_struct.mk mul
definition is_add_struct [instance] : algebra.add_struct nat
:= algebra.add_struct.mk add
definition to_nat (n : num) : nat
:= #algebra
num.rec nat.zero (λ n, pos_num.rec (succ zero) (λ n r, r + r) (λ n r, r + r + succ zero) n) n
end nat
namespace algebra
namespace semigroup
inductive semigroup_struct [class] (A : Type) : Type :=
mk : Π (mul : A → A → A), is_assoc mul → semigroup_struct A
definition mul {A : Type} (s : semigroup_struct A) (a b : A)
:= semigroup_struct.rec (fun f h, f) s a b
definition assoc {A : Type} (s : semigroup_struct A) : is_assoc (mul s)
:= semigroup_struct.rec (fun f h, h) s
definition is_mul_struct [instance] (A : Type) (s : semigroup_struct A) : mul_struct A
:= mul_struct.mk (mul s)
inductive semigroup : Type :=
mk : Π (A : Type), semigroup_struct A → semigroup
definition carrier [coercion] (g : semigroup)
:= semigroup.rec (fun c s, c) g
definition is_semigroup [instance] (g : semigroup) : semigroup_struct (carrier g)
:= semigroup.rec (fun c s, s) g
end semigroup
namespace monoid
check semigroup.mul
inductive monoid_struct [class] (A : Type) : Type :=
mk_monoid_struct : Π (mul : A → A → A) (id : A), is_assoc mul → is_id mul id → monoid_struct A
definition mul {A : Type} (s : monoid_struct A) (a b : A)
:= monoid_struct.rec (fun mul id a i, mul) s a b
definition assoc {A : Type} (s : monoid_struct A) : is_assoc (mul s)
:= monoid_struct.rec (fun mul id a i, a) s
open semigroup
definition is_semigroup_struct [instance] (A : Type) (s : monoid_struct A) : semigroup_struct A
:= semigroup_struct.mk (mul s) (assoc s)
inductive monoid : Type :=
mk_monoid : Π (A : Type), monoid_struct A → monoid
definition carrier [coercion] (m : monoid)
:= monoid.rec (fun c s, c) m
definition is_monoid [instance] (m : monoid) : monoid_struct (carrier m)
:= monoid.rec (fun c s, s) m
end monoid
end algebra
section
open algebra algebra.semigroup algebra.monoid
variable M : monoid
variables a b c : M
check a*b*c*a*b*c*a*b*a*b*c*a
check a*b
end