111 lines
3.2 KiB
Text
111 lines
3.2 KiB
Text
|
import standard
|
||
|
|
using num
|
||
|
|
|
||
|
|
abbreviation Type1 := Type.{1}
|
||
|
|
|
||
|
|
section
|
||
|
|
parameter {A : Type}
|
||
|
|
parameter f : A → A → A
|
||
|
|
parameter one : A
|
||
|
|
parameter inv : A → A
|
||
|
|
infixl `*`:75 := 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 (A : Type) : Type :=
|
||
|
|
| mk_mul_struct : (A → A → A) → mul_struct A
|
||
|
|
|
||
|
|
inductive add_struct (A : Type) : Type :=
|
||
|
|
| mk_add_struct : (A → A → A) → add_struct A
|
||
|
|
|
||
|
|
definition mul [inline] {A : Type} {s : mul_struct A} (a b : A)
|
||
|
|
:= mul_struct_rec (fun f, f) s a b
|
||
|
|
|
||
|
|
infixl `*`:75 := mul
|
||
|
|
|
||
|
|
definition add [inline] {A : Type} {s : add_struct A} (a b : A)
|
||
|
|
:= add_struct_rec (fun f, f) s a b
|
||
|
|
|
||
|
|
infixl `+`:65 := add
|
||
|
|
end
|
||
|
|
|
||
|
|
namespace nat
|
||
|
|
inductive nat : Type :=
|
||
|
|
| zero : nat
|
||
|
|
| succ : nat → nat
|
||
|
|
|
||
|
|
variable add : nat → nat → nat
|
||
|
|
variable mul : nat → nat → nat
|
||
|
|
|
||
|
|
definition is_mul_struct [inline] [instance] : algebra.mul_struct nat
|
||
|
|
:= algebra.mk_mul_struct mul
|
||
|
|
|
||
|
|
definition is_add_struct [inline] [instance] : algebra.add_struct nat
|
||
|
|
:= algebra.mk_add_struct add
|
||
|
|
|
||
|
|
definition to_nat (n : num) : nat
|
||
|
|
:= #algebra
|
||
|
|
num_rec zero (λ n, pos_num_rec (succ zero) (λ n r, r + r) (λ n r, r + r + succ zero) n) n
|
||
|
|
end
|
||
|
|
|
||
|
|
namespace algebra
|
||
|
|
namespace semigroup
|
||
|
|
inductive semigroup_struct (A : Type) : Type :=
|
||
|
|
| mk_semigroup_struct : Π (mul : A → A → A), is_assoc mul → semigroup_struct A
|
||
|
|
|
||
|
|
definition mul [inline] {A : Type} (s : semigroup_struct A) (a b : A)
|
||
|
|
:= semigroup_struct_rec (fun f h, f) s a b
|
||
|
|
|
||
|
|
definition assoc [inline] {A : Type} (s : semigroup_struct A) : is_assoc (mul s)
|
||
|
|
:= semigroup_struct_rec (fun f h, h) s
|
||
|
|
|
||
|
|
definition is_mul_struct [inline] [instance] (A : Type) (s : semigroup_struct A) : mul_struct A
|
||
|
|
:= mk_mul_struct (mul s)
|
||
|
|
|
||
|
|
inductive semigroup : Type :=
|
||
|
|
| mk_semigroup : Π (A : Type), semigroup_struct A → semigroup
|
||
|
|
|
||
|
|
definition carrier [inline] [coercion] (g : semigroup)
|
||
|
|
:= semigroup_rec (fun c s, c) g
|
||
|
|
|
||
|
|
definition is_semigroup [inline] [instance] (g : semigroup) : semigroup_struct (carrier g)
|
||
|
|
:= semigroup_rec (fun c s, s) g
|
||
|
|
end
|
||
|
|
|
||
|
|
namespace monoid
|
||
|
|
inductive monoid_struct (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 [inline] {A : Type} (s : monoid_struct A) (a b : A)
|
||
|
|
:= monoid_struct_rec (fun mul id a i, mul) s a b
|
||
|
|
|
||
|
|
definition assoc [inline] {A : Type} (s : monoid_struct A) : is_assoc (mul s)
|
||
|
|
:= monoid_struct_rec (fun mul id a i, a) s
|
||
|
|
|
||
|
|
using algebra.semigroup -- Fix: allow user to write just semigroup
|
||
|
|
definition is_semigroup_struct [inline] [instance] (A : Type) (s : monoid_struct A) : semigroup_struct A
|
||
|
|
:= mk_semigroup_struct (mul s) (assoc s)
|
||
|
|
|
||
|
|
inductive monoid : Type :=
|
||
|
|
| mk_monoid : Π (A : Type), monoid_struct A → monoid
|
||
|
|
|
||
|
|
definition carrier [inline] [coercion] (m : monoid)
|
||
|
|
:= monoid_rec (fun c s, c) m
|
||
|
|
|
||
|
|
definition is_monoid [inline] [instance] (m : monoid) : monoid_struct (carrier m)
|
||
|
|
:= monoid_rec (fun c s, s) m
|
||
|
|
end
|
||
|
|
end
|
||
|
|
|
||
|
|
section
|
||
|
|
using 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
|