lean2/tests/lean/run/group2.lean
2014-07-07 16:09:30 -07:00

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Text

import standard
using num
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
inductive group_struct (A : Type) : Type :=
| mk_group_struct : Π (mul : A → A → A) (one : A) (inv : A → A), is_assoc mul → is_id mul one → is_inv mul one inv → group_struct A
inductive group : Type :=
| mk_group : Π (A : Type), group_struct A → group
definition carrier (g : group) : Type
:= group_rec (λ c s, c) g
coercion carrier
definition group_to_struct [instance] (g : group) : group_struct (carrier g)
:= group_rec (λ (A : Type) (s : group_struct A), s) g
check group_struct
definition mul [inline] {A : Type} {s : group_struct A} (a b : A) : A
:= group_struct_rec (λ mul one inv h1 h2 h3, mul) s a b
infixl `*`:75 := mul
section
variable G1 : group
variable G2 : group
variables a b c : G2
variables d e : G1
check a * b * b
check d * e
end
variable G : group.{1}
variables a b : G
definition val : G := a*b
check val
variable pos_real : Type.{1}
variable rmul : pos_real → pos_real → pos_real
variable rone : pos_real
variable rinv : pos_real → pos_real
axiom H1 : is_assoc rmul
axiom H2 : is_id rmul rone
axiom H3 : is_inv rmul rone rinv
definition real_group_struct [inline] [instance] : group_struct pos_real
:= mk_group_struct rmul rone rinv H1 H2 H3
variables x y : pos_real
check x * y
theorem T (a b : pos_real): (rmul a b) = a*b
:= refl (rmul a b)