65 lines
1.6 KiB
Text
65 lines
1.6 KiB
Text
import standard
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using num
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section
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parameter {A : Type}
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parameter f : A → A → A
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parameter one : A
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parameter inv : A → A
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infixl `*`:75 := f
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postfix `^-1`:100 := inv
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definition is_assoc := ∀ a b c, (a*b)*c = a*b*c
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definition is_id := ∀ a, a*one = a
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definition is_inv := ∀ a, a*a^-1 = one
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end
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inductive group_struct (A : Type) : Type :=
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| 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
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inductive group : Type :=
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| mk_group : Π (A : Type), group_struct A → group
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definition carrier (g : group) : Type
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:= group_rec (λ c s, c) g
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coercion carrier
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definition group_to_struct [instance] (g : group) : group_struct (carrier g)
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:= group_rec (λ (A : Type) (s : group_struct A), s) g
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check group_struct
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definition mul [inline] {A : Type} {s : group_struct A} (a b : A) : A
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:= group_struct_rec (λ mul one inv h1 h2 h3, mul) s a b
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infixl `*`:75 := mul
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section
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variable G1 : group
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variable G2 : group
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variables a b c : G2
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variables d e : G1
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check a * b * b
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check d * e
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end
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variable G : group.{1}
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variables a b : G
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definition val : G := a*b
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check val
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variable pos_real : Type.{1}
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variable rmul : pos_real → pos_real → pos_real
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variable rone : pos_real
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variable rinv : pos_real → pos_real
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axiom H1 : is_assoc rmul
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axiom H2 : is_id rmul rone
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axiom H3 : is_inv rmul rone rinv
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definition real_group_struct [inline] [instance] : group_struct pos_real
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:= mk_group_struct rmul rone rinv H1 H2 H3
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variables x y : pos_real
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check x * y
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theorem T (a b : pos_real): (rmul a b) = a*b
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:= refl (rmul a b)
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