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