feat(library/init): add basic algebra

library/init/algebra.lean

@@ -0,0 +1,57 @@
+/-
+Copyright (c) 2016 Microsoft Corporation. All rights reserved.
+Released under Apache 2.0 license as described in the file LICENSE.
+Authors: Leonardo de Moura
+-/
+prelude
+import init.logic init.binary init.combinator init.meta.interactive
+
+universe variable u
+
+class semigroup (A : Type u) extends has_mul A :=
+(mul_assoc : ∀ a b c : A, a * b * c = a * (b * c))
+
+variables {A : Type u}
+
+@[simp] lemma mul_assoc [semigroup A] : ∀ a b c : A, a * b * c = a * (b * c) :=
+semigroup.mul_assoc
+
+class comm_semigroup (A : Type u) extends semigroup A :=
+(mul_comm : ∀ a b : A, a * b = b * a)
+
+@[simp] lemma mul_comm [comm_semigroup A] : ∀ a b : A, a * b = b * a :=
+comm_semigroup.mul_comm
+
+@[simp] lemma mul_left_comm [comm_semigroup A] : ∀ a b c : A, a * (b * c) = b * (a * c) :=
+left_comm mul mul_comm mul_assoc
+
+class left_cancel_semigroup (A : Type u) extends semigroup A :=
+(mul_left_cancel : ∀ a b c : A, a * b = a * c → b = c)
+
+lemma mul_left_cancel [left_cancel_semigroup A] {a b c : A} : a * b = a * c → b = c :=
+left_cancel_semigroup.mul_left_cancel a b c
+
+class right_cancel_semigroup (A : Type u) extends semigroup A :=
+(mul_right_cancel : ∀ a b c : A, a * b = c * b → a = c)
+
+lemma mul_right_cancel [right_cancel_semigroup A] {a b c : A} : a * b = c * b → a = c :=
+right_cancel_semigroup.mul_right_cancel a b c
+
+class monoid (A : Type u) extends semigroup A, has_one A :=
+(one_mul : ∀ a : A, 1 * a = a) (mul_one : ∀ a : A, a * 1 = a)
+
+@[simp] lemma one_mul [monoid A] : ∀ a : A, 1 * a = a :=
+monoid.one_mul
+
+@[simp] lemma mul_one [monoid A] : ∀ a : A, a * 1 = a :=
+monoid.mul_one
+
+class comm_monoid (A : Type u) extends monoid A, comm_semigroup A
+
+class group (A : Type u) extends monoid A, has_inv A :=
+(mul_left_inv : ∀ a : A, a⁻¹ * a = 1)
+
+@[simp] lemma mul_left_inv [group A] : ∀ a : A, a⁻¹ * a = 1 :=
+group.mul_left_inv
+
+class comm_group (A : Type u) extends group A, comm_monoid A

library/init/default.lean

@@ -12,4 +12,4 @@ import init.monad init.option init.state init.fin init.list init.char init.strin
 import init.monad_combinators init.set
 import init.timeit init.trace init.unsigned init.ordering init.list_classes init.coe
 import init.wf init.nat_div init.meta init.instances
-import init.sigma_lex init.simplifier init.id_locked
+import init.sigma_lex init.simplifier init.id_locked init.algebra

tests/lean/run/group.lean

@@ -1,38 +0,0 @@
-section
-  variable {A  : Type*}
-  variable f   : A → A → A
-  variable one : A
-  variable inv : A → A
-  local infixl `*`   := f
-  local 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 [class] 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
-
-inductive group : Type*
-| mk_group : Π (A : Type*), group_struct A → group
-
-definition carrier (g : group) : Type*
-:= group.rec (λ c s, c) g
-
-attribute [instance]
-definition group_to_struct (g : group) : group_struct (carrier g)
-:= group.rec (λ (A : Type*) (s : group_struct A), s) g
-
-check group_struct
-
-definition mul1 {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 `*` := mul1
-
-constant  G1 : group.{1}
-constant  G2 : group.{1}
-constants a b c : (carrier G2)
-constants d e : (carrier G1)
-check a * b * b
-check d * e

tests/lean/run/record10.lean

@@ -1,6 +1,3 @@
-structure [class] semigroup (A : Type) extends has_mul A :=
-(assoc : ∀ a b c, mul (mul a b) c = mul a (mul b c))
-
 print prefix semigroup
 
 print "======================="

tests/lean/run/simplifier_canonize_instances.lean

@@ -11,8 +11,6 @@ do (new_target, Heq) ← target >>= simplify failed [],
    try reflexivity
 
 universe l
-constants (group : Type* → Type)
-attribute [class] group
 
 constants (f₁ : Π (X : Type*) (X_grp : group X), X)
 constants (f₂ : Π (X : Type*) [X_grp : group X], X)

tests/lean/run/st_options.lean

@@ -1,59 +0,0 @@
-structure [class] semigroup (A : Type*) extends has_mul A :=
-(mul_assoc : ∀a b c, mul (mul a b) c = mul a (mul b c))
-
-structure [class] comm_semigroup (A : Type*) extends semigroup A :=
-(mul_comm : ∀a b, mul a b = mul b a)
-
-structure [class] left_cancel_semigroup (A : Type*) extends semigroup A :=
-(mul_left_cancel : ∀a b c, mul a b = mul a c → b = c)
-
-structure [class] right_cancel_semigroup (A : Type*) extends semigroup A :=
-(mul_right_cancel : ∀a b c, mul a b = mul c b → a = c)
-
-structure [class] add_semigroup (A : Type*) extends has_add A :=
-(add_assoc : ∀a b c, add (add a b) c = add a (add b c))
-
-structure [class] add_comm_semigroup (A : Type*) extends add_semigroup A :=
-(add_comm : ∀a b, add a b = add b a)
-
-structure [class] add_left_cancel_semigroup (A : Type*) extends add_semigroup A :=
-(add_left_cancel : ∀a b c, add a b = add a c → b = c)
-
-structure [class] add_right_cancel_semigroup (A : Type*) extends add_semigroup A :=
-(add_right_cancel : ∀a b c, add a b = add c b → a = c)
-
-structure [class] monoid (A : Type*) extends semigroup A, has_one A :=
-(one_mul : ∀a, mul one a = a) (mul_one : ∀a, mul a one = a)
-
-structure [class] comm_monoid (A : Type*) extends monoid A, comm_semigroup A
-
-structure [class] add_monoid (A : Type*) extends add_semigroup A, has_zero A :=
-(zero_add : ∀a, add zero a = a) (add_zero : ∀a, add a zero = a)
-
-structure [class] add_comm_monoid (A : Type*) extends add_monoid A, add_comm_semigroup A
-
-structure [class] group (A : Type*) extends monoid A, has_inv A :=
-(mul_left_inv : ∀a, mul (inv a) a = one)
-
-structure [class] comm_group (A : Type*) extends group A, comm_monoid A
-
-structure [class] add_group (A : Type*) extends add_monoid A, has_neg A :=
-(add_left_inv : ∀a, add (neg a) a = zero)
-
-structure [class] add_comm_group (A : Type*) extends add_group A, add_comm_monoid A
-
-structure [class] distrib (A : Type*) extends has_mul A, has_add A :=
-(left_distrib : ∀a b c, mul a (add b c) = add (mul a b) (mul a c))
-(right_distrib : ∀a b c, mul (add a b) c = add (mul a c) (mul b c))
-
-structure [class] mul_zero_class (A : Type*) extends has_mul A, has_zero A :=
-(zero_mul : ∀a, mul zero a = zero)
-(mul_zero : ∀a, mul a zero = zero)
-
-structure [class] zero_ne_one_class (A : Type*) extends has_zero A, has_one A :=
-(zero_ne_one : zero ≠ one)
-
-structure [class] semiring (A : Type*) extends add_comm_monoid A, monoid A, distrib A,
-    mul_zero_class A, zero_ne_one_class A
-
-set_option pp.implicit true

tests/lean/struct_class.lean

@@ -1,3 +1,6 @@
+prelude
+import init.core
+
 structure [class] point (A : Type*) (B : Type*) :=
 mk :: (x : A) (y : B)
 

tests/lean/struct_class.lean.expected.out

@@ -1,14 +1,7 @@
-alternative : (Type u → Type v) → Type (max (u+1) v)
-applicative : (Type u → Type v) → Type (max (u+1) v)
 decidable : Prop → Type
-functor : (Type u → Type v) → Type (max (u+1) v)
 has_add : Type u → Type (max 1 u)
 has_andthen : Type u → Type (max 1 u)
 has_append : Type u → Type (max 1 u)
-has_coe : Type u → Type v → Type (max 1 (imax u v))
-has_coe_t : Type u → Type v → Type (max 1 (imax u v))
-has_coe_to_fun : Type u → Type (max u (v+1))
-has_coe_to_sort : Type u → Type (max u (v+1))
 has_div : Type u → Type (max 1 u)
 has_dvd : Type u → Type (max 1 u)
 has_emptyc : Type u → Type (max 1 u)
@@ -16,45 +9,25 @@ has_insert : Type u → (Type u → Type v) → Type (max 1 (imax u v))
 has_inter : Type u → Type (max 1 u)
 has_inv : Type u → Type (max 1 u)
 has_le : Type u → Type (max 1 u)
-has_lift : Type u → Type v → Type (max 1 (imax u v))
-has_lift_t : Type u → Type v → Type (max 1 (imax u v))
 has_lt : Type u → Type (max 1 u)
 has_mem : Type u → (Type u → Type v) → Type (max 1 u v)
 has_mod : Type u → Type (max 1 u)
 has_mul : Type u → Type (max 1 u)
 has_neg : Type u → Type (max 1 u)
 has_one : Type u → Type (max 1 u)
-has_ordering : Type → Type
 has_sdiff : Type u → Type (max 1 u)
 has_sep : Type u → (Type u → Type v) → Type (max 1 (imax (max 1 u) v))
 has_sizeof : Type u → Type (max 1 u)
 has_ssubset : Type u → Type (max 1 u)
 has_sub : Type u → Type (max 1 u)
 has_subset : Type u → Type (max 1 u)
-has_to_format : Type u → Type (max 1 u)
-has_to_pexpr : Type u → Type (max 1 u)
-has_to_string : Type u → Type (max 1 u)
-has_to_tactic_format : Type → Type
 has_union : Type u → Type (max 1 u)
 has_zero : Type u → Type (max 1 u)
-inhabited : Type u → Type (max 1 u)
-is_associative : Π {A : Type u}, (A → A → A) → Type
-monad : (Type u → Type v) → Type (max (u+1) v)
-nonempty : Type u → Prop
 point : Type u_1 → Type u_2 → Type (max 1 u_1 u_2)
-setoid : Type u → Type (max 1 u)
-subsingleton : Type u → Prop
-alternative : (Type u → Type v) → Type (max (u+1) v)
-applicative : (Type u → Type v) → Type (max (u+1) v)
 decidable : Prop → Type
-functor : (Type u → Type v) → Type (max (u+1) v)
 has_add : Type u → Type (max 1 u)
 has_andthen : Type u → Type (max 1 u)
 has_append : Type u → Type (max 1 u)
-has_coe : Type u → Type v → Type (max 1 (imax u v))
-has_coe_t : Type u → Type v → Type (max 1 (imax u v))
-has_coe_to_fun : Type u → Type (max u (v+1))
-has_coe_to_sort : Type u → Type (max u (v+1))
 has_div : Type u → Type (max 1 u)
 has_dvd : Type u → Type (max 1 u)
 has_emptyc : Type u → Type (max 1 u)
@@ -62,31 +35,18 @@ has_insert : Type u → (Type u → Type v) → Type (max 1 (imax u v))
 has_inter : Type u → Type (max 1 u)
 has_inv : Type u → Type (max 1 u)
 has_le : Type u → Type (max 1 u)
-has_lift : Type u → Type v → Type (max 1 (imax u v))
-has_lift_t : Type u → Type v → Type (max 1 (imax u v))
 has_lt : Type u → Type (max 1 u)
 has_mem : Type u → (Type u → Type v) → Type (max 1 u v)
 has_mod : Type u → Type (max 1 u)
 has_mul : Type u → Type (max 1 u)
 has_neg : Type u → Type (max 1 u)
 has_one : Type u → Type (max 1 u)
-has_ordering : Type → Type
 has_sdiff : Type u → Type (max 1 u)
 has_sep : Type u → (Type u → Type v) → Type (max 1 (imax (max 1 u) v))
 has_sizeof : Type u → Type (max 1 u)
 has_ssubset : Type u → Type (max 1 u)
 has_sub : Type u → Type (max 1 u)
 has_subset : Type u → Type (max 1 u)
-has_to_format : Type u → Type (max 1 u)
-has_to_pexpr : Type u → Type (max 1 u)
-has_to_string : Type u → Type (max 1 u)
-has_to_tactic_format : Type → Type
 has_union : Type u → Type (max 1 u)
 has_zero : Type u → Type (max 1 u)
-inhabited : Type u → Type (max 1 u)
-is_associative : Π {A : Type u}, (A → A → A) → Type
-monad : (Type u → Type v) → Type (max (u+1) v)
-nonempty : Type u → Prop
 point : Type u_1 → Type u_2 → Type (max 1 u_1 u_2)
-setoid : Type u → Type (max 1 u)
-subsingleton : Type u → Prop