55fee26b36
For more information see: https://github.com/leanprover/lean/wiki/Refactoring-structures The new attribute [algebra] implements the [algebraic_class] described in the page above.
66 lines
2.8 KiB
Text
66 lines
2.8 KiB
Text
/-
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Copyright (c) 2017 Microsoft Corporation. All rights reserved.
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Released under Apache 2.0 license as described in the file LICENSE.
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Authors: Leonardo de Moura
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-/
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prelude
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import init.logic
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universes u
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@[algebra] class is_commutative (α : Type u) (op : α → α → α) : Prop :=
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(comm : ∀ a b, op a b = op b a)
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@[algebra] class is_associative (α : Type u) (op : α → α → α) : Prop :=
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(assoc : ∀ a b c, op (op a b) c = op a (op b c))
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@[algebra] class is_left_id (α : Type u) (op : α → α → α) (o : inout α) : Prop :=
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(left_id : ∀ a, op o a = a)
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@[algebra] class is_right_id (α : Type u) (op : α → α → α) (o : inout α) : Prop :=
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(right_id : ∀ a, op a o = a)
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@[algebra] class is_left_null (α : Type u) (op : α → α → α) (o : inout α) : Prop :=
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(left_null : ∀ a, op o a = o)
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@[algebra] class is_right_null (α : Type u) (op : α → α → α) (o : inout α) : Prop :=
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(right_null : ∀ a, op a o = o)
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@[algebra] class is_left_cancel (α : Type u) (op : α → α → α) : Prop :=
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(left_cancel : ∀ a b c, op a b = op a c → b = c)
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@[algebra] class is_right_cancel (α : Type u) (op : α → α → α) : Prop :=
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(right_cancel : ∀ a b c, op a b = op c b → a = c)
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@[algebra] class is_idempotent (α : Type u) (op : α → α → α) : Prop :=
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(idempotent : ∀ a, op a a = a)
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@[algebra] class is_left_distrib (α : Type u) (op₁ : α → α → α) (op₂ : inout α → α → α) : Prop :=
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(left_distrib : ∀ a b c, op₁ a (op₂ b c) = op₂ (op₁ a b) (op₁ a c))
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@[algebra] class is_right_distrib (α : Type u) (op₁ : α → α → α) (op₂ : inout α → α → α) : Prop :=
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(right_distrib : ∀ a b c, op₁ (op₂ a b) c = op₂ (op₁ a c) (op₁ b c))
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@[algebra] class is_left_inv (α : Type u) (op : α → α → α) (inv : inout α → α) (o : inout α) : Prop :=
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(left_inv : ∀ a, op (inv a) a = o)
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@[algebra] class is_right_inv (α : Type u) (op : α → α → α) (inv : inout α → α) (o : inout α) : Prop :=
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(right_inv : ∀ a, op a (inv a) = o)
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@[algebra] class is_cond_left_inv (α : Type u) (op : α → α → α) (inv : inout α → α) (o : inout α) (p : inout α → Prop) : Prop :=
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(left_inv : ∀ a, p a → op (inv a) a = o)
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@[algebra] class is_cond_right_inv (α : Type u) (op : α → α → α) (inv : inout α → α) (o : inout α) (p : inout α → Prop) : Prop :=
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(right_inv : ∀ a, p a → op a (inv a) = o)
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@[algebra] class is_distinct (α : Type u) (a : α) (b : α) : Prop :=
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(distinct : a ≠ b)
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/-
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-- The following type class doesn't seem very useful, a regular simp lemma should work for this.
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class is_inv (α : Type u) (β : Type v) (f : α → β) (g : inout β → α) : Prop :=
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(inv : ∀ a, g (f a) = a)
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-- The following one can also be handled using a regular simp lemma
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class is_idempotent (α : Type u) (f : α → α) : Prop :=
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(idempotent : ∀ a, f (f a) = f a)
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-/
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