/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Floris van Doorn
-/
prelude
import init.datatypes

notation `assume` binders `,` r:(scoped f, f) := r
notation `take`   binders `,` r:(scoped f, f) := r

structure has_zero [class] (A : Type) := (zero : A)
structure has_one [class] (A : Type) := (one : A)
structure has_add [class] (A : Type)  := (add : A → A → A)

definition zero [reducible] {A : Type} [s : has_zero A] : A := has_zero.zero A
definition one  [reducible] {A : Type} [s : has_one A] : A := has_one.one A
definition add  [reducible] {A : Type} [s : has_add A] : A → A → A := has_add.add
definition bit0 [reducible] {A : Type} [s  : has_add A] (a  : A)                 : A := add a a
definition bit1 [reducible] {A : Type} [s₁ : has_one A] [s₂ : has_add A] (a : A) : A := add (bit0 a) one

definition num_has_zero [reducible] [instance] : has_zero num :=
has_zero.mk num.zero

definition num_has_one [reducible] [instance] : has_one num :=
has_one.mk (num.pos pos_num.one)

definition pos_num_has_one [reducible] [instance] : has_one pos_num :=
has_one.mk (pos_num.one)

namespace pos_num
  open bool
  definition is_one (a : pos_num) : bool :=
  pos_num.rec_on a tt (λn r, ff) (λn r, ff)

  definition pred (a : pos_num) : pos_num :=
  pos_num.rec_on a one (λn r, bit0 n) (λn r, bool.rec_on (is_one n) (bit1 r) one)

  definition size (a : pos_num) : pos_num :=
  pos_num.rec_on a one (λn r, succ r) (λn r, succ r)

  definition add (a b : pos_num) : pos_num :=
  pos_num.rec_on a
    succ
    (λn f b, pos_num.rec_on b
      (succ (bit1 n))
      (λm r, succ (bit1 (f m)))
      (λm r, bit1 (f m)))
    (λn f b, pos_num.rec_on b
      (bit1 n)
      (λm r, bit1 (f m))
      (λm r, bit0 (f m)))
    b
end pos_num

definition pos_num_has_add [reducible] [instance] : has_add pos_num :=
has_add.mk pos_num.add

namespace num
  open pos_num

  definition add (a b : num) : num :=
  num.rec_on a b (λpa, num.rec_on b (pos pa) (λpb, pos (pos_num.add pa pb)))
end num

definition num_has_add [reducible] [instance] : has_add num :=
has_add.mk num.add

definition std.priority.default : num := 1000
definition std.priority.max     : num := 4294967295

/-
  Global declarations of right binding strength

  If a module reassigns these, it will be incompatible with other modules that adhere to these
  conventions.

  When hovering over a symbol, use "C-c C-k" to see how to input it.
-/
definition std.prec.max   : num := 1024 -- the strength of application, identifiers, (, [, etc.
definition std.prec.arrow : num := 25

/-
The next definition is "max + 10". It can be used e.g. for postfix operations that should
be stronger than application.
-/

definition std.prec.max_plus :=
num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ
  (num.succ std.prec.max)))))))))

/- Logical operations and relations -/

reserve prefix `¬`:40
reserve prefix `~`:40
reserve infixr ` ∧ `:35
reserve infixr ` /\ `:35
reserve infixr ` \/ `:30
reserve infixr ` ∨ `:30
reserve infix ` <-> `:20
reserve infix ` ↔ `:20
reserve infix ` = `:50
reserve infix ` ≠ `:50
reserve infix ` ≈ `:50
reserve infix ` ~ `:50
reserve infix ` ≡ `:50

reserve infixr ` ∘ `:60                   -- input with \comp
reserve postfix `⁻¹`:std.prec.max_plus  -- input with \sy or \-1 or \inv

reserve infixl ` ⬝ `:75
reserve infixr ` ▸ `:75

/- types and type constructors -/

reserve infixr ` ⊎ `:25
reserve infixr ` × `:30

/- arithmetic operations -/

reserve infixl ` + `:65
reserve infixl ` - `:65
reserve infixl ` * `:70
reserve infixl ` div `:70
reserve infixl ` mod `:70
reserve infixl ` / `:70
reserve prefix ` - `:100
reserve infix ` ^ `:80

reserve infix ` <= `:50
reserve infix ` ≤ `:50
reserve infix ` < `:50
reserve infix ` >= `:50
reserve infix ` ≥ `:50
reserve infix ` > `:50

/- boolean operations -/

reserve infixl ` && `:70
reserve infixl ` || `:65

/- set operations -/

reserve infix ` ∈ `:50
reserve infix ` ∉ `:50
reserve infixl ` ∩ `:70
reserve infixl ` ∪ `:65
reserve infix ` ⊆ `:50
reserve infix ` ⊇ `:50

/- other symbols -/

reserve infix ` ∣ `:50
reserve infixl ` ++ `:65
reserve infixr ` :: `:67