/-
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Authors: Leonardo de Moura, Jeremy Avigad, 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)
structure has_mul  [class] (A : Type) := (mul : A → A → A)
structure has_inv  [class] (A : Type) := (inv : A → A)
structure has_neg  [class] (A : Type) := (neg : A → A)
structure has_sub  [class] (A : Type) := (sub : A → A → A)
structure has_div  [class] (A : Type) := (div : A → A → A)
structure has_dvd  [class] (A : Type) := (dvd : A → A → Prop)
structure has_mod  [class] (A : Type) := (mod : A → A → A)
structure has_le   [class] (A : Type) := (le : A → A → Prop)
structure has_lt   [class] (A : Type) := (lt : A → A → Prop)

definition zero {A : Type} [s : has_zero A] : A            := has_zero.zero A
definition one  {A : Type} [s : has_one A]  : A            := has_one.one A
definition add  {A : Type} [s : has_add A]  : A → A → A    := has_add.add
definition mul  {A : Type} [s : has_mul A]  : A → A → A    := has_mul.mul
definition sub  {A : Type} [s : has_sub A]  : A → A → A    := has_sub.sub
definition div  {A : Type} [s : has_div A]  : A → A → A    := has_div.div
definition dvd  {A : Type} [s : has_dvd A]  : A → A → Prop := has_dvd.dvd
definition mod  {A : Type} [s : has_mod A]  : A → A → A    := has_mod.mod
definition neg  {A : Type} [s : has_neg A]  : A → A        := has_neg.neg
definition inv  {A : Type} [s : has_inv A]  : A → A        := has_inv.inv
definition le   {A : Type} [s : has_le A]   : A → A → Prop := has_le.le
definition lt   {A : Type} [s : has_lt A]   : A → A → Prop := has_lt.lt

definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Prop := le b a
definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Prop := lt b a
definition bit0 {A : Type} [s  : has_add A] (a  : A)                 : A := add a a
definition bit1 {A : Type} [s₁ : has_one A] [s₂ : has_add A] (a : A) : A := add (bit0 a) one

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

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

definition pos_num_has_one [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 [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 [instance] : has_add num :=
has_add.mk num.add

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

namespace nat
  protected definition prio := num.add std.priority.default 100

  protected definition add (a b : nat) : nat :=
  nat.rec a (λ b₁ r, succ r) b

  definition of_num (n : num) : nat :=
  num.rec zero
    (λ n, pos_num.rec (succ zero) (λ n r, nat.add (nat.add r r) (succ zero)) (λ n r, nat.add r r) n) n
end nat

attribute pos_num_has_add pos_num_has_one num_has_zero num_has_one num_has_add
          [instance] [priority nat.prio]

definition nat_has_zero [instance] [priority nat.prio] : has_zero nat :=
has_zero.mk nat.zero

definition nat_has_one [instance] [priority nat.prio] : has_one nat :=
has_one.mk (nat.succ (nat.zero))

definition nat_has_add [instance] [priority nat.prio] : has_add nat :=
has_add.mk nat.add

/-
  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
reserve infixr ` ▹ `:75

/- types and type constructors -/

reserve infixl ` ⊎ `:25
reserve infixl ` × `:30

/- arithmetic operations -/

reserve infixl ` + `:65
reserve infixl ` - `:65
reserve infixl ` * `:70
reserve infixl ` / `: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
reserve infix ` ' `:75   -- for the image of a set under a function
reserve infix ` '- `:75  -- for the preimage of a set under a function

/- other symbols -/

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

infix +    := add
infix *    := mul
infix -    := sub
infix /    := div
infix ∣    := dvd
infix %    := mod
prefix -   := neg
postfix ⁻¹ := inv
infix ≤    := le
infix ≥    := ge
infix <    := lt
infix >    := gt

notation [parsing_only] x ` +[`:65 A:0 `] `:0 y:65 := @add A _ x y
notation [parsing_only] x ` -[`:65 A:0 `] `:0 y:65 := @sub A _ x y
notation [parsing_only] x ` *[`:70 A:0 `] `:0 y:70 := @mul A _ x y
notation [parsing_only] x ` /[`:70 A:0 `] `:0 y:70 := @div A _ x y
notation [parsing_only] x ` ∣[`:70 A:0 `] `:0 y:70 := @dvd A _ x y
notation [parsing_only] x ` %[`:70 A:0 `] `:0 y:70 := @mod A _ x y
notation [parsing_only] x ` ≤[`:50 A:0 `] `:0 y:50 := @le A _ x y
notation [parsing_only] x ` ≥[`:50 A:0 `] `:0 y:50 := @ge A _ x y
notation [parsing_only] x ` <[`:50 A:0 `] `:0 y:50 := @lt A _ x y
notation [parsing_only] x ` >[`:50 A:0 `] `:0 y:50 := @gt A _ x y