lean2/library/init/reserved_notation.lean
2015-11-08 14:04:55 -08:00

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/-
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
-/
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_division [class] (A : Type) := (division : A → A → A)
structure has_divide [class] (A : Type) := (divide : A → A → A)
structure has_modulo [class] (A : Type) := (modulo : A → A → A)
structure has_dvd [class] (A : Type) := (dvd : A → A → Prop)
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 division {A : Type} [s : has_division A] : A → A → A := has_division.division
definition divide {A : Type} [s : has_divide A] : A → A → A := has_divide.divide
definition modulo {A : Type} [s : has_modulo A] : A → A → A := has_modulo.modulo
definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
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 [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
namespace nat
protected definition prio := num.add std.priority.default 100
protected definition add (a b : nat) : nat :=
nat.rec_on b a (λ b₁ r, succ r)
definition nat_has_zero [reducible] [instance] : has_zero nat :=
has_zero.mk nat.zero
definition nat_has_one [reducible] [instance] : has_one nat :=
has_one.mk (nat.succ (nat.zero))
definition nat_has_add [reducible] [instance] [priority nat.prio] : has_add nat :=
has_add.mk nat.add
definition of_num (n : num) : nat :=
num.rec zero
(λ n, pos_num.rec (succ zero) (λ n r, add (add r r) (succ zero)) (λ n r, add r r) n) n
end nat
/-
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 ` 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 ` :: `:65
infix + := add
infix * := mul
infix - := sub
infix / := division
infix div := divide
infix := dvd
infix mod := modulo
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 := @division A _ x y
notation [parsing_only] x ` div[`:70 A:0 `] `:0 y:70 := @divide A _ x y
notation [parsing_only] x ` mod[`:70 A:0 `] `:0 y:70 := @modulo 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