228 lines
7.5 KiB
Text
228 lines
7.5 KiB
Text
/-
|
||
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_mod [class] (A : Type) := (mod : A → A → A)
|
||
structure has_dvd.{l} [class] (A : Type.{l}) : Type.{l+1} := (dvd : A → A → Type.{l})
|
||
structure has_le.{l} [class] (A : Type.{l}) : Type.{l+1} := (le : A → A → Type.{l})
|
||
structure has_lt.{l} [class] (A : Type.{l}) : Type.{l+1} := (lt : A → A → Type.{l})
|
||
|
||
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 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 → Type := 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 → Type := has_le.le
|
||
definition lt {A : Type} [s : has_lt A] : A → A → Type := has_lt.lt
|
||
|
||
definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Type := le b a
|
||
definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Type := lt b a
|
||
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
|
||
|
||
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
|
||
|
||
definition nat_has_zero [reducible] [instance] [priority nat.prio] : has_zero nat :=
|
||
has_zero.mk nat.zero
|
||
|
||
definition nat_has_one [reducible] [instance] [priority nat.prio] : 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
|
||
|
||
/-
|
||
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 infixr ` ⊎ `:30
|
||
reserve infixr ` × `:35
|
||
|
||
/- 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
|
||
|
||
/- other symbols -/
|
||
|
||
reserve infix ` ∣ `:50
|
||
reserve infixl ` ++ `:65
|
||
reserve infixr ` :: `:67
|
||
|
||
/-
|
||
in the HoTT library we might not always want to overload the following notation,
|
||
so we put it in namespace algebra
|
||
-/
|
||
|
||
infix + := add
|
||
infix * := mul
|
||
infix - := sub
|
||
infix / := div
|
||
infix ∣ := dvd
|
||
infix % := mod
|
||
prefix - := neg
|
||
namespace algebra
|
||
postfix ⁻¹ := inv
|
||
end algebra
|
||
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
|