233 lines
7.6 KiB
Text
233 lines
7.6 KiB
Text
/-
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Copyright (c) 2014 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, Jeremy Avigad, Floris van Doorn
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-/
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prelude
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import init.datatypes
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notation `assume` binders `,` r:(scoped f, f) := r
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notation `take` binders `,` r:(scoped f, f) := r
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structure has_zero [class] (A : Type) := (zero : A)
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structure has_one [class] (A : Type) := (one : A)
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structure has_add [class] (A : Type) := (add : A → A → A)
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structure has_mul [class] (A : Type) := (mul : A → A → A)
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structure has_inv [class] (A : Type) := (inv : A → A)
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structure has_neg [class] (A : Type) := (neg : A → A)
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structure has_sub [class] (A : Type) := (sub : A → A → A)
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structure has_div [class] (A : Type) := (div : A → A → A)
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structure has_dvd [class] (A : Type) := (dvd : A → A → Prop)
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structure has_mod [class] (A : Type) := (mod : A → A → A)
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structure has_le [class] (A : Type) := (le : A → A → Prop)
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structure has_lt [class] (A : Type) := (lt : A → A → Prop)
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definition zero {A : Type} [s : has_zero A] : A := has_zero.zero A
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definition one {A : Type} [s : has_one A] : A := has_one.one A
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definition add {A : Type} [s : has_add A] : A → A → A := has_add.add
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definition mul {A : Type} [s : has_mul A] : A → A → A := has_mul.mul
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definition sub {A : Type} [s : has_sub A] : A → A → A := has_sub.sub
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definition div {A : Type} [s : has_div A] : A → A → A := has_div.div
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definition dvd {A : Type} [s : has_dvd A] : A → A → Prop := has_dvd.dvd
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definition mod {A : Type} [s : has_mod A] : A → A → A := has_mod.mod
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definition neg {A : Type} [s : has_neg A] : A → A := has_neg.neg
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definition inv {A : Type} [s : has_inv A] : A → A := has_inv.inv
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definition le {A : Type} [s : has_le A] : A → A → Prop := has_le.le
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definition lt {A : Type} [s : has_lt A] : A → A → Prop := has_lt.lt
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definition ge [reducible] {A : Type} [s : has_le A] (a b : A) : Prop := le b a
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definition gt [reducible] {A : Type} [s : has_lt A] (a b : A) : Prop := lt b a
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/-
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bit0 and bit1 are two auxiliary definition used when parsing numerals such as 13, 0, 26.
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The parser will generate the terms (bit1 (bit0 (bit1 one))), zero, and
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(bit0 (bit1 (bit0 (bit1 one)))). This works in any type with an addition, a zero and a one.
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More specifically, there must be type class instances for the classes for has_add, has_zero and
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has_one
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-/
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definition bit0 {A : Type} [s : has_add A] (a : A) : A := add a a
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definition bit1 {A : Type} [s₁ : has_one A] [s₂ : has_add A] (a : A) : A := add (bit0 a) one
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definition num_has_zero [instance] : has_zero num :=
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has_zero.mk num.zero
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definition num_has_one [instance] : has_one num :=
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has_one.mk (num.pos pos_num.one)
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definition pos_num_has_one [instance] : has_one pos_num :=
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has_one.mk (pos_num.one)
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namespace pos_num
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open bool
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definition is_one (a : pos_num) : bool :=
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pos_num.rec_on a tt (λn r, ff) (λn r, ff)
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definition pred (a : pos_num) : pos_num :=
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pos_num.rec_on a one (λn r, bit0 n) (λn r, bool.rec_on (is_one n) (bit1 r) one)
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definition size (a : pos_num) : pos_num :=
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pos_num.rec_on a one (λn r, succ r) (λn r, succ r)
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definition add (a b : pos_num) : pos_num :=
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pos_num.rec_on a
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succ
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(λn f b, pos_num.rec_on b
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(succ (bit1 n))
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(λm r, succ (bit1 (f m)))
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(λm r, bit1 (f m)))
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(λn f b, pos_num.rec_on b
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(bit1 n)
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(λm r, bit1 (f m))
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(λm r, bit0 (f m)))
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b
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end pos_num
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definition pos_num_has_add [instance] : has_add pos_num :=
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has_add.mk pos_num.add
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namespace num
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open pos_num
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definition add (a b : num) : num :=
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num.rec_on a b (λpa, num.rec_on b (pos pa) (λpb, pos (pos_num.add pa pb)))
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end num
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definition num_has_add [instance] : has_add num :=
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has_add.mk num.add
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definition std.priority.default : num := 1000
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definition std.priority.max : num := 4294967295
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namespace nat
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protected definition prio := num.add std.priority.default 100
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protected definition add (a b : nat) : nat :=
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nat.rec a (λ b₁ r, succ r) b
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definition of_num (n : num) : nat :=
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num.rec zero
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(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
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end nat
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attribute pos_num_has_add pos_num_has_one num_has_zero num_has_one num_has_add
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[instance] [priority nat.prio]
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definition nat_has_zero [instance] [priority nat.prio] : has_zero nat :=
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has_zero.mk nat.zero
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definition nat_has_one [instance] [priority nat.prio] : has_one nat :=
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has_one.mk (nat.succ (nat.zero))
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definition nat_has_add [instance] [priority nat.prio] : has_add nat :=
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has_add.mk nat.add
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/-
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Global declarations of right binding strength
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If a module reassigns these, it will be incompatible with other modules that adhere to these
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conventions.
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When hovering over a symbol, use "C-c C-k" to see how to input it.
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-/
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definition std.prec.max : num := 1024 -- the strength of application, identifiers, (, [, etc.
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definition std.prec.arrow : num := 25
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/-
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The next definition is "max + 10". It can be used e.g. for postfix operations that should
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be stronger than application.
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-/
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definition std.prec.max_plus :=
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num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ (num.succ
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(num.succ std.prec.max)))))))))
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/- Logical operations and relations -/
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reserve prefix `¬`:40
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reserve prefix `~`:40
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reserve infixr ` ∧ `:35
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reserve infixr ` /\ `:35
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reserve infixr ` \/ `:30
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reserve infixr ` ∨ `:30
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reserve infix ` <-> `:20
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reserve infix ` ↔ `:20
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reserve infix ` = `:50
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reserve infix ` ≠ `:50
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reserve infix ` ≈ `:50
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reserve infix ` ~ `:50
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reserve infix ` ≡ `:50
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reserve infixr ` ∘ `:60 -- input with \comp
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reserve postfix `⁻¹`:std.prec.max_plus -- input with \sy or \-1 or \inv
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reserve infixl ` ⬝ `:75
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reserve infixr ` ▸ `:75
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reserve infixr ` ▹ `:75
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/- types and type constructors -/
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reserve infixl ` ⊎ `:25
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reserve infixl ` × `:30
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/- arithmetic operations -/
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reserve infixl ` + `:65
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reserve infixl ` - `:65
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reserve infixl ` * `:70
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reserve infixl ` / `:70
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reserve infixl ` % `:70
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reserve prefix `-`:100
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reserve infix ` ^ `:80
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reserve infix ` <= `:50
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reserve infix ` ≤ `:50
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reserve infix ` < `:50
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reserve infix ` >= `:50
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reserve infix ` ≥ `:50
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reserve infix ` > `:50
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/- boolean operations -/
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reserve infixl ` && `:70
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reserve infixl ` || `:65
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/- set operations -/
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reserve infix ` ∈ `:50
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reserve infix ` ∉ `:50
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reserve infixl ` ∩ `:70
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reserve infixl ` ∪ `:65
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reserve infix ` ⊆ `:50
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reserve infix ` ⊇ `:50
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reserve infix ` ' `:75 -- for the image of a set under a function
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reserve infix ` '- `:75 -- for the preimage of a set under a function
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/- other symbols -/
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reserve infix ` ∣ `:50
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reserve infixl ` ++ `:65
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reserve infixr ` :: `:67
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infix + := add
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infix * := mul
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infix - := sub
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infix / := div
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infix ∣ := dvd
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infix % := mod
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prefix - := neg
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postfix ⁻¹ := inv
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infix ≤ := le
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infix ≥ := ge
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infix < := lt
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infix > := gt
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notation [parsing_only] x ` +[`:65 A:0 `] `:0 y:65 := @add A _ x y
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notation [parsing_only] x ` -[`:65 A:0 `] `:0 y:65 := @sub A _ x y
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notation [parsing_only] x ` *[`:70 A:0 `] `:0 y:70 := @mul A _ x y
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notation [parsing_only] x ` /[`:70 A:0 `] `:0 y:70 := @div A _ x y
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notation [parsing_only] x ` ∣[`:70 A:0 `] `:0 y:70 := @dvd A _ x y
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notation [parsing_only] x ` %[`:70 A:0 `] `:0 y:70 := @mod A _ x y
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notation [parsing_only] x ` ≤[`:50 A:0 `] `:0 y:50 := @le A _ x y
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notation [parsing_only] x ` ≥[`:50 A:0 `] `:0 y:50 := @ge A _ x y
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notation [parsing_only] x ` <[`:50 A:0 `] `:0 y:50 := @lt A _ x y
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notation [parsing_only] x ` >[`:50 A:0 `] `:0 y:50 := @gt A _ x y
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