155 lines
4.2 KiB
Text
155 lines
4.2 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, 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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definition zero [reducible] {A : Type} [s : has_zero A] : A := has_zero.zero A
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definition one [reducible] {A : Type} [s : has_one A] : A := has_one.one A
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definition add [reducible] {A : Type} [s : has_add A] : A → A → A := has_add.add
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definition bit0 [reducible] {A : Type} [s : has_add A] (a : A) : A := add a a
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definition bit1 [reducible] {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 [reducible] [instance] : has_zero num :=
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has_zero.mk num.zero
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definition num_has_one [reducible] [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 [reducible] [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 [reducible] [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 [reducible] [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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/-
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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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/- types and type constructors -/
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reserve infixr ` ⊎ `:25
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reserve infixr ` × `: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 ` div `:70
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reserve infixl ` mod `: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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/- 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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