286 lines
8.9 KiB
Text
286 lines
8.9 KiB
Text
/-
|
||
Copyright (c) 2014-2015 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, Jakob von Raumer
|
||
-/
|
||
|
||
prelude
|
||
import init.num init.wf
|
||
|
||
-- Empty type
|
||
-- ----------
|
||
|
||
namespace empty
|
||
|
||
protected theorem elim {A : Type} (H : empty) : A :=
|
||
empty.rec (λe, A) H
|
||
|
||
end empty
|
||
|
||
protected definition empty.has_decidable_eq [instance] : decidable_eq empty :=
|
||
take (a b : empty), decidable.inl (!empty.elim a)
|
||
|
||
-- Unit type
|
||
-- ---------
|
||
|
||
namespace unit
|
||
|
||
notation `⋆` := star
|
||
|
||
end unit
|
||
|
||
-- Sigma type
|
||
-- ----------
|
||
|
||
notation `Σ` binders `, ` r:(scoped P, sigma P) := r
|
||
|
||
namespace sigma
|
||
notation `⟨`:max t:(foldr `, ` (e r, mk e r)) `⟩`:0 := t --input ⟨ ⟩ as \< \>
|
||
|
||
namespace ops
|
||
postfix `.1`:(max+1) := pr1
|
||
postfix `.2`:(max+1) := pr2
|
||
abbreviation pr₁ := @pr1
|
||
abbreviation pr₂ := @pr2
|
||
end ops
|
||
end sigma
|
||
|
||
-- Sum type
|
||
-- --------
|
||
|
||
namespace sum
|
||
infixr ⊎ := sum
|
||
infixr + := sum
|
||
namespace low_precedence_plus
|
||
reserve infixr ` + `:25 -- conflicts with notation for addition
|
||
infixr ` + ` := sum
|
||
end low_precedence_plus
|
||
|
||
variables {a b c d : Type}
|
||
definition sum_of_sum_of_imp_of_imp (H₁ : a ⊎ b) (H₂ : a → c) (H₃ : b → d) : c ⊎ d :=
|
||
sum.rec_on H₁
|
||
(assume Ha : a, sum.inl (H₂ Ha))
|
||
(assume Hb : b, sum.inr (H₃ Hb))
|
||
|
||
definition sum_of_sum_of_imp_left (H₁ : a ⊎ c) (H : a → b) : b ⊎ c :=
|
||
sum.rec_on H₁
|
||
(assume H₂ : a, sum.inl (H H₂))
|
||
(assume H₂ : c, sum.inr H₂)
|
||
|
||
definition sum_of_sum_of_imp_right (H₁ : c ⊎ a) (H : a → b) : c ⊎ b :=
|
||
sum.rec_on H₁
|
||
(assume H₂ : c, sum.inl H₂)
|
||
(assume H₂ : a, sum.inr (H H₂))
|
||
end sum
|
||
|
||
-- Product type
|
||
-- ------------
|
||
|
||
abbreviation pair [constructor] := @prod.mk
|
||
|
||
namespace prod
|
||
|
||
-- notation for n-ary tuples
|
||
notation `(` h `, ` t:(foldl `,` (e r, prod.mk r e) h) `)` := t
|
||
infixr × := prod
|
||
|
||
namespace ops
|
||
infixr [parsing_only] * := prod
|
||
postfix `.1`:(max+1) := pr1
|
||
postfix `.2`:(max+1) := pr2
|
||
abbreviation pr₁ := @pr1
|
||
abbreviation pr₂ := @pr2
|
||
end ops
|
||
|
||
namespace low_precedence_times
|
||
|
||
reserve infixr ` * `:30 -- conflicts with notation for multiplication
|
||
infixr ` * ` := prod
|
||
|
||
end low_precedence_times
|
||
|
||
open prod.ops
|
||
|
||
definition flip [unfold 3] {A B : Type} (a : A × B) : B × A := pair (pr2 a) (pr1 a)
|
||
|
||
open well_founded
|
||
|
||
section
|
||
variables {A B : Type}
|
||
variable (Ra : A → A → Type)
|
||
variable (Rb : B → B → Type)
|
||
|
||
-- Lexicographical order based on Ra and Rb
|
||
inductive lex : A × B → A × B → Type :=
|
||
| left : ∀{a₁ b₁} a₂ b₂, Ra a₁ a₂ → lex (a₁, b₁) (a₂, b₂)
|
||
| right : ∀a {b₁ b₂}, Rb b₁ b₂ → lex (a, b₁) (a, b₂)
|
||
|
||
-- Relational product based on Ra and Rb
|
||
inductive rprod : A × B → A × B → Type :=
|
||
intro : ∀{a₁ b₁ a₂ b₂}, Ra a₁ a₂ → Rb b₁ b₂ → rprod (a₁, b₁) (a₂, b₂)
|
||
end
|
||
|
||
section
|
||
parameters {A B : Type}
|
||
parameters {Ra : A → A → Type} {Rb : B → B → Type}
|
||
local infix `≺`:50 := lex Ra Rb
|
||
|
||
definition lex.accessible {a} (aca : acc Ra a) (acb : ∀b, acc Rb b): ∀b, acc (lex Ra Rb) (a, b) :=
|
||
acc.rec_on aca
|
||
(λxa aca (iHa : ∀y, Ra y xa → ∀b, acc (lex Ra Rb) (y, b)),
|
||
λb, acc.rec_on (acb b)
|
||
(λxb acb
|
||
(iHb : ∀y, Rb y xb → acc (lex Ra Rb) (xa, y)),
|
||
acc.intro (xa, xb) (λp (lt : p ≺ (xa, xb)),
|
||
have aux : xa = xa → xb = xb → acc (lex Ra Rb) p, from
|
||
@prod.lex.rec_on A B Ra Rb (λp₁ p₂ h, pr₁ p₂ = xa → pr₂ p₂ = xb → acc (lex Ra Rb) p₁)
|
||
p (xa, xb) lt
|
||
(λa₁ b₁ a₂ b₂ (H : Ra a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
|
||
show acc (lex Ra Rb) (a₁, b₁), from
|
||
have Ra₁ : Ra a₁ xa, from eq.rec_on eq₂ H,
|
||
iHa a₁ Ra₁ b₁)
|
||
(λa b₁ b₂ (H : Rb b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ = xb),
|
||
show acc (lex Ra Rb) (a, b₁), from
|
||
have Rb₁ : Rb b₁ xb, from eq.rec_on eq₃ H,
|
||
have eq₂' : xa = a, from eq.rec_on eq₂ rfl,
|
||
eq.rec_on eq₂' (iHb b₁ Rb₁)),
|
||
aux rfl rfl)))
|
||
|
||
-- The lexicographical order of well founded relations is well-founded
|
||
definition lex.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (lex Ra Rb) :=
|
||
well_founded.intro (λp, destruct p (λa b, lex.accessible (Ha a) (well_founded.apply Hb) b))
|
||
|
||
-- Relational product is a subrelation of the lex
|
||
definition rprod.sub_lex : ∀ a b, rprod Ra Rb a b → lex Ra Rb a b :=
|
||
λa b H, prod.rprod.rec_on H (λ a₁ b₁ a₂ b₂ H₁ H₂, lex.left Rb a₂ b₂ H₁)
|
||
|
||
-- The relational product of well founded relations is well-founded
|
||
definition rprod.wf (Ha : well_founded Ra) (Hb : well_founded Rb) : well_founded (rprod Ra Rb) :=
|
||
subrelation.wf (rprod.sub_lex) (lex.wf Ha Hb)
|
||
|
||
end
|
||
|
||
end prod
|
||
|
||
/- logic using prod and sum -/
|
||
|
||
variables {a b c d : Type}
|
||
open prod sum unit
|
||
|
||
/- prod -/
|
||
|
||
definition not_prod_of_not_left (b : Type) (Hna : ¬a) : ¬(a × b) :=
|
||
assume H : a × b, absurd (pr1 H) Hna
|
||
|
||
definition not_prod_of_not_right (a : Type) {b : Type} (Hnb : ¬b) : ¬(a × b) :=
|
||
assume H : a × b, absurd (pr2 H) Hnb
|
||
|
||
definition prod.swap (H : a × b) : b × a :=
|
||
pair (pr2 H) (pr1 H)
|
||
|
||
definition prod_of_prod_of_imp_of_imp (H₁ : a × b) (H₂ : a → c) (H₃ : b → d) : c × d :=
|
||
by cases H₁ with aa bb; exact (H₂ aa, H₃ bb)
|
||
|
||
definition prod_of_prod_of_imp_left (H₁ : a × c) (H : a → b) : b × c :=
|
||
by cases H₁ with aa cc; exact (H aa, cc)
|
||
|
||
definition prod_of_prod_of_imp_right (H₁ : c × a) (H : a → b) : c × b :=
|
||
by cases H₁ with cc aa; exact (cc, H aa)
|
||
|
||
definition prod.comm : a × b ↔ b × a :=
|
||
iff.intro (λH, prod.swap H) (λH, prod.swap H)
|
||
|
||
definition prod.assoc : (a × b) × c ↔ a × (b × c) :=
|
||
iff.intro
|
||
(assume H, pair
|
||
(pr1 (pr1 H))
|
||
(pair (pr2 (pr1 H)) (pr2 H)))
|
||
(assume H, pair
|
||
(pair (pr1 H) (pr1 (pr2 H)))
|
||
(pr2 (pr2 H)))
|
||
|
||
definition prod_unit (a : Type) : a × unit ↔ a :=
|
||
iff.intro (assume H, pr1 H) (assume H, pair H star)
|
||
|
||
definition unit_prod (a : Type) : unit × a ↔ a :=
|
||
iff.intro (assume H, pr2 H) (assume H, pair star H)
|
||
|
||
definition prod_empty (a : Type) : a × empty ↔ empty :=
|
||
iff.intro (assume H, pr2 H) (assume H, !empty.elim H)
|
||
|
||
definition empty_prod (a : Type) : empty × a ↔ empty :=
|
||
iff.intro (assume H, pr1 H) (assume H, !empty.elim H)
|
||
|
||
definition prod_self (a : Type) : a × a ↔ a :=
|
||
iff.intro (assume H, pr1 H) (assume H, pair H H)
|
||
|
||
/- sum -/
|
||
|
||
definition not_sum (Hna : ¬a) (Hnb : ¬b) : ¬(a ⊎ b) :=
|
||
assume H : a ⊎ b, sum.rec_on H
|
||
(assume Ha, absurd Ha Hna)
|
||
(assume Hb, absurd Hb Hnb)
|
||
|
||
definition sum_of_sum_of_imp_of_imp (H₁ : a ⊎ b) (H₂ : a → c) (H₃ : b → d) : c ⊎ d :=
|
||
sum.rec_on H₁
|
||
(assume Ha : a, sum.inl (H₂ Ha))
|
||
(assume Hb : b, sum.inr (H₃ Hb))
|
||
|
||
definition sum_of_sum_of_imp_left (H₁ : a ⊎ c) (H : a → b) : b ⊎ c :=
|
||
sum.rec_on H₁
|
||
(assume H₂ : a, sum.inl (H H₂))
|
||
(assume H₂ : c, sum.inr H₂)
|
||
|
||
definition sum_of_sum_of_imp_right (H₁ : c ⊎ a) (H : a → b) : c ⊎ b :=
|
||
sum.rec_on H₁
|
||
(assume H₂ : c, sum.inl H₂)
|
||
(assume H₂ : a, sum.inr (H H₂))
|
||
|
||
definition sum.elim3 (H : a ⊎ b ⊎ c) (Ha : a → d) (Hb : b → d) (Hc : c → d) : d :=
|
||
sum.rec_on H Ha (assume H₂, sum.rec_on H₂ Hb Hc)
|
||
|
||
definition sum_resolve_right (H₁ : a ⊎ b) (H₂ : ¬a) : b :=
|
||
sum.rec_on H₁ (assume Ha, absurd Ha H₂) (assume Hb, Hb)
|
||
|
||
definition sum_resolve_left (H₁ : a ⊎ b) (H₂ : ¬b) : a :=
|
||
sum.rec_on H₁ (assume Ha, Ha) (assume Hb, absurd Hb H₂)
|
||
|
||
definition sum.swap (H : a ⊎ b) : b ⊎ a :=
|
||
sum.rec_on H (assume Ha, sum.inr Ha) (assume Hb, sum.inl Hb)
|
||
|
||
definition sum.comm : a ⊎ b ↔ b ⊎ a :=
|
||
iff.intro (λH, sum.swap H) (λH, sum.swap H)
|
||
|
||
definition sum.assoc : (a ⊎ b) ⊎ c ↔ a ⊎ (b ⊎ c) :=
|
||
iff.intro
|
||
(assume H, sum.rec_on H
|
||
(assume H₁, sum.rec_on H₁
|
||
(assume Ha, sum.inl Ha)
|
||
(assume Hb, sum.inr (sum.inl Hb)))
|
||
(assume Hc, sum.inr (sum.inr Hc)))
|
||
(assume H, sum.rec_on H
|
||
(assume Ha, (sum.inl (sum.inl Ha)))
|
||
(assume H₁, sum.rec_on H₁
|
||
(assume Hb, sum.inl (sum.inr Hb))
|
||
(assume Hc, sum.inr Hc)))
|
||
|
||
definition sum_unit (a : Type) : a ⊎ unit ↔ unit :=
|
||
iff.intro (assume H, star) (assume H, sum.inr H)
|
||
|
||
definition unit_sum (a : Type) : unit ⊎ a ↔ unit :=
|
||
iff.intro (assume H, star) (assume H, sum.inl H)
|
||
|
||
definition sum_empty (a : Type) : a ⊎ empty ↔ a :=
|
||
iff.intro
|
||
(assume H, sum.rec_on H (assume H1 : a, H1) (assume H1 : empty, !empty.elim H1))
|
||
(assume H, sum.inl H)
|
||
|
||
definition empty_sum (a : Type) : empty ⊎ a ↔ a :=
|
||
iff.intro
|
||
(assume H, sum.rec_on H (assume H1 : empty, !empty.elim H1) (assume H1 : a, H1))
|
||
(assume H, sum.inr H)
|
||
|
||
definition sum_self (a : Type) : a ⊎ a ↔ a :=
|
||
iff.intro
|
||
(assume H, sum.rec_on H (assume H1, H1) (assume H1, H1))
|
||
(assume H, sum.inl H)
|