95 lines
3.6 KiB
Text
95 lines
3.6 KiB
Text
/-
|
||
Copyright (c) 2014 Microsoft Corporation. All rights reserved.
|
||
Released under Apache 2.0 license as described in the file LICENSE.
|
||
Author: Leonardo de Moura, Jeremy Avigad
|
||
-/
|
||
prelude
|
||
import init.num init.wf
|
||
|
||
definition pair [constructor] := @prod.mk
|
||
notation A × B := prod A B
|
||
-- notation for n-ary tuples
|
||
notation `(` h `, ` t:(foldl `, ` (e r, prod.mk r e) h) `)` := t
|
||
|
||
namespace prod
|
||
notation [parsing-only] A * B := prod A B
|
||
namespace low_precedence_times
|
||
reserve infixr [parsing-only] `*`:30 -- conflicts with notation for multiplication
|
||
infixr `*` := prod
|
||
end low_precedence_times
|
||
|
||
notation `pr₁` := pr1
|
||
notation `pr₂` := pr2
|
||
|
||
namespace ops
|
||
postfix `.1`:(max+1) := pr1
|
||
postfix `.2`:(max+1) := pr2
|
||
end ops
|
||
|
||
definition destruct [reducible] := @prod.cases_on
|
||
|
||
section
|
||
variables {A B : Type}
|
||
lemma pr1.mk (a : A) (b : B) : pr1 (mk a b) = a := rfl
|
||
lemma pr2.mk (a : A) (b : B) : pr2 (mk a b) = b := rfl
|
||
lemma eta : ∀ (p : A × B), mk (pr1 p) (pr2 p) = p
|
||
| (a, b) := rfl
|
||
end
|
||
|
||
open well_founded
|
||
|
||
section
|
||
variables {A B : Type}
|
||
variable (Ra : A → A → Prop)
|
||
variable (Rb : B → B → Prop)
|
||
|
||
-- Lexicographical order based on Ra and Rb
|
||
inductive lex : A × B → A × B → Prop :=
|
||
| 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 → Prop :=
|
||
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 → Prop} {Rb : B → B → Prop}
|
||
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, 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
|