/
lex.lean
121 lines (104 loc) · 5.58 KB
/
lex.lean
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/-
Copyright (c) 2016 Microsoft Corporation. All rights reserved.
Released under Apache 2.0 license as described in the file LICENSE.
Author: Leonardo de Moura
-/
prelude
import init.data.sigma.basic init.meta
universes u v
namespace psigma
section
variables {α : Sort u} {β : α → Sort v}
variable (r : α → α → Prop)
variable (s : ∀ a, β a → β a → Prop)
-- Lexicographical order based on r and s
inductive lex : psigma β → psigma β → Prop
| left : ∀ {a₁ : α} (b₁ : β a₁) {a₂ : α} (b₂ : β a₂), r a₁ a₂ → lex ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
| right : ∀ (a : α) {b₁ b₂ : β a}, s a b₁ b₂ → lex ⟨a, b₁⟩ ⟨a, b₂⟩
end
section
open well_founded tactic
parameters {α : Sort u} {β : α → Sort v}
parameters {r : α → α → Prop} {s : Π a : α, β a → β a → Prop}
local infix `≺`:50 := lex r s
def lex_accessible {a} (aca : acc r a) (acb : ∀ a, well_founded (s a))
: ∀ (b : β a), acc (lex r s) ⟨a, b⟩ :=
acc.rec_on aca
(λ xa aca (iha : ∀ y, r y xa → ∀ b : β y, acc (lex r s) ⟨y, b⟩),
λ b : β xa, acc.rec_on (well_founded.apply (acb xa) b)
(λ xb acb
(ihb : ∀ (y : β xa), s xa y xb → acc (lex r s) ⟨xa, y⟩),
acc.intro ⟨xa, xb⟩ (λ p (lt : p ≺ ⟨xa, xb⟩),
have aux : xa = xa → xb == xb → acc (lex r s) p, from
@psigma.lex.rec_on α β r s (λ p₁ p₂, p₂.1 = xa → p₂.2 == xb → acc (lex r s) p₁)
p ⟨xa, xb⟩ lt
(λ (a₁ : α) (b₁ : β a₁) (a₂ : α) (b₂ : β a₂) (h : r a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b₂ == xb),
begin subst eq₂, exact iha a₁ h b₁ end)
(λ (a : α) (b₁ b₂ : β a) (h : s a b₁ b₂) (eq₂ : a = xa) (eq₃ : b₂ == xb),
begin
subst eq₂,
have new_eq₃ := eq_of_heq eq₃,
subst new_eq₃,
exact ihb b₁ h
end),
aux rfl (heq.refl xb))))
-- The lexicographical order of well founded relations is well-founded
def lex_wf (ha : well_founded r) (hb : ∀ x, well_founded (s x)) : well_founded (lex r s) :=
well_founded.intro $ λ ⟨a, b⟩, lex_accessible (well_founded.apply ha a) hb b
end
section
parameters {α : Sort u} {β : Sort v}
def lex_ndep (r : α → α → Prop) (s : β → β → Prop) :=
lex r (λ a : α, s)
def lex_ndep_wf {r : α → α → Prop} {s : β → β → Prop} (ha : well_founded r) (hb : well_founded s)
: well_founded (lex_ndep r s) :=
well_founded.intro $ λ ⟨a, b⟩, lex_accessible (well_founded.apply ha a) (λ x, hb) b
end
section
variables {α : Sort u} {β : Sort v}
variable (r : α → α → Prop)
variable (s : β → β → Prop)
-- Reverse lexicographical order based on r and s
inductive rev_lex : @psigma α (λ a, β) → @psigma α (λ a, β) → Prop
| left : ∀ {a₁ a₂ : α} (b : β), r a₁ a₂ → rev_lex ⟨a₁, b⟩ ⟨a₂, b⟩
| right : ∀ (a₁ : α) {b₁ : β} (a₂ : α) {b₂ : β}, s b₁ b₂ → rev_lex ⟨a₁, b₁⟩ ⟨a₂, b₂⟩
end
section
open well_founded tactic
parameters {α : Sort u} {β : Sort v}
parameters {r : α → α → Prop} {s : β → β → Prop}
local infix `≺`:50 := rev_lex r s
def rev_lex_accessible {b} (acb : acc s b) (aca : ∀ a, acc r a): ∀ a, acc (rev_lex r s) ⟨a, b⟩ :=
acc.rec_on acb
(λ xb acb (ihb : ∀ y, s y xb → ∀ a, acc (rev_lex r s) ⟨a, y⟩),
λ a, acc.rec_on (aca a)
(λ xa aca (iha : ∀ y, r y xa → acc (rev_lex r s) (mk y xb)),
acc.intro ⟨xa, xb⟩ (λ p (lt : p ≺ ⟨xa, xb⟩),
have aux : xa = xa → xb = xb → acc (rev_lex r s) p, from
@rev_lex.rec_on α β r s (λ p₁ p₂, fst p₂ = xa → snd p₂ = xb → acc (rev_lex r s) p₁)
p ⟨xa, xb⟩ lt
(λ a₁ a₂ b (h : r a₁ a₂) (eq₂ : a₂ = xa) (eq₃ : b = xb),
show acc (rev_lex r s) ⟨a₁, b⟩, from
have r₁ : r a₁ xa, from eq.rec_on eq₂ h,
have aux : acc (rev_lex r s) ⟨a₁, xb⟩, from iha a₁ r₁,
eq.rec_on (eq.symm eq₃) aux)
(λ a₁ b₁ a₂ b₂ (h : s b₁ b₂) (eq₂ : a₂ = xa) (eq₃ : b₂ = xb),
show acc (rev_lex r s) (mk a₁ b₁), from
have s₁ : s b₁ xb, from eq.rec_on eq₃ h,
ihb b₁ s₁ a₁),
aux rfl rfl)))
def rev_lex_wf (ha : well_founded r) (hb : well_founded s) : well_founded (rev_lex r s) :=
well_founded.intro $ λ ⟨a, b⟩, rev_lex_accessible (apply hb b) (well_founded.apply ha) a
end
section
def skip_left (α : Type u) {β : Type v} (s : β → β → Prop) : @psigma α (λ a, β) → @psigma α (λ a, β) → Prop :=
rev_lex empty_relation s
def skip_left_wf (α : Type u) {β : Type v} {s : β → β → Prop} (hb : well_founded s) : well_founded (skip_left α s) :=
rev_lex_wf empty_wf hb
def mk_skip_left {α : Type u} {β : Type v} {b₁ b₂ : β} {s : β → β → Prop}
(a₁ a₂ : α) (h : s b₁ b₂) : skip_left α s ⟨a₁, b₁⟩ ⟨a₂, b₂⟩ :=
rev_lex.right _ _ h
end
instance has_well_founded {α : Type u} {β : α → Type v} [s₁ : has_well_founded α] [s₂ : ∀ a, has_well_founded (β a)] : has_well_founded (psigma β) :=
{r := lex s₁.r (λ a, (s₂ a).r), wf := lex_wf s₁.wf (λ a, (s₂ a).wf)}
end psigma