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refactor(library/data): simplify definition of disjoint

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Leonardo de Moura 2015-04-11 14:04:09 -07:00
parent ee895e00dd
commit 54a2d9750e
2 changed files with 21 additions and 48 deletions
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16
library/data/finset/basic.lean
View file

@ -168,24 +168,14 @@ end erase
definition disjoint (s₁ s₂ : finset A) : Prop :=
quot.lift_on₂ s₁ s₂ (λ l₁ l₂, disjoint (elt_of l₁) (elt_of l₂))
(λ v₁ v₂ w₁ w₂ p₁ p₂, propext (iff.intro
(λ d₁ a, and.intro
(λ ainw₁ : a ∈ elt_of w₁,
(λ d₁ a (ainw₁ : a ∈ elt_of w₁),
have ainv₁ : a ∈ elt_of v₁, from mem_perm (perm.symm p₁) ainw₁,
have nainv₂ : a ∉ elt_of v₂, from disjoint_left d₁ ainv₁,
not_mem_perm p₂ nainv₂)
(λ ainw₂ : a ∈ elt_of w₂,
have ainv₂ : a ∈ elt_of v₂, from mem_perm (perm.symm p₂) ainw₂,
have nainv₁ : a ∉ elt_of v₁, from disjoint_right d₁ ainv₂,
not_mem_perm p₁ nainv₁))
(λ d₂ a, and.intro
(λ ainv₁ : a ∈ elt_of v₁,
(λ d₂ a (ainv₁ : a ∈ elt_of v₁),
have ainw₁ : a ∈ elt_of w₁, from mem_perm p₁ ainv₁,
have nainw₂ : a ∉ elt_of w₂, from disjoint_left d₂ ainw₁,
not_mem_perm (perm.symm p₂) nainw₂)
(λ ainv₂ : a ∈ elt_of v₂,
have ainw₂ : a ∈ elt_of w₂, from mem_perm p₂ ainv₂,
have nainw₁ : a ∉ elt_of w₁, from disjoint_right d₂ ainw₂,
not_mem_perm (perm.symm p₁) nainw₁))))
not_mem_perm (perm.symm p₂) nainw₂)))
theorem disjoint.comm {s₁ s₂ : finset A} : disjoint s₁ s₂ → disjoint s₂ s₁ :=
quot.induction_on₂ s₁ s₂ (λ l₁ l₂ d, list.disjoint.comm d)

39
library/data/list/set.lean
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@ -132,45 +132,34 @@ end erase
section disjoint
variable {A : Type}
definition disjoint (l₁ l₂ : list A) : Prop := ∀ a, (a ∈ l₁ → a ∉ l₂) ∧ (a ∈ l₂ → a ∉ l₁)
definition disjoint (l₁ l₂ : list A) : Prop := ∀ a, (a ∈ l₁ → a ∈ l₂ → false)
lemma disjoint_left {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ {a}, a ∈ l₁ → a ∉ l₂ :=
λ d a, and.elim_left (d a)
λ d a, d a
lemma disjoint_right {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ {a}, a ∈ l₂ → a ∉ l₁ :=
λ d a, and.elim_right (d a)
λ d a i₂ i₁, d a i₁ i₂
lemma disjoint.comm {l₁ l₂ : list A} : disjoint l₁ l₂ → disjoint l₂ l₁ :=
λ d a, and.intro
(λ ainl₂ : a ∈ l₂, disjoint_right d ainl₂)
(λ ainl₁ : a ∈ l₁, disjoint_left d ainl₁)
λ d a i₂ i₁, d a i₁ i₂
lemma disjoint_of_disjoint_cons_left {a : A} {l₁ l₂} : disjoint (a::l₁) l₂ → disjoint l₁ l₂ :=
λ d x, and.intro
(λ xinl₁ : x ∈ l₁, disjoint_left d (or.inr xinl₁))
(λ xinl₂ : x ∈ l₂,
have nxinal₁ : x ∉ a::l₁, from disjoint_right d xinl₂,
not_mem_of_not_mem nxinal₁)
λ d x xinl₁, disjoint_left d (or.inr xinl₁)
lemma disjoint_of_disjoint_cons_right {a : A} {l₁ l₂} : disjoint l₁ (a::l₂) → disjoint l₁ l₂ :=
λ d, disjoint.comm (disjoint_of_disjoint_cons_left (disjoint.comm d))
lemma disjoint_nil_left (l : list A) : disjoint [] l :=
λ a, and.intro
(λ ab : a ∈ nil, absurd ab !not_mem_nil)
(λ ainl : a ∈ l, !not_mem_nil)
λ a ab, absurd ab !not_mem_nil
lemma disjoint_nil_right (l : list A) : disjoint l [] :=
disjoint.comm (disjoint_nil_left l)
lemma disjoint_cons_of_not_mem_of_disjoint {a : A} {l₁ l₂} : a ∉ l₂ → disjoint l₁ l₂ → disjoint (a::l₁) l₂ :=
λ nainl₂ d x, and.intro
(λ xinal₁ : x ∈ a::l₁, or.elim (eq_or_mem_of_mem_cons xinal₁)
λ nainl₂ d x (xinal₁ : x ∈ a::l₁),
or.elim (eq_or_mem_of_mem_cons xinal₁)
(λ xeqa : x = a, xeqa⁻¹ ▸ nainl₂)
(λ xinl₁ : x ∈ l₁, disjoint_left d xinl₁))
(λ (xinl₂ : x ∈ l₂) (xinal₁ : x ∈ a::l₁), or.elim (eq_or_mem_of_mem_cons xinal₁)
(λ xeqa : x = a, absurd (xeqa ▸ xinl₂) nainl₂)
(λ xinl₁ : x ∈ l₁, absurd xinl₁ (disjoint_right d xinl₂)))
(λ xinl₁ : x ∈ l₁, disjoint_left d xinl₁)
lemma disjoint_of_disjoint_append_left_left : ∀ {l₁ l₂ l : list A}, disjoint (l₁++l₂) l → disjoint l₁ l
| [] l₂ l d := disjoint_nil_left l
@ -245,16 +234,10 @@ theorem disjoint_of_nodup_append : ∀ {l₁ l₂ : list A}, nodup (l₁++l₂)
have nxin : x ∉ xs++l₂, from not_mem_of_nodup_cons d₁,
have nxinl₂ : x ∉ l₂, from not_mem_of_not_mem_append_right nxin,
have dsj : disjoint xs l₂, from disjoint_of_nodup_append d₂,
(λ a, and.intro
(λ ainxxs : a ∈ x::xs,
λ a (ainxxs : a ∈ x::xs),
or.elim (eq_or_mem_of_mem_cons ainxxs)
(λ aeqx : a = x, aeqx⁻¹ ▸ nxinl₂)
(λ ainxs : a ∈ xs, disjoint_left dsj ainxs))
(λ ainl₂ : a ∈ l₂,
have nainxs : a ∉ xs, from disjoint_right dsj ainl₂,
assume ain : a ∈ x::xs, or.elim (eq_or_mem_of_mem_cons ain)
(λ aeqx : a = x, absurd (aeqx ▸ ainl₂) nxinl₂)
(λ ainxs : a ∈ xs, absurd ainxs nainxs)))
(λ ainxs : a ∈ xs, disjoint_left dsj ainxs)
theorem nodup_append_of_nodup_of_nodup_of_disjoint : ∀ {l₁ l₂ : list A}, nodup l₁ → nodup l₂ → disjoint l₁ l₂ → nodup (l₁++l₂)
| [] l₂ d₁ d₂ dsj := by rewrite [append_nil_left]; exact d₂