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feat(library/data/list/basic): add more theorems for disjoint predicate

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Leonardo de Moura 2015-04-03 22:55:16 -07:00
parent 23a1f5fa4b
commit f1b7021ed0
1 changed files with 36 additions and 6 deletions
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library/data/list/basic.lean
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@ -682,22 +682,22 @@ variable {A : Type}
definition disjoint (l₁ l₂ : list A) : Prop := ∀ a, (a ∈ l₁ → a ∉ l₂) ∧ (a ∈ l₂ → a ∉ l₁)
lemma disjoint_left {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ a, a ∈ l₁ → a ∉ l₂ :=
lemma disjoint_left {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ {a}, a ∈ l₁ → a ∉ l₂ :=
λ d a, and.elim_left (d a)
lemma disjoint_right {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ a, a ∈ l₂ → a ∉ l₁ :=
lemma disjoint_right {l₁ l₂ : list A} : disjoint l₁ l₂ → ∀ {a}, a ∈ l₂ → a ∉ l₁ :=
λ d a, and.elim_right (d a)
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 a ainl₂)
(λ ainl₁ : a ∈ l₁, disjoint_left d a ainl₁)
(λ ainl₂ : a ∈ l₂, disjoint_right d ainl₂)
(λ ainl₁ : a ∈ l₁, disjoint_left d ainl₁)
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 x (or.inr xinl₁))
(λ xinl₁ : x ∈ l₁, disjoint_left d (or.inr xinl₁))
(λ xinl₂ : x ∈ l₂,
have nxinal₁ : x ∉ a::l₁, from disjoint_right d x xinl₂,
have nxinal₁ : x ∉ a::l₁, from disjoint_right d xinl₂,
not_mem_of_not_mem nxinal₁)
lemma disjoint_of_disjoint_cons_right {a : A} {l₁ l₂} : disjoint l₁ (a::l₂) → disjoint l₁ l₂ :=
@ -710,6 +710,36 @@ lemma disjoint_nil_left (l : list A) : disjoint [] l :=
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 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 xinal₁
(λ xeqa : x = a, absurd (xeqa ▸ xinl₂) nainl₂)
(λ xinl₁ : x ∈ l₁, absurd xinl₁ (disjoint_right d xinl₂)))
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
| (x::xs) l₂ l d :=
have nxinl : x ∉ l, from disjoint_left d !mem_cons,
have d₁ : disjoint (xs++l₂) l, from disjoint_of_disjoint_cons_left d,
have d₂ : disjoint xs l, from disjoint_of_disjoint_append_left_left d₁,
disjoint_cons_of_not_mem_of_disjoint nxinl d₂
lemma disjoint_of_disjoint_append_left_right : ∀ {l₁ l₂ l : list A}, disjoint (l₁++l₂) l → disjoint l₂ l
| [] l₂ l d := d
| (x::xs) l₂ l d :=
have d₁ : disjoint (xs++l₂) l, from disjoint_of_disjoint_cons_left d,
disjoint_of_disjoint_append_left_right d₁
lemma disjoint_of_disjoint_append_right_left : ∀ {l₁ l₂ l : list A}, disjoint l (l₁++l₂) → disjoint l l₁ :=
λ l₁ l₂ l d, disjoint.comm (disjoint_of_disjoint_append_left_left (disjoint.comm d))
lemma disjoint_of_disjoint_append_right_right : ∀ {l₁ l₂ l : list A}, disjoint l (l₁++l₂) → disjoint l l₂ :=
λ l₁ l₂ l d, disjoint.comm (disjoint_of_disjoint_append_left_right (disjoint.comm d))
end disjoint
/- no duplicates predicate -/