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feat(library/data/list/basic): more nodup theorems

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Leonardo de Moura 2015-04-09 14:50:00 -07:00
parent b4611ba33d
commit 64173ddf93
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library/data/list/basic.lean
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@ -224,6 +224,11 @@ theorem not_mem_of_not_mem_append_left {x : T} {s t : list T} : x ∉ s++t → x
theorem not_mem_of_not_mem_append_right {x : T} {s t : list T} : x ∉ s++t → x ∉ t :=
λ nxinst xint, absurd (mem_append_of_mem_or_mem (or.inr xint)) nxinst
theorem not_mem_append {x : T} {s t : list T} : x ∉ s → x ∉ t → x ∉ s++t :=
λ nxins nxint xinst, or.elim (mem_or_mem_of_mem_append xinst)
(λ xins, absurd xins nxins)
(λ xint, absurd xint nxint)
local attribute mem [reducible]
local attribute append [reducible]
theorem mem_split {x : T} {l : list T} : x ∈ l → ∃s t : list T, l = s ++ (x::t) :=
@ -824,6 +829,63 @@ theorem nodup_of_nodup_append_right : ∀ {l₁ l₂ : list A}, nodup (l₁++l
| [] l₂ n := n
| (x::xs) l₂ n := nodup_of_nodup_append_right (nodup_of_nodup_cons n)
theorem disjoint_of_nodup_append : ∀ {l₁ l₂ : list A}, nodup (l₁++l₂) → disjoint l₁ l₂
| [] l₂ d := disjoint_nil_left l₂
| (x::xs) l₂ d :=
have d₁ : nodup (x::(xs++l₂)), from d,
have d₂ : nodup (xs++l₂), from nodup_of_nodup_cons d₁,
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,
or.elim (mem_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 (mem_or_mem_of_mem_cons ain)
(λ aeqx : a = x, absurd ainl₂ (aeqx⁻¹ ▸ nxinl₂))
(λ ainxs : a ∈ xs, absurd ainxs nainxs)))
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
| (x::xs) l₂ d₁ d₂ dsj :=
have dsj₁ : disjoint xs l₂, from disjoint_of_disjoint_cons_left dsj,
have ndxs : nodup xs, from nodup_of_nodup_cons d₁,
have ndxsl₂ : nodup (xs++l₂), from nodup_append_of_nodup_of_nodup_of_disjoint ndxs d₂ dsj₁,
have nxinxs : x ∉ xs, from not_mem_of_nodup_cons d₁,
have nxinl₂ : x ∉ l₂, from disjoint_left dsj !mem_cons,
have nxinxsl₂ : x ∉ xs++l₂, from not_mem_append nxinxs nxinl₂,
nodup_cons nxinxsl₂ ndxsl₂
theorem nodup_app_comm {l₁ l₂ : list A} (d : nodup (l₁++l₂)) : nodup (l₂++l₁) :=
have d₁ : nodup l₁, from nodup_of_nodup_append_left d,
have d₂ : nodup l₂, from nodup_of_nodup_append_right d,
have dsj : disjoint l₁ l₂, from disjoint_of_nodup_append d,
nodup_append_of_nodup_of_nodup_of_disjoint d₂ d₁ (disjoint.comm dsj)
theorem nodup_head {a : A} {l₁ l₂ : list A} (d : nodup (l₁++(a::l₂))) : nodup (a::(l₁++l₂)) :=
have d₁ : nodup (a::(l₂++l₁)), from nodup_app_comm d,
have d₂ : nodup (l₂++l₁), from nodup_of_nodup_cons d₁,
have d₃ : nodup (l₁++l₂), from nodup_app_comm d₂,
have nain : a ∉ l₂++l₁, from not_mem_of_nodup_cons d₁,
have nain₂ : a ∉ l₂, from not_mem_of_not_mem_append_left nain,
have nain₁ : a ∉ l₁, from not_mem_of_not_mem_append_right nain,
nodup_cons (not_mem_append nain₁ nain₂) d₃
theorem nodup_middle {a : A} {l₁ l₂ : list A} (d : nodup (a::(l₁++l₂))) : nodup (l₁++(a::l₂)) :=
have d₁ : nodup (l₁++l₂), from nodup_of_nodup_cons d,
have nain : a ∉ l₁++l₂, from not_mem_of_nodup_cons d,
have disj : disjoint l₁ l₂, from disjoint_of_nodup_append d₁,
have d₂ : nodup l₁, from nodup_of_nodup_append_left d₁,
have d₃ : nodup l₂, from nodup_of_nodup_append_right d₁,
have nain₂ : a ∉ l₂, from not_mem_of_not_mem_append_right nain,
have nain₁ : a ∉ l₁, from not_mem_of_not_mem_append_left nain,
have d₄ : nodup (a::l₂), from nodup_cons nain₂ d₃,
have disj₂ : disjoint l₁ (a::l₂), from disjoint.comm (disjoint_cons_of_not_mem_of_disjoint nain₁ (disjoint.comm disj)),
nodup_append_of_nodup_of_nodup_of_disjoint d₂ d₄ disj₂
theorem nodup_map {f : A → B} (inj : injective f) : ∀ {l : list A}, nodup l → nodup (map f l)
| [] n := begin rewrite [map_nil], apply nodup_nil end
| (x::xs) n :=
@ -917,6 +979,13 @@ theorem nodup_erase_dup [H : decidable_eq A] : ∀ l : list A, nodup (erase_dup
assert nin : a ∉ erase_dup l, from
assume ab : a ∈ erase_dup l, absurd (mem_of_mem_erase_dup ab) nainl,
by rewrite [erase_dup_cons_of_not_mem nainl]; exact (nodup_cons nin r))
theorem erase_dup_eq_of_nodup [H : decidable_eq A] : ∀ {l : list A}, nodup l → erase_dup l = l
| [] d := rfl
| (a::l) d :=
assert nainl : a ∉ l, from not_mem_of_nodup_cons d,
assert dl : nodup l, from nodup_of_nodup_cons d,
by rewrite [erase_dup_cons_of_not_mem nainl, erase_dup_eq_of_nodup dl]
end nodup
/- union -/