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feat(library/data/list/comb): add any/all theorems

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Leonardo de Moura 2015-04-10 07:01:14 -07:00
parent 60ae9f627c
commit 9d8b5aa347
1 changed files with 18 additions and 0 deletions
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18
library/data/list/comb.lean
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@ -107,6 +107,24 @@ foldr (λ a r, p a ∧ r) true l
definition any (p : A → Prop) (l : list A) : Prop :=
foldr (λ a r, p a ∨ r) false l
theorem all_nil (p : A → Prop) : all p [] = true
theorem all_cons (p : A → Prop) (a : A) (l : list A) : all p (a::l) = (p a ∧ all p l)
theorem of_mem_of_all {p : A → Prop} {a : A} : ∀ {l}, a ∈ l → all p l → p a
| [] h₁ h₂ := absurd h₁ !not_mem_nil
| (b::l) h₁ h₂ :=
or.elim (eq_or_mem_of_mem_cons h₁)
(λ aeqb : a = b,
by rewrite [all_cons at h₂, -aeqb at h₂]; exact (and.elim_left h₂))
(λ ainl : a ∈ l,
have allp : all p l, by rewrite [all_cons at h₂]; exact (and.elim_right h₂),
of_mem_of_all ainl allp)
theorem any_nil (p : A → Prop) : any p [] = false
theorem any_cons (p : A → Prop) (a : A) (l : list A) : any p (a::l) = (p a ∨ any p l)
definition decidable_all (p : A → Prop) [H : decidable_pred p] : ∀ l, decidable (all p l)
| [] := decidable_true
| (a :: l) :=