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feat(empty): define negation of types

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Floris van Doorn 2014-11-08 20:55:42 -05:00 committed by Leonardo de Moura
parent bf27a17dec
commit 780e949992
1 changed files with 20 additions and 0 deletions
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library/data/empty.lean
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@ -16,3 +16,23 @@ namespace empty
protected theorem subsingleton [instance] : subsingleton empty :=
subsingleton.intro (λ a b, !elim a)
end empty
definition tneg.tneg (A : Type) := A → empty
prefix `~` := tneg.tneg
namespace tneg
variables {A B : Type}
protected definition intro (H : A → empty) : ~A := H
protected definition elim (H1 : ~A) (H2 : A) : empty := H1 H2
protected definition empty : ~empty := λH : empty, H
definition tabsurd (H1 : A) (H2 : ~A) : B := !empty.elim (H2 H1)
definition tneg_tneg_intro (H : A) : ~~A := λH2 : ~A, tneg.elim H2 H
definition tmt (H1 : A → B) (H2 : ~B) : ~A := λHA : A, tabsurd (H1 HA) H2
definition tneg_pi_left {B : A → Type} (H : ~Πa, B a) : ~~A :=
λHnA : ~A, tneg.elim H (λHA : A, tabsurd HA HnA)
definition tneg_function_right (H : ~(A → B)) : ~B :=
λHB : B, tneg.elim H (λHA : A, HB)
end tneg