Nuprl Lemma : dm-neg-is-hom
∀[T:Type]. ∀[eq:EqDecider(T)].
(λx.¬(x) ∈ Hom(free-DeMorgan-lattice(T;eq);opposite-lattice(free-DeMorgan-lattice(T;eq))))
Proof
Definitions occuring in Statement :
dm-neg: ¬(x),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
opposite-lattice: opposite-lattice(L),
bounded-lattice-hom: Hom(l1;l2),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
member: t ∈ T,
lambda: λx.A[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
dm-neg: ¬(x),
lattice-point: Point(l),
record-select: r.x,
opposite-lattice: opposite-lattice(L),
mk-bounded-distributive-lattice: mk-bounded-distributive-lattice,
mk-bounded-lattice: mk-bounded-lattice(T;m;j;z;o),
record-update: r[x := v],
ifthenelse: if b then t else f fi ,
eq_atom: x =a y,
bfalse: ff,
btrue: tt,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
top: Top,
subtype_rel: A ⊆r B,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
bdd-distributive-lattice: BoundedDistributiveLattice
Lemmas referenced :
free-dl-point,
deq-fset_wf,
fset_wf,
union-deq_wf,
strong-subtype-deq-subtype,
assert_wf,
fset-antichain_wf,
strong-subtype-set2,
lattice-extend-is-hom,
opposite-lattice_wf,
free-DeMorgan-lattice_wf,
bounded-lattice-hom_wf,
free-dist-lattice_wf,
bdd-distributive-lattice_wf,
deq_wf,
fset-antichain-singleton,
fset-singleton_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
hypothesis,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
unionEquality,
hypothesisEquality,
applyEquality,
setEquality,
independent_isectElimination,
lambdaEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
because_Cache,
setElimination,
rename,
axiomEquality,
universeEquality,
inrEquality,
dependent_set_memberEquality,
inlEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)].
(\mlambda{}x.\mneg{}(x) \mmember{} Hom(free-DeMorgan-lattice(T;eq);opposite-lattice(free-DeMorgan-lattice(T;eq))))
Date html generated:
2020_05_20-AM-08_54_29
Last ObjectModification:
2015_12_28-PM-02_00_45
Theory : lattices
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