Nuprl Lemma : dm-neg-properties
∀[T:Type]. ∀[eq:EqDecider(T)].
((∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq))))
∧ (∀[x,y:Point(free-DeMorgan-lattice(T;eq))]. (¬(x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(free-DeMorgan-lattice(T;eq))))
∧ (¬(0) = 1 ∈ Point(free-DeMorgan-lattice(T;eq)))
∧ (¬(1) = 0 ∈ Point(free-DeMorgan-lattice(T;eq))))
Proof
Definitions occuring in Statement :
dm-neg: ¬(x),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
lattice-0: 0,
lattice-1: 1,
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
and: P ∧ Q,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
bounded-lattice-hom: Hom(l1;l2),
squash: ↓T,
lattice-hom: Hom(l1;l2),
and: P ∧ Q,
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,
cand: A c∧ B,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
uimplies: b supposing a,
top: Top,
guard: {T},
true: True
Lemmas referenced :
dm-neg-is-hom,
lattice-point_wf,
free-DeMorgan-lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
opposite-lattice-0,
opposite-lattice-1,
deq_wf,
squash_wf,
true_wf,
opposite-lattice_wf,
opposite-lattice-meet,
dm-neg_wf,
subtype_rel_weakening,
ext-eq_weakening,
opposite-lattice-join
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyLambdaEquality,
setElimination,
rename,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
productElimination,
cumulativity,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
universeEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
independent_pairFormation,
voidElimination,
voidEquality,
equalitySymmetry,
hyp_replacement,
equalityTransitivity,
natural_numberEquality
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)].
((\mforall{}[x,y:Point(free-DeMorgan-lattice(T;eq))]. (\mneg{}(x \mwedge{} y) = \mneg{}(x) \mvee{} \mneg{}(y)))
\mwedge{} (\mforall{}[x,y:Point(free-DeMorgan-lattice(T;eq))]. (\mneg{}(x \mvee{} y) = \mneg{}(x) \mwedge{} \mneg{}(y)))
\mwedge{} (\mneg{}(0) = 1)
\mwedge{} (\mneg{}(1) = 0))
Date html generated:
2020_05_20-AM-08_54_34
Last ObjectModification:
2017_07_28-AM-09_16_42
Theory : lattices
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