Nuprl Lemma : dm-neg-inc
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[i:T]. (¬(<i>) = <1-i> ∈ Point(free-DeMorgan-lattice(T;eq)))
Proof
Definitions occuring in Statement :
dm-neg: ¬(x),
dmopp: <1-i>,
dminc: <i>,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
lattice-point: Point(l),
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
dm-neg: ¬(x),
free-dl-inc: free-dl-inc(x),
free-dml-deq: free-dml-deq(T;eq),
dminc: <i>,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
top: Top,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
true: True,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
dmopp: <1-i>,
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
deq_wf,
free-dl-inc_wf,
union-deq_wf,
equal_wf,
squash_wf,
true_wf,
lattice-point_wf,
free-dist-lattice_wf,
lattice-extend-dl-inc,
opposite-lattice_wf,
free-DeMorgan-lattice_wf,
opposite-lattice-point,
free-dml-deq_wf,
subtype_rel-equal,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
dmopp_wf,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
hypothesisEquality,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
axiomEquality,
because_Cache,
extract_by_obid,
cumulativity,
universeEquality,
lambdaEquality,
unionElimination,
unionEquality,
inrEquality,
inlEquality,
applyEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
voidEquality,
instantiate,
productEquality,
independent_isectElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[i:T]. (\mneg{}(<i>) = ə-i>)
Date html generated:
2020_05_20-AM-08_54_48
Last ObjectModification:
2017_07_28-AM-09_16_52
Theory : lattices
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