Nuprl Lemma : dm-neg-neg
∀[T:Type]. ∀[eq:EqDecider(T)]. ∀[x:Point(free-DeMorgan-lattice(T;eq))].
(¬(¬(x)) = x ∈ Point(free-DeMorgan-lattice(T;eq)))
Proof
Definitions occuring in Statement :
dm-neg: ¬(x),
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,
all: ∀x:A. B[x],
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
implies: P ⇒ Q,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
cand: A c∧ B,
compose: f o g,
dminc: <i>,
dmopp: <1-i>,
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
free-dist-lattice-hom-unique,
union-deq_wf,
free-DeMorgan-lattice_wf,
free-dml-deq_wf,
free-dl-inc_wf,
lattice-point_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,
deq_wf,
free-dist-lattice_wf,
lattice-0_wf,
bdd-distributive-lattice_wf,
lattice-1_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
opposite-lattice_wf,
compose-bounded-lattice-hom,
bounded-lattice-hom_wf,
dm-neg-is-hom,
dm-neg-is-hom-opposite,
dminc_wf,
dmopp_wf,
squash_wf,
true_wf,
dm-neg_wf,
dm-neg-inc,
iff_weakening_equal,
dm-neg-opp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
unionEquality,
cumulativity,
hypothesisEquality,
isectElimination,
hypothesis,
lambdaEquality,
applyEquality,
sqequalRule,
instantiate,
productEquality,
universeEquality,
because_Cache,
independent_isectElimination,
isect_memberEquality,
axiomEquality,
independent_functionElimination,
applyLambdaEquality,
setElimination,
rename,
equalitySymmetry,
dependent_set_memberEquality,
independent_pairFormation,
productElimination,
independent_pairEquality,
equalityTransitivity,
functionExtensionality,
comment,
unionElimination,
natural_numberEquality,
imageElimination,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)]. \mforall{}[x:Point(free-DeMorgan-lattice(T;eq))]. (\mneg{}(\mneg{}(x)) = x)
Date html generated:
2020_05_20-AM-08_54_55
Last ObjectModification:
2017_07_28-AM-09_16_56
Theory : lattices
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