Nuprl Lemma : dm-neg-is-hom-opposite
∀[T:Type]. ∀[eq:EqDecider(T)].
(λx.¬(x) ∈ Hom(opposite-lattice(free-DeMorgan-lattice(T;eq));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 :
bounded-lattice-hom: Hom(l1;l2),
member: t ∈ T,
uall: ∀[x:A]. B[x],
top: Top,
cand: A c∧ B,
uimplies: b supposing a,
so_apply: x[s],
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
bdd-distributive-lattice: BoundedDistributiveLattice,
subtype_rel: A ⊆r B,
lattice-hom: Hom(l1;l2),
lattice-0: 0,
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,
lattice-1: 1,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
fset-singleton: {x},
cons: [a / b],
empty-fset: {},
nil: [],
it: ⋅
Lemmas referenced :
deq_wf,
opposite-lattice-join,
dm-neg-properties,
opposite-lattice-meet,
opposite-lattice-point,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
bounded-lattice-structure_wf,
subtype_rel_set,
free-DeMorgan-lattice_wf,
opposite-lattice_wf,
lattice-point_wf,
subtype_rel-equal,
dm-neg_wf,
void-list-equality,
nil_wf,
lattice-1_wf,
bdd-distributive-lattice_wf,
lattice-0_wf
Rules used in proof :
universeEquality,
because_Cache,
isect_memberEquality,
hypothesisEquality,
cumulativity,
thin,
isectElimination,
extract_by_obid,
equalitySymmetry,
equalityTransitivity,
axiomEquality,
sqequalRule,
hypothesis,
sqequalHypSubstitution,
dependent_set_memberEquality,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
functionExtensionality,
independent_pairEquality,
independent_pairFormation,
productElimination,
voidEquality,
voidElimination,
independent_isectElimination,
productEquality,
instantiate,
applyEquality,
lambdaEquality,
rename,
setElimination
Latex:
\mforall{}[T:Type]. \mforall{}[eq:EqDecider(T)].
(\mlambda{}x.\mneg{}(x) \mmember{} Hom(opposite-lattice(free-DeMorgan-lattice(T;eq));free-DeMorgan-lattice(T;eq)))
Date html generated:
2020_05_20-AM-08_54_39
Last ObjectModification:
2020_02_03-AM-11_40_03
Theory : lattices
Home
Index