Nuprl Lemma : dM-neg-properties
∀[I:fset(ℕ)]
((∀[x,y:Point(dM(I))]. (dm-neg(names(I);NamesDeq;x ∧ y) = ¬(x) ∨ ¬(y) ∈ Point(dM(I))))
∧ (∀[x,y:Point(dM(I))]. (dm-neg(names(I);NamesDeq;x ∨ y) = ¬(x) ∧ ¬(y) ∈ Point(dM(I))))
∧ (∀[x:Point(dM(I))]. (dm-neg(names(I);NamesDeq;¬(x)) = x ∈ Point(dM(I)))))
Proof
Definitions occuring in Statement :
dM: dM(I),
names-deq: NamesDeq,
names: names(I),
dm-neg: ¬(x),
lattice-join: a ∨ b,
lattice-meet: a ∧ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
prop: ℙ,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
lattice-point: Point(l),
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
all: ∀x:A. B[x],
top: Top,
eq_atom: x =a y,
ifthenelse: if b then t else f fi ,
bfalse: ff,
lattice-join: a ∨ b,
lattice-meet: a ∧ b
Lemmas referenced :
dm-neg-neg,
rec_select_update_lemma,
nat_wf,
fset_wf,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
equal_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
dM_wf,
lattice-point_wf,
names-deq_wf,
names_wf,
dm-neg-properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
productElimination,
independent_pairFormation,
sqequalRule,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
applyEquality,
instantiate,
lambdaEquality,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
dependent_functionElimination,
voidElimination,
voidEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]
((\mforall{}[x,y:Point(dM(I))]. (dm-neg(names(I);NamesDeq;x \mwedge{} y) = \mneg{}(x) \mvee{} \mneg{}(y)))
\mwedge{} (\mforall{}[x,y:Point(dM(I))]. (dm-neg(names(I);NamesDeq;x \mvee{} y) = \mneg{}(x) \mwedge{} \mneg{}(y)))
\mwedge{} (\mforall{}[x:Point(dM(I))]. (dm-neg(names(I);NamesDeq;\mneg{}(x)) = x)))
Date html generated:
2016_05_18-AM-11_57_35
Last ObjectModification:
2016_04_06-AM-09_49_57
Theory : cubical!type!theory
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