Nuprl Lemma : dM-to-FL-neg-is-1
∀[I:fset(ℕ)]
∀x:Point(free-DeMorgan-lattice(names(I);NamesDeq))
dM-to-FL(I;x) = 0 ∈ Point(face_lattice(I)) supposing dM-to-FL(I;¬(x)) = 1 ∈ Point(face_lattice(I))
Proof
Definitions occuring in Statement :
dM-to-FL: dM-to-FL(I;z),
face_lattice: face_lattice(I),
names-deq: NamesDeq,
names: names(I),
dm-neg: ¬(x),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
lattice-0: 0,
lattice-1: 1,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
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,
lattice-point: Point(l),
record-select: r.x,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq),
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,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
dM-to-FL-neg2,
lattice-point_wf,
free-DeMorgan-lattice_wf,
names_wf,
names-deq_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,
fset_wf,
nat_wf,
face_lattice_wf,
dM-to-FL_wf,
dm-neg_wf,
subtype_rel-equal,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
lattice-1_wf,
bdd-distributive-lattice_wf,
squash_wf,
true_wf,
iff_weakening_equal,
lattice-1-meet,
bdd-distributive-lattice-subtype-bdd-lattice
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
lambdaFormation,
dependent_functionElimination,
because_Cache,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
independent_isectElimination,
setElimination,
rename,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]
\mforall{}x:Point(free-DeMorgan-lattice(names(I);NamesDeq))
dM-to-FL(I;x) = 0 supposing dM-to-FL(I;\mneg{}(x)) = 1
Date html generated:
2017_10_05-AM-01_13_03
Last ObjectModification:
2017_07_28-AM-09_30_40
Theory : cubical!type!theory
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