Nuprl Lemma : dM-to-FL-neg
∀[I:fset(ℕ)]
∀x:Point(free-DeMorgan-lattice(names(I);NamesDeq)). (dM-to-FL(I;x) ∧ dM-to-FL(I;¬(x)) = 0 ∈ 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-meet: a ∧ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
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,
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,
guard: {T},
free-dml-deq: free-dml-deq(T;eq),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
all: ∀x:A. B[x],
lattice-point: Point(l),
record-select: r.x,
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,
top: Top,
lattice-meet: a ∧ b,
face_lattice: face_lattice(I),
face-lattice: face-lattice(T;eq),
free-dist-lattice-with-constraints: free-dist-lattice-with-constraints(T;eq;x.Cs[x]),
constrained-antichain-lattice: constrained-antichain-lattice(T;eq;P),
fset-constrained-ac-glb: glb(P;ac1;ac2),
fset-minimals: fset-minimals(x,y.less[x; y]; s),
fset-filter: {x ∈ s | P[x]},
filter: filter(P;l),
reduce: reduce(f;k;as),
list_ind: list_ind,
f-union: f-union(domeq;rngeq;s;x.g[x]),
list_accum: list_accum,
dM-to-FL: dM-to-FL(I;z),
lattice-extend: lattice-extend(L;eq;eqL;f;ac),
lattice-fset-join: \/(s),
fset-image: f"(s),
empty-fset: {},
nil: [],
it: ⋅,
lattice-0: 0,
not: ¬A,
false: False,
true: True,
squash: ↓T,
bdd-lattice: BoundedLattice,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
lattice-fset-meet: /\(s),
cand: A c∧ B,
dminc: <i>,
dmopp: <1-i>,
dM_opp: <1-x>,
dM_inc: <x>,
lattice-axioms: lattice-axioms(l),
bounded-lattice-axioms: bounded-lattice-axioms(l)
Lemmas referenced :
deq-implies,
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,
free-dml-deq_wf,
lattice-fset-meet_wf,
free-dist-lattice_wf,
union-deq_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
fset-image_wf,
subtype_rel_self,
deq_wf,
free-dl-inc_wf,
fset_wf,
deq-fset_wf,
nat_wf,
dM-basis,
subtype_rel-equal,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
face_lattice_wf,
dM-to-FL_wf,
dm-neg_wf,
lattice-0_wf,
free-dl-point,
istype-void,
fset-induction,
lattice-fset-join_wf,
sq_stable__equal,
fset-member_wf,
fset-singleton_wf,
squash_wf,
true_wf,
istype-universe,
decidable_wf,
bdd-lattice_wf,
fset-image-add,
iff_weakening_equal,
lattice-fset-join-union,
lattice-fset-join-singleton,
bdd-distributive-lattice-subtype-distributive-lattice,
dM-to-FL-properties,
dm-neg-properties,
distributive-lattice-distrib,
reduce_nil_lemma,
lattice-1_wf,
lattice-0-meet,
lattice-fset-meet-union,
lattice-fset-meet-singleton,
free-dma-point,
dminc_wf,
dmopp_wf,
dm-neg-inc,
dM-to-FL-inc,
dM-to-FL-opp,
FL-meet-0-1,
dm-neg-opp,
lattice-join-0,
lattice-meet-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality_alt,
productEquality,
cumulativity,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
because_Cache,
universeIsType,
independent_isectElimination,
independent_functionElimination,
unionEquality,
unionIsType,
lambdaFormation_alt,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
hyp_replacement,
applyLambdaEquality,
setElimination,
rename,
isect_memberEquality_alt,
voidElimination,
functionIsType,
equalityIsType1,
natural_numberEquality,
imageElimination,
universeEquality,
functionEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_pairFormation,
dependent_set_memberEquality_alt,
productIsType,
unionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]
\mforall{}x:Point(free-DeMorgan-lattice(names(I);NamesDeq)). (dM-to-FL(I;x) \mwedge{} dM-to-FL(I;\mneg{}(x)) = 0)
Date html generated:
2019_11_04-PM-05_34_04
Last ObjectModification:
2018_11_10-AM-09_35_34
Theory : cubical!type!theory
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