Nuprl Lemma : dM-to-FL-eq-1
∀[I:fset(ℕ)]. ∀[x:Point(dM(I))]. uiff(dM-to-FL(I;x) = 1 ∈ Point(face_lattice(I));x = 1 ∈ Point(dM(I)))
Proof
Definitions occuring in Statement :
dM-to-FL: dM-to-FL(I;z),
face_lattice: face_lattice(I),
dM: dM(I),
lattice-1: 1,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
so_apply: x[s],
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
all: ∀x:A. B[x],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
dM: dM(I),
top: Top,
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
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,
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
dminc: <i>,
dM_inc: <x>,
dmopp: <1-i>,
dM_opp: <1-x>,
empty-fset: {},
lattice-fset-join: \/(s),
not: ¬A,
false: False,
bdd-lattice: BoundedLattice,
or: P ∨ Q,
lattice-fset-meet: /\(s),
reduce: reduce(f;k;as),
list_ind: list_ind,
nil: [],
it: ⋅,
lattice-1: 1,
fset-singleton: {x},
cons: [a / b]
Lemmas referenced :
equal_wf,
lattice-point_wf,
face_lattice_wf,
subtype_rel_set,
bounded-lattice-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
dM-to-FL_wf,
lattice-1_wf,
bdd-distributive-lattice_wf,
dM_wf,
DeMorgan-algebra-structure_wf,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
DeMorgan-algebra-axioms_wf,
fset_wf,
nat_wf,
dM-hom-basis,
bdd-distributive-lattice-subtype-bdd-lattice,
fl-deq_wf,
dM-to-FL-is-hom,
subtype_rel-equal,
bounded-lattice-hom_wf,
free-DeMorgan-lattice_wf,
names_wf,
names-deq_wf,
squash_wf,
true_wf,
free-dma-hom-is-lattice-hom,
iff_weakening_equal,
dM-basis,
dM-point,
deq-implies,
free-dl-point,
deq-fset_wf,
union-deq_wf,
strong-subtype-deq-subtype,
assert_wf,
fset-antichain_wf,
strong-subtype-set2,
fset-induction,
lattice-fset-join_wf,
fset-image_wf,
lattice-fset-meet_wf,
dM_inc_wf,
dM_opp_wf,
dminc_wf,
dmopp_wf,
sq_stable__all,
sq_stable__equal,
reduce_nil_lemma,
lattice-0_wf,
fset-image-empty,
face-lattice-0-not-1,
fset-add_wf,
not_wf,
fset-member_wf,
all_wf,
decidable_wf,
bdd-lattice_wf,
fset-image-add,
fset-singleton_wf,
lattice-fset-join-union,
lattice-fset-join-singleton,
face_lattice-1-join-irreducible,
empty-fset_wf,
lattice-fset-meet-union,
lattice-fset-meet-singleton,
lattice-meet-eq-1,
dM-to-FL-inc,
false_wf,
fl1-not-1,
dM-to-FL-opp,
fl0-not-1,
lattice-join-1,
lattice-1-join,
dM-to-FL-properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
universeEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
productElimination,
independent_pairEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
voidElimination,
voidEquality,
unionEquality,
setEquality,
functionEquality,
unionElimination,
lambdaFormation,
equalityUniverse,
levelHypothesis,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[x:Point(dM(I))]. uiff(dM-to-FL(I;x) = 1;x = 1)
Date html generated:
2017_10_05-AM-01_12_20
Last ObjectModification:
2017_07_28-AM-09_30_23
Theory : cubical!type!theory
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