Nuprl Lemma : cubical-term-restriction-is-1
ā[Gamma:jā¢]. ā[phi:{Gamma ā¢ _:š½}]. ā[I:fset(ā)]. ā[rho:Gamma(I)]. ā[J:fset(ā)]. ā[f:J ā¶ I].
((phi(rho) = 1 ā Point(face_lattice(I)))
ā (phi(f(rho)) = 1 ā Point(face_lattice(J))))
Proof
Definitions occuring in Statement :
face-type: š½
,
cubical-term-at: u(a)
,
cubical-term: {X ā¢ _:A}
,
face_lattice: face_lattice(I)
,
cube-set-restriction: f(s)
,
I_cube: A(I)
,
cubical_set: CubicalSet
,
names-hom: I ā¶ J
,
fset: fset(T)
,
nat: ā
,
uall: ā[x:A]. B[x]
,
implies: P
ā Q
,
equal: s = t ā T
,
lattice-1: 1
,
lattice-point: Point(l)
Definitions unfolded in proof :
uall: ā[x:A]. B[x]
,
member: t ā T
,
implies: P
ā Q
,
squash: āT
,
prop: ā
,
subtype_rel: A ār B
,
bdd-distributive-lattice: BoundedDistributiveLattice
,
so_lambda: Ī»2x.t[x]
,
and: P ā§ Q
,
so_apply: x[s]
,
uimplies: b supposing a
,
true: True
,
guard: {T}
,
iff: P
āā Q
,
rev_implies: P
ā Q
,
cubical-type-at: A(a)
,
pi1: fst(t)
,
face-type: š½
,
constant-cubical-type: (X)
,
I_cube: A(I)
,
functor-ob: ob(F)
,
face-presheaf: š½
,
lattice-point: Point(l)
,
record-select: r.x
,
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)
,
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
Lemmas referenced :
equal_wf,
squash_wf,
true_wf,
istype-universe,
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,
lattice-meet_wf,
lattice-join_wf,
face-term-at-restriction-eq-1,
lattice-1_wf,
subtype_rel_self,
iff_weakening_equal,
cubical-term-at_wf,
face-type_wf,
names-hom_wf,
I_cube_wf,
fset_wf,
nat_wf,
cubical-term_wf,
cubical_set_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
applyEquality,
thin,
lambdaEquality_alt,
sqequalHypSubstitution,
imageElimination,
extract_by_obid,
isectElimination,
hypothesisEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
universeIsType,
instantiate,
universeEquality,
sqequalRule,
productEquality,
cumulativity,
isectEquality,
because_Cache,
independent_isectElimination,
setElimination,
rename,
inhabitedIsType,
natural_numberEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination,
equalityIstype,
dependent_functionElimination,
axiomEquality,
functionIsTypeImplies,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[Gamma:j\mvdash{}]. \mforall{}[phi:\{Gamma \mvdash{} \_:\mBbbF{}\}]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[rho:Gamma(I)]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[f:J {}\mrightarrow{} I].
((phi(rho) = 1) {}\mRightarrow{} (phi(f(rho)) = 1))
Date html generated:
2020_05_20-PM-02_51_59
Last ObjectModification:
2020_04_04-PM-05_06_41
Theory : cubical!type!theory
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