Nuprl Lemma : comp-nc-0-subset-I_cube
∀[I:fset(ℕ)]. ∀[i:{i:ℕ| ¬i ∈ I} ]. ∀[phi:𝔽(I)]. ∀J:fset(ℕ). ∀[f:I,phi(J)]. ((i0) ⋅ f ∈ I+i,s(phi)(J))
Proof
Definitions occuring in Statement :
cubical-subset: I,psi,
face-presheaf: 𝔽,
cube-set-restriction: f(s),
I_cube: A(I),
nc-0: (i0),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
member: t ∈ T,
set: {x:A| B[x]}
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
and: P ∧ Q,
uimplies: b supposing a,
subtype_rel: A ⊆r B,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
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,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
squash: ↓T,
bdd-distributive-lattice: BoundedDistributiveLattice,
true: True,
nat: ℕ,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q
Lemmas referenced :
cubical-subset-I_cube-member,
member-cubical-subset-I_cube,
add-name_wf,
cube-set-restriction_wf,
face-presheaf_wf,
nc-s_wf,
f-subset-add-name,
nh-comp_wf,
nc-0_wf,
name-morph-satisfies-comp,
subtype_rel_self,
fset_wf,
names_wf,
assert_wf,
fset-antichain_wf,
union-deq_wf,
names-deq_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
name-morph-satisfies_wf,
names-hom_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,
equal_wf,
lattice-meet_wf,
lattice-join_wf,
I_cube_wf,
cubical-subset_wf,
nat_wf,
set_wf,
not_wf,
fset-member_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
nh-id-right,
iff_weakening_equal,
squash_wf,
true_wf,
nh-comp-assoc,
s-comp-nc-0
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lambdaFormation,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
setElimination,
rename,
hypothesis,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
applyEquality,
sqequalRule,
setEquality,
unionEquality,
productEquality,
lambdaEquality,
hyp_replacement,
equalitySymmetry,
imageElimination,
equalityTransitivity,
instantiate,
cumulativity,
universeEquality,
natural_numberEquality,
imageMemberEquality,
baseClosed,
axiomEquality,
isect_memberEquality,
intEquality,
independent_functionElimination
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} ]. \mforall{}[phi:\mBbbF{}(I)].
\mforall{}J:fset(\mBbbN{}). \mforall{}[f:I,phi(J)]. ((i0) \mcdot{} f \mmember{} I+i,s(phi)(J))
Date html generated:
2017_10_05-AM-02_18_56
Last ObjectModification:
2017_07_28-AM-10_18_40
Theory : cubical!type!theory
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