Nuprl Lemma : extend-face-term-property
∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[u:{I,phi ⊢ _:𝔽}]. ∀[J:fset(ℕ)]. ∀[g:I,phi(J)].
((extend-face-term(I;phi;u))<g> = u(g) ∈ Point(face_lattice(J)))
Proof
Definitions occuring in Statement :
extend-face-term: extend-face-term(I;phi;u),
face-type: 𝔽,
cubical-term-at: u(a),
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
face-presheaf: 𝔽,
fl-morph: <f>,
face_lattice: face_lattice(I),
I_cube: A(I),
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
cubical-subset: I,psi,
cube-cat: CubeCat,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
all: ∀x:A. B[x],
top: Top,
extend-face-term: extend-face-term(I;phi;u),
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,
bdd-distributive-lattice: BoundedDistributiveLattice,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
uimplies: b supposing a,
pi2: snd(t),
implies: P ⇒ Q,
guard: {T},
cat-arrow: cat-arrow(C),
names-hom: I ⟶ J,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
name-morph-satisfies: (psi f) = 1,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
iff: P ⇐⇒ Q,
cubical-type-at: A(a),
face-type: 𝔽,
constant-cubical-type: (X),
rev_implies: P ⇐ Q,
bdd-lattice: BoundedLattice,
cubical-term: {X ⊢ _:A},
cubical-term-at: u(a),
sq_stable: SqStable(P),
lattice-le: a ≤ b,
label: ...$L... t,
nh-comp: g ⋅ f,
dma-lift-compose: dma-lift-compose(I;J;eqi;eqj;f;g),
compose: f o g,
cat-comp: cat-comp(C),
order: Order(T;x,y.R[x; y]),
anti_sym: AntiSym(T;x,y.R[x; y])
Lemmas referenced :
I_cube_pair_redex_lemma,
cat_arrow_triple_lemma,
face_lattice_components_wf,
subtype_rel_self,
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,
set_wf,
fset_wf,
names_wf,
assert_wf,
fset-disjoint_wf,
names-deq_wf,
lattice-fset-join_wf,
decidable__equal_face_lattice,
fset-image_wf,
product-deq_wf,
deq-fset_wf,
strong-subtype-deq-subtype,
pi1_wf_top,
pi2_wf,
strong-subtype-set2,
face_lattice-deq_wf,
irr_face_wf,
fset-subtype2,
fset-member_wf,
I_cube_wf,
cubical-subset_wf,
nat_wf,
cubical-term_wf,
face-type_wf,
face-presheaf_wf,
small_cubical_set_subtype,
name-morph-satisfies_wf,
names-hom_wf,
irr-face-morph_wf,
irr-face-morph-satisfies,
lattice-le_wf,
lattice-fset-join-is-lub,
bdd-distributive-lattice-subtype-bdd-lattice,
subtype_rel_product,
top_wf,
member-fset-image-iff,
fl-morph_wf,
lattice-hom-le,
squash_wf,
true_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
iff_weakening_equal,
lattice-1-le-iff,
lattice-hom-fset-join,
cubical-term-at_wf,
and_wf,
strong-subtype-set3,
subtype_rel_nested_set2,
all_wf,
decidable_wf,
bdd-lattice_wf,
name-morph-satisfies-fset-join,
cubical-type-at_wf_face-type,
lattice-hom_wf,
lattice-0_wf,
lattice-1_wf,
face-type-at,
lattice-hom-meet,
irr-face-morph-property,
lattice-1-meet,
nh-comp_wf,
face-type-ap-morph,
cubical_type_at_pair_lemma,
cubical_type_ap_morph_pair_lemma,
cubical-subset-restriction,
sq_stable__equal,
face_lattice-le,
lattice-meet-eq-1,
fl-morph-comp2,
cubical-subset-I_cube-member,
implies-nh-comp-satisfies,
cubical-type_wf,
cubical_set_wf,
nh-comp-assoc,
lattice-le-order,
bdd-distributive-lattice-subtype-lattice
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
sqequalRule,
extract_by_obid,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
setElimination,
rename,
hypothesisEquality,
applyEquality,
isectElimination,
instantiate,
lambdaEquality,
productEquality,
cumulativity,
because_Cache,
independent_isectElimination,
setEquality,
productElimination,
independent_functionElimination,
lambdaFormation,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality,
universeEquality,
functionEquality,
equalityUniverse,
levelHypothesis
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:\mBbbF{}(I)]. \mforall{}[u:\{I,phi \mvdash{} \_:\mBbbF{}\}]. \mforall{}[J:fset(\mBbbN{})]. \mforall{}[g:I,phi(J)].
((extend-face-term(I;phi;u))<g> = u(g))
Date html generated:
2018_05_23-AM-11_12_29
Last ObjectModification:
2018_04_09-PM-07_20_54
Theory : cubical!type!theory
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