Nuprl Lemma : extend-face-term-le
∀[I:fset(ℕ)]. ∀[phi:𝔽(I)]. ∀[u:{I,phi ⊢ _:𝔽}]. extend-face-term(I;phi;u) ≤ phi
Proof
Definitions occuring in Statement :
extend-face-term: extend-face-term(I;phi;u),
face-type: 𝔽,
cubical-term: {X ⊢ _:A},
cubical-subset: I,psi,
face-presheaf: 𝔽,
face_lattice: face_lattice(I),
I_cube: A(I),
lattice-le: a ≤ b,
fset: fset(T),
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
extend-face-term: extend-face-term(I;phi;u),
all: ∀x:A. B[x],
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,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
pi2: snd(t),
bdd-distributive-lattice: BoundedDistributiveLattice,
uimplies: b supposing a,
implies: P ⇒ Q,
top: Top,
guard: {T},
lattice-le: a ≤ b,
cubical-subset: I,psi,
names-cat: NamesCat,
rep-sub-sheaf: rep-sub-sheaf(C;X;P),
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),
sq_stable: SqStable(P),
rev_uimplies: rev_uimplies(P;Q),
order: Order(T;x,y.R[x; y]),
trans: Trans(T;x,y.E[x; y])
Lemmas referenced :
face_lattice_components_wf,
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,
set_wf,
fset-disjoint_wf,
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,
lattice-fset-join_wf,
bdd-distributive-lattice-subtype-bdd-lattice,
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,
cubical-term_wf,
cubical-subset_wf,
face-type_wf,
I_cube_wf,
face-presheaf_wf,
nat_wf,
I_cube_pair_redex_lemma,
cat_arrow_triple_lemma,
irr-face-morph_wf,
name-morph-satisfies_wf,
irr-face-morph-satisfies,
lattice-le_wf,
lattice-fset-join-is-lub,
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,
strong-subtype-set3,
cubical-term-at_wf,
sq_stable__equal,
lattice-le-order,
bdd-distributive-lattice-subtype-lattice,
lattice-meet-le
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
isectElimination,
setEquality,
unionEquality,
hypothesis,
because_Cache,
productEquality,
lambdaEquality,
productElimination,
instantiate,
cumulativity,
universeEquality,
independent_isectElimination,
independent_functionElimination,
lambdaFormation,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
setElimination,
rename,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
axiomEquality,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
independent_pairFormation,
imageMemberEquality,
baseClosed,
imageElimination,
natural_numberEquality
Latex:
\mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:\mBbbF{}(I)]. \mforall{}[u:\{I,phi \mvdash{} \_:\mBbbF{}\}]. extend-face-term(I;phi;u) \mleq{} phi
Date html generated:
2017_10_05-AM-07_33_49
Last ObjectModification:
2017_03_03-AM-00_56_55
Theory : cubical!type!theory
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