Nuprl Lemma : fl-morph-face-lattice-tube1
∀[J:fset(ℕ)]. ∀[k:names(J)]. ∀[I:fset(ℕ)]. ∀[phi:Point(face_lattice(I))]. ∀[j:ℕ]. ∀[g:J ⟶ I+j].
(face-lattice-tube(I;phi;j))<g> = (s(phi))<g> ∨ (k=0) ∨ (k=1) ∈ Point(face_lattice(J))
supposing (g j) = <k> ∈ Point(dM(J))
Proof
Definitions occuring in Statement :
face-lattice-tube: face-lattice-tube(I;phi;j),
face-presheaf: 𝔽,
fl-morph: <f>,
fl1: (x=1),
fl0: (x=0),
face_lattice: face_lattice(I),
cube-set-restriction: f(s),
nc-s: s,
add-name: I+i,
names-hom: I ⟶ J,
dM_inc: <x>,
dM: dM(I),
names: names(I),
lattice-join: a ∨ b,
lattice-point: Point(l),
fset: fset(T),
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
face-lattice-tube: face-lattice-tube(I;phi;j),
prop: ℙ,
subtype_rel: A ⊆r B,
DeMorgan-algebra: DeMorganAlgebra,
so_lambda: λ2x.t[x],
and: P ∧ Q,
guard: {T},
so_apply: x[s],
names-hom: I ⟶ J,
all: ∀x:A. B[x],
names: names(I),
nat: ℕ,
bdd-distributive-lattice: BoundedDistributiveLattice,
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,
I_cube: A(I),
functor-ob: ob(F),
pi1: fst(t),
face-presheaf: 𝔽,
bounded-lattice-hom: Hom(l1;l2),
lattice-hom: Hom(l1;l2),
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
dM: dM(I),
free-DeMorgan-algebra: free-DeMorgan-algebra(T;eq),
mk-DeMorgan-algebra: mk-DeMorgan-algebra(L;n),
free-DeMorgan-lattice: free-DeMorgan-lattice(T;eq),
free-dist-lattice: free-dist-lattice(T; eq)
Lemmas referenced :
equal_wf,
lattice-point_wf,
dM_wf,
subtype_rel_set,
DeMorgan-algebra-structure_wf,
lattice-structure_wf,
lattice-axioms_wf,
bounded-lattice-structure-subtype,
DeMorgan-algebra-structure-subtype,
subtype_rel_transitivity,
bounded-lattice-structure_wf,
bounded-lattice-axioms_wf,
uall_wf,
lattice-meet_wf,
lattice-join_wf,
DeMorgan-algebra-axioms_wf,
trivial-member-add-name1,
fset-member_wf,
nat_wf,
int-deq_wf,
strong-subtype-deq-subtype,
strong-subtype-set3,
le_wf,
strong-subtype-self,
add-name_wf,
dM_inc_wf,
names-hom_wf,
face_lattice_wf,
fset_wf,
names_wf,
cube-set-restriction_wf,
face-presheaf_wf,
nc-s_wf,
f-subset-add-name,
subtype_rel_self,
assert_wf,
fset-antichain_wf,
union-deq_wf,
names-deq_wf,
fset-all_wf,
fset-contains-none_wf,
face-lattice-constraints_wf,
fl0_wf,
fl1_wf,
fl-morph_wf,
bounded-lattice-hom_wf,
bdd-distributive-lattice_wf,
dM-to-FL_wf,
squash_wf,
true_wf,
neg-dM_inc,
iff_weakening_equal,
dM-to-FL-opp,
dm-neg_wf,
subtype_rel-equal,
free-DeMorgan-lattice_wf,
dM-to-FL-inc,
fl-morph-join,
fl-morph-fl0,
fl-morph-fl1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
sqequalRule,
instantiate,
lambdaEquality,
productEquality,
independent_isectElimination,
cumulativity,
universeEquality,
because_Cache,
dependent_functionElimination,
dependent_set_memberEquality,
intEquality,
natural_numberEquality,
isect_memberEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
setEquality,
unionEquality,
setElimination,
rename,
imageElimination,
hyp_replacement,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
productElimination,
independent_functionElimination
Latex:
\mforall{}[J:fset(\mBbbN{})]. \mforall{}[k:names(J)]. \mforall{}[I:fset(\mBbbN{})]. \mforall{}[phi:Point(face\_lattice(I))]. \mforall{}[j:\mBbbN{}]. \mforall{}[g:J {}\mrightarrow{} I+j].
(face-lattice-tube(I;phi;j))<g> = (s(phi))<g> \mvee{} (k=0) \mvee{} (k=1) supposing (g j) = <k>
Date html generated:
2017_10_05-AM-01_17_19
Last ObjectModification:
2017_07_28-AM-09_33_01
Theory : cubical!type!theory
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