Nuprl Lemma : satisfies-s-face-lattice-tube
∀I:fset(ℕ). ∀i:{i:ℕ| ¬i ∈ I} . ∀phi:𝔽(I). ∀j:{j:ℕ| ¬j ∈ I+i} . ∀K:fset(ℕ). ∀f:K ⟶ I+j+i.
((s(face-lattice-tube(I;phi;j)) f) = 1
⇐⇒ (s(phi) s ⋅ f) = 1 ∨ ((s ⋅ f j) = 0 ∈ Point(dM(K))) ∨ ((s ⋅ f j) = 1 ∈ Point(dM(K))))
Proof
Definitions occuring in Statement :
name-morph-satisfies: (psi f) = 1,
face-lattice-tube: face-lattice-tube(I;phi;j),
face-presheaf: 𝔽,
cube-set-restriction: f(s),
I_cube: A(I),
nc-s: s,
add-name: I+i,
nh-comp: g ⋅ f,
names-hom: I ⟶ J,
dM1: 1,
dM0: 0,
dM: dM(I),
lattice-point: Point(l),
fset-member: a ∈ s,
fset: fset(T),
int-deq: IntDeq,
nat: ℕ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
uimplies: b supposing a,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
I_cube: A(I),
functor-ob: functor-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,
so_lambda: λ2x.t[x],
so_apply: x[s],
rev_implies: P ⇐ Q,
DeMorgan-algebra: DeMorganAlgebra,
guard: {T},
names-hom: I ⟶ J,
names: names(I),
nat: ℕ,
or: P ∨ Q,
uiff: uiff(P;Q)
Lemmas referenced :
I_cube_wf,
not_wf,
set_wf,
face-presheaf_wf,
cube-set-restriction_wf,
names-deq_wf,
union-deq_wf,
fset-antichain_wf,
assert_wf,
names_wf,
fset_wf,
name-morph-satisfies-comp,
iff_wf,
dM1_wf,
dM0_wf,
strong-subtype-self,
le_wf,
strong-subtype-set3,
strong-subtype-deq-subtype,
int-deq_wf,
nat_wf,
fset-member_wf,
trivial-member-add-name1,
names-hom_wf,
DeMorgan-algebra-axioms_wf,
lattice-join_wf,
lattice-meet_wf,
uall_wf,
bounded-lattice-axioms_wf,
bounded-lattice-structure_wf,
subtype_rel_transitivity,
DeMorgan-algebra-structure-subtype,
bounded-lattice-structure-subtype,
lattice-axioms_wf,
lattice-structure_wf,
DeMorgan-algebra-structure_wf,
subtype_rel_set,
dM_wf,
lattice-point_wf,
equal_wf,
or_wf,
satisfies-face-lattice-tube,
face-lattice-constraints_wf,
fset-contains-none_wf,
fset-all_wf,
subtype_rel_self,
fl-morph-face-lattice-tube1,
name-morph-satisfies_wf,
f-subset-add-name,
nc-s_wf,
add-name_wf,
nh-comp_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
independent_pairFormation,
applyEquality,
sqequalRule,
setEquality,
productEquality,
lambdaEquality,
addLevel,
productElimination,
impliesFunctionality,
independent_functionElimination,
instantiate,
cumulativity,
universeEquality,
equalityTransitivity,
equalitySymmetry,
dependent_set_memberEquality,
intEquality,
natural_numberEquality,
orFunctionality,
unionEquality
Latex:
\mforall{}I:fset(\mBbbN{}). \mforall{}i:\{i:\mBbbN{}| \mneg{}i \mmember{} I\} . \mforall{}phi:\mBbbF{}(I). \mforall{}j:\{j:\mBbbN{}| \mneg{}j \mmember{} I+i\} . \mforall{}K:fset(\mBbbN{}). \mforall{}f:K {}\mrightarrow{} I+j+i.
((s(face-lattice-tube(I;phi;j)) f) = 1 \mLeftarrow{}{}\mRightarrow{} (s(phi) s \mcdot{} f) = 1 \mvee{} ((s \mcdot{} f j) = 0) \mvee{} ((s \mcdot{} f j) = 1))
Date html generated:
2016_05_18-PM-00_20_58
Last ObjectModification:
2016_02_05-PM-05_46_56
Theory : cubical!type!theory
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