Nuprl Lemma : nerve_box_edge_wf
∀[C:SmallCategory]. ∀[I:Cname List]. ∀[J:nameset(I) List]. ∀[x:nameset(I)]. ∀[i:ℕ2].
∀[box:open_box(cubical-nerve(C);I;J;x;i)]. ∀[y:nameset(I)]. ∀[c:{c:name-morph(I;[])| (c y) = 0 ∈ ℕ2} ].
nerve_box_edge(box;c;y) ∈ cat-arrow(C) nerve_box_label(box;c) nerve_box_label(box;flip(c;y))
supposing (∃j1∈J. (∃j2∈J. ¬(j1 = j2 ∈ Cname))) ∨ (((c x) = i ∈ ℕ2) ∧ (¬↑null(J)))
Proof
Definitions occuring in Statement :
nerve_box_edge: nerve_box_edge(box;c;y),
nerve_box_label: nerve_box_label(box;L),
cubical-nerve: cubical-nerve(X),
open_box: open_box(X;I;J;x;i),
name-morph-flip: flip(f;y),
name-morph: name-morph(I;J),
nameset: nameset(L),
coordinate_name: Cname,
cat-arrow: cat-arrow(C),
small-category: SmallCategory,
l_exists: (∃x∈L. P[x]),
null: null(as),
nil: [],
list: T List,
int_seg: {i..j-},
assert: ↑b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
apply: f a,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
or: P ∨ Q,
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
top: Top,
guard: {T},
so_lambda: λ2x.t[x],
nameset: nameset(L),
so_apply: x[s],
all: ∀x:A. B[x],
name-morph: name-morph(I;J),
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
rev_implies: P ⇐ Q,
decidable: Dec(P),
not: ¬A,
false: False
Lemmas referenced :
nerve_box_edge_wf2,
not_wf,
assert_wf,
null_wf3,
subtype_rel_list,
top_wf,
l_exists_wf,
nameset_wf,
equal_wf,
coordinate_name_wf,
l_member_wf,
or_wf,
int_seg_wf,
set_wf,
name-morph_wf,
nil_wf,
equal-wf-T-base,
open_box_wf,
cubical-nerve_wf,
list_wf,
small-category_wf,
l_exists_iff,
exists_wf,
decidable__equal-coordinate_name
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
setElimination,
rename,
dependent_set_memberEquality,
because_Cache,
hypothesis,
independent_isectElimination,
unionElimination,
inlFormation,
productEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
sqequalRule,
inrFormation,
setEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
lambdaFormation,
dependent_functionElimination,
natural_numberEquality,
baseClosed,
productElimination,
independent_functionElimination,
addLevel,
existsFunctionality,
independent_pairFormation,
andLevelFunctionality,
promote_hyp,
dependent_pairFormation
Latex:
\mforall{}[C:SmallCategory]. \mforall{}[I:Cname List]. \mforall{}[J:nameset(I) List]. \mforall{}[x:nameset(I)]. \mforall{}[i:\mBbbN{}2].
\mforall{}[box:open\_box(cubical-nerve(C);I;J;x;i)]. \mforall{}[y:nameset(I)]. \mforall{}[c:\{c:name-morph(I;[])| (c y) = 0\} ].
nerve\_box\_edge(box;c;y) \mmember{} cat-arrow(C) nerve\_box\_label(box;c) nerve\_box\_label(box;flip(c;y))
supposing (\mexists{}j1\mmember{}J. (\mexists{}j2\mmember{}J. \mneg{}(j1 = j2))) \mvee{} (((c x) = i) \mwedge{} (\mneg{}\muparrow{}null(J)))
Date html generated:
2017_10_05-PM-03_37_39
Last ObjectModification:
2017_07_28-AM-11_25_43
Theory : cubical!sets
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