Nuprl Lemma : get_face_unique
∀X:CubicalSet. ∀I:Cname List. ∀f:I-face(X;I). ∀J:nameset(I) List. ∀x:nameset(I). ∀i:ℕ2. ∀box:open_box(X;I;J;x;i).
((f ∈ box) ⇒ (get_face(dimension(f);direction(f);box) = f ∈ I-face(X;I)))
Proof
Definitions occuring in Statement :
get_face: get_face(y;c;box),
open_box: open_box(X;I;J;x;i),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
cubical-set: CubicalSet,
nameset: nameset(L),
coordinate_name: Cname,
l_member: (x ∈ l),
list: T List,
int_seg: {i..j-},
all: ∀x:A. B[x],
implies: P ⇒ Q,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
open_box: open_box(X;I;J;x;i),
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
nameset: nameset(L),
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
sq_stable: SqStable(P),
cand: A c∧ B,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
squash: ↓T,
l_all: (∀x∈L.P[x]),
int_seg: {i..j-},
nat: ℕ,
lelt: i ≤ j < k,
guard: {T},
coordinate_name: Cname,
int_upper: {i...},
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
I-face: I-face(X;I),
iff: P ⇐⇒ Q,
face-dimension: dimension(f),
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
face-name: face-name(f),
face-direction: direction(f),
less_than: a < b,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2]
Lemmas referenced :
l_member_wf,
I-face_wf,
open_box_wf,
subtype_rel_list,
nameset_wf,
coordinate_name_wf,
int_seg_wf,
list_wf,
cubical-set_wf,
decidable__nameset,
face-dimension_wf,
get_face_wf,
face-direction_wf,
sq_stable__and,
equal_wf,
equal-wf-base,
sq_stable__equal,
nat_properties,
int_seg_properties,
sq_stable__l_member,
decidable__equal-coordinate_name,
sq_stable__le,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
istype-le,
istype-less_than,
length_wf,
pi1_wf_top,
cons_wf,
cons_member,
subtype_base_sq,
set_subtype_base,
le_wf,
int_subtype_base,
decidable__equal_int_seg,
non_neg_length,
length_wf_nat,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
decidable__equal_int,
intformeq_wf,
itermSubtract_wf,
int_formula_prop_eq_lemma,
int_term_value_subtract_lemma,
face-name_wf,
subtract_wf,
nameset_subtype_base,
lelt_wf,
get_face-wf,
pairwise-implies,
not_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
inhabitedIsType,
hypothesis,
thin,
equalityIsType1,
hypothesisEquality,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
universeIsType,
introduction,
extract_by_obid,
isectElimination,
setElimination,
rename,
applyEquality,
independent_isectElimination,
lambdaEquality_alt,
sqequalRule,
natural_numberEquality,
unionElimination,
productElimination,
isect_memberEquality_alt,
because_Cache,
axiomEquality,
functionIsTypeImplies,
imageMemberEquality,
baseClosed,
imageElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
voidElimination,
productIsType,
hyp_replacement,
applyLambdaEquality,
independent_pairEquality,
instantiate,
cumulativity,
intEquality,
closedConclusion,
equalityIsType4,
productEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}I:Cname List. \mforall{}f:I-face(X;I). \mforall{}J:nameset(I) List. \mforall{}x:nameset(I). \mforall{}i:\mBbbN{}2.
\mforall{}box:open\_box(X;I;J;x;i).
((f \mmember{} box) {}\mRightarrow{} (get\_face(dimension(f);direction(f);box) = f))
Date html generated:
2019_11_05-PM-00_28_34
Last ObjectModification:
2018_11_08-PM-00_51_59
Theory : cubical!sets
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