Nuprl Lemma : adjacent-compatible-iff
∀[X:CubicalSet]
∀I:Cname List. ∀L:I-face(X;I) List.
uiff(adjacent-compatible(X;I;L);∀f1,f2:I-face(X;I).
((f1 ∈ L)
⇒ (f2 ∈ L)
⇒ (¬(dimension(f1) = dimension(f2) ∈ Cname))
⇒ ((dimension(f2):=direction(f2))(cube(f1))
= (dimension(f1):=direction(f1))(cube(f2))
∈ X(I-[dimension(f1); dimension(f2)]))))
Proof
Definitions occuring in Statement :
adjacent-compatible: adjacent-compatible(X;I;L),
face-cube: cube(f),
face-direction: direction(f),
face-dimension: dimension(f),
I-face: I-face(X;I),
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
face-map: (x:=i),
cname_deq: CnameDeq,
coordinate_name: Cname,
list-diff: as-bs,
l_member: (x ∈ l),
cons: [a / b],
nil: [],
list: T List,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
implies: P ⇒ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
nameset: nameset(L),
so_lambda: λ2x.t[x],
so_apply: x[s],
true: True,
squash: ↓T,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
l_member: (x ∈ l),
exists: ∃x:A. B[x],
cand: A c∧ B,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
adjacent-compatible: adjacent-compatible(X;I;L),
pairwise: (∀x,y∈L. P[x; y]),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
I-face: I-face(X;I),
face-compatible: face-compatible(X;I;f1;f2),
spreadn: spread3,
face-dimension: dimension(f),
pi1: fst(t),
face-cube: cube(f),
face-direction: direction(f),
pi2: snd(t),
not: ¬A,
coordinate_name: Cname,
int_upper: {i...},
false: False,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
not_wf,
equal_wf,
coordinate_name_wf,
face-dimension_wf,
nameset_wf,
l_member_wf,
I-face_wf,
adjacent-compatible_wf,
all_wf,
I-cube_wf,
list-diff_wf,
cname_deq_wf,
cons_wf,
nil_wf,
cube-set-restriction_wf,
face-cube_wf,
list_wf,
face-map_wf2,
face-direction_wf,
name-morph_wf,
subtype_rel_wf,
squash_wf,
true_wf,
list-diff2,
iff_weakening_equal,
subtype_rel_self,
list-diff2-sym,
decidable__lt,
lelt_wf,
length_wf,
face-compatible_wf,
nat_properties,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
intformnot_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_not_lemma,
le_wf,
int_formula_prop_wf,
subtype_base_sq,
list_subtype_base,
set_subtype_base,
int_subtype_base,
nat_wf,
decidable__equal_int,
intformless_wf,
int_formula_prop_less_lemma,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_seg_wf,
decidable__equal-coordinate_name,
select_wf,
select_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
sqequalRule,
dependent_functionElimination,
axiomEquality,
because_Cache,
functionEquality,
natural_numberEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
baseClosed,
independent_isectElimination,
productElimination,
independent_functionElimination,
unionElimination,
dependent_set_memberEquality,
hyp_replacement,
applyLambdaEquality,
dependent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
instantiate,
cumulativity
Latex:
\mforall{}[X:CubicalSet]
\mforall{}I:Cname List. \mforall{}L:I-face(X;I) List.
uiff(adjacent-compatible(X;I;L);\mforall{}f1,f2:I-face(X;I).
((f1 \mmember{} L)
{}\mRightarrow{} (f2 \mmember{} L)
{}\mRightarrow{} (\mneg{}(dimension(f1) = dimension(f2)))
{}\mRightarrow{} ((dimension(f2):=direction(f2))(cube(f1))
= (dimension(f1):=direction(f1))(cube(f2)))))
Date html generated:
2017_10_05-AM-10_18_01
Last ObjectModification:
2017_07_28-AM-11_20_32
Theory : cubical!sets
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