Nuprl Lemma : A-face-compatible-image
∀X:CubicalSet. ∀A:{X ⊢ _}. ∀I,K:Cname List. ∀f:name-morph(I;K). ∀alpha:X(I). ∀fc1,fc2:A-face(X;A;I;alpha).
((↑isname(f (fst(fc1))))
⇒ (↑isname(f (fst(fc2))))
⇒ A-face-compatible(X;A;I;alpha;fc1;fc2)
⇒ A-face-compatible(X;A;K;f(alpha);A-face-image(X;A;I;K;f;alpha;fc1);A-face-image(X;A;I;K;f;alpha;fc2)))
Proof
Definitions occuring in Statement :
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
A-face: A-face(X;A;I;alpha),
cubical-type: {X ⊢ _},
cube-set-restriction: f(s),
I-cube: X(I),
cubical-set: CubicalSet,
name-morph: name-morph(I;J),
isname: isname(z),
coordinate_name: Cname,
list: T List,
assert: ↑b,
pi1: fst(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
apply: f a
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
A-face: A-face(X;A;I;alpha),
A-face-compatible: A-face-compatible(X;A;I;alpha;f1;f2),
spreadn: spread3,
A-face-image: A-face-image(X;A;I;K;f;alpha;face),
pi1: fst(t),
uall: ∀[x:A]. B[x],
member: t ∈ T,
name-morph: name-morph(I;J),
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
not: ¬A,
subtype_rel: A ⊆r B,
guard: {T},
nameset: nameset(L),
false: False,
prop: ℙ,
top: Top,
coordinate_name: Cname,
int_upper: {i...},
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
cand: A c∧ B,
squash: ↓T,
int_seg: {i..j-},
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
true: True
Lemmas referenced :
assert-isname,
coordinate_name_wf,
istype-void,
A-face-compatible_wf,
istype-assert,
isname_wf,
pi1_wf_top,
nameset_wf,
A-face_wf,
I-cube_wf,
name-morph_wf,
list_wf,
cubical-type_wf,
cubical-set_wf,
subtype_base_sq,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
istype-le,
member-list-diff,
cname_deq_wf,
cons_wf,
nil_wf,
int_seg_properties,
full-omega-unsat,
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,
int_formula_prop_wf,
member_singleton,
l_member_wf,
list-diff_wf,
name-morph_subtype_remove1,
cubical-type-ap-morph-comp,
cube-set-restriction_wf,
face-map_wf2,
subtype_rel_self,
subtype_rel_wf,
squash_wf,
true_wf,
list-diff2,
iff_weakening_equal,
list-diff2-sym,
equal_wf,
istype-universe,
cubical-type-at_wf,
name-comp_wf,
cube-set-restriction-comp,
face-maps-commute,
list-diff-subset,
nameset_subtype,
face-map-comp,
name-comp-assoc,
subtype_rel-equal,
cubical-type-ap-morph_wf,
not_wf,
ext-eq_weakening,
subtype_rel_weakening,
face-map-comp2,
trivial-equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
setElimination,
hypothesis,
independent_isectElimination,
because_Cache,
functionIsType,
equalityIsType1,
universeIsType,
equalityTransitivity,
equalitySymmetry,
lambdaEquality_alt,
inhabitedIsType,
independent_pairEquality,
isect_memberEquality_alt,
voidElimination,
independent_functionElimination,
instantiate,
cumulativity,
intEquality,
closedConclusion,
natural_numberEquality,
dependent_functionElimination,
dependent_set_memberEquality_alt,
independent_pairFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
hyp_replacement,
equalityIsType4,
promote_hyp,
lambdaEquality,
universeEquality,
productEquality,
addLevel,
dependent_set_memberEquality,
voidEquality,
isect_memberEquality,
dependent_pairFormation,
lambdaFormation
Latex:
\mforall{}X:CubicalSet. \mforall{}A:\{X \mvdash{} \_\}. \mforall{}I,K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}alpha:X(I).
\mforall{}fc1,fc2:A-face(X;A;I;alpha).
((\muparrow{}isname(f (fst(fc1))))
{}\mRightarrow{} (\muparrow{}isname(f (fst(fc2))))
{}\mRightarrow{} A-face-compatible(X;A;I;alpha;fc1;fc2)
{}\mRightarrow{} A-face-compatible(X;A;K;f(alpha);A-face-image(X;A;I;K;f;alpha;fc1);...))
Date html generated:
2019_11_05-PM-00_27_59
Last ObjectModification:
2018_11_08-PM-02_38_28
Theory : cubical!sets
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