Nuprl Lemma : face-compatible-image
∀X:CubicalSet. ∀I,K:Cname List. ∀f:name-morph(I;K). ∀fc1,fc2:I-face(X;I).
((↑isname(f (fst(fc1))))
⇒ (↑isname(f (fst(fc2))))
⇒ face-compatible(X;I;fc1;fc2)
⇒ face-compatible(X;K;face-image(X;I;K;f;fc1);face-image(X;I;K;f;fc2)))
Proof
Definitions occuring in Statement :
face-image: face-image(X;I;K;f;face),
face-compatible: face-compatible(X;I;f1;f2),
I-face: I-face(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,
I-face: I-face(X;I),
face-compatible: face-compatible(X;I;f1;f2),
spreadn: spread3,
face-image: face-image(X;I;K;f;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,
nameset: nameset(L),
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
subtype_rel: A ⊆r B,
guard: {T},
top: Top,
false: False,
sq_type: SQType(T),
so_apply: x[s],
so_lambda: λ2x.t[x],
int_upper: {i...},
coordinate_name: Cname,
not: ¬A,
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,
so_apply: x[s1;s2;s3],
so_lambda: so_lambda(x,y,z.t[x; y; z]),
append: as @ bs
Lemmas referenced :
assert-isname,
decidable__equal-coordinate_name,
not_wf,
equal_wf,
coordinate_name_wf,
nameset_wf,
face-compatible_wf,
assert_wf,
isname_wf,
pi1_wf_top,
I-face_wf,
name-morph_wf,
list_wf,
cubical-set_wf,
int_subtype_base,
le_wf,
set_subtype_base,
subtype_base_sq,
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,
equal-wf-base-T,
member_singleton,
l_member_wf,
list-diff_wf,
name-morph_subtype_remove1,
squash_wf,
true_wf,
cube-set-restriction-comp,
subtype_rel_self,
iff_weakening_equal,
I-cube_wf,
list_ind_nil_lemma,
list_ind_cons_lemma,
face-map_wf2,
subtype_rel_wf,
list-diff-diff,
list-diff2,
list-diff2-sym,
list_subtype_base,
name-comp_wf,
cube-set-restriction_wf,
face-map-comp
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
sqequalRule,
rename,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
setElimination,
hypothesis,
independent_isectElimination,
because_Cache,
dependent_functionElimination,
unionElimination,
equalityTransitivity,
equalitySymmetry,
lambdaEquality,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
natural_numberEquality,
intEquality,
cumulativity,
instantiate,
independent_functionElimination,
dependent_set_memberEquality,
independent_pairFormation,
applyLambdaEquality,
imageMemberEquality,
baseClosed,
imageElimination,
approximateComputation,
dependent_pairFormation,
int_eqEquality,
hyp_replacement,
addLevel,
impliesFunctionality,
productEquality,
universeEquality
Latex:
\mforall{}X:CubicalSet. \mforall{}I,K:Cname List. \mforall{}f:name-morph(I;K). \mforall{}fc1,fc2:I-face(X;I).
((\muparrow{}isname(f (fst(fc1))))
{}\mRightarrow{} (\muparrow{}isname(f (fst(fc2))))
{}\mRightarrow{} face-compatible(X;I;fc1;fc2)
{}\mRightarrow{} face-compatible(X;K;face-image(X;I;K;f;fc1);face-image(X;I;K;f;fc2)))
Date html generated:
2018_05_23-PM-06_32_29
Last ObjectModification:
2018_05_16-PM-03_01_57
Theory : cubical!sets
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