Nuprl Lemma : no-retraction-case-1
∀[k:ℕ]. ∀[K:1-dim-complex]. (0 < ||K|| ⇒ (¬retraction(|K|;rn-prod-metric(k);|∂(K)|)))
Proof
Definitions occuring in Statement :
rat-cube-complex-polyhedron: |K|,
rn-prod-metric: rn-prod-metric(n),
retraction: retraction(X;d;A),
length: ||as||,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
not: ¬A,
implies: P ⇒ Q,
natural_number: $n,
rat-complex-boundary: ∂(K),
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
rat-cube-dimension: dim(c),
sq_stable: SqStable(P),
compose: f o g,
isl: isl(x),
map: map(f;as),
concat: concat(ll),
remove-repeats: remove-repeats(eq;L),
face-complex: face-complex(k;L),
list_ind: list_ind,
reduce: reduce(f;k;as),
filter: filter(P;l),
rat-complex-boundary: ∂(K),
rat-cube-sub-complex: rat-cube-sub-complex(P;L),
req: x = y,
meq: x ≡ y,
label: ...$L... t,
cons: [a / b],
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
nil: [],
select: L[n],
l_exists: (∃x∈L. P[x]),
stable: Stable{P},
rev_uimplies: rev_uimplies(P;Q),
req_int_terms: t1 ≡ t2,
nequal: a ≠ b ∈ T ,
assert: ↑b,
bnot: ¬bb,
bfalse: ff,
sq_type: SQType(T),
ifthenelse: if b then t else f fi ,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
in-rat-cube: in-rat-cube(k;p;c),
rccint: [l, u],
i-member: r ∈ I,
stable-union: Error :stable-union,
rev_implies: P ⇐ Q,
true: True,
l_member: (x ∈ l),
is-mfun: f:FUN(X;Y),
req-vec: req-vec(n;x;y),
iff: P ⇐⇒ Q,
pi2: snd(t),
l_all: (∀x∈L.P[x]),
no_repeats: no_repeats(T;l),
so_apply: x[s],
so_lambda: λ2x.t[x],
uiff: uiff(P;Q),
guard: {T},
cand: A c∧ B,
pi1: fst(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
rat-cube-complex-polyhedron: |K|,
prop: ℙ,
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
nat_plus: ℕ+,
uimplies: b supposing a,
rational-cube-complex: n-dim-complex,
exists: ∃x:A. B[x],
subtract: n - m,
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
le: A ≤ B,
nat: ℕ,
all: ∀x:A. B[x],
and: P ∧ Q,
retraction: retraction(X;d;A),
false: False,
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rat-cube-complex-polyhedron-closed,
bool_cases,
inhabited-rat-cube_wf,
sq_stable__assert,
inhabited-iff-in-rat-cube,
member_filter_2,
intersecting-0-dim-cubes,
in-rat-cube_functionality,
subtype_rel_transitivity,
stable__meq,
compose_wf,
meq-same,
meq-in-0-dim-cube,
l_exists_iff,
0-dim-complex-polyhedron-decidable,
rat-cube-complex-polyhedron-subtype,
filter-contains,
non_neg_length,
btrue_neq_bfalse,
btrue_wf,
null_nil_lemma,
nil_wf,
member-implies-null-eq-bfalse,
assert-rceq,
member_wf,
filter_wf5,
istype-assert,
assert_of_bnot,
assert_wf,
Error :filter-length-less,
rceq_wf,
bnot_wf,
rat-cube-sub-complex_wf,
rat-complex-boundary-remove1,
real-vec-sep_functionality,
not-real-vec-sep-refl,
meq_transitivity,
meq_inversion,
in-rat-cube-face,
req-vec_weakening,
in-0-dim-cube,
real-vec-sep-symmetry,
real-vec-sep_wf,
rat-cube-face_wf,
1-dim-cube-endpoints,
l_all_iff,
member-rat-complex-boundary,
int_term_value_subtract_lemma,
subtract_wf,
cons_wf,
int_term_value_add_lemma,
itermAdd_wf,
add-is-int-iff,
nat_plus_properties,
length_wf_nat,
add_nat_plus,
length_of_cons_lemma,
product_subtype_list,
istype-base,
stuck-spread,
length_of_nil_lemma,
list-cases,
rat-cube-complex-polyhedron-inhabited,
is-mfun_wf,
l_exists_wf,
subtract-1-ge-0,
ge_wf,
rleq_antisymmetry,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
not_wf,
false_wf,
stable__from_decidable,
qle_weakening_lt_qorder,
rleq-rat2real,
req_weakening,
extensional-discrete-real-fun-is-constant,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_sub_lemma,
int-to-real_wf,
real_polynomial_null,
req-iff-rsub-is-0,
itermSubtract_wf,
rccint_wf,
i-member_wf,
rleq_wf,
rleq_weakening,
neg_assert_of_eq_int,
assert-bnot,
bool_subtype_base,
bool_wf,
bool_cases_sqequal,
eqff_to_assert,
subtype_base_sq,
assert_of_eq_int,
eqtt_to_assert,
in-rat-cube_wf,
real_wf,
eq_int_wf,
ifthenelse_wf,
rat-cube-dimension-one,
iff_weakening_equal,
istype-universe,
true_wf,
squash_wf,
l_member_wf,
req-rat2real,
req_wf,
rat-cube-dimension-zero,
equal_wf,
rat-cube-dimension_wf,
equal-wf-base,
iff_weakening_uiff,
rationals_wf,
req-vec_transitivity,
req-vec_inversion,
int_formula_prop_eq_lemma,
intformeq_wf,
istype-false,
int_seg_subtype_nat,
int_subtype_base,
lelt_wf,
set_subtype_base,
decidable__equal_int,
meq-rn-prod-metric,
real-vec_wf,
metric-on-subtype,
meq_wf,
rational-interval_wf,
int_seg_wf,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
itermVar_wf,
intformle_wf,
intformand_wf,
decidable__le,
int_seg_properties,
select_wf,
rat2real_wf,
req-vec_wf,
istype-nat,
rational-cube_wf,
length_wf,
istype-less_than,
int_formula_prop_wf,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
istype-int,
itermConstant_wf,
intformless_wf,
intformnot_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
boundary-polyhedron-subtype,
rn-prod-metric_wf,
rat-cube-complex-polyhedron_wf,
retraction_wf,
rational-cube-complex_wf,
subtype_rel_self,
istype-le,
istype-void,
rat-complex-boundary_wf,
0-dim-complex-polyhedron
Rules used in proof :
productEquality,
inlFormation_alt,
inrFormation_alt,
hyp_replacement,
closedConclusion,
baseApply,
pointwiseFunctionality,
hypothesis_subsumption,
intWeakElimination,
unionIsType,
unionEquality,
setIsType,
cumulativity,
equalityElimination,
imageMemberEquality,
universeEquality,
instantiate,
productIsType,
independent_pairEquality,
baseClosed,
addEquality,
minusEquality,
applyLambdaEquality,
sqequalBase,
intEquality,
functionExtensionality,
equalityIstype,
functionEquality,
int_eqEquality,
imageElimination,
functionIsType,
equalitySymmetry,
equalityTransitivity,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
isect_memberEquality_alt,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
independent_isectElimination,
universeIsType,
independent_functionElimination,
because_Cache,
promote_hyp,
applyEquality,
hypothesis,
voidElimination,
sqequalRule,
independent_pairFormation,
natural_numberEquality,
dependent_set_memberEquality_alt,
isectElimination,
hypothesisEquality,
dependent_functionElimination,
extract_by_obid,
productElimination,
rename,
setElimination,
sqequalHypSubstitution,
thin,
lambdaFormation_alt,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[K:1-dim-complex]. (0 < ||K|| {}\mRightarrow{} (\mneg{}retraction(|K|;rn-prod-metric(k);|\mpartial{}(K)|)))
Date html generated:
2019_10_30-AM-10_13_59
Last ObjectModification:
2019_10_29-PM-04_33_11
Theory : real!vectors
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