Nuprl Lemma : no-retraction-case-0
∀[k:ℕ]. ∀[K:0-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 :
retraction: retraction(X;d;A),
so_apply: x[s],
so_lambda: λ2x.t[x],
prop: ℙ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_apply: x[s1;s2],
top: Top,
so_lambda: λ2x y.t[x; y],
it: ⋅,
nil: [],
all: ∀x:A. B[x],
uimplies: b supposing a,
select: L[n],
exists: ∃x:A. B[x],
l_exists: (∃x∈L. P[x]),
subtype_rel: A ⊆r B,
false: False,
not: ¬A,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
nat: ℕ,
rational-cube-complex: n-dim-complex,
rat-cube-complex-polyhedron: |K|,
member: t ∈ T,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_seg_wf,
exists_wf,
subtype_rel_sets_simple,
metric-on-subtype,
rn-prod-metric_wf,
in-rat-cube_wf,
not_wf,
rat-cube-complex-polyhedron_wf,
retraction_wf,
member-not,
rat-cube-complex-polyhedron-inhabited,
l_exists_wf_nil,
real-vec_wf,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_and_lemma,
istype-int,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
int_seg_properties,
length_of_nil_lemma,
istype-base,
stuck-spread,
istype-nat,
istype-le,
istype-void,
rational-cube-complex_wf,
rational-cube_wf,
length_wf,
istype-less_than,
rat-complex-boundary-0-dim
Rules used in proof :
productIsType,
promote_hyp,
closedConclusion,
applyEquality,
setEquality,
functionIsType,
setIsType,
because_Cache,
dependent_functionElimination,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
isect_memberEquality_alt,
independent_isectElimination,
baseClosed,
productElimination,
independent_functionElimination,
lambdaEquality_alt,
voidElimination,
sqequalRule,
independent_pairFormation,
dependent_set_memberEquality_alt,
universeIsType,
rename,
setElimination,
natural_numberEquality,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
lambdaFormation_alt,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[K:0-dim-complex]. (0 < ||K|| {}\mRightarrow{} (\mneg{}retraction(|K|;rn-prod-metric(k);|\mpartial{}(K)|)))
Date html generated:
2019_10_30-AM-10_13_27
Last ObjectModification:
2019_10_29-PM-01_34_03
Theory : real!vectors
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