Nuprl Lemma : rat-complex-diameter-bound
∀[k:ℕ]. ∀[K:ℚCube(k) List].
∀[x,y:ℝ^k].
mdist(rn-prod-metric(k);x;y) ≤ rat-complex-diameter(k;K)
supposing ¬¬(∃c:ℚCube(k). ((c ∈ K) ∧ in-rat-cube(k;y;c) ∧ in-rat-cube(k;x;c)))
supposing 0 < ||K||
Proof
Definitions occuring in Statement :
rat-complex-diameter: rat-complex-diameter(k;K),
in-rat-cube: in-rat-cube(k;p;c),
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rleq: x ≤ y,
l_member: (x ∈ l),
length: ||as||,
list: T List,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
natural_number: $n,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
so_apply: x[s],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
nat: ℕ,
rat-complex-diameter: rat-complex-diameter(k;K),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
subtype_rel: A ⊆r B,
squash: ↓T,
guard: {T},
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
true: True,
cand: A c∧ B,
l_member: (x ∈ l),
stable: Stable{P},
or: P ∨ Q,
false: False,
prop: ℙ,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
not: ¬A,
and: P ∧ Q,
le: A ≤ B,
all: ∀x:A. B[x],
rnonneg: rnonneg(x),
rleq: x ≤ y,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
int_seg_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermAdd_wf,
intformless_wf,
istype-le,
decidable__lt,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
int_seg_properties,
select_wf,
subtract_wf,
rmaximum_ub,
iff_weakening_equal,
subtype_rel_self,
real_wf,
true_wf,
squash_wf,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
rat-cube-diameter_wf,
rat-cube-diameter-bound,
minimal-not-not-excluded-middle,
minimal-double-negation-hyp-elim,
rleq_wf,
not_wf,
false_wf,
rat-complex-diameter_wf,
rn-prod-metric_wf,
mdist_wf,
stable__rleq,
istype-nat,
list_wf,
length_wf,
istype-less_than,
real-vec_wf,
istype-void,
in-rat-cube_wf,
l_member_wf,
rational-cube_wf,
le_witness_for_triv
Rules used in proof :
addEquality,
dependent_set_memberEquality_alt,
independent_pairFormation,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
rename,
setElimination,
universeEquality,
instantiate,
baseClosed,
imageMemberEquality,
imageElimination,
applyEquality,
unionElimination,
lambdaFormation_alt,
because_Cache,
unionIsType,
voidElimination,
independent_functionElimination,
functionEquality,
productEquality,
unionEquality,
natural_numberEquality,
isectIsTypeImplies,
isect_memberEquality_alt,
universeIsType,
productIsType,
functionIsType,
inhabitedIsType,
functionIsTypeImplies,
independent_isectElimination,
equalitySymmetry,
hypothesis,
equalityTransitivity,
productElimination,
isectElimination,
extract_by_obid,
hypothesisEquality,
thin,
dependent_functionElimination,
lambdaEquality_alt,
sqequalHypSubstitution,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[K:\mBbbQ{}Cube(k) List].
\mforall{}[x,y:\mBbbR{}\^{}k].
mdist(rn-prod-metric(k);x;y) \mleq{} rat-complex-diameter(k;K)
supposing \mneg{}\mneg{}(\mexists{}c:\mBbbQ{}Cube(k). ((c \mmember{} K) \mwedge{} in-rat-cube(k;y;c) \mwedge{} in-rat-cube(k;x;c)))
supposing 0 < ||K||
Date html generated:
2019_11_04-PM-04_43_53
Last ObjectModification:
2019_10_31-PM-00_01_29
Theory : real!vectors
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