Nuprl Lemma : fine-iter-subdiv
∀k:ℕ
∀[n:ℕ]
∀K:{K:n-dim-complex| 0 < ||K||} . ∀M:ℕ+.
∃j:ℕ
∀[x,y:ℝ^k].
mdist(rn-prod-metric(k);x;y) ≤ (r1/r(M))
supposing ¬¬(∃c:ℚCube(k). ((c ∈ K'^(j)) ∧ in-rat-cube(k;y;c) ∧ in-rat-cube(k;x;c)))
Proof
Definitions occuring in Statement :
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),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
l_member: (x ∈ l),
length: ||as||,
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
and: P ∧ Q,
set: {x:A| B[x]} ,
natural_number: $n,
rational-cube-complex: n-dim-complex,
rational-cube: ℚCube(k)
Definitions unfolded in proof :
req_int_terms: t1 ≡ t2,
rdiv: (x/y),
uiff: uiff(P;Q),
sq_type: SQType(T),
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
so_apply: x[s],
so_lambda: λ2x.t[x],
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rge: x ≥ y,
rev_uimplies: rev_uimplies(P;Q),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
nat_plus: ℕ+,
false: False,
prop: ℙ,
rational-cube-complex: n-dim-complex,
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
not: ¬A,
cand: A c∧ B,
exists: ∃x:A. B[x],
and: P ∧ Q,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x]
Lemmas referenced :
real_term_value_var_lemma,
real_term_value_const_lemma,
real_term_value_mul_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rmul-rinv3,
req_transitivity,
rleq_functionality,
itermSubtract_wf,
rinv_wf2,
rmul_preserves_rleq,
int_term_value_mul_lemma,
itermMultiply_wf,
int_formula_prop_le_lemma,
intformle_wf,
decidable__le,
exp_wf_nat_plus,
rleq-int-fractions,
nequal_wf,
subtype_base_sq,
exp_wf3,
int_nzero-rational,
int-subtype-rationals,
equal_functionality_wrt_subtype_rel2,
int_subtype_base,
le_wf,
set_subtype_base,
rationals_wf,
equal-wf-base,
not_functionality_wrt_implies,
rneq-int,
istype-le,
log-property,
log_wf,
r-archimedean,
rleq_weakening_equal,
rleq_functionality_wrt_implies,
implies_weakening_uimplies,
rat-complex-diameter-bound,
exp-positive-stronger,
exp_wf2,
rmul_wf,
rat-complex-diameter_wf,
rat-complex-iter-subdiv-pos-length,
istype-nat,
length_wf,
istype-less_than,
rational-cube-complex_wf,
nat_plus_wf,
rless_wf,
int_formula_prop_wf,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
istype-int,
itermVar_wf,
itermConstant_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_plus_properties,
nat_properties,
rless-int,
int-to-real_wf,
rdiv_wf,
rn-prod-metric_wf,
mdist_wf,
rleq_wf,
istype-void,
in-rat-cube_wf,
Error :rat-complex-iter-subdiv_wf,
l_member_wf,
rational-cube_wf,
real-vec_wf,
le_witness_for_triv,
rat-complex-iter-subdiv-diameter
Rules used in proof :
sqequalBase,
cumulativity,
instantiate,
intEquality,
baseApply,
multiplyEquality,
baseClosed,
imageMemberEquality,
equalityIstype,
dependent_set_memberEquality_alt,
setIsType,
independent_pairFormation,
voidElimination,
int_eqEquality,
approximateComputation,
unionElimination,
independent_functionElimination,
inrFormation_alt,
natural_numberEquality,
closedConclusion,
applyEquality,
rename,
setElimination,
productIsType,
functionIsType,
because_Cache,
universeIsType,
isectIsType,
dependent_pairFormation_alt,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
independent_isectElimination,
equalitySymmetry,
equalityTransitivity,
productElimination,
dependent_functionElimination,
lambdaEquality_alt,
isect_memberEquality_alt,
sqequalRule,
isect_memberFormation_alt,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
hypothesis,
lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}k:\mBbbN{}
\mforall{}[n:\mBbbN{}]
\mforall{}K:\{K:n-dim-complex| 0 < ||K||\} . \mforall{}M:\mBbbN{}\msupplus{}.
\mexists{}j:\mBbbN{}
\mforall{}[x,y:\mBbbR{}\^{}k].
mdist(rn-prod-metric(k);x;y) \mleq{} (r1/r(M))
supposing \mneg{}\mneg{}(\mexists{}c:\mBbbQ{}Cube(k). ((c \mmember{} K'\^{}(j)) \mwedge{} in-rat-cube(k;y;c) \mwedge{} in-rat-cube(k;x;c)))
Date html generated:
2019_11_04-PM-04_44_04
Last ObjectModification:
2019_10_31-PM-03_37_20
Theory : real!vectors
Home
Index