Nuprl Lemma : rat-complex-iter-subdiv-diameter
∀[k,n:ℕ]. ∀[K:{K:n-dim-complex| 0 < ||K||} ]. ∀[j:ℕ].
(rat-complex-diameter(k;K'^(j)) ≤ ((r1/r(2^j)) * rat-complex-diameter(k;K)))
Proof
Definitions occuring in Statement :
rat-complex-diameter: rat-complex-diameter(k;K),
rdiv: (x/y),
rleq: x ≤ y,
rmul: a * b,
int-to-real: r(n),
exp: i^n,
length: ||as||,
nat: ℕ,
less_than: a < b,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
natural_number: $n,
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
rge: x ≥ y,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
decidable: Dec(P),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rdiv: (x/y),
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
rat-complex-iter-subdiv: Error :rat-complex-iter-subdiv,
rational-cube-complex: n-dim-complex,
le: A ≤ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
prop: ℙ,
and: P ∧ Q,
top: Top,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
uimplies: b supposing a,
ge: i ≥ j ,
false: False,
implies: P ⇒ Q,
nat: ℕ,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rmul-rinv3,
rsub_wf,
rleq-implies-rleq,
rmul_preserves_rleq,
req_wf,
rleq_wf,
exp_step,
rmul-int-fractions,
req_functionality,
exp_wf_nat_plus,
mul_bounds_1b,
exp-positive-stronger,
rat-sub-div-diameter,
rleq_functionality_wrt_implies,
decidable__lt,
exp-positive,
exp_wf2,
rat-complex-subdiv_wf,
rat-complex-iter-subdiv-pos-length,
istype-le,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
intformnot_wf,
decidable__le,
subtract_wf,
Error :rat-complex-iter-subdiv_wf,
Error :rat-complex-subdiv-non-nil,
less_than_wf,
assert_wf,
iff_weakening_uiff,
assert-bnot,
bool_subtype_base,
bool_wf,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
primrec-unroll,
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-identity1,
rinv1,
rmul_functionality,
req_transitivity,
req_weakening,
rleq_functionality,
rleq_weakening_equal,
itermMultiply_wf,
itermSubtract_wf,
rinv_wf2,
rless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
rmul_wf,
rat-complex-diameter_wf,
exp0_lemma,
primrec0_lemma,
istype-nat,
rational-cube_wf,
length_wf,
rational-cube-complex_wf,
subtract-1-ge-0,
le_witness_for_triv,
istype-less_than,
ge_wf,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
istype-void,
int_formula_prop_and_lemma,
istype-int,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties
Rules used in proof :
multiplyEquality,
applyEquality,
dependent_set_memberEquality_alt,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
unionElimination,
baseClosed,
imageMemberEquality,
inrFormation_alt,
closedConclusion,
equalityIstype,
because_Cache,
setIsType,
isectIsTypeImplies,
inhabitedIsType,
functionIsTypeImplies,
equalitySymmetry,
equalityTransitivity,
productElimination,
universeIsType,
independent_pairFormation,
sqequalRule,
voidElimination,
isect_memberEquality_alt,
dependent_functionElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
natural_numberEquality,
lambdaFormation_alt,
intWeakElimination,
rename,
setElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[K:\{K:n-dim-complex| 0 < ||K||\} ]. \mforall{}[j:\mBbbN{}].
(rat-complex-diameter(k;K'\^{}(j)) \mleq{} ((r1/r(2\^{}j)) * rat-complex-diameter(k;K)))
Date html generated:
2019_11_04-PM-04_44_00
Last ObjectModification:
2019_10_31-PM-00_33_30
Theory : real!vectors
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