Nuprl Lemma : rat-sub-div-diameter
∀[k,n:ℕ]. ∀[K:n-dim-complex].
rat-complex-diameter(k;(K)') ≤ ((r1/r(2)) * rat-complex-diameter(k;K)) supposing 0 < ||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),
length: ||as||,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
natural_number: $n,
rat-complex-subdiv: (K)',
rational-cube-complex: n-dim-complex
Definitions unfolded in proof :
l_member: (x ∈ l),
nat_plus: ℕ+,
rev_uimplies: rev_uimplies(P;Q),
subtract: n - m,
cand: A c∧ B,
rnonneg: rnonneg(x),
rleq: x ≤ y,
so_apply: x[s],
uiff: uiff(P;Q),
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
decidable: Dec(P),
ge: i ≥ j ,
nat: ℕ,
le: A ≤ B,
lelt: i ≤ j < k,
int_seg: {i..j-},
so_lambda: λ2x.t[x],
rat-complex-diameter: rat-complex-diameter(k;K),
sq_stable: SqStable(P),
prop: ℙ,
true: True,
less_than': less_than'(a;b),
squash: ↓T,
less_than: a < b,
implies: P ⇒ Q,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
all: ∀x:A. B[x],
or: P ∨ Q,
guard: {T},
rneq: x ≠ y,
rational-cube-complex: n-dim-complex,
subtype_rel: A ⊆r B,
member: t ∈ T,
uimplies: b supposing a,
uall: ∀[x:A]. B[x]
Lemmas referenced :
rmaximum_ub,
real_wf,
rleq_wf,
rmul_comm,
istype-false,
rleq-int-fractions2,
rmul_functionality_wrt_rleq,
req_weakening,
rleq_functionality,
l_member_wf,
zero-add,
add-commutes,
add-swap,
add-associates,
less_than_wf,
select_member,
rat-half-cube-diameter,
member-rat-complex-subdiv2,
iff_weakening_equal,
subtype_rel_self,
istype-universe,
list_wf,
true_wf,
squash_wf,
le_wf,
le_witness_for_triv,
rmaximum_wf,
int_seg_wf,
false_wf,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermAdd_wf,
intformless_wf,
subtract-is-int-iff,
add-is-int-iff,
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,
istype-void,
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,
rat-cube-diameter_wf,
subtract_wf,
rmaximum-lub,
istype-nat,
rational-cube-complex_wf,
rational-cube_wf,
length_wf,
istype-less_than,
rless_wf,
rless-int,
int-to-real_wf,
rdiv_wf,
rmul_wf,
rat-complex-subdiv_wf,
rat-complex-diameter_wf,
sq_stable__rleq,
Error :rat-complex-subdiv-non-nil
Rules used in proof :
productEquality,
productIsType,
universeEquality,
instantiate,
functionIsTypeImplies,
addEquality,
baseApply,
promote_hyp,
pointwiseFunctionality,
dependent_set_memberEquality_alt,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
approximateComputation,
unionElimination,
equalitySymmetry,
equalityTransitivity,
equalityIstype,
lambdaFormation_alt,
inhabitedIsType,
imageElimination,
universeIsType,
baseClosed,
imageMemberEquality,
independent_pairFormation,
independent_functionElimination,
productElimination,
dependent_functionElimination,
inrFormation_alt,
natural_numberEquality,
closedConclusion,
sqequalRule,
because_Cache,
rename,
setElimination,
lambdaEquality_alt,
applyEquality,
hypothesis,
independent_isectElimination,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
cut,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k,n:\mBbbN{}]. \mforall{}[K:n-dim-complex].
rat-complex-diameter(k;(K)') \mleq{} ((r1/r(2)) * rat-complex-diameter(k;K)) supposing 0 < ||K||
Date html generated:
2019_11_04-PM-04_43_13
Last ObjectModification:
2019_10_31-AM-09_56_26
Theory : real!vectors
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