Nuprl Lemma : rat-cube-diameter-bound
∀[k:ℕ]. ∀[c:ℚCube(k)]. ∀[x,y:ℝ^k].
(mdist(rn-prod-metric(k);x;y) ≤ rat-cube-diameter(k;c)) supposing (in-rat-cube(k;y;c) and in-rat-cube(k;x;c))
Proof
Definitions occuring in Statement :
rat-cube-diameter: rat-cube-diameter(k;c),
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,
nat: ℕ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
rational-cube: ℚCube(k)
Definitions unfolded in proof :
rge: x ≥ y,
cand: A c∧ B,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
req_int_terms: t1 ≡ t2,
guard: {T},
rev_uimplies: rev_uimplies(P;Q),
uiff: uiff(P;Q),
rmetric: rmetric(),
in-rat-cube: in-rat-cube(k;p;c),
rnonneg: rnonneg(x),
rleq: x ≤ y,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m],
pi1: fst(t),
pi2: snd(t),
rational-interval: ℚInterval,
rational-cube: ℚCube(k),
so_apply: x[s],
prop: ℙ,
top: Top,
false: False,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
implies: P ⇒ Q,
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
all: ∀x:A. B[x],
ge: i ≥ j ,
squash: ↓T,
less_than: a < b,
le: A ≤ B,
and: P ∧ Q,
lelt: i ≤ j < k,
int_seg: {i..j-},
real-vec: ℝ^n,
metric: metric(X),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
nat: ℕ,
prod-metric: prod-metric(k;d),
mdist: mdist(d;x;y),
rn-prod-metric: rn-prod-metric(n),
rat-cube-diameter: rat-cube-diameter(k;c),
uimplies: b supposing a,
member: t ∈ T,
uall: ∀[x:A]. B[x]
Lemmas referenced :
radd_functionality_wrt_rleq,
rleq-implies-rleq,
rleq_weakening_equal,
rsub_functionality_wrt_rleq,
rleq_functionality_wrt_implies,
rleq_weakening,
rabs-difference-bound-rleq,
real_term_value_const_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_sub_lemma,
real_polynomial_null,
req-iff-rsub-is-0,
rmax-req,
req_weakening,
rleq_functionality,
rleq_transitivity,
radd_wf,
radd-preserves-rleq,
rabs_wf,
rleq_wf,
istype-nat,
rational-cube_wf,
real-vec_wf,
in-rat-cube_wf,
le_witness_for_triv,
rat2real_wf,
rsub_wf,
int-to-real_wf,
rmax_wf,
int_seg_wf,
istype-less_than,
istype-le,
int_term_value_subtract_lemma,
int_term_value_add_lemma,
int_formula_prop_less_lemma,
itermSubtract_wf,
itermAdd_wf,
intformless_wf,
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,
rmetric_wf,
subtract_wf,
rsum_functionality_wrt_rleq
Rules used in proof :
isectIsTypeImplies,
functionIsTypeImplies,
equalityIstype,
lambdaFormation_alt,
addEquality,
because_Cache,
productIsType,
universeIsType,
voidElimination,
isect_memberEquality_alt,
int_eqEquality,
dependent_pairFormation_alt,
independent_functionElimination,
approximateComputation,
independent_isectElimination,
unionElimination,
dependent_functionElimination,
independent_pairFormation,
imageElimination,
productElimination,
dependent_set_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
inhabitedIsType,
applyEquality,
lambdaEquality_alt,
hypothesis,
hypothesisEquality,
rename,
setElimination,
natural_numberEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
sqequalRule,
cut,
introduction,
isect_memberFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}[k:\mBbbN{}]. \mforall{}[c:\mBbbQ{}Cube(k)]. \mforall{}[x,y:\mBbbR{}\^{}k].
(mdist(rn-prod-metric(k);x;y) \mleq{} rat-cube-diameter(k;c)) supposing
(in-rat-cube(k;y;c) and
in-rat-cube(k;x;c))
Date html generated:
2019_10_31-AM-06_03_19
Last ObjectModification:
2019_10_31-AM-00_15_36
Theory : real!vectors
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