Nuprl Lemma : rn-prod-metric-le-max-metric
∀[n:ℕ]. rn-prod-metric(n) ≤ r(n)*max-metric(n)
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
rn-prod-metric: rn-prod-metric(n),
real-vec: ℝ^n,
metric-leq: d1 ≤ d2,
scale-metric: c*d,
int-to-real: r(n),
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
rn-prod-metric: rn-prod-metric(n),
metric-leq: d1 ≤ d2,
mdist: mdist(d;x;y),
prod-metric: prod-metric(k;d),
scale-metric: c*d,
rmetric: rmetric(),
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
prop: ℙ,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
subtract: n - m,
less_than': less_than'(a;b),
decidable: Dec(P),
or: P ∨ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
metric: metric(X),
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
guard: {T},
max-metric: max-metric(n),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
sq_stable: SqStable(P),
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
istype-less_than,
le_witness_for_triv,
real-vec_wf,
istype-le,
subtract-1-ge-0,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
istype-nat,
rsum-empty,
rmul-nonneg-case1,
int-to-real_wf,
max-metric_wf,
rleq_weakening_equal,
metric-nonneg,
rsum_wf,
subtract_wf,
rabs_wf,
rsub_wf,
int_seg_properties,
decidable__lt,
itermAdd_wf,
itermSubtract_wf,
int_term_value_add_lemma,
int_term_value_subtract_lemma,
int_seg_wf,
radd_wf,
rmul_wf,
real-vec-subtype,
rleq_functionality,
rsum-split-last,
req_weakening,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
primrec-unroll,
primrec_wf,
real_wf,
rmax_wf,
lt_int_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_wf,
bool_subtype_base,
assert-bnot,
iff_weakening_uiff,
assert_wf,
less_than_wf,
rleq-rmax,
req-int,
subtract-add-cancel,
decidable__equal_int,
intformeq_wf,
int_formula_prop_eq_lemma,
rleq-int,
rleq_wf,
req_functionality,
radd-int,
rmul_functionality,
rmul_preserves_rleq2,
sq_stable__rleq,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
rleq-implies-rleq,
real_term_value_add_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
intWeakElimination,
lambdaFormation_alt,
natural_numberEquality,
independent_isectElimination,
approximateComputation,
independent_functionElimination,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
dependent_functionElimination,
isect_memberEquality_alt,
voidElimination,
independent_pairFormation,
universeIsType,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
dependent_set_memberEquality_alt,
because_Cache,
unionElimination,
minusEquality,
imageMemberEquality,
baseClosed,
applyEquality,
imageElimination,
productIsType,
addEquality,
closedConclusion,
equalityElimination,
equalityIstype,
promote_hyp,
instantiate,
cumulativity
Latex:
\mforall{}[n:\mBbbN{}]. rn-prod-metric(n) \mleq{} r(n)*max-metric(n)
Date html generated:
2019_10_30-AM-08_37_46
Last ObjectModification:
2019_10_02-AM-11_03_13
Theory : reals
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