Nuprl Lemma : rn-metric-leq-rn-prod-metric
∀[n:ℕ]. rn-metric(n) ≤ rn-prod-metric(n)
Proof
Definitions occuring in Statement :
rn-prod-metric: rn-prod-metric(n),
rn-metric: rn-metric(n),
real-vec: ℝ^n,
metric-leq: d1 ≤ d2,
nat: ℕ,
uall: ∀[x:A]. B[x]
Definitions unfolded in proof :
rn-metric: rn-metric(n),
metric-leq: d1 ≤ d2,
mdist: mdist(d;x;y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
implies: P ⇒ Q,
real-vec-dist: d(x;y),
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
and: P ∧ Q,
uimplies: b supposing a,
nat: ℕ,
less_than': less_than'(a;b),
not: ¬A,
false: False,
rn-prod-metric: rn-prod-metric(n),
rmetric: rmetric(),
prod-metric: prod-metric(k;d),
real-vec-sub: X - Y,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
dot-product: x⋅y,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
subtract: n - m,
true: True,
so_lambda: λ2x.t[x],
so_apply: x[s],
real-vec: ℝ^n,
int_seg: {i..j-},
lelt: i ≤ j < k,
nat_plus: ℕ+,
rge: x ≥ y,
guard: {T},
req_int_terms: t1 ≡ t2,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
pointwise-rleq: x[k] ≤ y[k] for k ∈ [n,m]
Lemmas referenced :
square-rleq-implies,
real-vec-dist_wf,
mdist_wf,
real-vec_wf,
rn-prod-metric_wf,
mdist-nonneg,
le_witness_for_triv,
istype-nat,
rnexp_wf,
istype-void,
istype-le,
real-vec-norm_wf,
real-vec-sub_wf,
dot-product_wf,
rleq_functionality,
real-vec-norm-squared,
req_weakening,
nat_properties,
full-omega-unsat,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
istype-int,
int_formula_prop_and_lemma,
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,
subtract-1-ge-0,
decidable__le,
intformnot_wf,
int_formula_prop_not_lemma,
subtract_wf,
rnexp2-nonneg,
int-to-real_wf,
rsum-empty,
rsum_wf,
rabs_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,
real-vec-subtype,
rmul_wf,
itermMultiply_wf,
radd-preserves-rleq,
rminus_wf,
itermMinus_wf,
rnexp_functionality,
rsum-split-last,
dot-product-split-last,
rleq_functionality_wrt_implies,
radd_functionality_wrt_rleq,
rleq_weakening_equal,
radd_functionality,
rnexp2,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_add_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
req_inversion,
rabs-rnexp2,
real_term_value_minus_lemma,
rleq_wf,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
rsum_nonneg,
zero-rleq-rabs,
rmul_preserves_rleq2,
rmul-nonneg-case1,
rleq-int,
istype-false
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isectElimination,
hypothesisEquality,
hypothesis,
applyEquality,
lambdaEquality_alt,
setElimination,
rename,
inhabitedIsType,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
productElimination,
independent_isectElimination,
functionIsTypeImplies,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
voidElimination,
because_Cache,
equalityIstype,
intWeakElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
unionElimination,
minusEquality,
imageMemberEquality,
baseClosed,
imageElimination,
productIsType,
addEquality,
instantiate,
universeEquality
Latex:
\mforall{}[n:\mBbbN{}]. rn-metric(n) \mleq{} rn-prod-metric(n)
Date html generated:
2019_10_30-AM-08_37_16
Last ObjectModification:
2019_10_02-AM-11_02_55
Theory : reals
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