Nuprl Lemma : max-metric-sep
∀n:ℕ. ∀x,y:ℝ^n. (r0 < mdist(max-metric(n);x;y) ⇐⇒ ∃i:ℕn. x i ≠ y i)
Proof
Definitions occuring in Statement :
max-metric: max-metric(n),
real-vec: ℝ^n,
mdist: mdist(d;x;y),
rneq: x ≠ y,
rless: x < y,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
metric-leq: d1 ≤ d2,
scale-metric: c*d,
mdist: mdist(d;x;y),
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
rev_implies: P ⇐ Q,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
sq_type: SQType(T),
rn-metric: rn-metric(n),
rless: x < y,
sq_exists: ∃x:A [B[x]],
nat_plus: ℕ+,
ge: i ≥ j ,
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
real-vec: ℝ^n,
le: A ≤ B,
less_than': less_than'(a;b),
subtype_rel: A ⊆r B,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2
Lemmas referenced :
max-metric-leq-rn-metric,
rn-metric-leq-max-metric,
rless_transitivity1,
int-to-real_wf,
mdist_wf,
real-vec_wf,
max-metric_wf,
rn-metric_wf,
rless_wf,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
rmul_preserves_rless,
rless-int,
nat_plus_properties,
nat_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
intformle_wf,
istype-int,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
rn-metric-sep,
int_seg_wf,
rneq_wf,
istype-nat,
real-vec-dist_wf,
istype-le,
itermAdd_wf,
int_term_value_add_lemma,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
rless_functionality,
req_weakening,
real-vec-dist-dim0,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
sqequalRule,
independent_pairFormation,
hypothesis,
natural_numberEquality,
independent_functionElimination,
independent_isectElimination,
universeIsType,
setElimination,
rename,
unionElimination,
instantiate,
cumulativity,
intEquality,
equalityTransitivity,
equalitySymmetry,
productElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
promote_hyp,
productIsType,
applyEquality,
dependent_set_memberEquality_alt,
inhabitedIsType,
imageElimination
Latex:
\mforall{}n:\mBbbN{}. \mforall{}x,y:\mBbbR{}\^{}n. (r0 < mdist(max-metric(n);x;y) \mLeftarrow{}{}\mRightarrow{} \mexists{}i:\mBbbN{}n. x i \mneq{} y i)
Date html generated:
2019_10_30-AM-08_43_44
Last ObjectModification:
2019_10_02-AM-11_06_39
Theory : reals
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