Nuprl Lemma : meq-iff-mdist-rleq
∀[X:Type]. ∀[d:metric(X)]. ∀[x,y:X]. (x ≡ y ⇐⇒ ∀k:ℕ+. (mdist(d;x;y) ≤ (r1/r(k))))
Proof
Definitions occuring in Statement :
mdist: mdist(d;x;y),
meq: x ≡ y,
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
natural_number: $n,
universe: Type
Definitions unfolded in proof :
meq: x ≡ y,
mdist: mdist(d;x;y),
uall: ∀[x:A]. B[x],
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
rev_implies: P ⇐ Q,
nat_plus: ℕ+,
uimplies: b supposing a,
rneq: x ≠ y,
guard: {T},
or: P ∨ Q,
decidable: Dec(P),
not: ¬A,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
rleq: x ≤ y,
rnonneg: rnonneg(x),
le: A ≤ B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
rless: x < y,
sq_exists: ∃x:A [B[x]],
rge: x ≥ y,
req_int_terms: t1 ≡ t2
Lemmas referenced :
nat_plus_wf,
req_wf,
mdist_wf,
int-to-real_wf,
rleq_wf,
rdiv_wf,
rless-int,
nat_plus_properties,
decidable__lt,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformless_wf,
itermConstant_wf,
itermVar_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_wf,
rless_wf,
le_witness_for_triv,
req_witness,
metric_wf,
istype-universe,
rleq-int-fractions2,
decidable__le,
intformle_wf,
itermMultiply_wf,
int_formula_prop_le_lemma,
int_term_value_mul_lemma,
rleq_functionality,
req_weakening,
mdist-nonneg,
rleq_antisymmetry,
rleq-iff-all-rless,
real_wf,
small-reciprocal-real,
radd_wf,
rleq_functionality_wrt_implies,
rleq_weakening_rless,
rleq_weakening_equal,
rleq_weakening,
itermSubtract_wf,
itermAdd_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation_alt,
introduction,
cut,
independent_pairFormation,
lambdaFormation_alt,
universeIsType,
extract_by_obid,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
functionIsType,
closedConclusion,
setElimination,
rename,
because_Cache,
independent_isectElimination,
inrFormation_alt,
dependent_functionElimination,
productElimination,
independent_functionElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
lambdaEquality_alt,
int_eqEquality,
isect_memberEquality_alt,
voidElimination,
independent_pairEquality,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
isectIsTypeImplies,
instantiate,
universeEquality,
multiplyEquality,
setIsType
Latex:
\mforall{}[X:Type]. \mforall{}[d:metric(X)]. \mforall{}[x,y:X]. (x \mequiv{} y \mLeftarrow{}{}\mRightarrow{} \mforall{}k:\mBbbN{}\msupplus{}. (mdist(d;x;y) \mleq{} (r1/r(k))))
Date html generated:
2019_10_29-AM-10_59_41
Last ObjectModification:
2019_10_02-AM-09_41_10
Theory : reals
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