Nuprl Lemma : mtb-cantor-map-onto-common
∀[X:Type]
∀d:metric(X). ∀cmplt:mcomplete(X with d). ∀mtb:m-TB(X;d). ∀n:ℕ. ∀x,y:X.
((mdist(d;x;y) ≤ (r1/r(n + 1)))
⇒ (∃p,q:mtb-cantor(mtb)
((p = q ∈ (i:ℕn ⟶ ℕ(fst(mtb)) i)) ∧ mtb-cantor-map(d;cmplt;mtb;p) ≡ x ∧ mtb-cantor-map(d;cmplt;mtb;q) ≡ y)))
Proof
Definitions occuring in Statement :
mtb-cantor-map: mtb-cantor-map(d;cmplt;mtb;p),
mtb-cantor: mtb-cantor(mtb),
m-TB: m-TB(X;d),
mcomplete: mcomplete(M),
mk-metric-space: X with d,
mdist: mdist(d;x;y),
meq: x ≡ y,
metric: metric(X),
rdiv: (x/y),
rleq: x ≤ y,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
pi1: fst(t),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
uimplies: b supposing a,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
not: ¬A,
false: False,
mtb-cantor: mtb-cantor(mtb),
m-TB: m-TB(X;d),
so_lambda: λ2x.t[x],
pi1: fst(t),
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
so_apply: x[s],
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
prop: ℙ,
istype: istype(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
cand: A c∧ B,
rneq: x ≠ y,
mtb-cantor-map: mtb-cantor-map(d;cmplt;mtb;p),
metric: metric(X),
true: True,
m-k-regular: m-k-regular(d;k;s),
mtb-point-cantor: mtb-point-cantor(mtb;p),
mtb-seq: mtb-seq(mtb;s),
spreadn: spread3,
rev_uimplies: rev_uimplies(P;Q),
rge: x ≥ y,
rleq: x ≤ y,
rnonneg: rnonneg(x),
int_upper: {i...},
req_int_terms: t1 ≡ t2,
sq_stable: SqStable(P),
rat_term_to_real: rat_term_to_real(f;t),
rtermAdd: left "+" right,
rat_term_ind: rat_term_ind,
rtermDivide: num "/" denom,
rtermConstant: "const",
rtermVar: rtermVar(var),
pi2: snd(t),
mconverges-to: lim n→∞.x[n] = y,
sq_exists: ∃x:A [B[x]]
Lemmas referenced :
mtb-point-cantor-seq-regular,
m-regularize-of-regular,
mtb-seq_wf,
mtb-point-cantor_wf,
m-k-regular-monotone,
istype-void,
istype-le,
istype-false,
m-regularize-mcauchy,
subtype_rel_dep_function,
nat_wf,
int_seg_wf,
nat_plus_wf,
int_seg_properties,
nat_properties,
decidable__le,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
istype-int,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
int_seg_subtype_nat,
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,
istype-less_than,
intformless_wf,
int_formula_prop_less_lemma,
meq_wf,
mtb-cantor-map_wf,
rleq_wf,
mdist_wf,
rdiv_wf,
int-to-real_wf,
rless-int,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
rless_wf,
istype-nat,
m-TB_wf,
mcomplete_wf,
mk-metric-space_wf,
metric_wf,
istype-universe,
cauchy-mlimit-unique,
mconverges-to_wf,
squash_wf,
true_wf,
real_wf,
subtype_rel_self,
iff_weakening_equal,
mtb-seq-mtb-point-cantor-mconverges-to,
m-regularize_wf,
rleq_functionality_wrt_implies,
rleq_weakening_equal,
rleq_weakening,
le_witness_for_triv,
rleq-int-fractions,
int_upper_properties,
itermMultiply_wf,
int_term_value_mul_lemma,
rleq_transitivity,
istype-int_upper,
itermSubtract_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
mdist-triangle-inequality,
radd_wf,
radd_functionality_wrt_rleq,
upper_subtype_nat,
sq_stable__le,
rleq_functionality,
radd_functionality,
req_weakening,
mdist-symm,
assert-rat-term-eq2,
rtermAdd_wf,
rtermDivide_wf,
rtermConstant_wf,
rtermVar_wf,
rneq-int,
intformeq_wf,
int_formula_prop_eq_lemma,
set_subtype_base,
le_wf,
int_subtype_base,
real_term_value_add_lemma,
imax_wf,
imax_nat,
nat_plus_properties,
imax_ub
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
because_Cache,
hypothesis,
independent_isectElimination,
dependent_set_memberEquality_alt,
natural_numberEquality,
independent_pairFormation,
sqequalRule,
voidElimination,
setElimination,
rename,
productElimination,
lambdaEquality_alt,
applyEquality,
independent_pairEquality,
inhabitedIsType,
equalityIstype,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeIsType,
imageElimination,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
equalityElimination,
promote_hyp,
instantiate,
cumulativity,
functionExtensionality_alt,
productIsType,
functionIsType,
closedConclusion,
addEquality,
inrFormation_alt,
universeEquality,
functionExtensionality,
imageMemberEquality,
baseClosed,
functionIsTypeImplies,
multiplyEquality,
baseApply,
intEquality,
sqequalBase,
dependent_set_memberFormation_alt,
applyLambdaEquality,
inlFormation_alt
Latex:
\mforall{}[X:Type]
\mforall{}d:metric(X). \mforall{}cmplt:mcomplete(X with d). \mforall{}mtb:m-TB(X;d). \mforall{}n:\mBbbN{}. \mforall{}x,y:X.
((mdist(d;x;y) \mleq{} (r1/r(n + 1)))
{}\mRightarrow{} (\mexists{}p,q:mtb-cantor(mtb)
((p = q) \mwedge{} mtb-cantor-map(d;cmplt;mtb;p) \mequiv{} x \mwedge{} mtb-cantor-map(d;cmplt;mtb;q) \mequiv{} y)))
Date html generated:
2019_10_30-AM-07_05_38
Last ObjectModification:
2019_10_09-PM-03_18_32
Theory : reals
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