Nuprl Lemma : cantor-to-interval-onto-lemma
∀a,b:ℝ.
  ∀x:ℝ. ∀n:ℕ. ∀f:{f:ℕn ⟶ 𝔹| x ∈ [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))]} .
    ∃g:{g:ℕn + 1 ⟶ 𝔹| x ∈ [fst(cantor-interval(a;b;g;n + 1)), snd(cantor-interval(a;b;g;n + 1))]} . (g = f ∈ (ℕn ⟶ 𝔹))\000C 
  supposing a < b
Proof
Definitions occuring in Statement : 
cantor-interval: cantor-interval(a;b;f;n)
, 
rccint: [l, u]
, 
i-member: r ∈ I
, 
rless: x < y
, 
real: ℝ
, 
int_seg: {i..j-}
, 
nat: ℕ
, 
bool: 𝔹
, 
uimplies: b supposing a
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
set: {x:A| B[x]} 
, 
function: x:A ⟶ B[x]
, 
add: n + m
, 
natural_number: $n
, 
equal: s = t ∈ T
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
uimplies: b supposing a
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
prop: ℙ
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
and: P ∧ Q
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
exists: ∃x:A. B[x]
, 
false: False
, 
top: Top
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
subtype_rel: A ⊆r B
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
guard: {T}
, 
sq_type: SQType(T)
, 
nat: ℕ
, 
rless: x < y
, 
sq_exists: ∃x:A [B[x]]
, 
nat_plus: ℕ+
, 
pi1: fst(t)
, 
pi2: snd(t)
, 
le: A ≤ B
, 
less_than': less_than'(a;b)
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
uiff: uiff(P;Q)
, 
subtract: n - m
, 
true: True
, 
real: ℝ
, 
sq_stable: SqStable(P)
, 
squash: ↓T
, 
ge: i ≥ j 
, 
cantor-interval: cantor-interval(a;b;f;n)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
bfalse: ff
, 
bnot: ¬bb
, 
ifthenelse: if b then t else f fi 
, 
assert: ↑b
, 
nequal: a ≠ b ∈ T 
, 
int_nzero: ℤ-o
, 
i-member: r ∈ I
, 
rccint: [l, u]
Lemmas referenced : 
cantor-middle-third-lemma, 
rless_wf, 
real_wf, 
int_seg_properties, 
full-omega-unsat, 
intformand_wf, 
intformless_wf, 
itermVar_wf, 
itermConstant_wf, 
intformle_wf, 
istype-int, 
int_formula_prop_and_lemma, 
istype-void, 
int_formula_prop_less_lemma, 
int_term_value_var_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_le_lemma, 
int_formula_prop_wf, 
int_seg_wf, 
decidable__equal_int, 
subtract_wf, 
subtype_base_sq, 
set_subtype_base, 
lelt_wf, 
int_subtype_base, 
intformnot_wf, 
intformeq_wf, 
itermSubtract_wf, 
int_formula_prop_not_lemma, 
int_formula_prop_eq_lemma, 
int_term_value_subtract_lemma, 
decidable__le, 
decidable__lt, 
istype-le, 
istype-less_than, 
subtype_rel_self, 
nat_plus_properties, 
guard_wf, 
all_wf, 
bool_wf, 
isect_wf, 
i-member_wf, 
rccint_wf, 
cantor-interval_wf, 
sq_exists_wf, 
equal_wf, 
subtype_rel_function, 
int_seg_subtype, 
istype-false, 
not-le-2, 
condition-implies-le, 
minus-add, 
minus-one-mul, 
add-swap, 
minus-one-mul-top, 
add-mul-special, 
zero-mul, 
add-zero, 
add-associates, 
add-commutes, 
le-add-cancel, 
sq_stable__less_than, 
itermAdd_wf, 
int_term_value_add_lemma, 
primrec-wf2, 
nat_properties, 
istype-nat, 
sq_stable__i-member, 
pi1_wf_top, 
pi2_wf, 
eq_int_wf, 
eqtt_to_assert, 
assert_of_eq_int, 
member_rccint_lemma, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
primrec-unroll, 
add-subtract-cancel, 
subtype_rel_product, 
top_wf, 
cantor-interval-rless, 
btrue_wf, 
not-equal-2, 
rleq_wf, 
ifthenelse_wf, 
lt_int_wf, 
int-rdiv_wf, 
nequal_wf, 
radd_wf, 
int-rmul_wf, 
assert_of_lt_int, 
iff_weakening_uiff, 
assert_wf, 
less_than_wf, 
bfalse_wf, 
sq_stable__rleq, 
subtype_rel_set
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
lambdaFormation_alt, 
isect_memberFormation_alt, 
cut, 
introduction, 
extract_by_obid, 
sqequalHypSubstitution, 
dependent_functionElimination, 
thin, 
hypothesisEquality, 
because_Cache, 
universeIsType, 
isectElimination, 
hypothesis, 
inhabitedIsType, 
natural_numberEquality, 
setElimination, 
rename, 
productElimination, 
independent_isectElimination, 
approximateComputation, 
independent_functionElimination, 
dependent_pairFormation_alt, 
lambdaEquality_alt, 
int_eqEquality, 
isect_memberEquality_alt, 
voidElimination, 
sqequalRule, 
independent_pairFormation, 
unionElimination, 
applyEquality, 
instantiate, 
cumulativity, 
intEquality, 
equalityTransitivity, 
equalitySymmetry, 
applyLambdaEquality, 
dependent_set_memberEquality_alt, 
productIsType, 
hypothesis_subsumption, 
functionIsType, 
functionEquality, 
closedConclusion, 
equalityIstype, 
addEquality, 
productEquality, 
minusEquality, 
multiplyEquality, 
imageMemberEquality, 
baseClosed, 
imageElimination, 
setIsType, 
equalityElimination, 
promote_hyp, 
dependent_set_memberFormation_alt, 
independent_pairEquality, 
spreadEquality, 
sqequalBase, 
dependent_pairEquality_alt, 
functionExtensionality, 
hyp_replacement
Latex:
\mforall{}a,b:\mBbbR{}.
    \mforall{}x:\mBbbR{}.  \mforall{}n:\mBbbN{}.  \mforall{}f:\{f:\mBbbN{}n  {}\mrightarrow{}  \mBbbB{}|  x  \mmember{}  [fst(cantor-interval(a;b;f;n)),  snd(cantor-interval(a;b;f;n))]\}  .
        \mexists{}g:\{g:\mBbbN{}n  +  1  {}\mrightarrow{}  \mBbbB{}|  x  \mmember{}  [fst(cantor-interval(a;b;g;n  +  1)),  snd(cantor-interval(a;b;g;n  +  1))]\} 
          (g  =  f) 
    supposing  a  <  b
Date html generated:
2019_10_30-AM-07_40_35
Last ObjectModification:
2018_12_11-AM-11_12_08
Theory : reals
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