Nuprl Lemma : cantor-to-interval-req
∀a,b,x:ℝ. ∀f:ℕ ⟶ 𝔹.
((∀n:ℕ. (x ∈ [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))])) ⇒ (cantor-to-interval(a;b;f) = x))
Proof
Definitions occuring in Statement :
cantor-to-interval: cantor-to-interval(a;b;f),
cantor-interval: cantor-interval(a;b;f;n),
rccint: [l, u],
i-member: r ∈ I,
req: x = y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
pi1: fst(t),
pi2: snd(t),
all: ∀x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
uall: ∀[x:A]. B[x],
cantor-interval: cantor-interval(a;b;f;n),
i-member: r ∈ I,
rccint: [l, u],
top: Top,
pi1: fst(t),
pi2: snd(t),
guard: {T},
uimplies: b supposing a,
squash: ↓T,
sq_stable: SqStable(P),
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
sq_type: SQType(T),
cand: A c∧ B,
int_upper: {i...},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
rneq: x ≠ y,
less_than: a < b,
nat_plus: ℕ+,
uiff: uiff(P;Q),
req_int_terms: t1 ≡ t2,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
rbetween: x≤y≤z,
rge: x ≥ y
Lemmas referenced :
istype-false,
istype-le,
primrec0_lemma,
istype-void,
rleq_transitivity,
cantor-to-interval_wf1,
sq_stable__req,
unique-limit,
req_inversion,
istype-nat,
i-member_wf,
rccint_wf,
cantor-interval_wf,
subtype_rel_function,
nat_wf,
bool_wf,
int_seg_wf,
int_seg_subtype_nat,
subtype_rel_self,
real_wf,
common-limit-squeeze-ext,
int-rdiv_wf,
exp_wf3,
subtype_base_sq,
int_subtype_base,
istype-int,
nequal_wf,
int-rmul_wf,
exp_wf2,
rsub_wf,
cantor-interval-inclusion,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
itermAdd_wf,
itermConstant_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
intformand_wf,
int_formula_prop_and_lemma,
rleq_weakening_equal,
constant-limit,
req_weakening,
rnexp-converges-ext,
rdiv_wf,
int-to-real_wf,
rless-int,
rless_wf,
rabs_wf,
rleq-int-fractions2,
istype-less_than,
rless-int-fractions3,
rless_functionality,
rabs-of-nonneg,
rmul-limit,
rnexp_wf,
rmul_wf,
itermSubtract_wf,
itermMultiply_wf,
req-iff-rsub-is-0,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_mul_lemma,
real_term_value_var_lemma,
real_term_value_const_lemma,
converges-to_functionality,
not-equal-2,
exp_wf_nat_plus,
nat_plus_properties,
intformless_wf,
int_formula_prop_less_lemma,
exp-positive-stronger,
rneq_functionality,
rnexp-int,
req_functionality,
rmul_functionality,
rnexp-rdiv,
rdiv_functionality,
int-rdiv-req,
int-rmul-req,
rmul_preserves_req,
rinv_wf2,
req_transitivity,
rmul-rinv3,
rleq_weakening,
rleq_functionality_wrt_implies,
rsub_functionality_wrt_rleq,
rleq_functionality,
cantor-interval-length
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
dependent_set_memberEquality_alt,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
introduction,
extract_by_obid,
isectElimination,
because_Cache,
hypothesisEquality,
isect_memberEquality_alt,
voidElimination,
productElimination,
independent_isectElimination,
applyLambdaEquality,
setElimination,
rename,
imageMemberEquality,
baseClosed,
imageElimination,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
lambdaEquality_alt,
inhabitedIsType,
equalityIsType1,
functionIsType,
universeIsType,
applyEquality,
instantiate,
cumulativity,
intEquality,
equalityIsType4,
closedConclusion,
addEquality,
unionElimination,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
inrFormation_alt
Latex:
\mforall{}a,b,x:\mBbbR{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}.
((\mforall{}n:\mBbbN{}. (x \mmember{} [fst(cantor-interval(a;b;f;n)), snd(cantor-interval(a;b;f;n))]))
{}\mRightarrow{} (cantor-to-interval(a;b;f) = x))
Date html generated:
2019_10_30-AM-07_40_07
Last ObjectModification:
2018_11_14-AM-10_07_05
Theory : reals
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