Nuprl Lemma : cantor_to_interval_wf
∀[a:ℝ]. ∀[b:{b:ℝ| a ≤ b} ]. ∀[f:ℕ ⟶ 𝔹]. (cantor_to_interval(a;b;f) ∈ {x:ℝ| x ∈ [a, b]} )
Proof
Definitions occuring in Statement :
cantor_to_interval: cantor_to_interval(a;b;f),
rccint: [l, u],
i-member: r ∈ I,
rleq: x ≤ y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
top: Top,
and: P ∧ Q,
uiff: uiff(P;Q),
uimplies: b supposing a,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
prop: ℙ,
cantor_to_interval: cantor_to_interval(a;b;f),
guard: {T},
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
true: True,
rge: x ≥ y,
req_int_terms: t1 ≡ t2,
cand: A c∧ B,
rneq: x ≠ y,
or: P ∨ Q,
rev_uimplies: rev_uimplies(P;Q),
rdiv: (x/y),
subtype_rel: A ⊆r B
Lemmas referenced :
member_rccint_lemma,
trivial-rleq-radd,
int-to-real_wf,
rleq-int,
false_wf,
cantor-to-interval_wf,
radd_wf,
rleq_transitivity,
nat_wf,
set_wf,
real_wf,
rleq_wf,
equal_wf,
bool_wf,
rsub_wf,
itermSubtract_wf,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
req-iff-rsub-is-0,
rless-int,
rless_functionality_wrt_implies,
rleq_weakening_equal,
radd_functionality_wrt_rleq,
rsub_functionality_wrt_rleq,
rless_functionality,
req_weakening,
real_polynomial_null,
real_term_value_sub_lemma,
real_term_value_add_lemma,
real_term_value_const_lemma,
real_term_value_var_lemma,
rleq-implies-rleq,
rless_wf,
rmul_preserves_rleq2,
rleq_functionality,
rmul_wf,
rmul-zero-both,
rmul_comm,
rdiv_wf,
rmul_preserves_rleq,
itermMultiply_wf,
rinv_wf2,
req_transitivity,
radd_functionality,
rmul-rinv3,
real_term_value_mul_lemma,
rmul-identity1,
iff_weakening_equal,
req_wf,
squash_wf,
true_wf,
rinv-mul-as-rdiv,
radd-preserves-rleq,
rminus_wf,
itermMinus_wf,
radd-rminus-assoc,
real_term_value_minus_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
isectElimination,
setElimination,
rename,
hypothesisEquality,
natural_numberEquality,
productElimination,
independent_isectElimination,
independent_functionElimination,
independent_pairFormation,
lambdaFormation,
functionExtensionality,
applyEquality,
lambdaEquality,
productEquality,
equalityTransitivity,
equalitySymmetry,
axiomEquality,
functionEquality,
because_Cache,
imageMemberEquality,
baseClosed,
approximateComputation,
int_eqEquality,
intEquality,
addLevel,
levelHypothesis,
dependent_set_memberEquality,
inrFormation,
imageElimination,
universeEquality
Latex:
\mforall{}[a:\mBbbR{}]. \mforall{}[b:\{b:\mBbbR{}| a \mleq{} b\} ]. \mforall{}[f:\mBbbN{} {}\mrightarrow{} \mBbbB{}]. (cantor\_to\_interval(a;b;f) \mmember{} \{x:\mBbbR{}| x \mmember{} [a, b]\} )
Date html generated:
2018_05_22-PM-02_10_29
Last ObjectModification:
2017_10_06-AM-11_20_14
Theory : reals
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