Nuprl Lemma : cantor-to-interval_wf
∀a,b:ℝ. ∀f:ℕ ⟶ 𝔹. (cantor-to-interval(a;b;f) ∈ {x:ℝ| (a ≤ x) ∧ (x ≤ b)} ) supposing a ≤ b
Proof
Definitions occuring in Statement :
cantor-to-interval: cantor-to-interval(a;b;f),
rleq: x ≤ y,
real: ℝ,
nat: ℕ,
bool: 𝔹,
uimplies: b supposing a,
all: ∀x:A. B[x],
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
function: x:A ⟶ B[x]
Definitions unfolded in proof :
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
subtype_rel: A ⊆r B,
prop: ℙ,
uall: ∀[x:A]. B[x],
and: P ∧ Q,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
nat: ℕ,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
top: Top,
int_upper: {i...},
cantor-interval: cantor-interval(a;b;f;n),
pi1: fst(t),
pi2: snd(t),
guard: {T}
Lemmas referenced :
nat_wf,
bool_wf,
rleq_wf,
real_wf,
cantor-to-interval_wf1,
converges-to_wf,
cantor-interval_wf,
subtype_rel_dep_function,
int_seg_wf,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
pi1_wf_top,
equal_wf,
cantor-interval-inclusion,
le_wf,
int_upper_wf,
primrec0_lemma,
constant-rleq-limit,
rleq_transitivity,
pi2_wf,
rleq-limit-constant
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
isect_memberFormation,
introduction,
cut,
applyEquality,
hypothesis,
sqequalHypSubstitution,
sqequalRule,
functionEquality,
extract_by_obid,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
isectElimination,
independent_isectElimination,
functionExtensionality,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
setEquality,
natural_numberEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
voidElimination,
voidEquality,
independent_functionElimination
Latex:
\mforall{}a,b:\mBbbR{}. \mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. (cantor-to-interval(a;b;f) \mmember{} \{x:\mBbbR{}| (a \mleq{} x) \mwedge{} (x \mleq{} b)\} ) supposing a \mleq{} b
Date html generated:
2017_10_03-AM-09_54_24
Last ObjectModification:
2017_07_28-AM-08_03_46
Theory : reals
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