Nuprl Lemma : weak-continuity-principle-real
∀x:ℝ. ∀F:ℝ ⟶ 𝔹. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:{ℕ+| F x = F (G n)})
Proof
Definitions occuring in Statement :
real: ℝ
,
int_seg: {i..j-}
,
nat_plus: ℕ+
,
bool: 𝔹
,
all: ∀x:A. B[x]
,
sq_exists: ∃x:{A| B[x]}
,
set: {x:A| B[x]}
,
apply: f a
,
function: x:A ⟶ B[x]
,
natural_number: $n
,
int: ℤ
,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x]
,
member: t ∈ T
,
prop: ℙ
,
uall: ∀[x:A]. B[x]
,
nat_plus: ℕ+
,
real: ℝ
,
subtype_rel: A ⊆r B
,
so_lambda: λ2x.t[x]
,
so_apply: x[s]
,
uimplies: b supposing a
,
le: A ≤ B
,
and: P ∧ Q
,
less_than': less_than'(a;b)
,
false: False
,
not: ¬A
,
implies: P
⇒ Q
,
less_than: a < b
,
squash: ↓T
,
true: True
,
sq_exists: ∃x:{A| B[x]}
,
guard: {T}
,
iff: P
⇐⇒ Q
Lemmas referenced :
nat_plus_wf,
real_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat_plus,
false_wf,
subtype_rel_self,
bool_wf,
WCP_wf,
regularize-k-regular,
less_than_wf,
regularize_wf,
regular-int-seq_wf,
subtype_rel_sets,
set_wf,
squash_wf,
true_wf,
regularize-real,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
functionEquality,
cut,
introduction,
extract_by_obid,
hypothesis,
setEquality,
sqequalHypSubstitution,
isectElimination,
thin,
natural_numberEquality,
setElimination,
rename,
because_Cache,
intEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
independent_pairFormation,
dependent_functionElimination,
functionExtensionality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
dependent_set_memberFormation,
equalityUniverse,
levelHypothesis,
imageElimination,
universeEquality,
productElimination
Latex:
\mforall{}x:\mBbbR{}. \mforall{}F:\mBbbR{} {}\mrightarrow{} \mBbbB{}. \mforall{}G:n:\mBbbN{}\msupplus{} {}\mrightarrow{} \{y:\mBbbR{}| x = y\} . (\mexists{}n:\{\mBbbN{}\msupplus{}| F x = F (G n)\})
Date html generated:
2017_10_03-AM-09_09_11
Last ObjectModification:
2017_09_11-PM-05_02_30
Theory : reals
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