Nuprl Lemma : weak-continuity-principle-real-nat
∀x:ℝ. ∀F:ℝ ⟶ ℕ. ∀G:n:ℕ+ ⟶ {y:ℝ| x = y ∈ (ℕ+n ⟶ ℤ)} . (∃n:{ℕ+| ((F x) = (F (G n)) ∈ ℕ)})
Proof
Definitions occuring in Statement :
real: ℝ,
int_seg: {i..j-},
nat_plus: ℕ+,
nat: ℕ,
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,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
prop: ℙ,
real: ℝ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
false: False,
not: ¬A,
implies: P ⇒ Q,
guard: {T},
iff: P ⇐⇒ Q,
exists: ∃x:A. B[x],
sq_exists: ∃x:{A| B[x]}
Lemmas referenced :
weak-continuity-principle-nat+-int-nat,
real_wf,
regularize-k-regular,
less_than_wf,
regularize_wf,
nat_plus_wf,
regular-int-seq_wf,
subtype_rel_sets,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat_plus,
false_wf,
subtype_rel_self,
nat_wf,
exists_wf,
squash_wf,
true_wf,
regularize-real,
iff_weakening_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
lambdaEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
hypothesis,
dependent_set_memberEquality,
natural_numberEquality,
sqequalRule,
independent_pairFormation,
imageMemberEquality,
baseClosed,
isectElimination,
functionEquality,
because_Cache,
intEquality,
setElimination,
rename,
independent_isectElimination,
setEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
cumulativity,
universeEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}x:\mBbbR{}. \mforall{}F:\mBbbR{} {}\mrightarrow{} \mBbbN{}. \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_33
Last ObjectModification:
2017_09_11-PM-05_03_50
Theory : reals
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