Nuprl Lemma : weak-continuity-implies-strong-cantor-unique
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ
∃M:n:ℕ ⟶ (ℕn ⟶ 𝔹) ⟶ (ℕ?)
∀f:ℕ ⟶ 𝔹. ∃n:ℕ. (((M n f) = (inl (F f)) ∈ (ℕ?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ (m = n ∈ ℕ))))
Proof
Definitions occuring in Statement :
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
unit: Unit,
apply: f a,
function: x:A ⟶ B[x],
inl: inl x,
union: left + right,
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
and: P ∧ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
cand: A c∧ B,
squash: ↓T,
true: True,
guard: {T},
uiff: uiff(P;Q),
top: Top,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bfalse: ff,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
isl: isl(x),
sq_type: SQType(T),
btrue: tt,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
weak-continuity-implies-strong-cantor,
nat_wf,
bool_wf,
strong-continuity-test_wf,
int_seg_wf,
all_wf,
exists_wf,
equal_wf,
unit_wf2,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
assert_wf,
isl_wf,
decidable__assert,
strong-continuity-test-prop1,
decidable__lt,
assert_functionality_wrt_uiff,
squash_wf,
true_wf,
isr-not-isl,
subtype_rel_union,
top_wf,
decidable__equal_int,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
le_wf,
intformless_wf,
int_formula_prop_less_lemma,
not-isl-assert-isr,
strong-continuity-test-prop2,
and_wf,
btrue_wf,
subtype_base_sq,
bool_subtype_base,
iff_weakening_equal,
strong-continuity-test-prop3
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
functionEquality,
hypothesis,
dependent_pairFormation,
lambdaEquality,
isectElimination,
functionExtensionality,
applyEquality,
natural_numberEquality,
setElimination,
rename,
because_Cache,
sqequalRule,
productEquality,
unionEquality,
independent_isectElimination,
independent_pairFormation,
inlEquality,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_functionElimination,
imageElimination,
cumulativity,
universeEquality,
imageMemberEquality,
baseClosed,
isect_memberEquality,
voidElimination,
voidEquality,
int_eqEquality,
intEquality,
computeAll,
dependent_set_memberEquality,
applyLambdaEquality,
instantiate
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}
\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} (\mBbbN{}?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbB{}. \mexists{}n:\mBbbN{}. (((M n f) = (inl (F f))) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} (m = n))))
Date html generated:
2017_04_17-AM-10_00_15
Last ObjectModification:
2017_02_27-PM-05_53_10
Theory : continuity
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