Nuprl Lemma : fan+weak-continuity-implies-uniform-continuity
∀F:(ℕ ⟶ 𝔹) ⟶ ℕ. ⇃(∃n:ℕ. ∀f,g:ℕ ⟶ 𝔹. ((f = g ∈ (ℕn ⟶ 𝔹)) ⇒ ((F f) = (F g) ∈ ℕ)))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
bool: 𝔹,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
all: ∀x:A. B[x],
guard: {T},
exists: ∃x:A. B[x],
nat: ℕ,
not: ¬A,
false: False,
less_than': less_than'(a;b),
and: P ∧ Q,
le: A ≤ B,
uimplies: b supposing a,
so_apply: x[s],
subtype_rel: A ⊆r B,
prop: ℙ,
implies: P ⇒ Q,
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
squash: ↓T,
so_apply: x[s1;s2],
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
or: P ∨ Q,
decidable: Dec(P),
ge: i ≥ j ,
assert: ↑b,
bnot: ¬bb,
sq_type: SQType(T),
bfalse: ff,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
btrue: tt,
it: ⋅,
unit: Unit,
bool: 𝔹,
int_seg: {i..j-},
ext2Cantor: ext2Cantor(n;f;d),
less_than: a < b,
lelt: i ≤ j < k,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True
Lemmas referenced :
bool_wf,
nat_wf,
implies-quotient-true2,
trivial-quotient-true,
subtype_rel_self,
false_wf,
int_seg_subtype_nat,
subtype_rel_dep_function,
int_seg_wf,
equal_wf,
all_wf,
exists_wf,
strong-continuity2-implies-weak-skolem-cantor-nat,
btrue_wf,
ext2Cantor_wf,
le_wf,
fan_theorem,
int_formula_prop_wf,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
imax_nat,
imax_wf,
less_than_wf,
assert-bnot,
bool_subtype_base,
subtype_base_sq,
bool_cases_sqequal,
eqff_to_assert,
assert_of_lt_int,
eqtt_to_assert,
lt_int_wf,
int_formula_prop_less_lemma,
intformless_wf,
decidable__lt,
int_seg_properties,
imax_strict_ub,
imax_ub,
int_subtype_base,
set_subtype_base,
assert_wf,
assert_elim,
and_wf,
iff_imp_equal_bool,
lelt_wf,
iff_weakening_equal,
true_wf,
squash_wf,
equal_functionality_wrt_subtype_rel2
Rules used in proof :
hypothesis,
extract_by_obid,
introduction,
cut,
functionEquality,
lambdaFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
independent_functionElimination,
productElimination,
rename,
setElimination,
independent_pairFormation,
independent_isectElimination,
functionExtensionality,
applyEquality,
natural_numberEquality,
lambdaEquality,
sqequalRule,
because_Cache,
isectElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
baseClosed,
imageMemberEquality,
imageElimination,
voidEquality,
voidElimination,
isect_memberEquality,
intEquality,
int_eqEquality,
approximateComputation,
unionElimination,
applyLambdaEquality,
equalitySymmetry,
equalityTransitivity,
dependent_set_memberEquality,
dependent_pairFormation,
cumulativity,
instantiate,
promote_hyp,
equalityElimination,
inrFormation,
inlFormation,
levelHypothesis,
addLevel,
universeEquality
Latex:
\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbB{}) {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbB{}. ((f = g) {}\mRightarrow{} ((F f) = (F g))))
Date html generated:
2018_05_21-PM-01_19_52
Last ObjectModification:
2018_05_18-PM-04_07_39
Theory : continuity
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