Nuprl Lemma : weak-continuity-implies-strong1
(∀F:(ℕ ⟶ ℕ) ⟶ ℕ. ⇃(∃M:(ℕ ⟶ ℕ) ⟶ ℕ. ∀f,g:ℕ ⟶ ℕ. ((f = g ∈ (ℕM f ⟶ ℕ)) ⇒ ((F f) = (F g) ∈ ℕ))))
⇒ (∀F:(ℕ ⟶ ℕ) ⟶ ℕ
⇃(∃M:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ (ℕ?)
∀f:ℕ ⟶ ℕ
((∃n:ℕ. ((M n f) = (inl (F f)) ∈ (ℕ?))) ∧ (∀n:ℕ. (M n f) = (inl (F f)) ∈ (ℕ?) supposing ↑isl(M n f)))))
Proof
Definitions occuring in Statement :
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
uimplies: b supposing a,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
true: True,
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 :
implies: P ⇒ Q,
all: ∀x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
uimplies: b supposing a,
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
exists: ∃x:A. B[x],
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
ge: i ≥ j ,
decidable: Dec(P),
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
ext2Baire: ext2Baire(n;f;d),
int_seg: {i..j-},
lelt: i ≤ j < k,
less_than: a < b,
squash: ↓T,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
isl: isl(x),
true: True,
cand: A c∧ B,
quotient: x,y:A//B[x; y]
Lemmas referenced :
nat_wf,
all_wf,
quotient_wf,
exists_wf,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
true_wf,
equiv_rel_true,
implies-prop-truncation,
unit_wf2,
isect_wf,
assert_wf,
isl_wf,
le_int_wf,
ext2Baire_wf,
le_wf,
bool_wf,
eqtt_to_assert,
assert_of_le_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
imax_wf,
imax_nat,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
int_seg_properties,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
imax_strict_ub,
set_subtype_base,
int_subtype_base,
imax_ub,
decidable__equal_int,
prop-truncation-quot,
quotient-member-eq,
equal-wf-base,
member_wf,
squash_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
rename,
cut,
hypothesis,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
functionEquality,
introduction,
extract_by_obid,
isectElimination,
sqequalRule,
lambdaEquality,
because_Cache,
natural_numberEquality,
applyEquality,
functionExtensionality,
setElimination,
independent_isectElimination,
independent_pairFormation,
productElimination,
comment,
unionEquality,
productEquality,
inlEquality,
independent_functionElimination,
dependent_pairFormation,
dependent_set_memberEquality,
unionElimination,
equalityElimination,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
cumulativity,
voidElimination,
inrEquality,
axiomEquality,
applyLambdaEquality,
int_eqEquality,
intEquality,
isect_memberEquality,
voidEquality,
computeAll,
imageElimination,
inlFormation,
inrFormation,
isect_memberFormation,
pointwiseFunctionality,
pertypeElimination,
imageMemberEquality,
baseClosed
Latex:
(\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}M:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}. \mforall{}f,g:\mBbbN{} {}\mrightarrow{} \mBbbN{}. ((f = g) {}\mRightarrow{} ((F f) = (F g)))))
{}\mRightarrow{} (\mforall{}F:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{}
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (\mBbbN{}?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}
((\mexists{}n:\mBbbN{}. ((M n f) = (inl (F f)))) \mwedge{} (\mforall{}n:\mBbbN{}. (M n f) = (inl (F f)) supposing \muparrow{}isl(M n f)))))
Date html generated:
2017_04_17-AM-09_59_59
Last ObjectModification:
2017_02_27-PM-05_55_15
Theory : continuity
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