Nuprl Lemma : strong-continuity-rel-unique-pair
∀P:(ℕ ⟶ ℕ) ⟶ ℕ ⟶ ℙ. ∀F:∀f:ℕ ⟶ ℕ. ⇃(∃n:ℕ. (P f n)).
⇃(∃M:n:ℕ ⟶ s:(ℕn ⟶ ℕ) ⟶ (k:ℕn × (P ext2Baire(n;s;0) k)?)
∀f:ℕ ⟶ ℕ
∃n:ℕ
∃k:ℕn
∃p:P ext2Baire(n;f;0) k
(((M n f) = (inl <k, p>) ∈ (k:ℕn × (P ext2Baire(n;f;0) k)?)) ∧ (∀m:ℕ. ((↑isl(M m f)) ⇒ (m = n ∈ ℕ)))))
Proof
Definitions occuring in Statement :
ext2Baire: ext2Baire(n;f;d),
quotient: x,y:A//B[x; y],
int_seg: {i..j-},
nat: ℕ,
assert: ↑b,
isl: isl(x),
prop: ℙ,
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],
pair: <a, b>,
product: 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,
uall: ∀[x:A]. B[x],
nat: ℕ,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
guard: {T},
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
spreadn: spread4,
exposed-it: exposed-it,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
int_seg: {i..j-},
lelt: i ≤ j < k,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
cand: A c∧ B,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
isl: isl(x),
ext2Baire: ext2Baire(n;f;d),
true: True,
nequal: a ≠ b ∈ T
Lemmas referenced :
strong-continuity-rel-unique,
implies-quotient-true,
exists_wf,
int_seg_wf,
unit_wf2,
all_wf,
int_seg_subtype_nat,
false_wf,
equal_wf,
subtype_rel_dep_function,
nat_wf,
subtype_rel_self,
subtype_rel_union,
assert_wf,
isl_wf,
ext2Baire_wf,
le_wf,
quotient_wf,
true_wf,
equiv_rel_true,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
nat_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
lelt_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_seg_properties,
btrue_wf,
bfalse_wf,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
decidable__equal_int,
and_wf,
set_subtype_base,
int_subtype_base,
union_subtype_base,
unit_subtype_base,
decidable__le,
intformle_wf,
itermConstant_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
isectElimination,
functionEquality,
because_Cache,
natural_numberEquality,
setElimination,
rename,
hypothesis,
unionEquality,
sqequalRule,
lambdaEquality,
productEquality,
applyEquality,
functionExtensionality,
independent_isectElimination,
independent_pairFormation,
inlEquality,
dependent_set_memberEquality,
dependent_pairEquality,
independent_functionElimination,
productElimination,
cumulativity,
universeEquality,
dependent_pairFormation,
unionElimination,
equalityElimination,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
equalityTransitivity,
equalitySymmetry,
promote_hyp,
instantiate,
inrEquality,
axiomEquality,
applyLambdaEquality
Latex:
\mforall{}P:(\mBbbN{} {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} \mBbbN{} {}\mrightarrow{} \mBbbP{}. \mforall{}F:\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. \00D9(\mexists{}n:\mBbbN{}. (P f n)).
\00D9(\mexists{}M:n:\mBbbN{} {}\mrightarrow{} s:(\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} (k:\mBbbN{}n \mtimes{} (P ext2Baire(n;s;0) k)?)
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}
\mexists{}n:\mBbbN{}
\mexists{}k:\mBbbN{}n. \mexists{}p:P ext2Baire(n;f;0) k. (((M n f) = (inl <k, p>)) \mwedge{} (\mforall{}m:\mBbbN{}. ((\muparrow{}isl(M m f)) {}\mRightarrow{} (m = n)\000C))))
Date html generated:
2017_04_17-AM-10_02_44
Last ObjectModification:
2017_02_27-PM-05_54_46
Theory : continuity
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