Nuprl Lemma : second-countable-choice
∀[X:𝕌']. ∀[R:ℕ ⟶ (n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ X) ⟶ ℙ'].
((∀n:ℕ. ∃A:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ X. R[n;A]) ⇒ (∃B:n:ℕ ⟶ (ℕn ⟶ ℕ) ⟶ X. ∀n:ℕ. R[n;B_n]))
Proof
Definitions occuring in Statement :
predicate-shift: A_x,
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
function: x:A ⟶ B[x],
natural_number: $n,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
so_lambda: λ2x.t[x],
nat: ℕ,
so_apply: x[s1;s2],
so_apply: x[s],
all: ∀x:A. B[x],
pi1: fst(t),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
int_seg: {i..j-},
ge: i ≥ j ,
lelt: i ≤ j < k,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
bfalse: ff,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
int_upper: {i...},
decidable: Dec(P),
subtract: n - m,
predicate-shift: A_x,
seq-single: seq-single(t),
seq-append: seq-append(n;m;s1;s2),
subtype_rel: A ⊆r B,
less_than: a < b,
true: True,
squash: ↓T,
nequal: a ≠ b ∈ T
Lemmas referenced :
all_wf,
nat_wf,
exists_wf,
int_seg_wf,
equal_wf,
eq_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_eq_int,
false_wf,
le_wf,
int_seg_properties,
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
int_upper_subtype_nat,
nequal-le-implies,
zero-add,
int_upper_properties,
decidable__lt,
intformnot_wf,
int_formula_prop_not_lemma,
lelt_wf,
subtract_wf,
decidable__le,
itermSubtract_wf,
int_term_value_subtract_lemma,
add-member-int_seg2,
intformeq_wf,
itermAdd_wf,
int_formula_prop_eq_lemma,
int_term_value_add_lemma,
add-associates,
add-swap,
add-commutes,
lt_int_wf,
assert_of_lt_int,
less_than_wf,
add-subtract-cancel,
decidable__equal_int,
predicate-shift_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
hypothesis,
promote_hyp,
thin,
sqequalHypSubstitution,
productElimination,
instantiate,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
sqequalRule,
lambdaEquality,
functionEquality,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
applyEquality,
functionExtensionality,
because_Cache,
universeEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
unionElimination,
equalityElimination,
independent_isectElimination,
dependent_set_memberEquality,
independent_pairFormation,
int_eqEquality,
intEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
hypothesis_subsumption,
hyp_replacement,
addEquality,
minusEquality,
lessCases,
sqequalAxiom,
imageMemberEquality,
baseClosed,
imageElimination,
applyLambdaEquality
Latex:
\mforall{}[X:\mBbbU{}']. \mforall{}[R:\mBbbN{} {}\mrightarrow{} (n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} X) {}\mrightarrow{} \mBbbP{}'].
((\mforall{}n:\mBbbN{}. \mexists{}A:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} X. R[n;A]) {}\mRightarrow{} (\mexists{}B:n:\mBbbN{} {}\mrightarrow{} (\mBbbN{}n {}\mrightarrow{} \mBbbN{}) {}\mrightarrow{} X. \mforall{}n:\mBbbN{}. R[n;B\_n]))
Date html generated:
2017_04_17-AM-09_36_09
Last ObjectModification:
2017_02_27-PM-05_34_25
Theory : fan-theorem
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