Nuprl Lemma : unbounded-decidable-nset-infinite
∀K:Type. ((K ⊆r ℕ) ⇒ (∀l:ℕ. ((l ∈ K) ∨ (¬(l ∈ K)))) ⇒ (∀B:ℕ. ∃k:K. B < k) ⇒ (∃f:K ⟶ ℕ. Surj(K;ℕ;f)))
Proof
Definitions occuring in Statement :
surject: Surj(A;B;f),
nat: ℕ,
less_than: a < b,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
upto: upto(n),
uiff: uiff(P;Q),
nat_plus: ℕ+,
select: L[n],
l_member: (x ∈ l),
cons: [a / b],
surject: Surj(A;B;f),
respects-equality: respects-equality(S;T),
sq_type: SQType(T),
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
decidable: Dec(P),
top: Top,
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
istype: istype(T),
prop: ℙ,
less_than': less_than'(a;b),
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
uall: ∀[x:A]. B[x],
nat: ℕ,
false: False,
not: ¬A,
bfalse: ff,
rev_implies: P ⇐ Q,
true: True,
and: P ∧ Q,
iff: P ⇐⇒ Q,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
isl: isl(x),
or: P ∨ Q,
subtype_rel: A ⊆r B,
member: t ∈ T,
exists: ∃x:A. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
btrue_neq_bfalse,
member-implies-null-eq-bfalse,
null_nil_lemma,
no_repeats-subtype,
no_repeats_from-upto,
no_repeats_filter,
from-upto-member-nat,
member_filter_2,
subtype_rel_sets_simple,
from-upto_wf,
length-one-iff,
subtract-add-cancel,
length-append,
filter_append_sq,
zero-le-nat,
from-upto-split,
list_subtype_base,
length_wf,
cons_wf,
false_wf,
add-is-int-iff,
nat_plus_properties,
add_nat_plus,
length_of_cons_lemma,
product_subtype_list,
nil_wf,
length_of_nil_lemma,
list-cases,
member_filter,
member_upto,
exists_wf,
decidable__equal_int,
int_formula_prop_eq_lemma,
intformeq_wf,
subtype-respects-equality,
subtype_base_sq,
le_witness_for_triv,
decidable__lt,
int_term_value_subtract_lemma,
int_formula_prop_not_lemma,
itermSubtract_wf,
intformnot_wf,
decidable__le,
equal-wf-base,
less_than_wf,
primrec-wf2,
subtract_wf,
istype-le,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_le_lemma,
int_formula_prop_and_lemma,
intformless_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformle_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
istype-universe,
subtype_rel_wf,
istype-less_than,
surject_wf,
istype-false,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_list,
l_member_wf,
bool_wf,
subtype_rel_dep_function,
subtype_rel_transitivity,
upto_wf,
filter_wf5,
length_wf_nat,
istype-assert,
istype-void,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
istype-true,
istype-nat,
bfalse_wf,
btrue_wf,
nat_wf
Rules used in proof :
Error :isectIsTypeImplies,
axiomEquality,
Error :isect_memberFormation_alt,
pointwiseFunctionality,
applyLambdaEquality,
hypothesis_subsumption,
minusEquality,
baseClosed,
closedConclusion,
baseApply,
cumulativity,
promote_hyp,
imageElimination,
Error :dependent_set_memberEquality_alt,
productEquality,
functionEquality,
addEquality,
Error :isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
universeEquality,
instantiate,
Error :unionIsType,
Error :setIsType,
setElimination,
setEquality,
productElimination,
Error :productIsType,
Error :functionIsType,
sqequalBase,
independent_isectElimination,
intEquality,
isectElimination,
Error :universeIsType,
voidElimination,
natural_numberEquality,
independent_pairFormation,
because_Cache,
independent_functionElimination,
dependent_functionElimination,
Error :equalityIstype,
unionElimination,
extract_by_obid,
introduction,
equalitySymmetry,
equalityTransitivity,
Error :inhabitedIsType,
thin,
hypothesis,
hypothesisEquality,
sqequalHypSubstitution,
functionExtensionality,
sqequalRule,
applyEquality,
Error :lambdaEquality_alt,
Error :dependent_pairFormation_alt,
rename,
cut,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}K:Type
((K \msubseteq{}r \mBbbN{}) {}\mRightarrow{} (\mforall{}l:\mBbbN{}. ((l \mmember{} K) \mvee{} (\mneg{}(l \mmember{} K)))) {}\mRightarrow{} (\mforall{}B:\mBbbN{}. \mexists{}k:K. B < k) {}\mRightarrow{} (\mexists{}f:K {}\mrightarrow{} \mBbbN{}. Surj(K;\mBbbN{};f)))
Date html generated:
2019_06_20-PM-03_02_26
Last ObjectModification:
2019_06_13-PM-07_17_10
Theory : continuity
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