Nuprl Lemma : ccc-nset-iff-finite
∀K:Type. ((K ⊆r ℕ) ⇒ K ⇒ (CCC(K) ⇐⇒ finite(K)))
Proof
Definitions occuring in Statement :
contra-cc: CCC(T),
finite: finite(T),
nat: ℕ,
subtype_rel: A ⊆r B,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
universe: Type
Definitions unfolded in proof :
so_apply: x[s1;s2],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
less_than: a < b,
lelt: i ≤ j < k,
int_seg: {i..j-},
l_exists: (∃x∈L. P[x]),
satisfiable_int_formula: satisfiable_int_formula(fmla),
ge: i ≥ j ,
sq_type: SQType(T),
l_member: (x ∈ l),
true: True,
less_than': less_than'(a;b),
subtract: n - m,
uiff: uiff(P;Q),
decidable: Dec(P),
le: A ≤ B,
top: Top,
cons: [a / b],
or: P ∨ Q,
rev_implies: P ⇐ Q,
uimplies: b supposing a,
nat: ℕ,
guard: {T},
subtype_rel: A ⊆r B,
exists: ∃x:A. B[x],
prop: ℙ,
uall: ∀[x:A]. B[x],
false: False,
not: ¬A,
member: t ∈ T,
cand: A c∧ B,
ccc-nset: CCCNSet(K),
and: P ∧ Q,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
CCC-Sigma02-dns,
int_formula_prop_less_lemma,
intformless_wf,
int_seg_properties,
le_wf,
length_wf,
istype-less_than,
int_formula_prop_wf,
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,
istype-int,
itermVar_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
nat_properties,
select_wf,
int_subtype_base,
subtype_base_sq,
imax-list-ub,
subtype_rel_list,
imax-list-member,
le-add-cancel,
add-zero,
add-associates,
add_functionality_wrt_le,
add-commutes,
minus-one-mul-top,
zero-add,
minus-one-mul,
minus-add,
condition-implies-le,
not-lt-2,
istype-false,
decidable__lt,
length_wf_nat,
length_of_cons_lemma,
product_subtype_list,
nil_member,
length_of_nil_lemma,
list-cases,
finite-iff-listable,
istype-universe,
subtype_rel_wf,
CCC-finite,
contra-cc_wf,
bounded-ccc-nset-finite,
istype-void,
nat_wf,
subtype_rel_transitivity,
istype-nat,
istype-le,
finite_wf,
ccc-nset-not-not-finite
Rules used in proof :
imageElimination,
applyLambdaEquality,
hyp_replacement,
Error :dependent_set_memberEquality_alt,
int_eqEquality,
approximateComputation,
cumulativity,
Error :dependent_pairFormation_alt,
equalitySymmetry,
equalityTransitivity,
Error :equalityIstype,
minusEquality,
addEquality,
natural_numberEquality,
Error :isect_memberEquality_alt,
hypothesis_subsumption,
promote_hyp,
unionElimination,
universeEquality,
instantiate,
productElimination,
independent_isectElimination,
intEquality,
rename,
setElimination,
Error :lambdaEquality_alt,
applyEquality,
because_Cache,
Error :inhabitedIsType,
Error :productIsType,
Error :functionIsType,
sqequalRule,
isectElimination,
Error :universeIsType,
voidElimination,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
hypothesis,
cut,
independent_pairFormation,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution
Latex:
\mforall{}K:Type. ((K \msubseteq{}r \mBbbN{}) {}\mRightarrow{} K {}\mRightarrow{} (CCC(K) \mLeftarrow{}{}\mRightarrow{} finite(K)))
Date html generated:
2019_06_20-PM-03_03_04
Last ObjectModification:
2019_06_14-AM-10_08_41
Theory : continuity
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