Nuprl Lemma : bounded-ccc-nset-decidable
∀K:Type. (CCCNSet(K) ⇒ (∀B:ℕ. ((∀k:K. (k ≤ B)) ⇒ (∀l:ℕ. ((l ∈ K) ∨ (¬(l ∈ K)))))))
Proof
Definitions occuring in Statement :
ccc-nset: CCCNSet(K),
nat: ℕ,
le: A ≤ B,
all: ∀x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
or: P ∨ Q,
member: t ∈ T,
universe: Type
Definitions unfolded in proof :
lelt: i ≤ j < k,
int_seg: {i..j-},
weakly-decidable-nset: WD(K),
sq_type: SQType(T),
le: A ≤ B,
top: Top,
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
or: P ∨ Q,
decidable: Dec(P),
transparent-nset: Transparent(K),
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
so_apply: x[s],
so_lambda: λ2x.t[x],
exists: ∃x:A. B[x],
uimplies: b supposing a,
nat: ℕ,
guard: {T},
subtype_rel: A ⊆r B,
and: P ∧ Q,
ccc-nset: CCCNSet(K),
prop: ℙ,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
member: t ∈ T,
all: ∀x:A. B[x]
Lemmas referenced :
nat_properties,
set_subtype_base,
decidable__lt,
int_subtype_base,
subtype_base_sq,
int_formula_prop_eq_lemma,
intformeq_wf,
decidable__equal_int,
int_formula_prop_wf,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_subtract_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_and_lemma,
intformless_wf,
itermVar_wf,
itermSubtract_wf,
itermConstant_wf,
intformle_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__le,
le_weakening,
le_functionality,
zero-le-nat,
le_wf,
primrec-wf2,
istype-less_than,
istype-int,
subtract_wf,
nat_wf,
subtype_rel_transitivity,
istype-nat,
istype-le,
ccc-nset-transparent,
istype-universe,
ccc-nset_wf,
ccc-nset-minimum,
ccc-nset-weakly-decidable
Rules used in proof :
Error :inrFormation_alt,
sqequalBase,
Error :equalityIstype,
Error :inlFormation_alt,
cumulativity,
independent_pairFormation,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
approximateComputation,
unionElimination,
Error :dependent_set_memberEquality_alt,
equalitySymmetry,
equalityTransitivity,
Error :dependent_pairFormation_alt,
productEquality,
functionEquality,
Error :setIsType,
Error :productIsType,
because_Cache,
natural_numberEquality,
independent_isectElimination,
intEquality,
rename,
setElimination,
Error :lambdaEquality_alt,
applyEquality,
Error :inhabitedIsType,
Error :functionIsType,
sqequalRule,
productElimination,
universeEquality,
instantiate,
isectElimination,
Error :universeIsType,
independent_functionElimination,
hypothesisEquality,
thin,
dependent_functionElimination,
sqequalHypSubstitution,
hypothesis,
Error :lambdaFormation_alt,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
extract_by_obid,
introduction,
cut
Latex:
\mforall{}K:Type. (CCCNSet(K) {}\mRightarrow{} (\mforall{}B:\mBbbN{}. ((\mforall{}k:K. (k \mleq{} B)) {}\mRightarrow{} (\mforall{}l:\mBbbN{}. ((l \mmember{} K) \mvee{} (\mneg{}(l \mmember{} K)))))))
Date html generated:
2019_06_20-PM-03_02_32
Last ObjectModification:
2019_06_13-PM-04_14_12
Theory : continuity
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