Nuprl Lemma : pseudo-bounded-not-unbounded
∀[S:{S:Type| S ⊆r ℕ} ]. (pseudo-bounded(S) ⇒ (¬(∀B:ℕ. ∃n:{B...}. (n ∈ S))))
Proof
Definitions occuring in Statement :
pseudo-bounded: pseudo-bounded(S),
int_upper: {i...},
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
not: ¬A,
implies: P ⇒ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
false: False,
exists: ∃x:A. B[x],
int_upper: {i...},
prop: ℙ,
pi1: fst(t),
pseudo-bounded: pseudo-bounded(S),
guard: {T},
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
le: A ≤ B,
and: P ∧ Q
Lemmas referenced :
sq_stable__subtype_rel,
nat_wf,
set_subtype_base,
le_wf,
istype-int,
int_subtype_base,
equal-wf-base,
istype-universe,
int_upper_wf,
pseudo-bounded_wf,
subtype_rel_wf,
subtype_rel_self,
exists_wf,
nat_properties,
decidable__le,
full-omega-unsat,
intformnot_wf,
intformle_wf,
itermVar_wf,
int_formula_prop_not_lemma,
istype-void,
int_formula_prop_le_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
le_weakening2,
decidable__lt,
subtype_rel_transitivity,
int_upper_properties,
intformand_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setElimination,
rename,
hypothesisEquality,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
Error :lambdaFormation_alt,
baseApply,
closedConclusion,
applyEquality,
intEquality,
Error :lambdaEquality_alt,
natural_numberEquality,
independent_isectElimination,
because_Cache,
equalityTransitivity,
equalitySymmetry,
Error :universeIsType,
promote_hyp,
productElimination,
Error :functionIsType,
Error :productIsType,
Error :equalityIsType4,
voidElimination,
Error :setIsType,
universeEquality,
Error :dependent_pairFormation_alt,
functionExtensionality,
Error :inhabitedIsType,
Error :equalityIsType1,
dependent_functionElimination,
Error :dependent_set_memberEquality_alt,
unionElimination,
approximateComputation,
int_eqEquality,
Error :isect_memberEquality_alt,
applyLambdaEquality,
independent_pairFormation
Latex:
\mforall{}[S:\{S:Type| S \msubseteq{}r \mBbbN{}\} ]. (pseudo-bounded(S) {}\mRightarrow{} (\mneg{}(\mforall{}B:\mBbbN{}. \mexists{}n:\{B...\}. (n \mmember{} S))))
Date html generated:
2019_06_20-PM-02_51_57
Last ObjectModification:
2018_10_05-PM-11_12_49
Theory : continuity
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