Nuprl Lemma : pseudo-bounded-is-bounded
∀[S:{S:Type| S ⊆r ℕ} ]. ∀m:S. (pseudo-bounded(S) ⇒ ⇃(∃B:ℕ. ∀n:S. (n ≤ B)))
Proof
Definitions occuring in Statement :
pseudo-bounded: pseudo-bounded(S),
quotient: x,y:A//B[x; y],
nat: ℕ,
subtype_rel: A ⊆r B,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
true: True,
set: {x:A| B[x]} ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
member: t ∈ T,
sq_stable: SqStable(P),
squash: ↓T,
pseudo-bounded: pseudo-bounded(S),
exists: ∃x:A. B[x],
and: P ∧ Q,
prop: ℙ,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
nat: ℕ,
subtype_rel: A ⊆r B,
int_upper: {i...},
so_apply: x[s],
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
false: False,
not: ¬A,
pi1: fst(t),
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
top: Top,
int_seg: {i..j-},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b,
lelt: i ≤ j < k,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than: a < b
Lemmas referenced :
sq_stable__subtype_rel,
nat_wf,
weak-continuity-truncated,
subtype_rel_wf,
squash_wf,
pseudo-bounded_wf,
exists_wf,
all_wf,
int_upper_wf,
less_than_wf,
int_upper_subtype_nat,
subtype_rel_transitivity,
equal_wf,
int_seg_wf,
subtype_rel_dep_function,
int_seg_subtype_nat,
false_wf,
subtype_rel_self,
le_wf,
implies-quotient-true,
imax_wf,
imax_nat,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformeq_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_wf,
less_than'_wf,
ifthenelse_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
int_seg_properties,
intformless_wf,
int_formula_prop_less_lemma,
set_subtype_base,
int_subtype_base,
assert_wf,
bnot_wf,
not_wf,
imax_ub,
bool_cases,
iff_transitivity,
iff_weakening_uiff,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
because_Cache,
hypothesis,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
hypothesisEquality,
baseClosed,
imageElimination,
promote_hyp,
productElimination,
setElimination,
rename,
dependent_set_memberEquality,
independent_pairFormation,
productEquality,
cumulativity,
dependent_functionElimination,
independent_isectElimination,
setEquality,
universeEquality,
functionEquality,
applyEquality,
functionExtensionality,
lambdaEquality,
intEquality,
natural_numberEquality,
dependent_pairFormation,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
unionElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
independent_pairEquality,
axiomEquality,
equalityElimination,
instantiate,
inlFormation,
impliesFunctionality,
inrFormation
Latex:
\mforall{}[S:\{S:Type| S \msubseteq{}r \mBbbN{}\} ]. \mforall{}m:S. (pseudo-bounded(S) {}\mRightarrow{} \00D9(\mexists{}B:\mBbbN{}. \mforall{}n:S. (n \mleq{} B)))
Date html generated:
2017_04_17-AM-09_54_48
Last ObjectModification:
2017_02_27-PM-05_49_24
Theory : continuity
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