Nuprl Lemma : bounded-decidable-nset-finite
∀K:Type. ((K ⊆r ℕ) ⇒ (∀l:ℕ. ((l ∈ K) ∨ (¬(l ∈ K)))) ⇒ (∀B:ℕ. ((∀k:K. (k ≤ B)) ⇒ finite(K))))
Proof
Definitions occuring in Statement :
finite: finite(T),
nat: ℕ,
subtype_rel: A ⊆r B,
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 :
l_member: (x ∈ l),
true: True,
cand: A c∧ B,
squash: ↓T,
less_than: a < b,
bfalse: ff,
btrue: tt,
ifthenelse: if b then t else f fi ,
assert: ↑b,
less_than': less_than'(a;b),
isl: isl(x),
prop: ℙ,
top: Top,
exists: ∃x:A. B[x],
satisfiable_int_formula: satisfiable_int_formula(fmla),
le: A ≤ B,
decidable: Dec(P),
ge: i ≥ j ,
lelt: i ≤ j < k,
int_seg: {i..j-},
false: False,
not: ¬A,
so_apply: x[s],
so_lambda: λ2x.t[x],
or: P ∨ Q,
uimplies: b supposing a,
nat: ℕ,
guard: {T},
subtype_rel: A ⊆r B,
rev_implies: P ⇐ Q,
and: P ∧ Q,
iff: P ⇐⇒ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
all: ∀x:A. B[x]
Lemmas referenced :
select_wf,
length_wf,
change-equality-type,
subtype-base-respects-equality,
member_filter,
no_repeats-subtype,
no_repeats_upto,
int_seg_properties,
no_repeats_filter,
l_member_wf,
no_repeats_wf,
istype-assert,
istype-true,
assert_wf,
istype-false,
int_seg_subtype_nat,
int_seg_wf,
subtype_rel_list,
upto_wf,
bfalse_wf,
btrue_wf,
filter_type,
decidable__le,
member_upto,
istype-less_than,
int_formula_prop_wf,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
intformless_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
decidable__lt,
nat_properties,
zero-le-nat,
istype-universe,
subtype_rel_wf,
istype-void,
int_subtype_base,
istype-int,
le_wf,
set_subtype_base,
istype-nat,
nat_wf,
subtype_rel_transitivity,
istype-le,
finite-iff-listable
Rules used in proof :
promote_hyp,
imageElimination,
Error :setIsType,
setEquality,
equalityTransitivity,
functionExtensionality,
Error :productIsType,
voidElimination,
Error :isect_memberEquality_alt,
int_eqEquality,
Error :dependent_pairFormation_alt,
approximateComputation,
unionElimination,
addEquality,
dependent_functionElimination,
independent_pairFormation,
Error :dependent_set_memberEquality_alt,
universeEquality,
instantiate,
equalitySymmetry,
sqequalBase,
because_Cache,
natural_numberEquality,
Error :equalityIstype,
Error :unionIsType,
independent_isectElimination,
intEquality,
Error :inhabitedIsType,
rename,
setElimination,
Error :lambdaEquality_alt,
applyEquality,
Error :universeIsType,
Error :functionIsType,
sqequalRule,
independent_functionElimination,
productElimination,
hypothesis,
hypothesisEquality,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
introduction,
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{}. ((\mforall{}k:K. (k \mleq{} B)) {}\mRightarrow{} finite(K))))
Date html generated:
2019_06_20-PM-03_02_18
Last ObjectModification:
2019_06_13-PM-03_57_08
Theory : continuity
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