Nuprl Lemma : Kripke2b
∀a:ℕ ⟶ ℕ. (is-absolutely-free{i:l}(a) 
⇒ init0(a) 
⇒ increasing-sequence(a) 
⇒ (¬(∀m:ℕ. ∃n:ℕ. ((a n) ≥ m ))))
Proof
Definitions occuring in Statement : 
is-absolutely-free: is-absolutely-free{i:l}(f)
, 
init0: init0(a)
, 
increasing-sequence: increasing-sequence(a)
, 
nat: ℕ
, 
ge: i ≥ j 
, 
all: ∀x:A. B[x]
, 
exists: ∃x:A. B[x]
, 
not: ¬A
, 
implies: P 
⇒ Q
, 
apply: f a
, 
function: x:A ⟶ B[x]
Definitions unfolded in proof : 
all: ∀x:A. B[x]
, 
implies: P 
⇒ Q
, 
not: ¬A
, 
false: False
, 
exists: ∃x:A. B[x]
, 
member: t ∈ T
, 
uall: ∀[x:A]. B[x]
, 
subtype_rel: A ⊆r B
, 
nat: ℕ
, 
ge: i ≥ j 
, 
prop: ℙ
, 
is-absolutely-free: is-absolutely-free{i:l}(f)
, 
uimplies: b supposing a
, 
le: A ≤ B
, 
and: P ∧ Q
, 
less_than': less_than'(a;b)
, 
iff: P 
⇐⇒ Q
, 
rev_implies: P 
⇐ Q
, 
decidable: Dec(P)
, 
or: P ∨ Q
, 
satisfiable_int_formula: satisfiable_int_formula(fmla)
, 
top: Top
, 
guard: {T}
, 
so_lambda: λ2x.t[x]
, 
so_apply: x[s]
, 
sq_type: SQType(T)
, 
zero-seq: 0s
, 
int_seg: {i..j-}
, 
lelt: i ≤ j < k
, 
kripke2b-baire-seq: kripke2b-baire-seq(a;x;F)
, 
bool: 𝔹
, 
unit: Unit
, 
it: ⋅
, 
btrue: tt
, 
uiff: uiff(P;Q)
, 
ifthenelse: if b then t else f fi 
, 
bfalse: ff
, 
bnot: ¬bb
, 
assert: ↑b
, 
nequal: a ≠ b ∈ T 
, 
true: True
, 
init0: init0(a)
, 
cantor2baire-aux: cantor2baire-aux(a;n)
, 
cantor2baire: cantor2baire(a)
, 
change-from1: change-from1(a)
, 
pi1: fst(t)
, 
label: ...$L... t
, 
squash: ↓T
, 
min-inc-seq: min-inc-seq(a;n;k)
, 
baire_eq_from: baire_eq_from(a;k)
Lemmas referenced : 
istype-nat, 
ge_wf, 
increasing-sequence_wf, 
init0_wf, 
is-absolutely-free_wf, 
nat_wf, 
not-quotient-true, 
equal_wf, 
int_seg_wf, 
subtype_rel_function, 
int_seg_subtype_nat, 
istype-false, 
subtype_rel_self, 
add_nat_wf, 
decidable__le, 
full-omega-unsat, 
intformnot_wf, 
intformle_wf, 
itermConstant_wf, 
istype-int, 
int_formula_prop_not_lemma, 
istype-void, 
int_formula_prop_le_lemma, 
int_term_value_constant_lemma, 
int_formula_prop_wf, 
istype-le, 
nat_properties, 
intformand_wf, 
itermVar_wf, 
itermAdd_wf, 
intformeq_wf, 
int_formula_prop_and_lemma, 
int_term_value_var_lemma, 
int_term_value_add_lemma, 
int_formula_prop_eq_lemma, 
false_wf, 
strong-continuity2-implies-uniform-continuity2-nat, 
kripke2b-baire-seq_wf, 
decidable__lt, 
intformless_wf, 
int_formula_prop_less_lemma, 
decidable__equal_nat, 
subtype_base_sq, 
set_subtype_base, 
le_wf, 
int_subtype_base, 
init0-implies-eq-upto1-zero-seq, 
zero-seq_wf, 
baire2cantor_wf, 
change-from1_wf, 
int_seg_properties, 
eq-finite-seqs_wf, 
cantor2baire_wf, 
eqtt_to_assert, 
eqff_to_assert, 
bool_cases_sqequal, 
bool_wf, 
bool_subtype_base, 
assert-bnot, 
neg_assert_of_eq_int, 
assert_of_eq_int, 
eq_int_wf, 
primrec1_lemma, 
subtype_rel_wf, 
istype-less_than, 
decidable__equal_int, 
eq-finite-seqs-iff-eq-upto, 
min-inc-seq_wf, 
iff_weakening_equal, 
squash_wf, 
true_wf, 
istype-universe, 
baire2cantor2baire, 
pi1_wf, 
min-increasing-sequence_wf, 
min-increasing-sequence-prop2, 
baire-diff-from_wf, 
implies-eq-upto-baire2cantor, 
eq-upto-baire-diff-from, 
increasing-baire-diff-from, 
init0-baire-diff-from, 
baire-diff-from-diff, 
baire_eq_from_wf, 
eq-upto-baire-eq-from, 
less_than_wf, 
assert_wf, 
iff_weakening_uiff, 
assert_of_lt_int, 
lt_int_wf, 
increasing-sequence-prop1
Rules used in proof : 
sqequalSubstitution, 
sqequalTransitivity, 
computationStep, 
sqequalReflexivity, 
Error :lambdaFormation_alt, 
cut, 
thin, 
because_Cache, 
hypothesis, 
sqequalHypSubstitution, 
independent_functionElimination, 
voidElimination, 
sqequalRule, 
Error :functionIsType, 
introduction, 
extract_by_obid, 
Error :productIsType, 
Error :universeIsType, 
isectElimination, 
applyEquality, 
hypothesisEquality, 
Error :lambdaEquality_alt, 
setElimination, 
rename, 
Error :inhabitedIsType, 
dependent_functionElimination, 
functionEquality, 
productEquality, 
natural_numberEquality, 
independent_isectElimination, 
independent_pairFormation, 
productElimination, 
Error :equalityIstype, 
Error :dependent_set_memberEquality_alt, 
addEquality, 
unionElimination, 
approximateComputation, 
Error :dependent_pairFormation_alt, 
Error :isect_memberEquality_alt, 
equalityTransitivity, 
equalitySymmetry, 
applyLambdaEquality, 
pointwiseFunctionality, 
promote_hyp, 
int_eqEquality, 
Error :functionExtensionality_alt, 
Error :setIsType, 
instantiate, 
cumulativity, 
intEquality, 
closedConclusion, 
equalityElimination, 
int_eqReduceFalseSq, 
int_eqReduceTrueSq, 
hyp_replacement, 
baseApply, 
sqequalBase, 
baseClosed, 
functionExtensionality, 
setEquality, 
imageElimination, 
imageMemberEquality, 
universeEquality
Latex:
\mforall{}a:\mBbbN{}  {}\mrightarrow{}  \mBbbN{}
    (is-absolutely-free\{i:l\}(a)
    {}\mRightarrow{}  init0(a)
    {}\mRightarrow{}  increasing-sequence(a)
    {}\mRightarrow{}  (\mneg{}(\mforall{}m:\mBbbN{}.  \mexists{}n:\mBbbN{}.  ((a  n)  \mgeq{}  m  ))))
Date html generated:
2019_06_20-PM-03_07_58
Last ObjectModification:
2018_12_06-PM-11_57_04
Theory : continuity
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