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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