Nuprl Lemma : nat-star-retract-property
∀s:ℕ ⟶ ℕ. (∃n:ℕ. 0 < s n ⇐⇒ ∃n:ℕ. 0 < nat-star-retract(s) n)
Proof
Definitions occuring in Statement :
nat-star-retract: nat-star-retract(s),
nat: ℕ,
less_than: a < b,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
natural_number: $n
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
exists: ∃x:A. B[x],
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
so_apply: x[s],
rev_implies: P ⇐ Q,
nat-star: ℕ*,
guard: {T},
int_seg: {i..j-},
lelt: i ≤ j < k,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
decidable: Dec(P),
or: P ∨ Q,
le: A ≤ B,
less_than': less_than'(a;b),
nat: ℕ,
ge: i ≥ j ,
nat-star-retract: nat-star-retract(s),
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,
less_than: a < b,
squash: ↓T
Lemmas referenced :
exists_wf,
nat_wf,
less_than_wf,
nat-star-retract_wf,
nat-star_wf,
int_seg_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformless_wf,
itermVar_wf,
itermConstant_wf,
intformle_wf,
int_formula_prop_and_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_le_lemma,
int_formula_prop_wf,
int_seg_wf,
decidable__equal_int,
subtract_wf,
int_seg_subtype,
false_wf,
decidable__le,
intformnot_wf,
itermSubtract_wf,
intformeq_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
int_formula_prop_eq_lemma,
le_wf,
all_wf,
int_seg_subtype_nat,
decidable__lt,
lelt_wf,
set_wf,
primrec-wf2,
nat_properties,
itermAdd_wf,
int_term_value_add_lemma,
decidable__exists_int_seg,
bl-exists_wf,
upto_wf,
l_member_wf,
lt_int_wf,
bool_wf,
eqtt_to_assert,
assert-bl-exists,
l_exists_functionality,
assert_wf,
iff_weakening_uiff,
subtype_rel_set,
assert_of_lt_int,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
l_exists_wf,
l_exists_iff,
bnot_wf,
not_wf,
bool_cases,
iff_transitivity,
assert_of_bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
sqequalHypSubstitution,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
sqequalRule,
lambdaEquality,
natural_numberEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
because_Cache,
setElimination,
rename,
functionEquality,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
unionElimination,
addLevel,
equalityTransitivity,
equalitySymmetry,
applyLambdaEquality,
levelHypothesis,
hypothesis_subsumption,
dependent_set_memberEquality,
addEquality,
independent_functionElimination,
instantiate,
setEquality,
equalityElimination,
promote_hyp,
cumulativity,
imageElimination,
impliesFunctionality
Latex:
\mforall{}s:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (\mexists{}n:\mBbbN{}. 0 < s n \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. 0 < nat-star-retract(s) n)
Date html generated:
2017_04_17-AM-09_55_20
Last ObjectModification:
2017_02_27-PM-05_49_45
Theory : continuity
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