Nuprl Lemma : mul-initial-seg-property2
∀f:ℕ ⟶ ℕ. (∃n:ℕ. ((f n) = 0 ∈ ℤ) ⇐⇒ ∃n:ℕ. ∀m:ℕ. (mul-initial-seg(f) m) = 0 ∈ ℤ supposing n ≤ m)
Proof
Definitions occuring in Statement :
mul-initial-seg: mul-initial-seg(f),
nat: ℕ,
uimplies: b supposing a,
le: 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,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
nat: ℕ,
so_apply: x[s],
rev_implies: P ⇐ Q,
uimplies: b supposing a,
exists: ∃x:A. B[x],
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
not: ¬A,
top: Top,
cand: A c∧ B,
mul-initial-seg: mul-initial-seg(f),
upto: upto(n),
from-upto: [n, m),
ifthenelse: if b then t else f fi ,
lt_int: i <z j,
bfalse: ff,
sq_type: SQType(T),
guard: {T},
true: True,
squash: ↓T,
nat_plus: ℕ+
Lemmas referenced :
exists_wf,
nat_wf,
equal-wf-T-base,
all_wf,
isect_wf,
le_wf,
mul-initial-seg_wf,
nat_properties,
decidable__le,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermAdd_wf,
itermVar_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_add_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
mul-initial-seg-property,
decidable__lt,
intformless_wf,
int_formula_prop_less_lemma,
less_than_wf,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
set_wf,
primrec-wf2,
map_nil_lemma,
reduce_nil_lemma,
subtype_base_sq,
int_subtype_base,
false_wf,
equal-wf-base,
equal_wf,
squash_wf,
true_wf,
mul-initial-seg-step,
iff_weakening_equal,
int_entire
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesis,
sqequalRule,
lambdaEquality,
intEquality,
applyEquality,
functionExtensionality,
hypothesisEquality,
setElimination,
rename,
baseClosed,
because_Cache,
functionEquality,
productElimination,
dependent_pairFormation,
dependent_set_memberEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
independent_isectElimination,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
isect_memberFormation,
independent_functionElimination,
productEquality,
instantiate,
addLevel,
cumulativity,
levelHypothesis,
promote_hyp,
imageElimination,
universeEquality,
equalityUniverse,
imageMemberEquality
Latex:
\mforall{}f:\mBbbN{} {}\mrightarrow{} \mBbbN{}. (\mexists{}n:\mBbbN{}. ((f n) = 0) \mLeftarrow{}{}\mRightarrow{} \mexists{}n:\mBbbN{}. \mforall{}m:\mBbbN{}. (mul-initial-seg(f) m) = 0 supposing n \mleq{} m)
Date html generated:
2018_05_21-PM-08_38_05
Last ObjectModification:
2017_07_26-PM-06_02_23
Theory : general
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