Nuprl Lemma : least-factor_wf
∀[n:ℤ]. least-factor(n) ∈ {p:ℕ| 1 < p ∧ prime(p) ∧ (p | n) ∧ (∀q:ℕ. (prime(q) ⇒ (q | n) ⇒ (p ≤ q)))} supposing 1 < |n\000C|
Proof
Definitions occuring in Statement :
least-factor: least-factor(n),
prime: prime(a),
divides: b | a,
absval: |i|,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
nat: ℕ,
all: ∀x:A. B[x],
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
false: False,
top: Top,
and: P ∧ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
assoced: a ~ b,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
less_than: a < b,
squash: ↓T,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
le: A ≤ B,
less_than': less_than'(a;b),
ge: i ≥ j ,
least-factor: least-factor(n),
subtract: n - m,
cand: A c∧ B,
so_lambda: λ2x.t[x],
so_apply: x[s],
atomic: atomic(a),
reducible: reducible(a),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
gt: i > j,
sq_type: SQType(T),
nat_plus: ℕ+,
true: True,
divides: b | a,
prime: prime(a)
Lemmas referenced :
less_than_wf,
absval_wf,
nat_wf,
divides_iff_rem_zero,
satisfiable-full-omega-tt,
intformand_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
equal-wf-base,
int_subtype_base,
nequal_wf,
absval_assoced,
decidable__le,
subtract_wf,
intformnot_wf,
intformle_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_subtract_lemma,
decidable__lt,
lelt_wf,
assert_of_eq_int,
itermAdd_wf,
int_term_value_add_lemma,
subtract-add-cancel,
assert_wf,
eq_int_wf,
int_seg_properties,
int_seg_subtype_nat,
false_wf,
nat_properties,
mu_wf,
mu-property,
add-nat,
le_wf,
add-subtract-cancel,
add-associates,
add-swap,
add-commutes,
zero-add,
divides_wf,
equal-wf-T-base,
not_wf,
prime_wf,
all_wf,
atomic_imp_prime,
assoced_wf,
reducible_wf,
assoced_nelim,
absval_ifthenelse,
int_nzero_properties,
int_nzero_wf,
lt_int_wf,
bool_wf,
le_int_wf,
bnot_wf,
itermMinus_wf,
int_term_value_minus_lemma,
assoced_functionality_wrt_assoced,
assoced_weakening,
uiff_transitivity,
eqtt_to_assert,
assert_of_lt_int,
eqff_to_assert,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
equal_wf,
itermMultiply_wf,
int_term_value_mul_lemma,
subtype_base_sq,
neg_mul_arg_bounds,
gt_wf,
decidable__equal_int,
mul_preserves_lt,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
le-add-cancel,
divides_reflexivity,
divides_functionality_wrt_assoced,
equal-wf-base-T,
divides_transitivity
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
hypothesis,
sqequalRule,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
extract_by_obid,
isectElimination,
thin,
natural_numberEquality,
hypothesisEquality,
applyEquality,
lambdaEquality,
setElimination,
rename,
isect_memberEquality,
because_Cache,
intEquality,
dependent_functionElimination,
dependent_set_memberEquality,
lambdaFormation,
independent_isectElimination,
dependent_pairFormation,
int_eqEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
baseApply,
closedConclusion,
baseClosed,
productElimination,
independent_functionElimination,
unionElimination,
imageElimination,
remainderEquality,
addEquality,
applyLambdaEquality,
promote_hyp,
addLevel,
impliesFunctionality,
productEquality,
functionEquality,
minusEquality,
levelHypothesis,
equalityElimination,
instantiate,
cumulativity,
inrFormation,
inlFormation,
multiplyEquality,
impliesLevelFunctionality
Latex:
\mforall{}[n:\mBbbZ{}]
least-factor(n) \mmember{} \{p:\mBbbN{}| 1 < p \mwedge{} prime(p) \mwedge{} (p | n) \mwedge{} (\mforall{}q:\mBbbN{}. (prime(q) {}\mRightarrow{} (q | n) {}\mRightarrow{} (p \mleq{} q)))\} su\000Cpposing 1 < |n|
Date html generated:
2018_05_21-PM-06_58_55
Last ObjectModification:
2017_07_26-PM-05_00_24
Theory : general
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