Nuprl Lemma : proper-divisor_wf
∀[n:ℕ+]. (proper-divisor(n) ∈ Dec(∃n1:ℤ [(n1 < n ∧ (2 ≤ n1) ∧ (n1 | n))]))
Proof
Definitions occuring in Statement :
proper-divisor: proper-divisor(n),
divides: b | a,
nat_plus: ℕ+,
less_than: a < b,
decidable: Dec(P),
uall: ∀[x:A]. B[x],
le: A ≤ B,
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
member: t ∈ T,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
proper-divisor: proper-divisor(n),
int_upper: {i...},
le: A ≤ B,
and: P ∧ Q,
less_than': less_than'(a;b),
false: False,
not: ¬A,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
nat_plus: ℕ+,
so_apply: x[s],
sq_exists: ∃x:A [B[x]],
all: ∀x:A. B[x],
decidable: Dec(P),
or: P ∨ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
nat: ℕ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
cand: A c∧ B,
divides: b | a,
subtype_rel: A ⊆r B,
sq_type: SQType(T),
int_nzero: ℤ-o,
true: True,
nequal: a ≠ b ∈ T ,
has-value: (a)↓,
less_than: a < b,
squash: ↓T,
ge: i ≥ j ,
exp: i^n,
eq_int: (i =z j),
subtract: n - m
Lemmas referenced :
trial-division_wf,
cons_wf,
int_upper_wf,
false_wf,
le_wf,
nil_wf,
sq_exists_wf,
less_than_wf,
divides_wf,
top_wf,
equal_wf,
nat_plus_wf,
not_wf,
lt_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_lt_int,
eq_int_wf,
assert_of_eq_int,
iff_transitivity,
bnot_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
le_int_wf,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
proper-divisor-aux_wf,
nat_plus_properties,
decidable__lt,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformless_wf,
itermVar_wf,
itermMultiply_wf,
itermConstant_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_less_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_term_value_constant_lemma,
int_formula_prop_wf,
intformle_wf,
int_formula_prop_le_lemma,
intformeq_wf,
int_formula_prop_eq_lemma,
decidable__equal_int,
int_subtype_base,
set_wf,
subtype_base_sq,
equal-wf-base,
divides_iff_div_exact,
true_wf,
nequal_wf,
value-type-has-value,
iroot-property,
nat_plus_subtype_nat,
decidable__le,
iroot_wf,
nat_wf,
nat_properties,
decidable__or,
or_wf,
intformor_wf,
int_formula_prop_or_lemma,
primrec-unroll,
primrec1_lemma,
itermAdd_wf,
int_term_value_add_lemma,
exp_wf2,
mul-distributes,
mul-distributes-right,
add-associates,
mul-commutes,
mul-swap,
one-mul,
add-swap,
add-commutes,
two-mul,
mul_preserves_lt,
not-lt-2,
less-iff-le,
add_functionality_wrt_le,
zero-add,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
natural_numberEquality,
hypothesis,
dependent_set_memberEquality,
sqequalRule,
independent_pairFormation,
lambdaFormation,
because_Cache,
unionEquality,
intEquality,
lambdaEquality,
productEquality,
setElimination,
rename,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
axiomEquality,
inlEquality,
equalityElimination,
baseClosed,
productElimination,
independent_isectElimination,
impliesFunctionality,
multiplyEquality,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
baseApply,
closedConclusion,
applyEquality,
inrEquality,
setEquality,
instantiate,
cumulativity,
promote_hyp,
addLevel,
callbyvalueReduce,
imageMemberEquality,
applyLambdaEquality,
addEquality,
imageElimination
Latex:
\mforall{}[n:\mBbbN{}\msupplus{}]. (proper-divisor(n) \mmember{} Dec(\mexists{}n1:\mBbbZ{} [(n1 < n \mwedge{} (2 \mleq{} n1) \mwedge{} (n1 | n))]))
Date html generated:
2018_05_21-PM-08_16_08
Last ObjectModification:
2017_07_26-PM-05_50_20
Theory : general
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