Nuprl Lemma : divisor-in-range
∀n:{2...}
∀[k:ℕ]
∀i:{1...}. ∀j:{i..i + k-}.
(∃m:ℤ [(m < n ∧ (2 ≤ m) ∧ (m | n))]) ∨ (¬(∃m:ℤ [((2 ≤ m) ∧ (i ≤ m) ∧ (m ≤ j) ∧ (m | n))])) supposing j < n
Proof
Definitions occuring in Statement :
divides: b | a,
int_upper: {i...},
int_seg: {i..j-},
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
add: n + m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
member: t ∈ T,
int_upper: {i...},
nat: ℕ,
uimplies: b supposing a,
int_seg: {i..j-},
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
implies: P ⇒ Q,
nat_plus: ℕ+,
le: A ≤ B,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
subtype_rel: A ⊆r B,
less_than': less_than'(a;b),
true: True,
sq_stable: SqStable(P),
squash: ↓T,
subtract: n - m,
lelt: i ≤ j < k,
guard: {T},
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
top: Top,
sq_type: SQType(T),
sq_exists: ∃x:A [B[x]],
gcd: gcd(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bnot: ¬bb,
assert: ↑b,
nequal: a ≠ b ∈ T ,
cand: A c∧ B,
gcd_p: GCD(a;b;y),
iseg_product: iseg_product(i;j),
int_nzero: ℤ-o,
less_than: a < b
Lemmas referenced :
uniform-comp-nat-induction,
all_wf,
int_upper_wf,
int_seg_wf,
isect_wf,
less_than_wf,
or_wf,
sq_exists_wf,
le_wf,
divides_wf,
istype-int,
not_wf,
nat_wf,
member-less_than,
iseg_product_rem_wf,
decidable__lt,
istype-false,
not-lt-2,
add_functionality_wrt_le,
add-commutes,
zero-add,
le-add-cancel,
int_seg_subtype_nat,
decidable__le,
not-le-2,
sq_stable__le,
condition-implies-le,
minus-add,
minus-one-mul,
minus-one-mul-top,
add-associates,
int_seg_properties,
nat_properties,
int_upper_properties,
full-omega-unsat,
intformand_wf,
intformnot_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
int_formula_prop_and_lemma,
istype-void,
int_formula_prop_not_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_wf,
intformless_wf,
itermAdd_wf,
int_formula_prop_less_lemma,
int_term_value_add_lemma,
set-value-type,
equal_wf,
int-value-type,
better-gcd_wf,
subtype_base_sq,
int_subtype_base,
better-gcd-gcd,
upper_subtype_nat,
iseg_product_wf,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
intformeq_wf,
int_formula_prop_eq_lemma,
eqff_to_assert,
set_subtype_base,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
gcd_wf,
istype-universe,
iseg_product_rem_property,
iff_weakening_equal,
rem_rem_to_rem,
not-equal-2,
gcd_com,
gcd_sat_pred,
combinations-step,
subtract_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
lelt_wf,
decidable__equal_int,
itermMultiply_wf,
int_term_value_mul_lemma,
divisors_bound,
gcd-non-neg,
gcd_is_divisor_2,
squash_wf,
true_wf,
subtype_rel_self,
div_rem_sum,
nequal_wf,
rem_bounds_1,
add-is-int-iff,
multiply-is-int-iff,
false_wf,
gcd-positive,
divides-combinations,
pdivisor_bound,
one_divs_any
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
sqequalRule,
lambdaEquality_alt,
natural_numberEquality,
hypothesis,
setElimination,
rename,
because_Cache,
addEquality,
intEquality,
productEquality,
hypothesisEquality,
universeIsType,
independent_functionElimination,
isect_memberFormation_alt,
independent_isectElimination,
dependent_set_memberEquality_alt,
productElimination,
dependent_functionElimination,
unionElimination,
independent_pairFormation,
voidElimination,
applyEquality,
imageMemberEquality,
baseClosed,
imageElimination,
minusEquality,
approximateComputation,
dependent_pairFormation_alt,
int_eqEquality,
isect_memberEquality_alt,
productIsType,
cutEval,
equalityTransitivity,
equalitySymmetry,
equalityIsType1,
inhabitedIsType,
instantiate,
cumulativity,
isectIsType,
functionIsType,
unionIsType,
equalityElimination,
equalityIsType2,
baseApply,
closedConclusion,
promote_hyp,
universeEquality,
remainderEquality,
inlFormation_alt,
dependent_set_memberFormation_alt,
equalityIsType4,
divideEquality,
pointwiseFunctionality,
inrFormation_alt,
applyLambdaEquality
Latex:
\mforall{}n:\{2...\}
\mforall{}[k:\mBbbN{}]
\mforall{}i:\{1...\}. \mforall{}j:\{i..i + k\msupminus{}\}.
(\mexists{}m:\mBbbZ{} [(m < n \mwedge{} (2 \mleq{} m) \mwedge{} (m | n))]) \mvee{} (\mneg{}(\mexists{}m:\mBbbZ{} [((2 \mleq{} m) \mwedge{} (i \mleq{} m) \mwedge{} (m \mleq{} j) \mwedge{} (m | n))]))
supposing j < n
Date html generated:
2019_10_15-AM-11_17_39
Last ObjectModification:
2018_10_09-PM-02_12_13
Theory : general
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