Nuprl Lemma : divisor-test_wf
∀[n:ℕ]. ∀[i:ℕ+]. ∀[j:ℤ].
(divisor-test(n;i;j) ∈ {n1:ℤ| n1 < n ∧ (2 ≤ n1) ∧ (n1 | n)} ∨ (gcd(n;iseg_product(i;j)) = 1 ∈ ℤ)) supposing ((i ≤ j) \000Cand j < n)
Proof
Definitions occuring in Statement :
divisor-test: divisor-test(n;i;j),
iseg_product: iseg_product(i;j),
divides: b | a,
gcd: gcd(a;b),
nat_plus: ℕ+,
nat: ℕ,
less_than: a < b,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
le: A ≤ B,
or: P ∨ Q,
and: P ∧ Q,
member: t ∈ T,
set: {x:A| B[x]} ,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat: ℕ,
implies: P ⇒ Q,
false: False,
ge: i ≥ j ,
uimplies: b supposing a,
satisfiable_int_formula: satisfiable_int_formula(fmla),
exists: ∃x:A. B[x],
not: ¬A,
top: Top,
and: P ∧ Q,
prop: ℙ,
guard: {T},
nat_plus: ℕ+,
int_seg: {i..j-},
lelt: i ≤ j < k,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
le: A ≤ B,
less_than': less_than'(a;b),
divisor-test: divisor-test(n;i;j),
has-value: (a)↓,
so_lambda: λ2x.t[x],
so_apply: x[s],
sq_type: SQType(T),
less_than: a < b,
gcd: gcd(a;b),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
squash: ↓T,
nequal: a ≠ b ∈ T ,
true: True,
exposed-bfalse: exposed-bfalse,
cand: A c∧ B,
iseg_product: iseg_product(i;j),
int_nzero: ℤ-o,
subtract: n - m,
coprime: CoPrime(a,b)
Lemmas referenced :
nat_properties,
satisfiable-full-omega-tt,
intformand_wf,
intformle_wf,
itermConstant_wf,
itermVar_wf,
intformless_wf,
int_formula_prop_and_lemma,
int_formula_prop_le_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_formula_prop_less_lemma,
int_formula_prop_wf,
ge_wf,
less_than_wf,
le_wf,
subtract_wf,
nat_plus_wf,
nat_wf,
int_seg_wf,
int_seg_properties,
decidable__le,
intformnot_wf,
itermSubtract_wf,
int_formula_prop_not_lemma,
int_term_value_subtract_lemma,
decidable__equal_int,
int_seg_subtype,
false_wf,
intformeq_wf,
int_formula_prop_eq_lemma,
value-type-has-value,
set-value-type,
int-value-type,
iseg_product_rem_wf,
less_than_transitivity1,
less_than_transitivity2,
le_weakening2,
nat_plus_properties,
decidable__lt,
itermAdd_wf,
int_term_value_add_lemma,
lelt_wf,
subtype_base_sq,
int_subtype_base,
iseg_product_wf,
better-gcd-gcd,
eq_int_wf,
bool_wf,
uiff_transitivity,
equal-wf-T-base,
assert_wf,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
bnot_wf,
not_wf,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
gcd_wf,
equal_wf,
iseg_product_rem_property,
iff_weakening_equal,
rem_rem_to_rem,
gcd_com,
lt_int_wf,
assert_of_lt_int,
le_int_wf,
assert_functionality_wrt_uiff,
bnot_of_lt_int,
assert_of_le_int,
gcd_is_divisor_1,
divides_wf,
equal-wf-base,
true_wf,
set_subtype_base,
combinations-step,
itermMultiply_wf,
int_term_value_mul_lemma,
combinations_wf_int,
divisors_bound,
gcd_is_divisor_2,
div_rem_sum,
nequal_wf,
rem_bounds_1,
add-is-int-iff,
multiply-is-int-iff,
or_wf,
set_wf,
not-lt-2,
less-iff-le,
condition-implies-le,
add-associates,
minus-one-mul,
add-commutes,
minus-one-mul-top,
add-mul-special,
zero-mul,
zero-add,
minus-add,
add-swap,
add_functionality_wrt_le,
add-zero,
two-mul,
le-add-cancel,
iseg_product-split,
gcd_sat_gcd_p,
gcd_p_wf,
squash_wf,
coprime_prod,
gcd_unique,
assoced_elim,
gcd-positive
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
intWeakElimination,
natural_numberEquality,
independent_isectElimination,
dependent_pairFormation,
lambdaEquality,
int_eqEquality,
intEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
computeAll,
independent_functionElimination,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
because_Cache,
productElimination,
unionElimination,
applyEquality,
applyLambdaEquality,
hypothesis_subsumption,
dependent_set_memberEquality,
callbyvalueReduce,
addEquality,
instantiate,
cumulativity,
equalityElimination,
baseClosed,
impliesFunctionality,
imageElimination,
remainderEquality,
imageMemberEquality,
inlEquality,
productEquality,
divideEquality,
addLevel,
multiplyEquality,
pointwiseFunctionality,
promote_hyp,
baseApply,
closedConclusion,
minusEquality,
inrEquality,
setEquality,
universeEquality,
isect_memberFormation
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[i:\mBbbN{}\msupplus{}]. \mforall{}[j:\mBbbZ{}].
(divisor-test(n;i;j) \mmember{} \{n1:\mBbbZ{}| n1 < n \mwedge{} (2 \mleq{} n1) \mwedge{} (n1 | n)\} \mvee{} (gcd(n;iseg\_product(i;j)) = 1)) sup\000Cposing
((i \mleq{} j) and
j < n)
Date html generated:
2018_05_21-PM-08_15_08
Last ObjectModification:
2017_07_26-PM-05_49_42
Theory : general
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