Nuprl Lemma : gcd-list-property
∀L:ℤ List+
((∃R:ℤ List. (L = gcd-list(L) * R ∈ (ℤ List))) ∧ (∃S:ℤ List. ((||S|| = ||L|| ∈ ℤ) ∧ (gcd-list(L) = S ⋅ L ∈ ℤ))))
Proof
Definitions occuring in Statement :
int-vec-mul: a * as,
integer-dot-product: as ⋅ bs,
gcd-list: gcd-list(L),
listp: A List+,
length: ||as||,
list: T List,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
listp: A List+,
member: t ∈ T,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
cons: [a / b],
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
false: False,
and: P ∧ Q,
gcd-list: gcd-list(L),
uimplies: b supposing a,
top: Top,
subtype_rel: A ⊆r B,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
so_lambda: λ2x.t[x],
prop: ℙ,
so_apply: x[s],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
cand: A c∧ B,
int-vec-mul: a * as,
true: True,
not: ¬A,
uiff: uiff(P;Q),
guard: {T},
sq_type: SQType(T),
ge: i ≥ j ,
subtract: n - m,
le: A ≤ B,
nat: ℕ
Lemmas referenced :
list-cases,
product_subtype_list,
listp_wf,
length_of_nil_lemma,
int-valueall-type,
tl_wf,
cons_wf,
hd_wf,
length_cons_ge_one,
subtype_rel_list,
top_wf,
better-gcd_wf,
reduce_hd_cons_lemma,
reduce_tl_cons_lemma,
length_of_cons_lemma,
eager-accum-list_accum,
list_induction,
all_wf,
exists_wf,
list_wf,
equal-wf-base,
list_subtype_base,
int_subtype_base,
list_accum_nil_lemma,
list_accum_cons_lemma,
nil_wf,
map_cons_lemma,
map_nil_lemma,
squash_wf,
true_wf,
mul-commutes,
one-mul,
length-singleton,
int_dot_cons_lemma,
int_dot_nil_left_lemma,
add-zero,
list_accum_wf,
better-gcd-properties,
equal_wf,
null_nil_lemma,
btrue_wf,
and_wf,
null_wf,
null_cons_lemma,
bfalse_wf,
btrue_neq_bfalse,
cons_one_one,
int-vec-mul_wf,
subtype_base_sq,
mul-swap,
non_neg_length,
length_wf_nat,
nat_wf,
subtype_rel-equal,
base_wf,
le_antisymmetry_iff,
length_wf,
condition-implies-le,
minus-add,
minus-one-mul,
zero-add,
minus-one-mul-top,
add-commutes,
add_functionality_wrt_le,
add-associates,
le-add-cancel,
subtract_wf,
add-swap,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
nat_properties,
mul-distributes,
integer-dot-product_wf,
mul-associates
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
setElimination,
thin,
rename,
cut,
intEquality,
introduction,
extract_by_obid,
isectElimination,
hypothesis,
dependent_functionElimination,
hypothesisEquality,
unionElimination,
promote_hyp,
hypothesis_subsumption,
productElimination,
sqequalRule,
imageElimination,
voidElimination,
because_Cache,
independent_isectElimination,
isect_memberEquality,
voidEquality,
applyEquality,
lambdaEquality,
productEquality,
baseApply,
closedConclusion,
baseClosed,
independent_functionElimination,
independent_pairFormation,
dependent_pairFormation,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
multiplyEquality,
dependent_set_memberEquality,
applyLambdaEquality,
equalityUniverse,
levelHypothesis,
instantiate,
cumulativity,
sqequalIntensionalEquality,
addEquality,
minusEquality
Latex:
\mforall{}L:\mBbbZ{} List\msupplus{}
((\mexists{}R:\mBbbZ{} List. (L = gcd-list(L) * R)) \mwedge{} (\mexists{}S:\mBbbZ{} List. ((||S|| = ||L||) \mwedge{} (gcd-list(L) = S \mcdot{} L))))
Date html generated:
2017_09_29-PM-05_51_49
Last ObjectModification:
2017_05_31-PM-03_09_49
Theory : omega
Home
Index