Nuprl Lemma : gcd-properties
∀a,b:ℤ.
((∃c:ℤ. ((c * gcd(a;b)) = a ∈ ℤ)) ∧ (∃d:ℤ. ((d * gcd(a;b)) = b ∈ ℤ)) ∧ (∃s,t:ℤ. (gcd(a;b) = ((s * a) + (t * b)) ∈ ℤ)))
Proof
Definitions occuring in Statement :
gcd: gcd(a;b),
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
add: n + m,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
int_seg: {i..j-},
false: False,
lelt: i ≤ j < k,
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
guard: {T},
uimplies: b supposing a,
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
less_than: a < b,
squash: ↓T,
cand: A c∧ B,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
le: A ≤ B,
less_than': less_than'(a;b),
true: True,
sq_type: SQType(T),
nat: ℕ,
prop: ℙ,
exists: ∃x:A. B[x],
sq_stable: SqStable(P),
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,
int_nzero: ℤ-o,
nequal: a ≠ b ∈ T ,
ge: i ≥ j ,
nat_plus: ℕ+
Lemmas referenced :
less_than_transitivity1,
less_than_irreflexivity,
int_seg_wf,
decidable__int_equal,
subtract_wf,
subtype_base_sq,
set_subtype_base,
int_subtype_base,
decidable__le,
istype-false,
not-le-2,
less-iff-le,
le_antisymmetry_iff,
condition-implies-le,
minus-add,
minus-minus,
minus-one-mul,
add-swap,
minus-one-mul-top,
add-commutes,
zero-add,
add_functionality_wrt_le,
add-associates,
add-zero,
le-add-cancel,
decidable__lt,
not-lt-2,
le-add-cancel-alt,
istype-le,
istype-less_than,
subtype_rel_self,
int_seg_properties,
absval_wf,
not-equal-2,
le_wf,
equal-wf-base,
istype-int,
primrec-wf2,
sq_stable__le,
add-mul-special,
zero-mul,
istype-nat,
absval-non-neg,
le_reflexive,
eq_int_wf,
eqtt_to_assert,
assert_of_eq_int,
eqff_to_assert,
bool_cases_sqequal,
bool_wf,
bool_subtype_base,
assert-bnot,
neg_assert_of_eq_int,
one-mul,
rem_bounds_z,
nequal_wf,
easy-member-int_seg,
istype-sqequal,
remainder_wfa,
not-equal-implies-less,
two-mul,
mul-distributes-right,
omega-shadow,
mul-distributes,
mul-commutes,
mul-associates,
nat_properties,
div_rem_sum,
equal_wf,
squash_wf,
true_wf,
istype-universe,
add_functionality_wrt_eq,
divide_wfa,
rem_to_div,
iff_weakening_equal,
mul-swap
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
thin,
sqequalHypSubstitution,
setElimination,
rename,
productElimination,
hypothesis,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
universeIsType,
dependent_functionElimination,
unionElimination,
applyEquality,
sqequalRule,
instantiate,
because_Cache,
dependent_set_memberEquality_alt,
independent_pairFormation,
imageElimination,
equalityTransitivity,
equalitySymmetry,
addEquality,
minusEquality,
Error :memTop,
productIsType,
promote_hyp,
hypothesis_subsumption,
lambdaEquality_alt,
functionIsType,
functionEquality,
intEquality,
productEquality,
baseApply,
closedConclusion,
baseClosed,
setIsType,
inhabitedIsType,
imageMemberEquality,
multiplyEquality,
equalityElimination,
dependent_pairFormation_alt,
equalityIstype,
cumulativity,
sqequalBase,
universeEquality
Latex:
\mforall{}a,b:\mBbbZ{}.
((\mexists{}c:\mBbbZ{}. ((c * gcd(a;b)) = a))
\mwedge{} (\mexists{}d:\mBbbZ{}. ((d * gcd(a;b)) = b))
\mwedge{} (\mexists{}s,t:\mBbbZ{}. (gcd(a;b) = ((s * a) + (t * b)))))
Date html generated:
2020_05_19-PM-09_36_09
Last ObjectModification:
2020_01_08-PM-04_39_18
Theory : int_1
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