Nuprl Lemma : gcd-reduce-ext
∀p,q:ℤ. ∃g:ℕ. ∃a,b,x,y:ℤ. ((p = (a * g) ∈ ℤ) ∧ (q = (b * g) ∈ ℤ) ∧ (((x * a) + (y * b)) = 1 ∈ ℤ))
Proof
Definitions occuring in Statement :
nat: ℕ,
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
and: P ∧ Q,
multiply: n * m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
member: t ∈ T,
remainder: n rem m,
divide: n ÷ m,
subtract: n - m,
so_apply: x[s1;s2],
natrec: natrec,
genrec: genrec,
genrec-ap: genrec-ap,
spreadn: spread7,
gcd-reduce,
decidable__equal_int,
decidable__int_equal,
uall: ∀[x:A]. B[x],
so_lambda: so_lambda(x,y,z,w.t[x; y; z; w]),
so_apply: x[s1;s2;s3;s4],
so_lambda: λ2x.t[x],
top: Top,
so_apply: x[s],
uimplies: b supposing a,
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T,
so_lambda: λ2x y.t[x; y]
Lemmas referenced :
gcd-reduce,
lifting-strict-int_eq,
istype-void,
strict4-decide,
lifting-strict-spread,
has-value_wf_base,
istype-base,
is-exception_wf,
decidable__equal_int,
decidable__int_equal
Rules used in proof :
introduction,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
instantiate,
extract_by_obid,
hypothesis,
sqequalRule,
thin,
sqequalHypSubstitution,
equalityTransitivity,
equalitySymmetry,
isectElimination,
baseClosed,
Error :isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
Error :lambdaFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
Error :universeIsType,
baseApply,
closedConclusion,
hypothesisEquality,
callbyvalueExceptionCases,
Error :inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
Error :inlFormation_alt,
Error :inhabitedIsType,
sqequalSqle,
divergentSqle,
callbyvalueSpread,
productElimination,
sqleReflexivity,
Error :equalityIstype,
dependent_functionElimination,
independent_functionElimination,
spreadExceptionCases,
axiomSqleEquality
Latex:
\mforall{}p,q:\mBbbZ{}. \mexists{}g:\mBbbN{}. \mexists{}a,b,x,y:\mBbbZ{}. ((p = (a * g)) \mwedge{} (q = (b * g)) \mwedge{} (((x * a) + (y * b)) = 1))
Date html generated:
2019_06_20-PM-02_27_15
Last ObjectModification:
2019_03_10-PM-02_40_50
Theory : num_thy_1
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