Nuprl Lemma : chinese-remainder2-extract
∀r,s,a,b:ℤ. Dec(∃x:ℤ [((x ≡ a mod r) ∧ (x ≡ b mod s))])
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
decidable: Dec(P),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
int: ℤ
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,
sign: sign(x),
ifthenelse: if b then t else f fi ,
le_int: i ≤z j,
bnot: ¬bb,
lt_int: i <z j,
btrue: tt,
it: ⋅,
bfalse: ff,
eq_int: (i =z j),
genrec-ap: genrec-ap,
spreadn: spread7,
chinese-remainder2,
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 y.t[x; y],
top: Top,
uimplies: b supposing a,
so_lambda: λ2x.t[x],
so_apply: x[s],
strict4: strict4(F),
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
has-value: (a)↓,
prop: ℙ,
or: P ∨ Q,
squash: ↓T
Lemmas referenced :
chinese-remainder2,
lifting-strict-spread,
istype-void,
strict4-spread,
lifting-strict-int_eq,
strict4-decide,
has-value_wf_base,
istype-base,
is-exception_wf,
istype-universe,
value-type-has-value,
int-value-type,
lifting-strict-decide,
lifting-strict-less,
gcd-reduce,
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,
isect_memberEquality_alt,
voidElimination,
independent_isectElimination,
independent_pairFormation,
lambdaFormation_alt,
callbyvalueCallbyvalue,
callbyvalueReduce,
universeIsType,
baseApply,
closedConclusion,
hypothesisEquality,
callbyvalueExceptionCases,
inrFormation_alt,
imageMemberEquality,
imageElimination,
exceptionSqequal,
inlFormation_alt,
inhabitedIsType,
sqequalSqle,
divergentSqle,
callbyvalueSpread,
productElimination,
sqleReflexivity,
equalityIstype,
dependent_functionElimination,
independent_functionElimination,
spreadExceptionCases,
axiomSqleEquality,
callbyvalueIntEq,
intEquality,
int_eqExceptionCases,
callbyvalueMultiply,
multiplyExceptionCases,
because_Cache
Latex:
\mforall{}r,s,a,b:\mBbbZ{}. Dec(\mexists{}x:\mBbbZ{} [((x \mequiv{} a mod r) \mwedge{} (x \mequiv{} b mod s))])
Date html generated:
2019_10_15-AM-11_16_52
Last ObjectModification:
2019_06_26-PM-03_39_28
Theory : general
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