Nuprl Lemma : chinese-remainder2
∀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 :
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
and: P ∧ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
subtype_rel: A ⊆r B,
sq_exists: ∃x:A [B[x]],
decidable: Dec(P),
not: ¬A,
or: P ∨ Q,
cand: A c∧ B,
guard: {T},
uimplies: b supposing a,
prop: ℙ,
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
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,
sq_type: SQType(T),
eqmod: a ≡ b mod m,
divides: b | a,
ge: i ≥ j ,
satisfiable_int_formula: satisfiable_int_formula(fmla),
false: False,
top: Top,
int_nzero: ℤ-o,
has-value: (a)↓,
true: True,
squash: ↓T
Lemmas referenced :
gcd-reduce,
eq_int_wf,
bool_wf,
equal-wf-T-base,
assert_wf,
equal-wf-base,
int_subtype_base,
eqmod_weakening,
eqmod_wf,
false_wf,
bnot_wf,
not_wf,
subtract_wf,
uiff_transitivity,
eqtt_to_assert,
assert_of_eq_int,
iff_transitivity,
iff_weakening_uiff,
eqff_to_assert,
assert_of_bnot,
equal_wf,
subtype_base_sq,
nat_properties,
decidable__equal_int,
satisfiable-full-omega-tt,
intformand_wf,
intformnot_wf,
intformeq_wf,
itermVar_wf,
itermConstant_wf,
itermMultiply_wf,
int_formula_prop_and_lemma,
int_formula_prop_not_lemma,
int_formula_prop_eq_lemma,
int_term_value_var_lemma,
int_term_value_constant_lemma,
int_term_value_mul_lemma,
int_formula_prop_wf,
itermSubtract_wf,
int_term_value_subtract_lemma,
div_rem_sum,
nequal_wf,
add-is-int-iff,
multiply-is-int-iff,
itermAdd_wf,
int_term_value_add_lemma,
value-type-has-value,
int-value-type,
equal-wf-base-T,
eqmod-zero,
eqmod_functionality_wrt_eqmod,
add_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod,
mul-swap,
mul-commutes,
zero-mul,
add-zero,
squash_wf,
true_wf,
mul_assoc,
minus_functionality_wrt_eq,
iff_weakening_equal,
itermMinus_wf,
int_term_value_minus_lemma,
minus_functionality_wrt_eqmod,
minus-one-mul,
mul-associates,
divides_iff_rem_zero
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
intEquality,
isectElimination,
setElimination,
rename,
hypothesis,
natural_numberEquality,
equalityTransitivity,
equalitySymmetry,
baseClosed,
because_Cache,
sqequalRule,
baseApply,
closedConclusion,
applyEquality,
inlEquality,
dependent_set_memberEquality,
independent_pairFormation,
independent_isectElimination,
productEquality,
functionEquality,
setEquality,
inrEquality,
lambdaEquality,
remainderEquality,
unionElimination,
equalityElimination,
independent_functionElimination,
impliesFunctionality,
promote_hyp,
instantiate,
cumulativity,
dependent_pairFormation,
int_eqEquality,
isect_memberEquality,
voidElimination,
voidEquality,
computeAll,
pointwiseFunctionality,
divideEquality,
callbyvalueReduce,
addEquality,
multiplyEquality,
imageElimination,
universeEquality,
imageMemberEquality,
minusEquality
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:
2018_05_21-PM-08_11_50
Last ObjectModification:
2017_07_26-PM-05_47_07
Theory : general
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