Nuprl Lemma : chinese-remainder1
∀r:ℤ. ∀s:{s':ℤ| CoPrime(r,s')} . ∀a,b:ℤ. (∃x:ℤ [((x ≡ a mod r) ∧ (x ≡ b mod s))])
Proof
Definitions occuring in Statement :
eqmod: a ≡ b mod m,
coprime: CoPrime(a,b),
all: ∀x:A. B[x],
sq_exists: ∃x:A [B[x]],
and: P ∧ Q,
set: {x:A| B[x]} ,
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],
prop: ℙ,
nat: ℕ,
decidable: Dec(P),
or: P ∨ Q,
uimplies: b supposing a,
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
cand: A c∧ B,
sq_exists: ∃x:A [B[x]],
false: False,
satisfiable_int_formula: satisfiable_int_formula(fmla),
not: ¬A,
ge: i ≥ j ,
so_apply: x[s],
so_lambda: λ2x.t[x],
subtype_rel: A ⊆r B,
divides: b | a,
eqmod: a ≡ b mod m,
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
true: True,
squash: ↓T,
nequal: a ≠ b ∈ T ,
int_nzero: ℤ-o,
subtract: n - m,
uiff: uiff(P;Q),
sq_stable: SqStable(P),
assoced: a ~ b,
less_than': less_than'(a;b),
le: A ≤ B,
gcd_p: GCD(a;b;y),
coprime: CoPrime(a,b)
Lemmas referenced :
gcd-reduce,
coprime_wf,
istype-int,
decidable__equal_int,
subtype_base_sq,
int_subtype_base,
mul-commutes,
one-mul,
eqmod_wf,
eqmod_refl,
int_formula_prop_wf,
int_term_value_add_lemma,
int_term_value_constant_lemma,
int_term_value_var_lemma,
int_term_value_mul_lemma,
int_formula_prop_eq_lemma,
int_formula_prop_not_lemma,
int_formula_prop_and_lemma,
itermAdd_wf,
itermConstant_wf,
itermVar_wf,
itermMultiply_wf,
intformeq_wf,
intformnot_wf,
intformand_wf,
full-omega-unsat,
nat_properties,
set_subtype_base,
subtract_wf,
int_term_value_subtract_lemma,
itermSubtract_wf,
iff_weakening_equal,
subtype_rel_self,
mul_assoc,
istype-universe,
true_wf,
squash_wf,
equal_wf,
int_entire_a,
div_rem_sum,
nequal_wf,
remainder_wfa,
mul-distributes,
add-associates,
minus-one-mul,
mul-swap,
mul-associates,
zero-mul,
zero-add,
add-zero,
eqmod-zero,
eqmod_functionality_wrt_eqmod,
add_functionality_wrt_eqmod,
multiply_functionality_wrt_eqmod,
eqmod_weakening,
subtract_functionality_wrt_eqmod,
add-is-int-iff,
multiply-is-int-iff,
false_wf,
divide_wfa,
sq_stable__eqmod,
one_divs_any,
istype-le,
istype-void,
assoced_nelim,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation_alt,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
hypothesisEquality,
setElimination,
rename,
hypothesis,
productElimination,
inhabitedIsType,
setIsType,
universeIsType,
isectElimination,
equalityTransitivity,
equalitySymmetry,
unionElimination,
instantiate,
cumulativity,
intEquality,
independent_isectElimination,
independent_functionElimination,
because_Cache,
Error :memTop,
sqequalRule,
natural_numberEquality,
productIsType,
independent_pairFormation,
dependent_set_memberFormation_alt,
voidElimination,
int_eqEquality,
lambdaEquality_alt,
dependent_pairFormation_alt,
approximateComputation,
sqequalBase,
applyEquality,
baseClosed,
closedConclusion,
baseApply,
equalityIstype,
multiplyEquality,
promote_hyp,
imageMemberEquality,
universeEquality,
imageElimination,
addEquality,
dependent_set_memberEquality_alt,
pointwiseFunctionality
Latex:
\mforall{}r:\mBbbZ{}. \mforall{}s:\{s':\mBbbZ{}| CoPrime(r,s')\} . \mforall{}a,b:\mBbbZ{}. (\mexists{}x:\mBbbZ{} [((x \mequiv{} a mod r) \mwedge{} (x \mequiv{} b mod s))])
Date html generated:
2020_05_20-AM-08_13_21
Last ObjectModification:
2020_01_09-AM-00_05_14
Theory : general
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