### 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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