### Nuprl Lemma : special-mod4-decomp-unique

`∀m:ℤ. ∃!k:ℤ. ∃b:{-2..3-}. ((m = ((4 * k) + b) ∈ ℤ) ∧ ((|b| = 2 ∈ ℤ) `` (↑isEven(k))))`

Proof

Definitions occuring in Statement :  isEven: `isEven(n)` absval: `|i|` int_seg: `{i..j-}` assert: `↑b` exists!: `∃!x:T. P[x]` all: `∀x:A. B[x]` exists: `∃x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` multiply: `n * m` add: `n + m` minus: `-n` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` member: `t ∈ T` uall: `∀[x:A]. B[x]` so_lambda: `λ2x.t[x]` and: `P ∧ Q` subtype_rel: `A ⊆r B` int_seg: `{i..j-}` so_apply: `x[s]` uimplies: `b supposing a` prop: `ℙ` implies: `P `` Q` exists!: `∃!x:T. P[x]` exists: `∃x:A. B[x]` cand: `A c∧ B` divides: `b | a` guard: `{T}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` not: `¬A` satisfiable_int_formula: `satisfiable_int_formula(fmla)` false: `False` top: `Top` sq_type: `SQType(T)` subtract: `n - m` absval: `|i|` uiff: `uiff(P;Q)` same-parity: `same-parity(n;m)` ifthenelse: `if b then t else f fi ` btrue: `tt` bfalse: `ff` isEven: `isEven(n)` modulus: `a mod n` eq_int: `(i =z j)` assert: `↑b` int_nzero: `ℤ-o` true: `True` nequal: `a ≠ b ∈ T ` iff: `P `⇐⇒` Q`
Lemmas referenced :  set_wf int_seg_wf equal-wf-base int_subtype_base set_subtype_base lelt_wf assert_wf isEven_wf special-mod4-decomp_wf set-value-type equal_wf product-value-type istype-int assert_witness subtract_wf int_seg_properties decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermSubtract_wf itermVar_wf itermMultiply_wf itermConstant_wf itermAdd_wf int_formula_prop_and_lemma istype-void int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_term_value_add_lemma int_formula_prop_wf intformless_wf intformle_wf int_formula_prop_less_lemma int_formula_prop_le_lemma subtype_base_sq isEven-add bool_cases bool_wf bool_subtype_base eqtt_to_assert eqff_to_assert assert_of_bnot decidable__le decidable__lt le_wf less_than_wf divides_iff_rem_zero nequal_wf int_seg_subtype_special int_seg_cases
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin productEquality intEquality minusEquality natural_numberEquality hypothesis sqequalRule lambdaEquality_alt productElimination baseApply closedConclusion baseClosed hypothesisEquality applyEquality inhabitedIsType independent_isectElimination functionEquality because_Cache productIsType universeIsType cutEval dependent_set_memberEquality_alt equalityTransitivity equalitySymmetry equalityIsType1 setElimination rename dependent_pairFormation_alt independent_pairFormation independent_functionElimination equalityIsType4 functionIsType dependent_functionElimination unionElimination approximateComputation int_eqEquality isect_memberEquality_alt voidElimination instantiate cumulativity promote_hyp callbyvalueReduce sqleReflexivity hypothesis_subsumption applyLambdaEquality

Latex:
\mforall{}m:\mBbbZ{}.  \mexists{}!k:\mBbbZ{}.  \mexists{}b:\{-2..3\msupminus{}\}.  ((m  =  ((4  *  k)  +  b))  \mwedge{}  ((|b|  =  2)  {}\mRightarrow{}  (\muparrow{}isEven(k))))

Date html generated: 2019_10_15-AM-11_26_53
Last ObjectModification: 2018_10_09-PM-00_14_32

Theory : general

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