### Nuprl Lemma : sparse-signed-rep-lemma1

`∀m:ℤ. (∃p:ℤ × {-2..3-} [let k,b = p in (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` all: `∀x:A. B[x]` sq_exists: `∃x:A [B[x]]` implies: `P `` Q` and: `P ∧ Q` spread: spread def product: `x:A × B[x]` 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]` int_nzero: `ℤ-o` true: `True` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` uimplies: `b supposing a` sq_type: `SQType(T)` guard: `{T}` false: `False` prop: `ℙ` so_lambda: `λ2x.t[x]` so_apply: `x[s]` top: `Top` pi2: `snd(t)` pi1: `fst(t)` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` and: `P ∧ Q` sq_exists: `∃x:A [B[x]]` int_seg: `{i..j-}` lelt: `i ≤ j < k` absval: `|i|` cand: `A c∧ B` subtype_rel: `A ⊆r B` nat: `ℕ` uiff: `uiff(P;Q)` ifthenelse: `if b then t else f fi ` btrue: `tt` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` bfalse: `ff` same-parity: `same-parity(n;m)` bool: `𝔹` unit: `Unit` it: `⋅` bnot: `¬bb` assert: `↑b` isOdd: `isOdd(n)` eq_int: `(i =z j)` modulus: `a mod n` remainder: `n rem m`
Lemmas referenced :  divrem_wf subtype_base_sq int_subtype_base nequal_wf set-value-type equal_wf product-value-type divrem-sq pi2_wf pi1_wf_top istype-void div_rem_sum decidable__equal_int full-omega-unsat intformand_wf intformnot_wf intformeq_wf itermVar_wf itermAdd_wf itermMultiply_wf itermConstant_wf istype-int int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_eq_lemma int_term_value_var_lemma int_term_value_add_lemma int_term_value_mul_lemma int_term_value_constant_lemma int_formula_prop_wf decidable__le intformle_wf int_formula_prop_le_lemma decidable__lt intformless_wf int_formula_prop_less_lemma istype-le istype-less_than set_subtype_base lelt_wf istype-assert isEven_wf subtract_wf itermSubtract_wf int_term_value_subtract_lemma rem_bounds_absval absval_ifthenelse decidable__assert lt_int_wf assert_wf bnot_wf not_wf less_than_wf itermMinus_wf int_term_value_minus_lemma absval_wf bool_cases bool_wf bool_subtype_base eqtt_to_assert assert_of_lt_int eqff_to_assert iff_transitivity iff_weakening_uiff assert_of_bnot int_seg_properties isEven-add bool_cases_sqequal assert-bnot subtract-elim equal-wf-base le_int_wf le_wf uiff_transitivity assert_functionality_wrt_uiff bnot_of_lt_int assert_of_le_int
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation_alt cut productEquality intEquality thin introduction extract_by_obid sqequalHypSubstitution isectElimination hypothesisEquality dependent_set_memberEquality_alt natural_numberEquality instantiate cumulativity independent_isectElimination hypothesis dependent_functionElimination equalityTransitivity equalitySymmetry independent_functionElimination voidElimination equalityIstype inhabitedIsType baseClosed sqequalBase universeIsType cutEval sqequalRule lambdaEquality_alt setElimination rename productElimination applyLambdaEquality independent_pairEquality isect_memberEquality_alt because_Cache unionElimination approximateComputation dependent_pairFormation_alt int_eqEquality independent_pairFormation dependent_set_memberFormation_alt addEquality minusEquality productIsType callbyvalueReduce sqleReflexivity applyEquality baseApply closedConclusion functionIsType hyp_replacement equalityElimination promote_hyp

Latex:
\mforall{}m:\mBbbZ{}.  (\mexists{}p:\mBbbZ{}  \mtimes{}  \{-2..3\msupminus{}\}  [let  k,b  =  p  in  (m  =  ((4  *  k)  +  b))  \mwedge{}  ((|b|  =  2)  {}\mRightarrow{}  (\muparrow{}isEven(k)))])

Date html generated: 2019_10_15-AM-11_26_17
Last ObjectModification: 2019_06_26-PM-04_34_09

Theory : general

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