### Nuprl Lemma : div_reduce_inequality

`∀[a:ℤ]`
`  ((∀n:ℕ+. ∀x:ℤ.  uiff(0 ≤ (a + (n * x));0 ≤ ((a ÷↓ n) + x)))`
`  ∧ (∀n:{...-1}. ∀x:ℤ.  uiff(0 ≤ (a + (n * x));0 ≤ ((a ÷↓ (-n)) + ((-1) * x)))))`

Proof

Definitions occuring in Statement :  div_floor: `a ÷↓ n` int_lower: `{...i}` nat_plus: `ℕ+` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` all: `∀x:A. B[x]` and: `P ∧ Q` multiply: `n * m` add: `n + m` minus: `-n` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` and: `P ∧ Q` all: `∀x:A. B[x]` uiff: `uiff(P;Q)` uimplies: `b supposing a` le: `A ≤ B` cand: `A c∧ B` subtype_rel: `A ⊆r B` nat_plus: `ℕ+` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` implies: `P `` Q` nequal: `a ≠ b ∈ T ` not: `¬A` false: `False` guard: `{T}` decidable: `Dec(P)` or: `P ∨ Q` top: `Top` rev_uimplies: `rev_uimplies(P;Q)` ge: `i ≥ j ` int_lower: `{...i}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` prop: `ℙ` squash: `↓T`
Lemmas referenced :  le_witness_for_triv istype-int nat_plus_wf int_lower_wf div_floor_bounds subtype_rel_sets less_than_wf nequal_wf istype-less_than less_than_transitivity1 le_weakening less_than_irreflexivity int_subtype_base istype-le div_floor_wf decidable__le not-le-2 mul_preserves_le nat_plus_subtype_nat istype-void add_functionality_wrt_lt le_reflexive add-associates multiply-is-int-iff set_subtype_base add-is-int-iff mul-distributes mul-commutes one-mul zero-mul add-commutes add-swap mul-distributes-right le_functionality add_functionality_wrt_le int_lower_properties subtract_wf decidable__lt minus-one-mul-top istype-false not-lt-2 minus-le condition-implies-le minus-zero add-zero zero-add le-add-cancel-alt minus-one-mul add-mul-special not-equal-2 minus-add minus-minus le-add-cancel2 uiff_wf squash_wf true_wf le_wf mul-associates mul-swap
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut sqequalRule sqequalHypSubstitution productElimination thin independent_pairEquality Error :lambdaEquality_alt,  dependent_functionElimination hypothesisEquality Error :isect_memberEquality_alt,  isectElimination extract_by_obid equalityTransitivity hypothesis equalitySymmetry independent_isectElimination Error :isectIsTypeImplies,  Error :inhabitedIsType,  Error :functionIsTypeImplies,  Error :lambdaFormation_alt,  Error :universeIsType,  independent_pairFormation minusEquality natural_numberEquality applyEquality intEquality because_Cache closedConclusion setElimination rename Error :setIsType,  independent_functionElimination voidElimination Error :equalityIsType4,  baseClosed addEquality multiplyEquality unionElimination baseApply Error :dependent_set_memberEquality_alt,  Error :inlFormation_alt,  Error :inrFormation_alt,  hyp_replacement imageElimination universeEquality imageMemberEquality

Latex:
\mforall{}[a:\mBbbZ{}]
((\mforall{}n:\mBbbN{}\msupplus{}.  \mforall{}x:\mBbbZ{}.    uiff(0  \mleq{}  (a  +  (n  *  x));0  \mleq{}  ((a  \mdiv{}\mdownarrow{}  n)  +  x)))
\mwedge{}  (\mforall{}n:\{...-1\}.  \mforall{}x:\mBbbZ{}.    uiff(0  \mleq{}  (a  +  (n  *  x));0  \mleq{}  ((a  \mdiv{}\mdownarrow{}  (-n))  +  ((-1)  *  x)))))

Date html generated: 2019_06_20-AM-11_25_54
Last ObjectModification: 2018_10_27-AM-11_46_24

Theory : arithmetic

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