### Nuprl Lemma : div_2_to_1

`∀[a:{...0}]. ∀[b:ℕ+].  ((a ÷ b) = (-((-a) ÷ b)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_lower: `{...i}` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` divide: `n ÷ m` minus: `-n` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` nat_plus: `ℕ+` int_lower: `{...i}` uimplies: `b supposing a` all: `∀x:A. B[x]` top: `Top` int_nzero: `ℤ-o` and: `P ∧ Q` le: `A ≤ B` cand: `A c∧ B` less_than: `a < b` squash: `↓T` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` false: `False` guard: `{T}` subtype_rel: `A ⊆r B` prop: `ℙ` uiff: `uiff(P;Q)` nat: `ℕ` exists: `∃x:A. B[x]` true: `True` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` less_than': `less_than'(a;b)` sq_type: `SQType(T)` subtract: `n - m`
Lemmas referenced :  nat_plus_properties int_lower_properties add_functionality_wrt_le le_reflexive nat_plus_wf int_lower_wf istype-void minus-one-mul zero-add add-mul-special zero-mul div_unique3 less_than_transitivity1 minus-one-mul-top le_weakening less_than_irreflexivity int_subtype_base nequal_wf rem-zero div_rem_sum rem_bounds_1 istype-le istype-less_than absval_wf less_than_wf absval-minus iff_weakening_equal rem_bounds_absval mul-distributes one-mul mul-commutes mul-associates equal_wf mul_preserves_eq le_wf false_wf minus-zero le_antisymmetry subtype_base_sq rem-sign le-add-cancel add-zero mul-distributes-right add-swap add-commutes add-associates minus-add condition-implies-le less-iff-le add_functionality_wrt_lt
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin hypothesisEquality hypothesis setElimination rename natural_numberEquality minusEquality because_Cache independent_isectElimination dependent_functionElimination Error :universeIsType,  sqequalRule Error :isect_memberEquality_alt,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  multiplyEquality voidElimination Error :dependent_set_memberEquality_alt,  independent_pairFormation productElimination imageElimination Error :lambdaFormation_alt,  equalityTransitivity equalitySymmetry independent_functionElimination Error :equalityIstype,  applyEquality baseClosed sqequalBase intEquality divideEquality Error :dependent_pairFormation_alt,  remainderEquality Error :productIsType,  Error :lambdaEquality_alt,  baseApply closedConclusion Error :functionIsType,  imageMemberEquality lambdaFormation voidEquality isect_memberEquality lambdaEquality cumulativity instantiate addEquality dependent_set_memberEquality

Latex:
\mforall{}[a:\{...0\}].  \mforall{}[b:\mBbbN{}\msupplus{}].    ((a  \mdiv{}  b)  =  (-((-a)  \mdiv{}  b)))

Date html generated: 2019_06_20-AM-11_25_00
Last ObjectModification: 2019_01_04-PM-06_13_14

Theory : arithmetic

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