### Nuprl Lemma : divide-le

`∀[a:ℕ+]. ∀[b,x:ℤ].  uiff(b ≤ (a * x);adjust_div(b;a) ≤ x)`

Proof

Definitions occuring in Statement :  adjust_div: `adjust_div(b;a)` nat_plus: `ℕ+` uiff: `uiff(P;Q)` uall: `∀[x:A]. B[x]` le: `A ≤ B` multiply: `n * m` int: `ℤ`
Definitions unfolded in proof :  adjust_div: `adjust_div(b;a)` member: `t ∈ T` uall: `∀[x:A]. B[x]` nat_plus: `ℕ+` nequal: `a ≠ b ∈ T ` not: `¬A` implies: `P `` Q` false: `False` guard: `{T}` uimplies: `b supposing a` all: `∀x:A. B[x]` prop: `ℙ` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` uiff: `uiff(P;Q)` and: `P ∧ Q` less_than: `a < b` less_than': `less_than'(a;b)` top: `Top` true: `True` squash: `↓T` le: `A ≤ B` bfalse: `ff` exists: `∃x:A. B[x]` or: `P ∨ Q` sq_type: `SQType(T)` bnot: `¬bb` ifthenelse: `if b then t else f fi ` assert: `↑b` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` subtype_rel: `A ⊆r B` int_nzero: `ℤ-o` so_lambda: `λ2x.t[x]` so_apply: `x[s]` decidable: `Dec(P)` subtract: `n - m` nat: `ℕ` int_lower: `{...i}` ge: `i ≥ j ` gt: `i > j`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity cut introduction extract_by_obid sqequalHypSubstitution isectElimination thin natural_numberEquality remainderEquality because_Cache setElimination rename hypothesis lambdaFormation hypothesisEquality independent_isectElimination dependent_functionElimination independent_functionElimination voidElimination intEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry productElimination sqequalRule lessCases isect_memberFormation sqequalAxiom isect_memberEquality independent_pairFormation voidEquality imageMemberEquality baseClosed imageElimination independent_pairEquality lambdaEquality axiomEquality dependent_pairFormation promote_hyp instantiate cumulativity impliesFunctionality applyEquality setEquality baseApply closedConclusion multiplyEquality addEquality divideEquality minusEquality sqequalIntensionalEquality dependent_set_memberEquality

Latex:
\mforall{}[a:\mBbbN{}\msupplus{}].  \mforall{}[b,x:\mBbbZ{}].    uiff(b  \mleq{}  (a  *  x);adjust\_div(b;a)  \mleq{}  x)

Date html generated: 2017_04_14-AM-07_19_44
Last ObjectModification: 2017_02_27-PM-02_54_03

Theory : arithmetic

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