### Nuprl Lemma : rleq2-iff-rnonneg2

`∀[x,y:ℕ+ ⟶ ℤ].  (rleq2(x;y) `⇐⇒` rnonneg2(reg-seq-add(y;-(x))))`

Proof

Definitions occuring in Statement :  rleq2: `rleq2(x;y)` rnonneg2: `rnonneg2(x)` rminus: `-(x)` reg-seq-add: `reg-seq-add(x;y)` nat_plus: `ℕ+` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` function: `x:A ⟶ B[x]` int: `ℤ`
Definitions unfolded in proof :  rminus: `-(x)` reg-seq-add: `reg-seq-add(x;y)` uall: `∀[x:A]. B[x]` iff: `P `⇐⇒` Q` and: `P ∧ Q` implies: `P `` Q` rleq2: `rleq2(x;y)` rnonneg2: `rnonneg2(x)` all: `∀x:A. B[x]` member: `t ∈ T` exists: `∃x:A. B[x]` subtract: `n - m` nat_plus: `ℕ+` prop: `ℙ` so_lambda: `λ2x.t[x]` int_upper: `{i...}` le: `A ≤ B` guard: `{T}` uimplies: `b supposing a` so_apply: `x[s]` rev_implies: `P `` Q`
Lemmas referenced :  int_upper_wf all_wf le_wf less_than_transitivity1 less_than_wf nat_plus_wf rleq2_wf subtract_wf rnonneg2_wf
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation independent_pairFormation lambdaFormation sqequalHypSubstitution cut hypothesis dependent_functionElimination thin hypothesisEquality productElimination dependent_pairFormation lemma_by_obid isectElimination setElimination rename lambdaEquality multiplyEquality minusEquality natural_numberEquality addEquality applyEquality dependent_set_memberEquality because_Cache independent_isectElimination functionEquality intEquality

Latex:
\mforall{}[x,y:\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{}].    (rleq2(x;y)  \mLeftarrow{}{}\mRightarrow{}  rnonneg2(reg-seq-add(y;-(x))))

Date html generated: 2016_05_18-AM-07_15_16
Last ObjectModification: 2015_12_28-AM-00_44_30

Theory : reals

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