### Nuprl Lemma : arith-example1

`∀x,y,z:ℤ.`
`  ((((x - 1) ≤ y) ∧ (y ≤ (x + 1)))`
`  `` (((x - 1) ≤ z) ∧ (z ≤ (x + 1)))`
`  `` (((y - 1) ≤ z) ∧ (z ≤ (y + 1)))`
`  `` (((x = y ∈ ℤ) ∨ (x = z ∈ ℤ)) ∨ (y = z ∈ ℤ)))`

Proof

Definitions occuring in Statement :  le: `A ≤ B` all: `∀x:A. B[x]` implies: `P `` Q` or: `P ∨ Q` and: `P ∧ Q` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` member: `t ∈ T` prop: `ℙ` uall: `∀[x:A]. B[x]` subtype_rel: `A ⊆r B` or: `P ∨ Q` le: `A ≤ B` uimplies: `b supposing a` subtract: `n - m` top: `Top` uiff: `uiff(P;Q)` not: `¬A` less_than': `less_than'(a;b)` true: `True` false: `False` iff: `P `⇐⇒` Q` decidable: `Dec(P)` rev_implies: `P `` Q` guard: `{T}`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin productEquality cut introduction extract_by_obid isectElimination hypothesisEquality natural_numberEquality hypothesis addEquality intEquality applyEquality sqequalRule because_Cache unionElimination independent_isectElimination lambdaEquality isect_memberEquality voidElimination voidEquality minusEquality independent_functionElimination dependent_functionElimination inlFormation independent_pairFormation inrFormation addLevel orFunctionality

Latex:
\mforall{}x,y,z:\mBbbZ{}.
((((x  -  1)  \mleq{}  y)  \mwedge{}  (y  \mleq{}  (x  +  1)))
{}\mRightarrow{}  (((x  -  1)  \mleq{}  z)  \mwedge{}  (z  \mleq{}  (x  +  1)))
{}\mRightarrow{}  (((y  -  1)  \mleq{}  z)  \mwedge{}  (z  \mleq{}  (y  +  1)))
{}\mRightarrow{}  (((x  =  y)  \mvee{}  (x  =  z))  \mvee{}  (y  =  z)))

Date html generated: 2017_04_14-AM-07_16_27
Last ObjectModification: 2017_02_27-PM-02_51_28

Theory : arithmetic

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