`∀n:ℤ. (((n + 1) mod 2) = if (n mod 2 =z 0) then 1 else 0 fi  ∈ ℤ)`

Proof

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

Latex:
\mforall{}n:\mBbbZ{}.  (((n  +  1)  mod  2)  =  if  (n  mod  2  =\msubz{}  0)  then  1  else  0  fi  )

Date html generated: 2018_07_25-PM-01_27_40
Last ObjectModification: 2018_06_07-AM-10_00_47

Theory : arithmetic

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