Nuprl Lemma : div_4_to_1

`∀[a:ℕ]. ∀[b:{...-1}].  ((a ÷ b) = (-(a ÷ -b)) ∈ ℤ)`

Proof

Definitions occuring in Statement :  int_lower: `{...i}` nat: `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` int_lower: `{...i}` nat: `ℕ` uimplies: `b supposing a` all: `∀x:A. B[x]` top: `Top` le: `A ≤ B` and: `P ∧ Q` nequal: `a ≠ b ∈ T ` uiff: `uiff(P;Q)` rev_uimplies: `rev_uimplies(P;Q)` decidable: `Dec(P)` or: `P ∨ Q` iff: `P `⇐⇒` Q` not: `¬A` rev_implies: `P `` Q` implies: `P `` Q` false: `False` subtract: `n - m` less_than': `less_than'(a;b)` true: `True` int_nzero: `ℤ-o` prop: `ℙ` ge: `i ≥ j ` nat_plus: `ℕ+` subtype_rel: `A ⊆r B` exists: `∃x:A. B[x]` guard: `{T}` squash: `↓T` sq_type: `SQType(T)`
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity Error :isect_memberFormation_alt,  introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin minusEquality natural_numberEquality hypothesisEquality hypothesis setElimination rename because_Cache independent_isectElimination dependent_functionElimination Error :universeIsType,  sqequalRule Error :isect_memberEquality_alt,  axiomEquality Error :isectIsTypeImplies,  Error :inhabitedIsType,  voidElimination multiplyEquality productElimination addEquality unionElimination Error :inlFormation_alt,  independent_pairFormation Error :lambdaFormation_alt,  Error :inrFormation_alt,  independent_functionElimination Error :dependent_set_memberEquality_alt,  intEquality divideEquality equalityTransitivity equalitySymmetry applyEquality Error :lambdaEquality_alt,  Error :dependent_pairFormation_alt,  remainderEquality Error :productIsType,  Error :equalityIstype,  baseApply closedConclusion baseClosed sqequalBase Error :functionIsType,  hyp_replacement imageElimination imageMemberEquality instantiate cumulativity

Latex:
\mforall{}[a:\mBbbN{}].  \mforall{}[b:\{...-1\}].    ((a  \mdiv{}  b)  =  (-(a  \mdiv{}  -b)))

Date html generated: 2019_06_20-AM-11_25_04
Last ObjectModification: 2019_01_03-PM-04_06_01

Theory : arithmetic

Home Index