Nuprl Lemma : rem-sign
∀[a:ℤ]. ∀[n:ℤ-o]. (((0 ≤ a) ⇒ (0 ≤ (a rem n))) ∧ (0 < a rem n ⇒ 0 < a) ∧ (a rem n < 0 ⇒ a < 0))
Proof
Definitions occuring in Statement :
int_nzero: ℤ-o,
less_than: a < b,
uall: ∀[x:A]. B[x],
le: A ≤ B,
implies: P ⇒ Q,
and: P ∧ Q,
remainder: n rem m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
int_nzero: ℤ-o,
decidable: Dec(P),
or: P ∨ Q,
and: P ∧ Q,
cand: A c∧ B,
implies: P ⇒ Q,
uall: ∀[x:A]. B[x],
nat: ℕ,
prop: ℙ,
nat_plus: ℕ+,
le: A ≤ B,
nequal: a ≠ b ∈ T ,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
false: False,
uiff: uiff(P;Q),
uimplies: b supposing a,
less_than': less_than'(a;b),
true: True,
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
int_lower: {...i},
rev_uimplies: rev_uimplies(P;Q),
guard: {T},
gt: i > j,
ge: i ≥ j
Lemmas referenced :
decidable__le,
rem_bounds_1,
le_wf,
decidable__lt,
false_wf,
not-lt-2,
not-equal-2,
add_functionality_wrt_le,
zero-add,
add-zero,
le-add-cancel,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
less_than_wf,
equal_wf,
rem_bounds_4,
not-le-2,
int_nzero_wf,
minus-one-mul,
minus-one-mul-top,
minus-minus,
add-swap,
add-associates,
le-add-cancel2,
or_wf,
less_than'_wf,
member-less_than,
rem_bounds_2,
not-gt-2,
less_than_transitivity1,
less_than_irreflexivity,
rem_bounds_3
Rules used in proof :
cut,
lemma_by_obid,
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
dependent_functionElimination,
thin,
natural_numberEquality,
setElimination,
rename,
hypothesisEquality,
hypothesis,
unionElimination,
lambdaFormation,
isectElimination,
dependent_set_memberEquality,
productElimination,
independent_pairFormation,
voidElimination,
independent_functionElimination,
independent_isectElimination,
addEquality,
sqequalRule,
minusEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidEquality,
because_Cache,
remainderEquality,
intEquality,
inlFormation,
inrFormation,
addLevel,
orFunctionality,
isect_memberFormation,
introduction,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a:\mBbbZ{}]. \mforall{}[n:\mBbbZ{}\msupminus{}\msupzero{}]. (((0 \mleq{} a) {}\mRightarrow{} (0 \mleq{} (a rem n))) \mwedge{} (0 < a rem n {}\mRightarrow{} 0 < a) \mwedge{} (a rem n < 0 {}\mRightarrow{} a < 0))
Date html generated:
2016_05_13-PM-03_34_47
Last ObjectModification:
2015_12_26-AM-09_43_57
Theory : arithmetic
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