Nuprl Lemma : div_bounds_4
∀[a:{...0}]. ∀[n:{...-1}]. (0 ≤ (a ÷ n))
Proof
Definitions occuring in Statement :
int_lower: {...i},
uall: ∀[x:A]. B[x],
le: A ≤ B,
divide: n ÷ m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
int_lower: {...i},
nequal: a ≠ b ∈ T ,
all: ∀x:A. B[x],
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
uimplies: b supposing a,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
div_bounds_3,
less_than'_wf,
not-equal-2,
decidable__le,
le_wf,
false_wf,
not-le-2,
condition-implies-le,
add-associates,
add-commutes,
add-swap,
zero-add,
minus-add,
minus-zero,
add_functionality_wrt_le,
le-add-cancel,
or_wf,
int_lower_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
divideEquality,
setElimination,
rename,
natural_numberEquality,
independent_isectElimination,
addEquality,
unionElimination,
inlFormation,
independent_pairFormation,
lambdaFormation,
voidElimination,
inrFormation,
applyEquality,
isect_memberEquality,
voidEquality,
intEquality,
minusEquality,
independent_functionElimination,
addLevel,
orFunctionality,
axiomEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a:\{...0\}]. \mforall{}[n:\{...-1\}]. (0 \mleq{} (a \mdiv{} n))
Date html generated:
2016_05_13-PM-03_35_18
Last ObjectModification:
2015_12_26-AM-09_43_19
Theory : arithmetic
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