Nuprl Lemma : div_bounds_3
∀[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
, 
int_lower: {...i}
, 
le: A ≤ B
, 
and: P ∧ Q
, 
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
, 
iff: P 
⇐⇒ Q
, 
not: ¬A
, 
rev_implies: P 
⇐ Q
, 
implies: P 
⇒ Q
, 
false: False
, 
subtract: n - m
, 
top: Top
, 
less_than': less_than'(a;b)
, 
true: True
, 
int_nzero: ℤ-o
, 
prop: ℙ
, 
nat: ℕ
, 
subtype_rel: A ⊆r B
, 
cand: A c∧ B
, 
sq_type: SQType(T)
, 
guard: {T}
, 
gt: i > j
Lemmas referenced : 
int_lower_properties, 
not-equal-2, 
decidable__le, 
istype-le, 
istype-false, 
not-le-2, 
istype-void, 
condition-implies-le, 
add-associates, 
add-commutes, 
add-swap, 
zero-add, 
minus-add, 
minus-zero, 
add_functionality_wrt_le, 
le-add-cancel, 
le_witness_for_triv, 
int_lower_wf, 
div_rem_sum, 
nequal_wf, 
rem_bounds_3, 
mul_preserves_le, 
add-zero, 
minus-minus, 
minus-one-mul, 
minus-one-mul-top, 
subtract_wf, 
le_reflexive, 
add-is-int-iff, 
int_subtype_base, 
mul-distributes-right, 
add-mul-special, 
mul_preserves_eq, 
mul-distributes, 
mul-commutes, 
mul-associates, 
one-mul, 
subtype_base_sq, 
zero-mul, 
add_functionality_wrt_lt, 
multiply-is-int-iff, 
less_than_transitivity1, 
less_than_irreflexivity
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, 
remainderEquality, 
because_Cache, 
productElimination, 
dependent_functionElimination, 
independent_isectElimination, 
addEquality, 
unionElimination, 
Error :inlFormation_alt, 
independent_pairFormation, 
Error :lambdaFormation_alt, 
voidElimination, 
Error :inrFormation_alt, 
sqequalRule, 
Error :isect_memberEquality_alt, 
independent_functionElimination, 
divideEquality, 
equalityTransitivity, 
equalitySymmetry, 
Error :universeIsType, 
Error :isectIsTypeImplies, 
Error :inhabitedIsType, 
Error :dependent_set_memberEquality_alt, 
intEquality, 
baseClosed, 
baseApply, 
closedConclusion, 
applyEquality, 
multiplyEquality, 
instantiate, 
cumulativity
Latex:
\mforall{}[a:\{...0\}].  \mforall{}[n:\{...-1\}].    (0  \mleq{}  (a  \mdiv{}  n))
Date html generated:
2019_06_20-AM-11_24_43
Last ObjectModification:
2019_01_02-AM-11_31_17
Theory : arithmetic
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