Nuprl Lemma : div_bounds_1
∀[a:ℕ]. ∀[n:ℕ+]. (0 ≤ (a ÷ n))
Proof
Definitions occuring in Statement :
nat_plus: ℕ+,
nat: ℕ,
uall: ∀[x:A]. B[x],
le: A ≤ B,
divide: n ÷ m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
nat: ℕ,
le: A ≤ B,
and: P ∧ Q,
not: ¬A,
implies: P ⇒ Q,
false: False,
nequal: a ≠ b ∈ T ,
guard: {T},
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
int_nzero: ℤ-o,
ge: i ≥ j ,
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
uiff: uiff(P;Q),
subtract: n - m,
subtype_rel: A ⊆r B,
top: Top,
less_than': less_than'(a;b),
true: True,
cand: A c∧ B,
sq_type: SQType(T)
Lemmas referenced :
zero-mul,
add-mul-special,
le_transitivity,
subtype_base_sq,
le_reflexive,
multiply-is-int-iff,
mul-commutes,
le-add-cancel,
add_functionality_wrt_le,
minus-one-mul-top,
minus-one-mul,
int_subtype_base,
add-is-int-iff,
zero-add,
add-associates,
add-commutes,
add-swap,
minus-minus,
minus-add,
add-zero,
minus-zero,
condition-implies-le,
not-le-2,
false_wf,
le_weakening2,
mul_preserves_le,
decidable__le,
less_than_wf,
le_wf,
rem_bounds_1,
nequal_wf,
div_rem_sum,
nat_wf,
nat_plus_wf,
equal_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
less_than'_wf,
nat_properties,
nat_plus_properties
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
sqequalRule,
productElimination,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
because_Cache,
divideEquality,
lambdaFormation,
natural_numberEquality,
independent_isectElimination,
independent_functionElimination,
voidElimination,
intEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
isect_memberEquality,
dependent_set_memberEquality,
unionElimination,
minusEquality,
independent_pairFormation,
addEquality,
applyEquality,
voidEquality,
baseApply,
closedConclusion,
baseClosed,
multiplyEquality,
instantiate,
cumulativity
Latex:
\mforall{}[a:\mBbbN{}]. \mforall{}[n:\mBbbN{}\msupplus{}]. (0 \mleq{} (a \mdiv{} n))
Date html generated:
2016_05_13-PM-03_35_04
Last ObjectModification:
2016_01_14-PM-06_40_02
Theory : arithmetic
Home
Index