Nuprl Lemma : div_bounds_2
∀[a:{...-1}]. ∀[n:ℕ+]. ((a ÷ n) ≤ 0)
Proof
Definitions occuring in Statement :
int_lower: {...i},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
le: A ≤ B,
divide: n ÷ m,
minus: -n,
natural_number: $n
Definitions unfolded in proof :
true: True,
less_than': less_than'(a;b),
top: Top,
subtract: n - m,
uiff: uiff(P;Q),
rev_implies: P ⇐ Q,
iff: P ⇐⇒ Q,
or: P ∨ Q,
decidable: Dec(P),
and: P ∧ Q,
le: A ≤ B,
int_lower: {...i},
guard: {T},
false: False,
not: ¬A,
nequal: a ≠ b ∈ T ,
implies: P ⇒ Q,
all: ∀x:A. B[x],
prop: ℙ,
uimplies: b supposing a,
so_apply: x[s],
so_lambda: λ2x.t[x],
int_nzero: ℤ-o,
nat_plus: ℕ+,
subtype_rel: A ⊆r B,
member: t ∈ T,
uall: ∀[x:A]. B[x],
ge: i ≥ j ,
rev_uimplies: rev_uimplies(P;Q),
gt: i > j,
squash: ↓T,
sq_stable: SqStable(P),
sq_type: SQType(T)
Lemmas referenced :
int_lower_wf,
nat_plus_wf,
less_than'_wf,
le-add-cancel2,
add_functionality_wrt_le,
zero-add,
minus-zero,
minus-add,
add-commutes,
condition-implies-le,
not-le-2,
false_wf,
decidable__le,
le_wf,
rem_bounds_2,
int_subtype_base,
equal-wf-base,
equal_wf,
less_than_irreflexivity,
le_weakening,
less_than_transitivity1,
nequal_wf,
less_than_wf,
subtype_rel_sets,
div_rem_sum,
add-associates,
minus-one-mul-top,
add-swap,
minus-one-mul,
add-zero,
nat_plus_subtype_nat,
mul_preserves_le,
multiply-is-int-iff,
mul-commutes,
le_functionality,
zero-mul,
add-mul-special,
one-mul,
set_subtype_base,
le_reflexive,
add_functionality_wrt_lt,
less_than_transitivity2,
minus-is-int-iff,
add-is-int-iff,
less-iff-le,
sq_stable_from_decidable,
subtype_base_sq
Rules used in proof :
equalitySymmetry,
equalityTransitivity,
axiomEquality,
divideEquality,
independent_pairEquality,
voidEquality,
isect_memberEquality,
addEquality,
independent_pairFormation,
unionElimination,
productElimination,
minusEquality,
baseClosed,
voidElimination,
independent_functionElimination,
dependent_functionElimination,
lambdaFormation,
setEquality,
rename,
setElimination,
independent_isectElimination,
hypothesis,
natural_numberEquality,
lambdaEquality,
intEquality,
sqequalRule,
applyEquality,
hypothesisEquality,
because_Cache,
thin,
isectElimination,
sqequalHypSubstitution,
extract_by_obid,
cut,
introduction,
isect_memberFormation,
sqequalReflexivity,
computationStep,
sqequalTransitivity,
sqequalSubstitution,
closedConclusion,
baseApply,
remainderEquality,
multiplyEquality,
imageElimination,
imageMemberEquality,
cumulativity,
instantiate
Latex:
\mforall{}[a:\{...-1\}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a \mdiv{} n) \mleq{} 0)
Date html generated:
2017_09_29-PM-05_47_11
Last ObjectModification:
2017_07_31-PM-04_18_52
Theory : arithmetic
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