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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