Nuprl Lemma : div_2_to_1
∀[a:{...0}]. ∀[b:ℕ+]. ((a ÷ b) = (-((-a) ÷ b)) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_lower: {...i},
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
divide: n ÷ m,
minus: -n,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
nat_plus: ℕ+,
int_lower: {...i},
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
int_nzero: ℤ-o,
and: P ∧ Q,
le: A ≤ B,
cand: A c∧ B,
less_than: a < b,
squash: ↓T,
nequal: a ≠ b ∈ T ,
not: ¬A,
implies: P ⇒ Q,
false: False,
guard: {T},
subtype_rel: A ⊆r B,
prop: ℙ,
uiff: uiff(P;Q),
nat: ℕ,
exists: ∃x:A. B[x],
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
less_than': less_than'(a;b),
sq_type: SQType(T),
subtract: n - m
Lemmas referenced :
nat_plus_properties,
int_lower_properties,
add_functionality_wrt_le,
le_reflexive,
nat_plus_wf,
int_lower_wf,
istype-void,
minus-one-mul,
zero-add,
add-mul-special,
zero-mul,
div_unique3,
less_than_transitivity1,
minus-one-mul-top,
le_weakening,
less_than_irreflexivity,
int_subtype_base,
nequal_wf,
rem-zero,
div_rem_sum,
rem_bounds_1,
istype-le,
istype-less_than,
absval_wf,
less_than_wf,
absval-minus,
iff_weakening_equal,
rem_bounds_absval,
mul-distributes,
one-mul,
mul-commutes,
mul-associates,
equal_wf,
mul_preserves_eq,
le_wf,
false_wf,
minus-zero,
le_antisymmetry,
subtype_base_sq,
rem-sign,
le-add-cancel,
add-zero,
mul-distributes-right,
add-swap,
add-commutes,
add-associates,
minus-add,
condition-implies-le,
less-iff-le,
add_functionality_wrt_lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
setElimination,
rename,
natural_numberEquality,
minusEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
Error :universeIsType,
sqequalRule,
Error :isect_memberEquality_alt,
axiomEquality,
Error :isectIsTypeImplies,
Error :inhabitedIsType,
multiplyEquality,
voidElimination,
Error :dependent_set_memberEquality_alt,
independent_pairFormation,
productElimination,
imageElimination,
Error :lambdaFormation_alt,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
Error :equalityIstype,
applyEquality,
baseClosed,
sqequalBase,
intEquality,
divideEquality,
Error :dependent_pairFormation_alt,
remainderEquality,
Error :productIsType,
Error :lambdaEquality_alt,
baseApply,
closedConclusion,
Error :functionIsType,
imageMemberEquality,
lambdaFormation,
voidEquality,
isect_memberEquality,
lambdaEquality,
cumulativity,
instantiate,
addEquality,
dependent_set_memberEquality
Latex:
\mforall{}[a:\{...0\}]. \mforall{}[b:\mBbbN{}\msupplus{}]. ((a \mdiv{} b) = (-((-a) \mdiv{} b)))
Date html generated:
2019_06_20-AM-11_25_00
Last ObjectModification:
2019_01_04-PM-06_13_14
Theory : arithmetic
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