Nuprl Lemma : div_3_to_1
∀[a:{...0}]. ∀[b:{...-1}]. ((a ÷ b) = ((-a) ÷ -b) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_lower: {...i},
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,
int_lower: {...i},
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
subtype_rel: A ⊆r B,
subtract: n - m,
int_nzero: ℤ-o,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
guard: {T},
less_than': less_than'(a;b),
true: True,
nat: ℕ,
nat_plus: ℕ+,
exists: ∃x:A. B[x],
cand: A c∧ B,
squash: ↓T,
sq_type: SQType(T),
less_than: a < b
Lemmas referenced :
int_lower_properties,
add_functionality_wrt_le,
le_reflexive,
int_lower_wf,
subtract_wf,
minus-one-mul-top,
minus-one-mul,
zero-add,
add-mul-special,
zero-mul,
add-swap,
add-associates,
add-zero,
div_unique3,
not-equal-2,
decidable__le,
le_wf,
false_wf,
not-le-2,
condition-implies-le,
add-commutes,
minus-add,
minus-zero,
le-add-cancel,
or_wf,
nequal_wf,
minus-minus,
le-add-cancel2,
rem-zero,
div_rem_sum,
rem_bounds_1,
decidable__lt,
not-lt-2,
minus-le,
le-add-cancel-alt,
less_than_wf,
absval_wf,
equal-wf-base,
squash_wf,
true_wf,
absval_sym,
nat_wf,
iff_weakening_equal,
equal_wf,
absval_pos,
absval_neg,
mul_preserves_eq,
mul-associates,
one-mul,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
mul-distributes,
mul-swap,
mul-commutes,
subtype_base_sq,
int_subtype_base,
le_antisymmetry,
add_functionality_wrt_lt,
decidable__int_equal
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
minusEquality,
natural_numberEquality,
hypothesisEquality,
hypothesis,
setElimination,
rename,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
sqequalRule,
isect_memberEquality,
axiomEquality,
multiplyEquality,
voidElimination,
voidEquality,
applyEquality,
lambdaEquality,
intEquality,
dependent_set_memberEquality,
productElimination,
addEquality,
unionElimination,
inlFormation,
independent_pairFormation,
lambdaFormation,
inrFormation,
independent_functionElimination,
addLevel,
orFunctionality,
divideEquality,
dependent_pairFormation,
productEquality,
baseApply,
closedConclusion,
baseClosed,
functionEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
imageMemberEquality,
universeEquality,
hyp_replacement,
remainderEquality,
instantiate,
cumulativity
Latex:
\mforall{}[a:\{...0\}]. \mforall{}[b:\{...-1\}]. ((a \mdiv{} b) = ((-a) \mdiv{} -b))
Date html generated:
2017_04_14-AM-07_18_14
Last ObjectModification:
2017_02_27-PM-02_54_16
Theory : arithmetic
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