Nuprl Lemma : rem_3_to_1
∀[a:{...0}]. ∀[n:{...-1}]. ((a rem n) = (-(-a rem -n)) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_lower: {...i},
uall: ∀[x:A]. B[x],
remainder: n rem 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,
squash: ↓T,
prop: ℙ,
int_nzero: ℤ-o,
le: A ≤ B,
and: P ∧ Q,
nequal: a ≠ b ∈ T ,
subtype_rel: A ⊆r B,
uiff: uiff(P;Q),
rev_uimplies: rev_uimplies(P;Q),
decidable: Dec(P),
or: P ∨ Q,
iff: P ⇐⇒ Q,
not: ¬A,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
false: False,
guard: {T},
subtract: n - m,
less_than': less_than'(a;b),
true: True
Lemmas referenced :
int_lower_properties,
int_lower_wf,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
equal_wf,
squash_wf,
true_wf,
rem_to_div,
not-equal-2,
minus-one-mul-top,
decidable__le,
le_wf,
false_wf,
not-le-2,
condition-implies-le,
add-associates,
add-commutes,
add-swap,
zero-add,
minus-add,
minus-zero,
le-add-cancel,
or_wf,
nequal_wf,
minus-one-mul,
minus-minus,
le-add-cancel2,
iff_weakening_equal,
add-mul-special,
zero-mul,
add-zero,
div_3_to_1,
mul-associates,
one-mul,
mul-commutes,
mul-swap
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,
sqequalRule,
isect_memberEquality,
axiomEquality,
because_Cache,
independent_isectElimination,
dependent_functionElimination,
voidElimination,
voidEquality,
multiplyEquality,
applyEquality,
lambdaEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
intEquality,
dependent_set_memberEquality,
productElimination,
addEquality,
unionElimination,
inlFormation,
independent_pairFormation,
lambdaFormation,
inrFormation,
independent_functionElimination,
addLevel,
orFunctionality,
imageMemberEquality,
baseClosed,
divideEquality
Latex:
\mforall{}[a:\{...0\}]. \mforall{}[n:\{...-1\}]. ((a rem n) = (-(-a rem -n)))
Date html generated:
2017_04_14-AM-07_18_26
Last ObjectModification:
2017_02_27-PM-02_53_06
Theory : arithmetic
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