Nuprl Lemma : rem_2_to_1
∀[a:{...0}]. ∀[n:ℕ+]. ((a rem n) = (-(-a rem n)) ∈ ℤ)
Proof
Definitions occuring in Statement :
int_lower: {...i},
nat_plus: ℕ+,
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},
subtype_rel: A ⊆r B,
nat_plus: ℕ+,
int_nzero: ℤ-o,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
prop: ℙ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
nequal: a ≠ b ∈ T ,
not: ¬A,
false: False,
guard: {T},
true: True,
squash: ↓T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
subtract: n - m,
top: Top
Lemmas referenced :
nat_plus_wf,
int_lower_wf,
subtype_rel_sets,
less_than_wf,
nequal_wf,
less_than_transitivity1,
le_weakening,
less_than_irreflexivity,
equal_wf,
equal-wf-base,
int_subtype_base,
squash_wf,
true_wf,
rem_to_div,
iff_weakening_equal,
subtract_wf,
minus-one-mul,
mul-associates,
minus-one-mul-top,
mul-commutes,
one-mul,
minus-add,
minus-minus,
div_2_to_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
axiomEquality,
because_Cache,
natural_numberEquality,
intEquality,
setElimination,
rename,
applyEquality,
lambdaEquality,
independent_isectElimination,
setEquality,
lambdaFormation,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
baseClosed,
minusEquality,
imageElimination,
equalityTransitivity,
equalitySymmetry,
universeEquality,
imageMemberEquality,
productElimination,
multiplyEquality,
divideEquality,
voidEquality,
addEquality
Latex:
\mforall{}[a:\{...0\}]. \mforall{}[n:\mBbbN{}\msupplus{}]. ((a rem n) = (-(-a rem n)))
Date html generated:
2017_04_14-AM-07_18_22
Last ObjectModification:
2017_02_27-PM-02_52_32
Theory : arithmetic
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