Nuprl Lemma : le-iff-nonneg
∀[x,y:ℤ]. uiff(x ≤ y;0 ≤ (y - x))
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
subtract: n - m,
natural_number: $n,
int: ℤ
Definitions unfolded in proof :
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
member: t ∈ T,
not: ¬A,
implies: P ⇒ Q,
false: False,
subtract: n - m,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
top: Top,
prop: ℙ,
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
le: A ≤ B
Lemmas referenced :
minus-one-mul,
minus-add,
minus-minus,
zero-add,
minus-one-mul-top,
add-commutes,
less_than_wf,
subtract_wf,
not_wf,
less-iff-positive,
uiff_wf,
iff_weakening_uiff,
le_wf,
not-lt,
less_than'_wf
Rules used in proof :
cut,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
lambdaFormation,
thin,
sqequalHypSubstitution,
hypothesis,
independent_functionElimination,
sqequalRule,
lemma_by_obid,
isectElimination,
hypothesisEquality,
applyEquality,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
minusEquality,
addEquality,
natural_numberEquality,
dependent_functionElimination,
addLevel,
productElimination,
independent_isectElimination,
cumulativity,
independent_pairEquality,
axiomEquality,
equalityTransitivity,
equalitySymmetry,
intEquality
Latex:
\mforall{}[x,y:\mBbbZ{}]. uiff(x \mleq{} y;0 \mleq{} (y - x))
Date html generated:
2016_05_13-PM-03_29_53
Last ObjectModification:
2015_12_26-AM-09_47_34
Theory : arithmetic
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