Nuprl Lemma : less-iff-positive
∀[x,y:ℤ]. uiff(x < y;0 < y - x)
Proof
Definitions occuring in Statement :
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
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,
prop: ℙ,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
subtract: n - m,
top: Top
Lemmas referenced :
less_than_wf,
subtract_wf,
member-less_than,
add-monotonic,
add-inverse,
zero-add,
add-associates,
minus-one-mul,
add-swap,
add-commutes,
add-mul-special,
zero-mul
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
natural_numberEquality,
because_Cache,
intEquality,
sqequalRule,
productElimination,
independent_pairEquality,
isect_memberEquality,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
minusEquality,
inlFormation,
dependent_functionElimination,
independent_functionElimination,
voidElimination,
voidEquality,
multiplyEquality
Latex:
\mforall{}[x,y:\mBbbZ{}]. uiff(x < y;0 < y - x)
Date html generated:
2019_06_20-AM-11_22_32
Last ObjectModification:
2018_08_17-PM-00_02_42
Theory : arithmetic
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