Nuprl Lemma : minus_mono_wrt_lt
∀[i,j:ℤ]. uiff(i > j;-i < -j)
Proof
Definitions occuring in Statement :
less_than: a < b,
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
gt: i > j,
minus: -n,
int: ℤ
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
prop: ℙ,
gt: i > j,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
subtract: n - m,
nat_plus: ℕ+,
less_than: a < b,
squash: ↓T,
less_than': less_than'(a;b),
true: True,
implies: P ⇒ Q,
le: A ≤ B,
not: ¬A,
false: False,
decidable: Dec(P),
or: P ∨ Q
Lemmas referenced :
minus-one-mul,
gt_wf,
member-less_than,
less_than_wf,
less-iff-le,
add_functionality_wrt_le,
subtract_wf,
le_reflexive,
minus-one-mul-top,
add-associates,
one-mul,
add-swap,
add-commutes,
add-mul-special,
two-mul,
mul-distributes-right,
zero-mul,
zero-add,
not-lt-2,
mul-associates,
add-zero,
omega-shadow,
mul-distributes,
le-add-cancel,
decidable__lt
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
independent_pairFormation,
sqequalRule,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
hypothesis,
independent_isectElimination,
minusEquality,
productElimination,
independent_pairEquality,
isect_memberEquality,
because_Cache,
equalityTransitivity,
equalitySymmetry,
intEquality,
dependent_functionElimination,
addEquality,
natural_numberEquality,
multiplyEquality,
applyEquality,
lambdaEquality,
voidElimination,
voidEquality,
dependent_set_memberEquality,
imageMemberEquality,
baseClosed,
independent_functionElimination,
unionElimination
Latex:
\mforall{}[i,j:\mBbbZ{}]. uiff(i > j;-i < -j)
Date html generated:
2018_05_21-PM-00_02_08
Last ObjectModification:
2018_05_19-AM-07_13_14
Theory : arithmetic
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