Nuprl Lemma : test-arith
∀[x,y,z:ℤ]. (((y + 1) ≤ x)
⇒ ((z + 1) ≤ y)
⇒ ((x + (-1)) ≤ z)
⇒ False)
Proof
Definitions occuring in Statement :
uall: ∀[x:A]. B[x]
,
le: A ≤ B
,
implies: P
⇒ Q
,
false: False
,
add: n + m
,
minus: -n
,
natural_number: $n
,
int: ℤ
Definitions unfolded in proof :
member: t ∈ T
,
uall: ∀[x:A]. B[x]
,
prop: ℙ
,
implies: P
⇒ Q
,
false: False
,
uimplies: b supposing a
,
subtract: n - m
,
top: Top
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
le: A ≤ B
,
not: ¬A
,
less_than': less_than'(a;b)
,
true: True
Lemmas referenced :
le_wf,
condition-implies-le,
minus-add,
minus-one-mul,
add-swap,
add-commutes,
minus-minus,
add_functionality_wrt_le,
add-associates,
le-add-cancel
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
addEquality,
hypothesisEquality,
minusEquality,
natural_numberEquality,
hypothesis,
intEquality,
because_Cache,
isect_memberFormation,
introduction,
lambdaFormation,
sqequalRule,
lambdaEquality,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
independent_isectElimination,
voidEquality,
multiplyEquality,
productElimination,
independent_functionElimination
Latex:
\mforall{}[x,y,z:\mBbbZ{}]. (((y + 1) \mleq{} x) {}\mRightarrow{} ((z + 1) \mleq{} y) {}\mRightarrow{} ((x + (-1)) \mleq{} z) {}\mRightarrow{} False)
Date html generated:
2016_05_13-PM-03_31_52
Last ObjectModification:
2015_12_26-AM-09_45_59
Theory : arithmetic
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