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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