Nuprl Lemma : arith-example1
∀x,y,z:ℤ.
((((x - 1) ≤ y) ∧ (y ≤ (x + 1)))
⇒ (((x - 1) ≤ z) ∧ (z ≤ (x + 1)))
⇒ (((y - 1) ≤ z) ∧ (z ≤ (y + 1)))
⇒ (((x = y ∈ ℤ) ∨ (x = z ∈ ℤ)) ∨ (y = z ∈ ℤ)))
Proof
Definitions occuring in Statement :
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
subtract: n - m,
add: n + m,
natural_number: $n,
int: ℤ,
equal: s = t ∈ T
Definitions unfolded in proof :
all: ∀x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
member: t ∈ T,
prop: ℙ,
uall: ∀[x:A]. B[x],
subtype_rel: A ⊆r B,
or: P ∨ Q,
le: A ≤ B,
uimplies: b supposing a,
subtract: n - m,
top: Top,
uiff: uiff(P;Q),
not: ¬A,
less_than': less_than'(a;b),
true: True,
false: False,
iff: P ⇐⇒ Q,
decidable: Dec(P),
rev_implies: P ⇐ Q,
guard: {T}
Lemmas referenced :
le_wf,
subtract_wf,
equal-wf-base,
int_subtype_base,
false_wf,
or_wf,
condition-implies-le,
add-associates,
minus-one-mul,
add-swap,
minus-one-mul-top,
add_functionality_wrt_le,
add-commutes,
le-add-cancel2,
minus-add,
minus-minus,
le-add-cancel,
decidable__int_equal,
not-equal-2
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalHypSubstitution,
productElimination,
thin,
productEquality,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
natural_numberEquality,
hypothesis,
addEquality,
intEquality,
applyEquality,
sqequalRule,
because_Cache,
unionElimination,
independent_isectElimination,
lambdaEquality,
isect_memberEquality,
voidElimination,
voidEquality,
minusEquality,
independent_functionElimination,
dependent_functionElimination,
inlFormation,
independent_pairFormation,
inrFormation,
addLevel,
orFunctionality
Latex:
\mforall{}x,y,z:\mBbbZ{}.
((((x - 1) \mleq{} y) \mwedge{} (y \mleq{} (x + 1)))
{}\mRightarrow{} (((x - 1) \mleq{} z) \mwedge{} (z \mleq{} (x + 1)))
{}\mRightarrow{} (((y - 1) \mleq{} z) \mwedge{} (z \mleq{} (y + 1)))
{}\mRightarrow{} (((x = y) \mvee{} (x = z)) \mvee{} (y = z)))
Date html generated:
2017_04_14-AM-07_16_27
Last ObjectModification:
2017_02_27-PM-02_51_28
Theory : arithmetic
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