Nuprl Lemma : le-add-cancel
∀[c,t:ℤ]. uiff((c + t) ≤ t;c ≤ 0)
Proof
Definitions occuring in Statement :
uiff: uiff(P;Q),
uall: ∀[x:A]. B[x],
le: A ≤ B,
add: 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,
le: A ≤ B,
not: ¬A,
implies: P ⇒ Q,
false: False,
prop: ℙ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
top: Top,
sq_type: SQType(T),
guard: {T}
Lemmas referenced :
zero-add,
add-inverse,
add-zero,
zero-mul,
add-mul-special,
minus-one-mul,
add-associates,
add-is-int-iff,
int_subtype_base,
subtype_base_sq,
add_functionality_wrt_le,
le_reflexive,
less_than'_wf,
le_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
independent_pairFormation,
isect_memberFormation,
introduction,
cut,
sqequalRule,
sqequalHypSubstitution,
productElimination,
thin,
independent_pairEquality,
lambdaEquality,
dependent_functionElimination,
hypothesisEquality,
because_Cache,
axiomEquality,
equalityTransitivity,
hypothesis,
equalitySymmetry,
lemma_by_obid,
isectElimination,
addEquality,
voidElimination,
natural_numberEquality,
intEquality,
isect_memberEquality,
minusEquality,
independent_isectElimination,
instantiate,
baseApply,
closedConclusion,
baseClosed,
applyEquality,
voidEquality,
independent_functionElimination
Latex:
\mforall{}[c,t:\mBbbZ{}]. uiff((c + t) \mleq{} t;c \mleq{} 0)
Date html generated:
2016_05_13-PM-03_31_09
Last ObjectModification:
2016_01_14-PM-06_41_09
Theory : arithmetic
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