Nuprl Lemma : rleq2-iff-rnonneg2
∀[x,y:ℕ+ ⟶ ℤ]. (rleq2(x;y) ⇐⇒ rnonneg2(reg-seq-add(y;-(x))))
Proof
Definitions occuring in Statement :
rleq2: rleq2(x;y),
rnonneg2: rnonneg2(x),
rminus: -(x),
reg-seq-add: reg-seq-add(x;y),
nat_plus: ℕ+,
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
function: x:A ⟶ B[x],
int: ℤ
Definitions unfolded in proof :
rminus: -(x),
reg-seq-add: reg-seq-add(x;y),
uall: ∀[x:A]. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
rleq2: rleq2(x;y),
rnonneg2: rnonneg2(x),
all: ∀x:A. B[x],
member: t ∈ T,
exists: ∃x:A. B[x],
subtract: n - m,
nat_plus: ℕ+,
prop: ℙ,
so_lambda: λ2x.t[x],
int_upper: {i...},
le: A ≤ B,
guard: {T},
uimplies: b supposing a,
so_apply: x[s],
rev_implies: P ⇐ Q
Lemmas referenced :
int_upper_wf,
all_wf,
le_wf,
less_than_transitivity1,
less_than_wf,
nat_plus_wf,
rleq2_wf,
subtract_wf,
rnonneg2_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
independent_pairFormation,
lambdaFormation,
sqequalHypSubstitution,
cut,
hypothesis,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
dependent_pairFormation,
lemma_by_obid,
isectElimination,
setElimination,
rename,
lambdaEquality,
multiplyEquality,
minusEquality,
natural_numberEquality,
addEquality,
applyEquality,
dependent_set_memberEquality,
because_Cache,
independent_isectElimination,
functionEquality,
intEquality
Latex:
\mforall{}[x,y:\mBbbN{}\msupplus{} {}\mrightarrow{} \mBbbZ{}]. (rleq2(x;y) \mLeftarrow{}{}\mRightarrow{} rnonneg2(reg-seq-add(y;-(x))))
Date html generated:
2016_05_18-AM-07_15_16
Last ObjectModification:
2015_12_28-AM-00_44_30
Theory : reals
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