Nuprl Lemma : std-simplex-properties
∀[n:ℕ]. ∀[v:Δ(n)]. ((∀i:ℕn + 1. (r0 ≤ (v i))) ∧ (Σ{v i | 0≤i≤n} = r1))
Proof
Definitions occuring in Statement :
std-simplex: Δ(n),
rsum: Σ{x[k] | n≤k≤m},
rleq: x ≤ y,
req: x = y,
int-to-real: r(n),
int_seg: {i..j-},
nat: ℕ,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
add: n + m,
natural_number: $n
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
and: P ∧ Q,
cand: A c∧ B,
all: ∀x:A. B[x],
std-simplex: Δ(n),
int_seg: {i..j-},
lelt: i ≤ j < k,
le: A ≤ B,
real-vec: ℝ^n,
sq_stable: SqStable(P),
implies: P ⇒ Q,
squash: ↓T,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
rleq: x ≤ y,
rnonneg: rnonneg(x),
uimplies: b supposing a,
guard: {T}
Lemmas referenced :
sq_stable__rleq,
int-to-real_wf,
int_seg_wf,
sq_stable__req,
rsum_wf,
le_witness_for_triv,
req_witness,
std-simplex_wf,
istype-nat
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
introduction,
cut,
lambdaFormation_alt,
sqequalHypSubstitution,
setElimination,
thin,
rename,
extract_by_obid,
isectElimination,
productElimination,
hypothesis,
applyEquality,
hypothesisEquality,
independent_functionElimination,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
universeIsType,
natural_numberEquality,
addEquality,
independent_pairFormation,
lambdaEquality_alt,
independent_pairEquality,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
functionIsTypeImplies,
inhabitedIsType,
because_Cache,
isect_memberEquality_alt,
isectIsTypeImplies
Latex:
\mforall{}[n:\mBbbN{}]. \mforall{}[v:\mDelta{}(n)]. ((\mforall{}i:\mBbbN{}n + 1. (r0 \mleq{} (v i))) \mwedge{} (\mSigma{}\{v i | 0\mleq{}i\mleq{}n\} = r1))
Date html generated:
2019_10_30-AM-11_30_32
Last ObjectModification:
2019_07_31-AM-11_45_49
Theory : real!vectors
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