Nuprl Lemma : bag-summation_functionality_wrt_le
∀[T:Type]. ∀[b:bag(T)]. ∀[f,g:{x:T| x ↓∈ b} ⟶ ℤ].
Σ(x∈b). f[x] ≤ Σ(x∈b). g[x] supposing ∀x:T. (x ↓∈ b ⇒ (f[x] ≤ g[x]))
Proof
Definitions occuring in Statement :
bag-member: x ↓∈ bs,
bag-summation: Σ(x∈b). f[x],
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
implies: P ⇒ Q,
set: {x:A| B[x]} ,
lambda: λx.A[x],
function: x:A ⟶ B[x],
add: n + m,
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
all: ∀x:A. B[x],
uimplies: b supposing a,
implies: P ⇒ Q,
sq_stable: SqStable(P),
squash: ↓T,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
bag_wf,
le_wf,
all_wf,
set_wf,
sq_stable__bag-member,
bag-subtype,
bag-member_wf,
bag-summation_functionality_wrt_le_1
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
cut,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
hypothesis,
cumulativity,
dependent_functionElimination,
equalityTransitivity,
equalitySymmetry,
independent_isectElimination,
lambdaFormation,
setElimination,
rename,
independent_functionElimination,
introduction,
sqequalRule,
imageMemberEquality,
baseClosed,
imageElimination,
lambdaEquality,
functionEquality,
because_Cache,
applyEquality,
dependent_set_memberEquality,
intEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[b:bag(T)]. \mforall{}[f,g:\{x:T| x \mdownarrow{}\mmember{} b\} {}\mrightarrow{} \mBbbZ{}].
\mSigma{}(x\mmember{}b). f[x] \mleq{} \mSigma{}(x\mmember{}b). g[x] supposing \mforall{}x:T. (x \mdownarrow{}\mmember{} b {}\mRightarrow{} (f[x] \mleq{} g[x]))
Date html generated:
2016_05_15-PM-02_58_17
Last ObjectModification:
2016_01_16-AM-08_38_00
Theory : bags
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