Nuprl Lemma : summand-le-lsum
∀[T:Type]. ∀[L:T List]. ∀[f:{x:T| (x ∈ L)} ⟶ ℤ].
∀x:{x:T| (x ∈ L)} . (f[x] ≤ Σ(f[x] | x ∈ L)) supposing ∀x:{x:T| (x ∈ L)} . (0 ≤ f[x])
Proof
Definitions occuring in Statement :
lsum: Σ(f[x] | x ∈ L),
l_member: (x ∈ l),
list: T List,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
le: A ≤ B,
all: ∀x:A. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
natural_number: $n,
int: ℤ,
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
all: ∀x:A. B[x],
prop: ℙ,
le: A ≤ B,
and: P ∧ Q,
so_apply: x[s],
sq_type: SQType(T),
implies: P ⇒ Q,
guard: {T},
lsum: Σ(f[x] | x ∈ L),
squash: ↓T,
true: True
Lemmas referenced :
summand-le-l_sum,
l_member_wf,
le_witness_for_triv,
istype-le,
istype-int,
list_wf,
istype-universe,
list-subtype,
subtype_base_sq,
int_subtype_base,
l_sum_wf,
squash_wf,
true_wf,
map_wf,
eta_conv
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
independent_isectElimination,
lambdaFormation_alt,
dependent_functionElimination,
setIsType,
universeIsType,
sqequalRule,
lambdaEquality_alt,
productElimination,
equalityTransitivity,
equalitySymmetry,
functionIsTypeImplies,
inhabitedIsType,
functionIsType,
natural_numberEquality,
applyEquality,
instantiate,
universeEquality,
cumulativity,
intEquality,
independent_functionElimination,
imageElimination,
setEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[T:Type]. \mforall{}[L:T List]. \mforall{}[f:\{x:T| (x \mmember{} L)\} {}\mrightarrow{} \mBbbZ{}].
\mforall{}x:\{x:T| (x \mmember{} L)\} . (f[x] \mleq{} \mSigma{}(f[x] | x \mmember{} L)) supposing \mforall{}x:\{x:T| (x \mmember{} L)\} . (0 \mleq{} f[x])
Date html generated:
2020_05_19-PM-09_48_40
Last ObjectModification:
2020_01_23-PM-00_57_17
Theory : list_1
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