Nuprl Lemma : bag-reduce_wf
∀[T:Type]. ∀[zero:T]. ∀[f:T ⟶ T ⟶ T].
(∀[bs:bag(T)]. (bag-reduce(x,y.f[x;y];zero;bs) ∈ T)) supposing (Assoc(T;λx,y. f[x;y]) and Comm(T;λx,y. f[x;y]))
Proof
Definitions occuring in Statement :
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
bag: bag(T),
comm: Comm(T;op),
assoc: Assoc(T;op),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
member: t ∈ T,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bag-reduce: bag-reduce(x,y.f[x; y];zero;bs),
prop: ℙ,
so_apply: x[s1;s2],
assoc: Assoc(T;op),
infix_ap: x f y,
comm: Comm(T;op),
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
true: True,
top: Top,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
guard: {T},
subtype_rel: A ⊆r B,
label: ...$L... t,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
so_apply: x[s1;s2;s3]
Lemmas referenced :
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
bag_wf,
assoc_wf,
comm_wf,
permutation-invariant,
reduce_wf,
squash_wf,
true_wf,
cons_wf,
reduce_cons_lemma,
iff_weakening_equal,
list_induction,
all_wf,
append_wf,
nil_wf,
list_ind_nil_lemma,
reduce_nil_lemma,
list_ind_cons_lemma
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
sqequalHypSubstitution,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaFormation,
rename,
dependent_functionElimination,
independent_functionElimination,
productEquality,
axiomEquality,
isect_memberEquality,
lambdaEquality,
applyEquality,
functionExtensionality,
functionEquality,
universeEquality,
addLevel,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
levelHypothesis,
voidElimination,
voidEquality,
independent_isectElimination
Latex:
\mforall{}[T:Type]. \mforall{}[zero:T]. \mforall{}[f:T {}\mrightarrow{} T {}\mrightarrow{} T].
(\mforall{}[bs:bag(T)]. (bag-reduce(x,y.f[x;y];zero;bs) \mmember{} T)) supposing
(Assoc(T;\mlambda{}x,y. f[x;y]) and
Comm(T;\mlambda{}x,y. f[x;y]))
Date html generated:
2017_10_01-AM-08_48_08
Last ObjectModification:
2017_07_26-PM-04_32_22
Theory : bags
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