Nuprl Lemma : bag-accum_wf
∀[T,S:Type]. ∀[init:S]. ∀[f:S ⟶ T ⟶ S]. ∀[bs:bag(T)].
bag-accum(v,x.f[v;x];init;bs) ∈ S supposing ∀v:S. ∀x,y:T. (f[f[v;y];x] = f[f[v;x];y] ∈ S)
Proof
Definitions occuring in Statement :
bag-accum: bag-accum(v,x.f[v; x];init;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
member: t ∈ T,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
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-accum: bag-accum(v,x.f[v; x];init;bs),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
squash: ↓T,
true: True,
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
append: as @ bs,
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
subtype_rel: A ⊆r B,
guard: {T}
Lemmas referenced :
list_wf,
permutation_wf,
equal_wf,
equal-wf-base,
all_wf,
bag_wf,
permutation-invariant,
list_accum_wf,
squash_wf,
true_wf,
cons_wf,
list_induction,
append_wf,
nil_wf,
list_ind_nil_lemma,
list_accum_cons_lemma,
list_accum_nil_lemma,
list_ind_cons_lemma,
iff_weakening_equal
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,
lambdaEquality,
applyEquality,
functionExtensionality,
isect_memberEquality,
functionEquality,
universeEquality,
addLevel,
hyp_replacement,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
levelHypothesis,
voidElimination,
voidEquality,
independent_isectElimination
Latex:
\mforall{}[T,S:Type]. \mforall{}[init:S]. \mforall{}[f:S {}\mrightarrow{} T {}\mrightarrow{} S]. \mforall{}[bs:bag(T)].
bag-accum(v,x.f[v;x];init;bs) \mmember{} S supposing \mforall{}v:S. \mforall{}x,y:T. (f[f[v;y];x] = f[f[v;x];y])
Date html generated:
2017_10_01-AM-08_48_12
Last ObjectModification:
2017_07_26-PM-04_32_25
Theory : bags
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