Nuprl Lemma : bag-splits-permutation1
∀T:Type. ∀L:T List. ∀a,b:T. permutation(bag(T) × bag(T);bag-splits([a; [b / L]]);bag-splits([b; [a / L]]))
Proof
Definitions occuring in Statement :
bag-splits: bag-splits(b),
bag: bag(T),
permutation: permutation(T;L1;L2),
cons: [a / b],
list: T List,
all: ∀x:A. B[x],
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
bag-splits: bag-splits(b),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
member: t ∈ T,
top: Top,
so_apply: x[s1;s2;s3],
uall: ∀[x:A]. B[x],
implies: P ⇒ Q,
bag-append: as + bs,
bag-map: bag-map(f;bs),
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
compose: f o g,
pi1: fst(t),
pi2: snd(t),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_ind_cons_lemma,
bag-splits_wf_list,
list_wf,
bag_wf,
equal_wf,
append_assoc_sq,
map_append_sq,
bag-append_wf,
single-bag_wf,
pi1_wf,
pi2_wf,
append_functionality_wrt_permutation,
map_wf,
append_wf,
permutation_weakening,
bag-append-assoc-comm,
iff_weakening_equal,
map-map,
permutation_wf,
compose_wf,
squash_wf,
true_wf,
permutation-rotate
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
cumulativity,
hypothesisEquality,
isectElimination,
productEquality,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
universeEquality,
comment,
lambdaEquality,
independent_pairEquality,
productElimination,
functionEquality,
because_Cache,
functionExtensionality,
applyEquality,
independent_isectElimination,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed,
hyp_replacement,
applyLambdaEquality
Latex:
\mforall{}T:Type. \mforall{}L:T List. \mforall{}a,b:T.
permutation(bag(T) \mtimes{} bag(T);bag-splits([a; [b / L]]);bag-splits([b; [a / L]]))
Date html generated:
2017_10_01-AM-08_59_54
Last ObjectModification:
2017_07_26-PM-04_41_54
Theory : bags
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