Nuprl Lemma : sub-bag-map-equal
∀[T,U:Type]. ∀[b1,b2:bag(T)]. ∀[f:T ⟶ U].
(b1 = b2 ∈ bag(T)) supposing (sub-bag(T;b2;b1) and sub-bag(U;bag-map(f;b1);bag-map(f;b2)))
Proof
Definitions occuring in Statement :
sub-bag: sub-bag(T;as;bs),
bag-map: bag-map(f;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
sub-bag: sub-bag(T;as;bs),
exists: ∃x:A. B[x],
member: t ∈ T,
uall: ∀[x:A]. B[x],
true: True,
squash: ↓T,
prop: ℙ,
subtype_rel: A ⊆r B,
uimplies: b supposing a,
bag-append: as + bs,
bag-map: bag-map(f;bs),
empty-bag: {},
top: Top,
all: ∀x:A. B[x],
guard: {T},
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
uiff: uiff(P;Q),
bag-null: bag-null(bs)
Lemmas referenced :
bag_wf,
bag_to_squash_list,
equal_wf,
bag-map_wf,
bag-append_wf,
list-subtype-bag,
map_append_sq,
equal-wf-T-base,
bag-append-empty,
bag-subtype-list,
sub-bag_wf,
squash_wf,
true_wf,
iff_weakening_equal,
append_assoc,
map_wf,
subtype_rel_list,
top_wf,
bag-append-cancel,
nil_wf,
append_wf,
bag-append-eq-empty,
assert-bag-null,
bag-map-null
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
cut,
introduction,
extract_by_obid,
isectElimination,
hypothesisEquality,
hypothesis,
equalityTransitivity,
equalitySymmetry,
because_Cache,
natural_numberEquality,
imageElimination,
promote_hyp,
hyp_replacement,
applyLambdaEquality,
cumulativity,
functionExtensionality,
applyEquality,
rename,
independent_isectElimination,
lambdaEquality,
sqequalRule,
isect_memberEquality,
voidElimination,
voidEquality,
baseClosed,
dependent_functionElimination,
functionEquality,
universeEquality,
isect_memberFormation,
axiomEquality,
imageMemberEquality,
independent_functionElimination,
equalityElimination,
independent_pairFormation
Latex:
\mforall{}[T,U:Type]. \mforall{}[b1,b2:bag(T)]. \mforall{}[f:T {}\mrightarrow{} U].
(b1 = b2) supposing (sub-bag(T;b2;b1) and sub-bag(U;bag-map(f;b1);bag-map(f;b2)))
Date html generated:
2017_10_01-AM-09_05_16
Last ObjectModification:
2017_07_26-PM-04_45_20
Theory : bags
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