Nuprl Lemma : bag-count-map
∀[T1,T2:Type]. ∀[f:T1 ⟶ T2]. ∀[eq1:EqDecider(T1)]. ∀[eq2:EqDecider(T2)]. ∀[x:T2]. ∀[bs:bag(T1)]. ∀[g:T2 ⟶ T1].
(#x in bag-map(f;bs)) ~ (#g x in bs) supposing (∀x:T2. ((f (g x)) = x ∈ T2)) ∧ (∀x:T1. ((g (f x)) = x ∈ T1))
Proof
Definitions occuring in Statement :
bag-count: (#x in bs),
bag-map: bag-map(f;bs),
bag: bag(T),
deq: EqDecider(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
and: P ∧ Q,
nat: ℕ,
so_lambda: λ2x.t[x],
so_apply: x[s],
squash: ↓T,
all: ∀x:A. B[x],
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q,
implies: P ⇒ Q,
true: True,
sq_type: SQType(T),
prop: ℙ,
inject: Inj(A;B;f)
Lemmas referenced :
bag-count-ap-map,
subtype_base_sq,
nat_wf,
set_subtype_base,
le_wf,
int_subtype_base,
bag-count_wf,
equal_wf,
iff_weakening_equal,
all_wf,
bag_wf,
deq_wf,
squash_wf,
true_wf
Rules used in proof :
cut,
introduction,
extract_by_obid,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
hypothesis,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
productElimination,
instantiate,
cumulativity,
independent_isectElimination,
sqequalRule,
intEquality,
lambdaEquality,
natural_numberEquality,
applyEquality,
functionExtensionality,
imageElimination,
because_Cache,
dependent_functionElimination,
equalitySymmetry,
imageMemberEquality,
baseClosed,
equalityTransitivity,
independent_functionElimination,
sqequalAxiom,
productEquality,
isect_memberEquality,
functionEquality,
universeEquality,
lambdaFormation
Latex:
\mforall{}[T1,T2:Type]. \mforall{}[f:T1 {}\mrightarrow{} T2]. \mforall{}[eq1:EqDecider(T1)]. \mforall{}[eq2:EqDecider(T2)]. \mforall{}[x:T2]. \mforall{}[bs:bag(T1)].
\mforall{}[g:T2 {}\mrightarrow{} T1].
(\#x in bag-map(f;bs)) \msim{} (\#g x in bs) supposing (\mforall{}x:T2. ((f (g x)) = x)) \mwedge{} (\mforall{}x:T1. ((g (f x)) = x))
Date html generated:
2018_05_21-PM-09_46_05
Last ObjectModification:
2017_07_26-PM-06_29_56
Theory : bags_2
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