Nuprl Lemma : bag-map-trivial
∀[A:Type]. ∀[as:bag(A)]. ∀[f:A ⟶ A]. bag-map(f;as) = as ∈ bag(A) supposing ∀x:A. ((f x) = x ∈ A)
Proof
Definitions occuring in Statement :
bag-map: bag-map(f;bs),
bag: bag(T),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bag: bag(T),
quotient: x,y:A//B[x; y],
and: P ∧ Q,
member: t ∈ T,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
implies: P ⇒ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s1;s2],
uimplies: b supposing a,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s],
bag-map: bag-map(f;bs),
squash: ↓T,
true: True,
subtype_rel: A ⊆r B,
guard: {T},
iff: P ⇐⇒ Q,
rev_implies: P ⇐ Q
Lemmas referenced :
list_wf,
quotient-member-eq,
permutation_wf,
permutation-equiv,
equal_wf,
bag_wf,
bag-map_wf,
equal-wf-base,
all_wf,
squash_wf,
true_wf,
trivial_map,
iff_weakening_equal,
l_member_wf,
list-subtype-bag
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
pointwiseFunctionalityForEquality,
because_Cache,
sqequalRule,
pertypeElimination,
cut,
productElimination,
thin,
equalityTransitivity,
hypothesis,
equalitySymmetry,
introduction,
extract_by_obid,
isectElimination,
cumulativity,
hypothesisEquality,
lambdaFormation,
rename,
lambdaEquality,
independent_isectElimination,
dependent_functionElimination,
independent_functionElimination,
hyp_replacement,
applyLambdaEquality,
functionExtensionality,
applyEquality,
productEquality,
functionEquality,
universeEquality,
isect_memberFormation,
isect_memberEquality,
axiomEquality,
imageElimination,
natural_numberEquality,
imageMemberEquality,
baseClosed
Latex:
\mforall{}[A:Type]. \mforall{}[as:bag(A)]. \mforall{}[f:A {}\mrightarrow{} A]. bag-map(f;as) = as supposing \mforall{}x:A. ((f x) = x)
Date html generated:
2017_10_01-AM-08_46_04
Last ObjectModification:
2017_07_26-PM-04_31_06
Theory : bags
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