Nuprl Lemma : map-l-union-1
∀[T:Type]. ∀[f:T ⟶ T]. ∀[eq:EqDecider(T)]. ∀[as,bs:T List].
map(f;as ⋃ bs) ~ map(f;as) ⋃ map(f;bs) supposing Inj({x:T| (x ∈ as ⋃ bs)} ;T;f)
Proof
Definitions occuring in Statement :
l-union: as ⋃ bs,
l_member: (x ∈ l),
map: map(f;as),
list: T List,
deq: EqDecider(T),
inject: Inj(A;B;f),
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
set: {x:A| B[x]} ,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
prop: ℙ,
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
all: ∀x:A. B[x]
Lemmas referenced :
inject_wf,
l_member_wf,
l-union_wf,
subtype_rel_dep_function,
set_wf,
list_wf,
deq_wf,
map-l-union
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
setEquality,
hypothesisEquality,
applyEquality,
sqequalRule,
lambdaEquality,
independent_isectElimination,
setElimination,
rename,
because_Cache,
lambdaFormation,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry,
functionEquality,
universeEquality
Latex:
\mforall{}[T:Type]. \mforall{}[f:T {}\mrightarrow{} T]. \mforall{}[eq:EqDecider(T)]. \mforall{}[as,bs:T List].
map(f;as \mcup{} bs) \msim{} map(f;as) \mcup{} map(f;bs) supposing Inj(\{x:T| (x \mmember{} as \mcup{} bs)\} ;T;f)
Date html generated:
2016_05_14-PM-03_24_58
Last ObjectModification:
2015_12_26-PM-06_22_08
Theory : decidable!equality
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