Nuprl Lemma : mon_for_map
∀g:IAbMonoid. ∀A,B:Type. ∀e:A ⟶ B. ∀f:B ⟶ |g|. ∀as:A List.
((For{g} y ∈ map(e;as). f[y]) = (For{g} x ∈ as. f[e x]) ∈ |g|)
Proof
Definitions occuring in Statement :
mon_for: For{g} x ∈ as. f[x],
map: map(f;as),
list: T List,
so_apply: x[s],
all: ∀x:A. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
iabmonoid: IAbMonoid,
grp_car: |g|
Definitions unfolded in proof :
all: ∀x:A. B[x],
uall: ∀[x:A]. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
iabmonoid: IAbMonoid,
imon: IMonoid,
so_apply: x[s],
implies: P ⇒ Q,
top: Top,
infix_ap: x f y,
prop: ℙ
Lemmas referenced :
list_induction,
equal_wf,
grp_car_wf,
mon_for_wf,
map_wf,
list_wf,
map_nil_lemma,
mon_for_nil_lemma,
grp_id_wf,
map_cons_lemma,
mon_for_cons_lemma,
grp_op_wf,
iabmonoid_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
setElimination,
rename,
hypothesis,
dependent_functionElimination,
applyEquality,
independent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
because_Cache,
functionEquality,
universeEquality
Latex:
\mforall{}g:IAbMonoid. \mforall{}A,B:Type. \mforall{}e:A {}\mrightarrow{} B. \mforall{}f:B {}\mrightarrow{} |g|. \mforall{}as:A List.
((For\{g\} y \mmember{} map(e;as). f[y]) = (For\{g\} x \mmember{} as. f[e x]))
Date html generated:
2016_05_16-AM-07_36_40
Last ObjectModification:
2015_12_28-PM-05_45_35
Theory : list_2
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