Nuprl Lemma : l_all-map
∀[T,A:Type]. ∀as:T List. ∀f:T ⟶ A. ∀P:A ⟶ ℙ. ((∀x∈map(f;as).P[x]) ⇐⇒ (∀x∈as.P[f x]))
Proof
Definitions occuring in Statement :
l_all: (∀x∈L.P[x]),
map: map(f;as),
list: T List,
uall: ∀[x:A]. B[x],
prop: ℙ,
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
member: t ∈ T,
so_lambda: λ2x.t[x],
so_apply: x[s],
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
guard: {T}
Lemmas referenced :
l_all_iff,
map_wf,
l_member_wf,
member_map,
equal_wf,
l_all_wf,
list_wf,
member-map,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
independent_pairFormation,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
thin,
hypothesisEquality,
dependent_functionElimination,
cumulativity,
functionExtensionality,
applyEquality,
hypothesis,
sqequalRule,
lambdaEquality,
setElimination,
rename,
setEquality,
productElimination,
independent_functionElimination,
dependent_pairFormation,
productEquality,
because_Cache,
functionEquality,
universeEquality,
hyp_replacement,
equalitySymmetry,
dependent_set_memberEquality,
equalityTransitivity,
applyLambdaEquality
Latex:
\mforall{}[T,A:Type]. \mforall{}as:T List. \mforall{}f:T {}\mrightarrow{} A. \mforall{}P:A {}\mrightarrow{} \mBbbP{}. ((\mforall{}x\mmember{}map(f;as).P[x]) \mLeftarrow{}{}\mRightarrow{} (\mforall{}x\mmember{}as.P[f x]))
Date html generated:
2017_04_17-AM-07_30_33
Last ObjectModification:
2017_02_27-PM-04_08_06
Theory : list_1
Home
Index