Nuprl Lemma : member-product-map
∀[A,B,C:Type].
∀F:A ⟶ B ⟶ C. ∀as:A List. ∀bs:B List. ∀c:C.
((c ∈ product-map(F;as;bs)) ⇐⇒ ∃a:A. ((a ∈ as) ∧ (∃b:B. ((b ∈ bs) ∧ (c = (F a b) ∈ C)))))
Proof
Definitions occuring in Statement :
product-map: product-map(F;as;bs),
l_member: (x ∈ l),
list: T List,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
product-map: product-map(F;as;bs),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
and: P ∧ Q,
so_apply: x[s],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
top: Top,
concat: concat(ll),
iff: P ⇐⇒ Q,
false: False,
rev_implies: P ⇐ Q,
or: P ∨ Q,
cand: A c∧ B,
guard: {T}
Lemmas referenced :
list_induction,
all_wf,
list_wf,
iff_wf,
l_member_wf,
concat_wf,
map_wf,
exists_wf,
equal_wf,
map_nil_lemma,
reduce_nil_lemma,
false_wf,
nil_member,
nil_wf,
map_cons_lemma,
or_wf,
member-map,
member_append,
append_wf,
concat-cons,
cons_member,
cons_wf,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalRule,
sqequalReflexivity,
sqequalTransitivity,
computationStep,
isect_memberFormation,
lambdaFormation,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
applyEquality,
functionExtensionality,
productEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
productElimination,
addLevel,
allFunctionality,
impliesFunctionality,
existsFunctionality,
andLevelFunctionality,
existsLevelFunctionality,
rename,
orFunctionality,
functionEquality,
universeEquality,
levelHypothesis,
promote_hyp,
unionElimination,
dependent_pairFormation,
inlFormation,
inrFormation,
equalitySymmetry,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination,
equalityTransitivity
Latex:
\mforall{}[A,B,C:Type].
\mforall{}F:A {}\mrightarrow{} B {}\mrightarrow{} C. \mforall{}as:A List. \mforall{}bs:B List. \mforall{}c:C.
((c \mmember{} product-map(F;as;bs)) \mLeftarrow{}{}\mRightarrow{} \mexists{}a:A. ((a \mmember{} as) \mwedge{} (\mexists{}b:B. ((b \mmember{} bs) \mwedge{} (c = (F a b))))))
Date html generated:
2017_04_14-AM-09_27_16
Last ObjectModification:
2017_02_27-PM-04_01_01
Theory : list_1
Home
Index