Nuprl Lemma : member_list_accum_l_subset
∀[T:Type]
∀f:(T List) ⟶ T ⟶ (T List). ∀L,a:T List. ∀x:T.
((∀a:T List. ∀x:T. l_subset(T;a;f[a;x]))
⇒ ((x ∈ a) ∨ (∃z:T. ((z ∈ L) ∧ (∀l:T List. (x ∈ f[l;z])))))
⇒ (x ∈ accumulate (with value a and list item x):
f[a;x]
over list:
L
with starting value:
a)))
Proof
Definitions occuring in Statement :
l_subset: l_subset(T;as;bs),
l_member: (x ∈ l),
list_accum: list_accum,
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
or: P ∨ Q,
and: P ∧ Q,
function: x:A ⟶ B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
member: t ∈ T,
so_lambda: λ2x.t[x],
prop: ℙ,
implies: P ⇒ Q,
so_apply: x[s1;s2],
or: P ∨ Q,
exists: ∃x:A. B[x],
and: P ∧ Q,
so_lambda: λ2x y.t[x; y],
so_apply: x[s],
uimplies: b supposing a,
not: ¬A,
false: False,
guard: {T},
l_subset: l_subset(T;as;bs),
iff: P ⇐⇒ Q,
cand: A c∧ B
Lemmas referenced :
list_induction,
list_wf,
l_subset_wf,
l_member_wf,
list_accum_wf,
list_accum_nil_lemma,
null_nil_lemma,
btrue_wf,
member-implies-null-eq-bfalse,
nil_wf,
btrue_neq_bfalse,
list_accum_cons_lemma,
cons_wf,
istype-universe,
cons_member
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation_alt,
lambdaFormation_alt,
cut,
thin,
introduction,
extract_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality_alt,
functionEquality,
hypothesis,
applyEquality,
unionEquality,
productEquality,
inhabitedIsType,
universeIsType,
independent_functionElimination,
dependent_functionElimination,
Error :memTop,
unionElimination,
productElimination,
because_Cache,
independent_isectElimination,
equalityTransitivity,
equalitySymmetry,
voidElimination,
unionIsType,
productIsType,
functionIsType,
rename,
inlFormation_alt,
instantiate,
universeEquality,
hyp_replacement,
applyLambdaEquality,
inrFormation_alt,
dependent_pairFormation_alt,
independent_pairFormation
Latex:
\mforall{}[T:Type]
\mforall{}f:(T List) {}\mrightarrow{} T {}\mrightarrow{} (T List). \mforall{}L,a:T List. \mforall{}x:T.
((\mforall{}a:T List. \mforall{}x:T. l\_subset(T;a;f[a;x]))
{}\mRightarrow{} ((x \mmember{} a) \mvee{} (\mexists{}z:T. ((z \mmember{} L) \mwedge{} (\mforall{}l:T List. (x \mmember{} f[l;z])))))
{}\mRightarrow{} (x \mmember{} accumulate (with value a and list item x):
f[a;x]
over list:
L
with starting value:
a)))
Date html generated:
2020_05_20-AM-08_06_38
Last ObjectModification:
2020_01_17-AM-11_19_52
Theory : general
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