Nuprl Lemma : member_list_accum_l_subset2
∀[T:Type]
∀f:(T List) ⟶ T ⟶ (T List). ∀L,a:T List. ∀x:T.
((∀a:T List. ∀x:T. l_subset(T;f[a;x];[x / a]))
⇒ (x ∈ accumulate (with value a and list item x):
f[a;x]
over list:
L
with starting value:
a))
⇒ ((x ∈ a) ∨ (x ∈ L)))
Proof
Definitions occuring in Statement :
l_subset: l_subset(T;as;bs),
l_member: (x ∈ l),
list_accum: list_accum,
cons: [a / b],
list: T List,
uall: ∀[x:A]. B[x],
so_apply: x[s1;s2],
all: ∀x:A. B[x],
implies: P ⇒ Q,
or: 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],
implies: P ⇒ Q,
prop: ℙ,
so_apply: x[s1;s2],
so_apply: x[s],
so_lambda: λ2x y.t[x; y],
or: P ∨ Q,
top: Top,
l_subset: l_subset(T;as;bs),
iff: P ⇐⇒ Q,
and: P ∧ Q,
guard: {T},
rev_implies: P ⇐ Q
Lemmas referenced :
list_induction,
all_wf,
list_wf,
l_subset_wf,
cons_wf,
l_member_wf,
list_accum_wf,
or_wf,
list_accum_nil_lemma,
nil_wf,
list_accum_cons_lemma,
cons_member,
equal_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
thin,
lemma_by_obid,
sqequalHypSubstitution,
isectElimination,
hypothesisEquality,
sqequalRule,
lambdaEquality,
cumulativity,
hypothesis,
because_Cache,
functionEquality,
applyEquality,
independent_functionElimination,
dependent_functionElimination,
isect_memberEquality,
voidElimination,
voidEquality,
inlFormation,
rename,
unionElimination,
productElimination,
inrFormation,
universeEquality
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;f[a;x];[x / a]))
{}\mRightarrow{} (x \mmember{} accumulate (with value a and list item x):
f[a;x]
over list:
L
with starting value:
a))
{}\mRightarrow{} ((x \mmember{} a) \mvee{} (x \mmember{} L)))
Date html generated:
2016_05_15-PM-03_43_52
Last ObjectModification:
2015_12_27-PM-01_19_35
Theory : general
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