Nuprl Lemma : member-insert-by
∀[T:Type]
∀eq,r:T ⟶ T ⟶ 𝔹.
∀x:T. ∀L:T List. ∀z:T. ((z ∈ insert-by(eq;r;x;L)) ⇐⇒ (z = x ∈ T) ∨ (z ∈ L))
supposing ∀a,b:T. (↑(eq a b) ⇐⇒ a = b ∈ T)
Proof
Definitions occuring in Statement :
insert-by: insert-by(eq;r;x;l),
l_member: (x ∈ l),
list: T List,
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
or: P ∨ Q,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
uimplies: b supposing a,
member: t ∈ T,
iff: P ⇐⇒ Q,
and: P ∧ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
so_lambda: λ2x.t[x],
so_apply: x[s],
insert-by: insert-by(eq;r;x;l),
so_lambda: so_lambda(x,y,z.t[x; y; z]),
top: Top,
so_apply: x[s1;s2;s3],
or: P ∨ Q,
false: False,
guard: {T},
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
ifthenelse: if b then t else f fi ,
bfalse: ff
Lemmas referenced :
assert_wf,
assert_witness,
equal_wf,
list_induction,
all_wf,
iff_wf,
l_member_wf,
insert-by_wf,
or_wf,
list_wf,
list_ind_nil_lemma,
false_wf,
nil_member,
nil_wf,
member_singleton,
cons_wf,
list_ind_cons_lemma,
bool_wf,
cons_member,
equal-wf-T-base,
bnot_wf,
not_wf,
eqtt_to_assert,
uiff_transitivity,
eqff_to_assert,
assert_of_bnot,
and_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
lambdaFormation,
cut,
introduction,
sqequalRule,
sqequalHypSubstitution,
lambdaEquality,
dependent_functionElimination,
thin,
hypothesisEquality,
productElimination,
independent_pairEquality,
axiomEquality,
hypothesis,
extract_by_obid,
isectElimination,
applyEquality,
functionExtensionality,
cumulativity,
independent_functionElimination,
rename,
because_Cache,
isect_memberEquality,
voidElimination,
voidEquality,
independent_pairFormation,
inlFormation,
unionElimination,
addLevel,
impliesFunctionality,
orFunctionality,
functionEquality,
universeEquality,
equalityTransitivity,
equalitySymmetry,
inrFormation,
baseClosed,
equalityElimination,
independent_isectElimination,
hyp_replacement,
dependent_set_memberEquality,
applyLambdaEquality,
setElimination
Latex:
\mforall{}[T:Type]
\mforall{}eq,r:T {}\mrightarrow{} T {}\mrightarrow{} \mBbbB{}.
\mforall{}x:T. \mforall{}L:T List. \mforall{}z:T. ((z \mmember{} insert-by(eq;r;x;L)) \mLeftarrow{}{}\mRightarrow{} (z = x) \mvee{} (z \mmember{} L))
supposing \mforall{}a,b:T. (\muparrow{}(eq a b) \mLeftarrow{}{}\mRightarrow{} a = b)
Date html generated:
2017_04_17-AM-08_26_08
Last ObjectModification:
2017_02_27-PM-04_48_20
Theory : list_1
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