Nuprl Lemma : member-ite
∀b:𝔹. ∀[T:Type]. ∀A,B:T List. ∀x:T. ((x ∈ if b then A else B fi ) ⇐⇒ ((↑b) ∧ (x ∈ A)) ∨ ((¬↑b) ∧ (x ∈ B)))
Proof
Definitions occuring in Statement :
l_member: (x ∈ l),
list: T List,
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
or: P ∨ Q,
and: P ∧ Q,
universe: Type
Definitions unfolded in proof :
all: ∀x:A. B[x],
member: t ∈ T,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uall: ∀[x:A]. B[x],
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
assert: ↑b,
iff: P ⇐⇒ Q,
or: P ∨ Q,
cand: A c∧ B,
prop: ℙ,
rev_implies: P ⇐ Q,
not: ¬A,
true: True,
false: False,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb
Latex:
\mforall{}b:\mBbbB{}
\mforall{}[T:Type]
\mforall{}A,B:T List. \mforall{}x:T. ((x \mmember{} if b then A else B fi ) \mLeftarrow{}{}\mRightarrow{} ((\muparrow{}b) \mwedge{} (x \mmember{} A)) \mvee{} ((\mneg{}\muparrow{}b) \mwedge{} (x \mmember{} B)))
Date html generated:
2016_05_16-AM-10_23_11
Last ObjectModification:
2015_12_28-PM-09_20_26
Theory : new!event-ordering
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