Nuprl Lemma : conditional-ifthenelse
∀[T,V:Type]. ∀[A,B:T ⟶ 𝔹]. ∀[f:{x:T| ↑(A x)} ⟶ V]. ∀[g:{x:T| ↑(B x)} ⟶ V].
((λx.if A x then f x else g x fi ) = [λx.(↑(A x))? f : g] ∈ ({x:T| (↑(A x)) ∨ (↑(B x))} ⟶ V))
Proof
Definitions occuring in Statement :
conditional: [P? f : g],
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
or: P ∨ Q,
set: {x:A| B[x]} ,
apply: f a,
lambda: λx.A[x],
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ T,
bool-decider: bool-decider(b)
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
prop: ℙ,
conditional: [P? f : g],
or: P ∨ Q,
top: Top,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
bfalse: ff,
exists: ∃x:A. B[x],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Latex:
\mforall{}[T,V:Type]. \mforall{}[A,B:T {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:\{x:T| \muparrow{}(A x)\} {}\mrightarrow{} V]. \mforall{}[g:\{x:T| \muparrow{}(B x)\} {}\mrightarrow{} V].
((\mlambda{}x.if A x then f x else g x fi ) = [\mlambda{}x.(\muparrow{}(A x))? f : g])
Date html generated:
2016_05_16-AM-10_15_35
Last ObjectModification:
2015_12_28-PM-09_28_48
Theory : new!event-ordering
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