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)
Lemmas :
set_wf,
or_wf,
assert_wf,
branch-ifthenelse,
bool_wf,
eqtt_to_assert,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
\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:
2015_07_17-AM-09_01_35
Last ObjectModification:
2015_01_27-PM-00_56_40
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