Nuprl Lemma : apply-ifthenelse-pi1-eq
∀[T1,U1,T2,U2:Type]. ∀[x:T1 × U1]. ∀[y:T2 × U2]. ∀[b:𝔹].
((fst(if b then x else y fi )) = if b then fst(x) else fst(y) fi ∈ if b then T1 else T2 fi )
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
pi1: fst(t),
product: x:A × B[x],
universe: Type,
equal: s = t ∈ T
Definitions unfolded in proof :
bool: 𝔹,
ifthenelse: if b then t else f fi ,
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
pi1_wf,
bool_wf
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
unionElimination,
thin,
sqequalRule,
cut,
lemma_by_obid,
isectElimination,
hypothesisEquality,
lambdaEquality,
hypothesis,
because_Cache,
productEquality,
universeEquality,
isect_memberFormation,
introduction,
isect_memberEquality,
axiomEquality
Latex:
\mforall{}[T1,U1,T2,U2:Type]. \mforall{}[x:T1 \mtimes{} U1]. \mforall{}[y:T2 \mtimes{} U2]. \mforall{}[b:\mBbbB{}].
((fst(if b then x else y fi )) = if b then fst(x) else fst(y) fi )
Date html generated:
2016_05_13-PM-04_01_32
Last ObjectModification:
2015_12_26-AM-10_48_58
Theory : bool_1
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