Nuprl Lemma : ifthenelse_sqequal
∀[a,x1,y1,x2,y2:Base].
if a then x1 else y1 fi ~ if a then x2 else y2 fi
supposing ((∃z:Base. (a ~ inl z)) ⇒ (x1 ~ x2)) ∧ ((∃z:Base. (a ~ inr z )) ⇒ (y1 ~ y2))
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
exists: ∃x:A. B[x],
implies: P ⇒ Q,
and: P ∧ Q,
inr: inr x ,
inl: inl x,
base: Base,
sqequal: s ~ t
Definitions unfolded in proof :
and: P ∧ Q,
uall: ∀[x:A]. B[x],
member: t ∈ T,
uimplies: b supposing a,
implies: P ⇒ Q,
prop: ℙ,
so_lambda: λ2x.t[x],
so_apply: x[s]
Lemmas referenced :
and_wf,
base_wf,
exists_wf,
ifthenelse_sqle
Rules used in proof :
sqequalHypSubstitution,
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
productElimination,
thin,
sqequalSqle,
cut,
lemma_by_obid,
isectElimination,
sqequalRule,
baseApply,
closedConclusion,
baseClosed,
hypothesisEquality,
independent_isectElimination,
independent_pairFormation,
lambdaFormation,
independent_functionElimination,
hypothesis,
sqleReflexivity,
lambdaEquality,
sqequalIntensionalEquality,
because_Cache,
functionEquality,
isect_memberFormation,
introduction,
sqequalAxiom,
isect_memberEquality,
equalityTransitivity,
equalitySymmetry
Latex:
\mforall{}[a,x1,y1,x2,y2:Base].
if a then x1 else y1 fi \msim{} if a then x2 else y2 fi
supposing ((\mexists{}z:Base. (a \msim{} inl z)) {}\mRightarrow{} (x1 \msim{} x2)) \mwedge{} ((\mexists{}z:Base. (a \msim{} inr z )) {}\mRightarrow{} (y1 \msim{} y2))
Date html generated:
2016_05_13-PM-03_45_20
Last ObjectModification:
2016_01_14-PM-07_06_33
Theory : computation
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