Nuprl Lemma : decide-ite
∀[b:𝔹]. ∀[x,y,f,g:Top].
(case if b then x else y fi of inl(a) => f[a] | inr(a) => g[a] ~ if b
then case x of inl(a) => f[a] | inr(a) => g[a]
else case y of inl(a) => f[a] | inr(a) => g[a]
fi )
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
so_apply: x[s],
decide: case b of inl(x) => s[x] | inr(y) => t[y],
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
all: ∀x:A. B[x],
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
uiff: uiff(P;Q),
and: P ∧ Q,
uimplies: b supposing a,
ifthenelse: if b then t else f fi ,
bfalse: ff,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False
Lemmas referenced :
bool_wf,
eqtt_to_assert,
top_wf,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesisEquality,
thin,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
productElimination,
independent_isectElimination,
sqequalRule,
sqequalAxiom,
isect_memberEquality,
because_Cache,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination
Latex:
\mforall{}[b:\mBbbB{}]. \mforall{}[x,y,f,g:Top].
(case if b then x else y fi of inl(a) => f[a] | inr(a) => g[a] \msim{} if b
then case x of inl(a) => f[a] | inr(a) => g[a]
else case y of inl(a) => f[a] | inr(a) => g[a]
fi )
Date html generated:
2018_05_21-PM-08_00_23
Last ObjectModification:
2017_07_26-PM-05_37_15
Theory : general
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