Nuprl Lemma : branch-ifthenelse
∀[b:𝔹]. ∀[P,Q:Top]. (if x:↑b then P else Q fi ~ if b then P else Q fi )
Proof
Definitions occuring in Statement :
branch: if p:P then A[p] else B fi ,
bool-decider: bool-decider(b),
assert: ↑b,
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
sqequal: s ~ t
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
member: t ∈ T,
branch: if p:P then A[p] else B fi ,
all: ∀x:A. B[x],
implies: P ⇒ Q,
decidable: Dec(P),
or: P ∨ Q,
prop: ℙ,
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],
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
false: False,
not: ¬A
Lemmas referenced :
top_wf,
bool_wf,
bool-decider_wf,
decidable_wf,
assert_wf,
equal_wf,
eqtt_to_assert,
eqff_to_assert,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
isect_memberFormation,
introduction,
cut,
hypothesis,
sqequalAxiom,
extract_by_obid,
sqequalRule,
sqequalHypSubstitution,
isect_memberEquality,
isectElimination,
thin,
hypothesisEquality,
because_Cache,
lambdaFormation,
unionElimination,
equalityTransitivity,
equalitySymmetry,
dependent_functionElimination,
independent_functionElimination,
equalityElimination,
productElimination,
independent_isectElimination,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
voidElimination
Latex:
\mforall{}[b:\mBbbB{}]. \mforall{}[P,Q:Top]. (if x:\muparrow{}b then P else Q fi \msim{} if b then P else Q fi )
Date html generated:
2017_10_01-AM-09_12_56
Last ObjectModification:
2017_07_26-PM-04_48_35
Theory : general
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