Nuprl Lemma : ifthenelse-as-band-bnot
∀[b:𝔹]. ∀[x:Top]. (if b then ff else x fi ~ (¬bb) ∧b x)
Proof
Definitions occuring in Statement :
band: p ∧b q,
bnot: ¬bb,
ifthenelse: if b then t else f fi ,
bfalse: ff,
bool: 𝔹,
uall: ∀[x:A]. B[x],
top: Top,
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 ,
bnot: ¬bb,
bfalse: ff,
band: p ∧b q,
exists: ∃x:A. B[x],
prop: ℙ,
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
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,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
because_Cache,
voidElimination,
isect_memberEquality
Latex:
\mforall{}[b:\mBbbB{}]. \mforall{}[x:Top]. (if b then ff else x fi \msim{} (\mneg{}\msubb{}b) \mwedge{}\msubb{} x)
Date html generated:
2018_05_21-PM-00_03_53
Last ObjectModification:
2018_01_28-PM-02_28_11
Theory : bool_1
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