Nuprl Lemma : fun_thru_ite
∀[S,T:Type]. ∀[f:S ⟶ T]. ∀[b:𝔹]. ∀[p,q:S]. ((f if b then p else q fi ) = if b then f p else f q fi ∈ T)
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
equal: s = t ∈ 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,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
Error :isect_memberFormation_alt,
introduction,
cut,
hypothesisEquality,
thin,
extract_by_obid,
hypothesis,
lambdaFormation,
sqequalHypSubstitution,
unionElimination,
equalityElimination,
isectElimination,
because_Cache,
productElimination,
independent_isectElimination,
sqequalRule,
applyEquality,
dependent_pairFormation,
promote_hyp,
dependent_functionElimination,
instantiate,
cumulativity,
equalityTransitivity,
equalitySymmetry,
independent_functionElimination,
voidElimination,
Error :inhabitedIsType,
isect_memberEquality,
axiomEquality,
Error :universeIsType,
Error :functionIsType,
functionEquality,
universeEquality
Latex:
\mforall{}[S,T:Type]. \mforall{}[f:S {}\mrightarrow{} T]. \mforall{}[b:\mBbbB{}]. \mforall{}[p,q:S].
((f if b then p else q fi ) = if b then f p else f q fi )
Date html generated:
2019_06_20-AM-11_31_43
Last ObjectModification:
2018_09_26-AM-11_28_10
Theory : bool_1
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