Nuprl Lemma : iflift_1
∀[A,B:Type]. ∀[c:𝔹]. ∀[f:A ⟶ B]. ∀[x,y:A]. (f[if c then x else y fi ] = if c then f[x] else f[y] fi ∈ B)
Proof
Definitions occuring in Statement :
ifthenelse: if b then t else f fi ,
bool: 𝔹,
uall: ∀[x:A]. B[x],
so_apply: x[s],
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 ,
so_apply: x[s],
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,
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,
inhabitedIsType,
isect_memberEquality,
axiomEquality,
universeIsType,
functionIsType,
functionEquality,
universeEquality
Latex:
\mforall{}[A,B:Type]. \mforall{}[c:\mBbbB{}]. \mforall{}[f:A {}\mrightarrow{} B]. \mforall{}[x,y:A].
(f[if c then x else y fi ] = if c then f[x] else f[y] fi )
Date html generated:
2019_10_15-AM-10_46_34
Last ObjectModification:
2018_09_27-AM-09_41_13
Theory : basic
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