Nuprl Lemma : fpf-cap-single
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[x,y:A]. ∀[v,z:Top]. (x : v(y)?z ~ if eq x y then v else z fi )
Proof
Definitions occuring in Statement :
fpf-single: x : v
,
fpf-cap: f(x)?z
,
deq: EqDecider(T)
,
ifthenelse: if b then t else f fi
,
uall: ∀[x:A]. B[x]
,
top: Top
,
apply: f a
,
universe: Type
,
sqequal: s ~ t
Definitions unfolded in proof :
fpf-single: x : v
,
fpf-cap: f(x)?z
,
fpf-ap: f(x)
,
fpf-dom: x ∈ dom(f)
,
pi1: fst(t)
,
pi2: snd(t)
,
all: ∀x:A. B[x]
,
member: t ∈ T
,
top: Top
,
deq: EqDecider(T)
,
implies: P
⇒ Q
,
bool: 𝔹
,
unit: Unit
,
it: ⋅
,
btrue: tt
,
uall: ∀[x:A]. B[x]
,
uiff: uiff(P;Q)
,
and: P ∧ Q
,
uimplies: b supposing a
,
eqof: eqof(d)
,
bor: p ∨bq
,
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
,
not: ¬A
Lemmas referenced :
deq_member_cons_lemma,
deq_member_nil_lemma,
bool_wf,
eqtt_to_assert,
safe-assert-deq,
eqff_to_assert,
equal_wf,
bool_cases_sqequal,
subtype_base_sq,
bool_subtype_base,
assert-bnot,
top_wf,
deq_wf
Rules used in proof :
sqequalSubstitution,
sqequalTransitivity,
computationStep,
sqequalReflexivity,
sqequalRule,
cut,
introduction,
extract_by_obid,
sqequalHypSubstitution,
dependent_functionElimination,
thin,
isect_memberEquality,
voidElimination,
voidEquality,
hypothesis,
applyEquality,
setElimination,
rename,
hypothesisEquality,
lambdaFormation,
unionElimination,
equalityElimination,
isectElimination,
equalityTransitivity,
equalitySymmetry,
productElimination,
independent_isectElimination,
because_Cache,
dependent_pairFormation,
promote_hyp,
instantiate,
cumulativity,
independent_functionElimination,
universeEquality,
isect_memberFormation,
sqequalAxiom
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[x,y:A]. \mforall{}[v,z:Top]. (x : v(y)?z \msim{} if eq x y then v else z fi )
Date html generated:
2018_05_21-PM-09_25_11
Last ObjectModification:
2018_02_09-AM-10_20_57
Theory : finite!partial!functions
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