Nuprl Lemma : fpf-vals-nil
∀[A:Type]. ∀[eq:EqDecider(A)]. ∀[B:A ⟶ Type]. ∀[P:A ⟶ 𝔹]. ∀[f:x:A fp-> B[x]]. ∀[a:A].
(fpf-vals(eq;P;f) ~ []) supposing ((∀b:A. (↑(P b) ⇐⇒ b = a ∈ A)) and (¬↑a ∈ dom(f)))
Proof
Definitions occuring in Statement :
fpf-vals: fpf-vals(eq;P;f),
fpf-dom: x ∈ dom(f),
fpf: a:A fp-> B[a],
nil: [],
deq: EqDecider(T),
assert: ↑b,
bool: 𝔹,
uimplies: b supposing a,
uall: ∀[x:A]. B[x],
so_apply: x[s],
all: ∀x:A. B[x],
iff: P ⇐⇒ Q,
not: ¬A,
apply: f a,
function: x:A ⟶ B[x],
universe: Type,
sqequal: s ~ t,
equal: s = t ∈ T
Definitions unfolded in proof :
fpf-vals: fpf-vals(eq;P;f),
let: let,
fpf: a:A fp-> B[a],
pi1: fst(t),
pi2: snd(t),
member: t ∈ T,
uall: ∀[x:A]. B[x],
so_lambda: λ2x.t[x],
so_apply: x[s],
subtype_rel: A ⊆r B,
uimplies: b supposing a,
all: ∀x:A. B[x],
top: Top,
prop: ℙ,
implies: P ⇒ Q,
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
iff: P ⇐⇒ Q,
and: P ∧ Q,
rev_implies: P ⇐ Q,
ifthenelse: if b then t else f fi ,
bfalse: ff,
not: ¬A,
false: False,
uiff: uiff(P;Q),
exists: ∃x:A. B[x],
or: P ∨ Q,
sq_type: SQType(T),
guard: {T},
bnot: ¬bb,
assert: ↑b,
fpf-dom: x ∈ dom(f),
cons: [a / b]
Latex:
\mforall{}[A:Type]. \mforall{}[eq:EqDecider(A)]. \mforall{}[B:A {}\mrightarrow{} Type]. \mforall{}[P:A {}\mrightarrow{} \mBbbB{}]. \mforall{}[f:x:A fp-> B[x]]. \mforall{}[a:A].
(fpf-vals(eq;P;f) \msim{} []) supposing ((\mforall{}b:A. (\muparrow{}(P b) \mLeftarrow{}{}\mRightarrow{} b = a)) and (\mneg{}\muparrow{}a \mmember{} dom(f)))
Date html generated:
2016_05_16-AM-11_24_45
Last ObjectModification:
2015_12_29-AM-09_24_12
Theory : event-ordering
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