Nuprl Lemma : pairs-fpf_property
∀[A,B:Type].
∀eq1:EqDecider(A). ∀eq2:EqDecider(B). ∀L:(A × B) List.
(no_repeats(A;fpf-domain(fpf(L)))
∧ (∀a:A. ((a ∈ fpf-domain(fpf(L))) ⇐⇒ ∃b:B. (<a, b> ∈ L)))
∧ ∀a∈dom(fpf(L)). l=fpf(L)(a) ⇒ no_repeats(B;l) ∧ (∀b:B. ((b ∈ l) ⇐⇒ (<a, b> ∈ L))))
Proof
Definitions occuring in Statement :
pairs-fpf: fpf(L),
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
fpf-domain: fpf-domain(f),
no_repeats: no_repeats(T;l),
l_member: (x ∈ l),
list: T List,
deq: EqDecider(T),
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
exists: ∃x:A. B[x],
iff: P ⇐⇒ Q,
and: P ∧ Q,
pair: <a, b>,
product: x:A × B[x],
universe: Type
Definitions unfolded in proof :
uall: ∀[x:A]. B[x],
all: ∀x:A. B[x],
and: P ∧ Q,
cand: A c∧ B,
member: t ∈ T,
pairs-fpf: fpf(L),
fpf-domain: fpf-domain(f),
pi1: fst(t),
subtype_rel: A ⊆r B,
so_lambda: λ2x.t[x],
so_apply: x[s],
uimplies: b supposing a,
top: Top,
iff: P ⇐⇒ Q,
implies: P ⇒ Q,
prop: ℙ,
rev_implies: P ⇐ Q,
exists: ∃x:A. B[x],
squash: ↓T,
pi2: snd(t),
true: True,
fpf-all: ∀x∈dom(f). v=f(x) ⇒ P[x; v],
fpf-ap: f(x),
eqof: eqof(d),
deq: EqDecider(T),
bool: 𝔹,
unit: Unit,
it: ⋅,
btrue: tt,
ifthenelse: if b then t else f fi ,
uiff: uiff(P;Q),
bfalse: ff,
not: ¬A,
false: False,
or: P ∨ Q,
guard: {T},
sq_type: SQType(T),
bnot: ¬bb,
assert: ↑b
Latex:
\mforall{}[A,B:Type].
\mforall{}eq1:EqDecider(A). \mforall{}eq2:EqDecider(B). \mforall{}L:(A \mtimes{} B) List.
(no\_repeats(A;fpf-domain(fpf(L)))
\mwedge{} (\mforall{}a:A. ((a \mmember{} fpf-domain(fpf(L))) \mLeftarrow{}{}\mRightarrow{} \mexists{}b:B. (<a, b> \mmember{} L)))
\mwedge{} \mforall{}a\mmember{}dom(fpf(L)). l=fpf(L)(a) {}\mRightarrow{} no\_repeats(B;l) \mwedge{} (\mforall{}b:B. ((b \mmember{} l) \mLeftarrow{}{}\mRightarrow{} (<a, b> \mmember{} L))))
Date html generated:
2016_05_16-AM-11_36_56
Last ObjectModification:
2016_01_17-PM-03_50_17
Theory : event-ordering
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